Calculus
Calculus
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Table of Contents
1. Calculus.................................................. ........................25Precalculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Basics of Differentiation
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7
Applications of Derivatives
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 8
Important Theorems
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 8
Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Basics of Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Integration techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Applications of Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Parametric and Polar Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Parametric Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Polar Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Sequences and Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Series and calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Multivariable and Differential Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
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Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Further Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Formal Theory of Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Acknowledgements and Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2. Contributing................................................ ......................34Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Templates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
{{Calculus/Top Nav}}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
{{Calculus/TOC}}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
{{Calculus/Def}}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
{{Calculus/Stub}}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
On Inclusiveness vs. Exclusiveness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
TODO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3. Introduction................................................. .....................37What is calculus?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Why learn calculus?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
What is involved in learning calculus?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
What you should know before using this text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Scope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4. Precalculus.................................................. ....................41Exercises.................................................. ....................42
Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Convert to interval notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
State the following intervals using inequalities. . . . . . . . . . . . . . . . . . . . . . . 42
Which one of the following is a true statement?. . . . . . . . . . . . . . . . . . . . . . 43
Evaluate the following expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Simplify the following. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Factor the following expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
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Simplify the following. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Decomposition of functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Graphing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Solutions.................................................. .....................49Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5. Algebra................................................. ..........................50Rules of arithmetic and algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Interval notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Exponents and radicals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Factoring and roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Simplifying rational expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6. Functions.................................................. .......................57Classical understanding of functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Modern understanding of functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
The vertical line test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Important functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Example functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Manipulating functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Composition of functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Domain and Range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
One-to-one Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Inverse functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7. Graphing linear functions............................................... .......72
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Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Slope-intercept form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Point-slope form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Calculating slope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Two-point form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
8. Limits.................................................. ............................75An Introduction to Limits............................................... .....76
Intuitive Look. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Informal definition of a limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Limit rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
The Squeeze Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Finding limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Using limit notation to describe asymptotes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Key application of limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
External links. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Exercises.................................................. ....................92Limits with Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Basic Limit Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
One Sided Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Two Sided Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Limits to Infinity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Limits of Piece Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Contents.................................................. .....................97Solutions.................................................. .....................98
Basic Limit Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Harder Limit Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
9. Finite Limits................................................. ...................100Informal Finite Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
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One-Sided Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
10. Infinite Limits................................................ .................103Informal infinite limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Limits at infinity of rational functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Infinity is not a number.............................................. .......107Addition breaks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Reinterpret formulas that use
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Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
11. Continuity................................................ .....................112Defining Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Discontinuities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Removable Discontinuities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Jump Discontinuities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
One-Sided Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Intermediate Value Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Application: bisection method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
12. Formal Definition of the Limit............................................ ..117Formal Definition of the Limit at Infinity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Formal Definition of a Limit Being Infinity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
13. Differentiation................................................ .................127Basics of Differentiation
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Important Theorems
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Differentiation Defined................................................. ....129What is differentiation?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
The Definition of Slope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Of a Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Of a Graph of a Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
The Rate of Change of a Function at a Point. . . . . . . . . . . . . . . . . . . . . . . . . . 135
The Definition of the Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Understanding the Derivative Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Differentiation rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Derivative of a Constant Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Derivative of a Linear Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Constant multiple and addition rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
The Power Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Derivatives of polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Exercises................................................. ....................148Find The Derivative By Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Prove Differentiation Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Find The Derivative By Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
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Power Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Product Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Quotient Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Chain Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Exponentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Logarithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Trig Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
More Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Implicit Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Logarithmic Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Equation of Tangent Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Higher Order Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Relative Extrema. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Range of Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Absolute Extrema. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Determine Intervals of Change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Determine Intervals of Concavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Word Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Graphing Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Contents.................................................. ....................158Basics of Differentiation
See Picture License Information Here
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Applications of Derivatives
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Important Theorems
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Solutions.................................................. ...................160Find The Derivative By Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Prove Differentiation Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Proof of the Derivative of a Constant Function. . . . . . . . . . . . . . . . . . . . . . 162
Proof of the Derivative of a Linear Function. . . . . . . . . . . . . . . . . . . . . . . . . 163
Proof of the Constant Multiple Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Proof of the Addition and Subtraction Rules. . . . . . . . . . . . . . . . . . . . . . . . 164
Find The Derivative By Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Implicit Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Higher Order Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
14. Product and Quotient Rules.............................................. .167Product Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Application, proof of the power rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Quotient rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
15. Derivatives of Trigonometric Functions..................................17116. Chain Rule................................................. ...................17517. More Differentiation Rules............................................... ..177
External links. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
18. Higher Order Derivatives............................................. ......178Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
19. Implicit differentiation................................................ ........181Explicit differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
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Implicit differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Uses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Implicit Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Inverse Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
20. Derivatives of Exponential and Logarithm Functions..................187Exponential Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Logarithm Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
Logarithmic differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
21. Extrema and Points of Inflection..........................................192The Extremum Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
22. Newton's Method................................................ ............196Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
See Also. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
23. Related Rates................................................. ...............200Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Related Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Common Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Filling Tank. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Problem Set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Solution Set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
24. Kinematics.................................................. ..................207Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
Average Velocity and Acceleration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
Instantaneous Velocity and Acceleration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
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Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
25. Optimization.................................................. ................211Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Volume Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Sales Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
26. Euler's Method................................................ ...............215Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
27. Extreme Value Theorem............................................... .....216First Derivative Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Second Derivative Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
28. Rolle's Theorem................................................. ............224Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
29. Mean Value Theorem for Functions......................................226Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Differentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
Definition of Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Cauchy's Mean Value Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
30. Integration................................................ .....................231Basics of Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Integration techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
Applications of Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
Exercises................................................. ....................233Set One: Sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Set Two: Integration of Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Indefinite Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Integration by parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
Contents................................................... ...................235Basics of Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
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Integration techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
Applications of Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
Solutions.................................................. ...................237Solutions to Set One. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Solutions to Set Two. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Solutions to Set Three. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
31. Indefinite integral.............................................. ..............239Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Indefinite integral identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
Basic Properties of Indefinite Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
Indefinite integrals of Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
Integral of the Inverse function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
Integral of the Exponential function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
Integral of Sine and Cosine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
The Substitution Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Preliminary Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Generalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
General Substitution Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
Integration by Parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
Preliminary Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
General Integration by Parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
32. Definite integral.............................................. ................251Definition of the Definite Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
Independence of Variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Left and Right Handed Riemann Sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
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Basic Properties of the Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
The Constant Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
The addition and subtraction rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
The Comparison Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
Linearity with respect to endpoints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Even and odd functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
33. Fundamental Theorem of Calculus.......................................266Fundamental Theorem of Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
Statement of the Fundamental Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Integration of Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
34. Infinite Sums................................................. .................273Exact Integrals as Limits of Sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
35. Recognizing Derivatives and the Substitution Rule....................276Recognizing Derivatives and Reversing Derivative Rules. . . . . . . . . . . . . . . . . . . 276
Integration by Substitution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Integrating with the derivative present. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Proof of the substitution rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
36. Integration by Parts........................................... ...............281Integration by Parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
With definite integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
37. Integration by Complexifying............................................ ...28538. Partial Fraction Decomposition........................................... .286
Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
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39. Trigonometric Substitution............................................. ....290Sine substitution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
Tangent substitution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
Secant substitution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
40. Tangent Half Angle............................................... ...........295Alternate Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
41. Trigonometric Integrals............................................ .........300Powers of Sine and Cosine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
Powers of Tan and Secant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
More trigonometric combinations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
42. Reduction Formula.............................................. ............30643. Irrational Functions................................................ ..........308
Type 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
Type 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
Type 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
Type 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
Type 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
44. Numerical Approximations............................................... ..311Riemann Sum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Right Rectangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Left Rectangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Trapezoidal Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
Simpson's Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
Further reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
45. Improper integrals............................................... ............313L'Hopital's Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
Improper Integrals with Infinite Limits of Integration. . . . . . . . . . . . . . . . . . . . . . . . . 316
Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
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Improper Integrals with a Finite Number Discontinuities. . . . . . . . . . . . . . . . . . . . 320
Definition of improper integrals with a single discontinuity. . . . . . . . . . . . . . . . 320
Definition: Improper integrals with finite number of discontinuities. . . . . . . 322
Comparison Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
An extension of the comparison theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
46. Area.................................................. ..........................327Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
Area between two curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
47. Volume................................................. .......................330Formal Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
Example 1: A right cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
Example 2: A right circular cone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
Example 3: A sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
Extension to Non-trivial Solids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
48. Volume of solids of revolution........................................... ...336Revolution solids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
Calculating the volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
49. Arc length................................................. ....................338Definition (Length of a Curve). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
The Arclength Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
Arclength of a parametric curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
50. Surface area................................................ ..................342Definition (Surface of Revolution). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
The Surface Area Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
51. Work................................................ ...........................34552. Centre of mass................................................ ...............34653. Parametric and Polar Equations..........................................347
Parametric Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
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Polar Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
54. Parametric Introduction.............................................. .......348Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
Forms of Parametric Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
Parametric Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
Vector Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
Equalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
Converting Parametric Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
55. Parametric Differentiation............................................ ......351Taking Derivatives of Parametric Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
Slope of Parametric Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
Concavity of Parametric Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
56. Parametric Integration............................................ ..........354Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
57. Polar Introduction................................................ ............355Plotting points with polar coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
Converting between polar and Cartesian coordinates. . . . . . . . . . . . . . . . . . . . 357
Polar equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
Circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Polar rose. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
Archimedean spiral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
Conic sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
58. Polar Differentiation.............................................. ...........362Differential calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
59. Polar Integration.............................................. ...............363Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
Proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
Integral calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
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Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
An interesting example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
60. Sequences and Series............................................... .......367Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
Series and calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
Exercises................................................. ....................369Hints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
Answers only. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
Full solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
61. Sequences.................................................. ..................376Examples and notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
Types and properties of sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
Sequences in analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
62. Series................................................. .........................379Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
Absolute convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
Ratio test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
Integral test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
Limit comparison test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
Alternating series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
Geometric series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
Telescoping series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
63. Taylor series.............................................. ....................388Taylor Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
Derivation/why this works. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
List of Taylor series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
Multiple dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
History. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
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Constructing a Taylor Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
Generalized Mean Value Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
64. Power series............................................. .....................396Motivations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
An example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
Radius of convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
An example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
Another example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
Abstraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
Differentiation and Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
Further reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
65. Multivariable and differential calculus....................................40066. Vectors................................................. .......................402
Two-Dimensional Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
Component Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
Magnitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
Direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
Vector Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
Applications of Scalar Multiplication and Dot Product. . . . . . . . . . . . . . . . . . . . 406
Polar coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
Three-Dimensional Coordinates and Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
Basic definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
Three dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
Cylindrical coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
Spherical coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
Cross Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
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Triple Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
Three-Dimensional Lines and Planes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
Vector-Valued Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
Limits, Derivatives, and Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
Velocity, Acceleration, Curvature, and a brief mention of the Binormal. . . 418
67. Lines and Planes in Space.............................................. ...420Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
Parametric Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
Line in Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
Plane in Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
Vector Equation (of a Plane in Space, or of a Line in a Plane). . . . . . . . . . . . . . 421
Scalar Equation (of a Plane in Space, or of a Line in a Plane). . . . . . . . . . . . . . 422
68. Multivariable calculus.............................................. .........423
Topology in Rn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
Lengths and distances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
Open and closed balls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
Boundary points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
Curves and parameterizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
Collision and intersection points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
Continuity and differentiability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
Tangent vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
Different parameterizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
Limits and continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
Special note about limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
Differentiable functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
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Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
Rules of taking Jacobians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
Alternate notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
Directional derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
Gradient vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
Divergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
Curl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
Product and chain rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
Second order differentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
Riemann sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
Iterated integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
Order of integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
Parametric integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
Line integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
Surface and Volume Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
Gauss's divergence theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
Stokes' curl theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
69. Ordinary differential equations............................................453Notations and terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
Terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
Some simple differential equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
Basic first order DEs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
Separable equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
Homogeneous equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
Linear equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
Exact equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
Basic second and higher order ODE's. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
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Reducible ODE's. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
Linear ODEs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466
70. Partial differential equations............................................. ..472Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
First order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
Special cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
Quasilinear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
Fully non-linear PDEs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
Second order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
Elliptic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
Hyperbolic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
Parabolic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
71. Multivariable and differential calculus:Exercises.......................49672. Extensions.................................................. ..................497
Further Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
Formal Theory of Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
73. Systems of ordinary differential equations..............................498First order systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
74. Real numbers................................................ ................501Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
Constructing the Real Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
Properties of Real Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
75. Complex numbers.............................................. .............506Equality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
Notation and operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
The field of complex numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
The complex plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
Polar form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
Conversion from the polar form to the Cartesian form. . . . . . . . . . . . . . . . . . . . 509
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Conversion from the Cartesian form to the polar form. . . . . . . . . . . . . . . . . . . . 509
Notation of the polar form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
Multiplication, division, exponentiation, and root extraction in the polarform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
Absolute value, conjugation and distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
Complex fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
Matrix representation of complex numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
76. References................................................. ...................51577. Table of Trigonometry........................................... ............516
Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
Pythagorean Identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
Double Angle Identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
Angle Sum Identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
Product-to-sum identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518
78. Summation notation................................................. ........519Common summations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
79. Tables of Derivatives.......................................... ..............521General Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
Powers and Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
Exponential and Logarithmic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
Inverse Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
Hyperbolic and Inverse Hyperbolic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
80. Tables of Integrals........................................... ................524Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
Powers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
Basic Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
Reciprocal Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
Inverse Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
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Exponential and Logarithmic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
Inverse Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
81. Acknowledgements................................................ ..........528Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
Contributors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
Other Calculus Textbooks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
82. Choosing delta................................................. ..............53083. More Differentation Rules............................................... ...533
External links. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
84. Mean Value Theorem............................................... ........534Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
Differentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536
Definition of Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
Cauchy's Mean Value Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
85. Infinite series................................................ .................539Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
Series and calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
GNU Free Documentation License. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .540
List of Contributors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .548
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Calculus
Welcome to the Wikibook ofCalculus
A printable versionof Calculus is available.()
SeePic-tureLi-censeInfor-ma-tionHere
This wikibook aims to be a qualitycalculus textbook through which users may master thediscipline. Standard topics such aslimits, differentiationandintegrationare covered as well asseveral others. Pleasecontributewherever you feel the need.
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Calculus• Introduction• Contributing
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Precalculus
• Algebra
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• Functions
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• Graphing linear functions
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• Exercises
Limits
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• Calculus/Limits/An Introduction to Limits
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• Finite Limits
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• Infinite Limits
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• Continuity
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• Formal Definition of the Limit
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• Calculus/Limits/Exercises
Differentiation
Basics of Differentiation
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• Differentiation Defined• Product and Quotient Rules• Derivatives of Trigonometric Functions• Chain Rule• More differentiation rules- More rules for differentiation• Higher Order derivatives- An introduction to second power derivatives• Implicit Differentiation• Derivatives of Exponential and Logarithm Functions• Exercises
Applications of Derivatives
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• Extrema and Points of Inflection• Newton's Method• Related Rates• Kinematics• Optimization• Euler's Method• Exercises
Important Theorems
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• Extreme Value Theorem• Rolle's Theorem• Mean Value Theorem
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Integration
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Basics of Integration
• Indefinite integral
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• Definite integral
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• Fundamental Theorem of Calculus
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Integration techniques
• Infinite Sums• Derivative Rules and the Substitution Rule• Integration by Parts• Complexifying• Rational Functions by Partial Fraction Decomposition• Trigonometric Substitutions• Tangent Half Angle Substitution• Trigonometric Integrals• Reduction Formula• Irrational Functions• Numerical Approximations• Integration techniques
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• Improper integrals
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• Exercises
Applications of Integration
• Area• Volume• Volume of solids of revolution• Arc length• Surface area• Work• Centre of mass• Pressure and force• Probability
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Parametric and Polar Equations
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Parametric Equations
• Introduction to Parametric Equations• Differentiation and Parametric Equations• Integration and Parametric Equations
Polar Equations
• Introduction to Polar Equations• Differentiation and Polar Equations• Integration and Polar Equations
Sequences and Series
Basics
• Sequences• Series
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Series and calculus
• Taylor series• Power series• Exercises
Multivariable and Differential Calculus
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• Vectors
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• Lines and Planes in Space
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• Multivariable Calculus
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• Ordinary Differential Equations
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• Partial Differential Equations
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• Exercises
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Extensions
Further Analysis
• Systems of Ordinary Differential Equations
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Formal Theory of Calculus
• Real numbers
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• Complex numbers
References
• Table of Trigonometry• Summation notation• Tables of Derivatives• Tables of Integrals
Acknowledgements and Further Reading
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Contributing
Precalculus→Calculus← Introduction
Contributing
Notes on contributing to theCalculustextbook. This is theWikibooks:Local manuals ofstylefor theCalculustextbook.
Structure
The structure currently being implemented is as follows:
• The book is divided into major sections (these sections may incorporate material thatmight comprise two or more chapters in a standard text).
• These sections and their articles are listed on themain page. Some articles actually spanmore than one page and if a user clicks on the link to them they will be taken to thesection's contents page where they can select which page of the article they would liketo view.
• Each section has a page with its complete, detailed contents listed plus a link to thatsection's page(s) of Exercises and Problems. Section pages should incorporate moreactual material such as descriptions of articles and discussion of the chapter's importantpoints or important/unusual prerequisites.
• Pages of Exercises are divided into sets (by type of problem) of sequentially numberedexercises or problems. Each page of exercises links to a corresponding page of solu-tions.NamingConvention:Calculus/[Section-Name]/Exercises(ex.Calculus/Differenti-ation/Exercises)
• Pages of solutions are organized based on their corresponding page of exercises (exercis-es set one has a corresponding set one of solutions, etc.) Solutions should be completeand logical. Particularly important steps should be noted and decribed in words as wellas being shown symbolicly.Naming Convention: Calculus/[Section-Name]/Solutions(ex. Calculus/Differentiation/Solutions)
• All content pages and section pages should include at the end {{Calculus/TOC}} whichis a quick-navigation template.
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Templates
SeeCategory:Calculus (book)/Templatesfor a list of allCalculustextbook templates.
{{Calculus/Top Nav}}
{{ Calculus/Top Nav |Limits|Infinite Limits}} produces this navigationbox:
Infinite Limits →Calculus← Limits
Contributing
All Calculuscontent pages should include this at the top of the page.
{{Calculus/TOC}}
{{ Calculus/TOC }} produces this navigation box:
All Calculuscontent pages should include this at the bottom of the page. This also adds thepage toCategory:Calculus (book).
{{Calculus/Def}}
{{ Calculus/Def |text=My definition here.}} produces a box for importanttext:
My definition here.Use this to introduce significant new definitions and concepts. SeeCalculus/Limits#Informal
definition of a limit, where the informal definition of a limit is inside such a box.
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{{Calculus/Stub}}
{{ Calculus/Stub }} produces a stub notice, signifying that the given page or sectionis too short.
On Inclusiveness vs. Exclusiveness
An Extensions section is included for further topics. Any study beyond fairly basic calculusshould be in this section. We should aim to include more than is necessary and ensure that ourreaders are aware of this. The book should be structured so that J. Random Student can find arigorous course in the essentials of Calculus but also include further study of the topic that readerscan pick and choose topics from, as their interests warrant.
TODO
• ADD MORE, ORGANIZE MORE.• Provide answer for Differentation exercise on the absolute value function.• Provide a more in-depth Precalculus section.• Create problem sets (preferrably original, rigorous and realistic).• Create answer sets.• Move existing problems from within the articles to the appropriate Exercises page.
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Introduction
Contributing→Calculus
Introduction
Wikipediahas related information at
SeePic-
Calculus
tureLi-censeInfor-ma-tionHere
What is calculus?
Calculus is the branch of mathematics dealing with instantaneous rates of change of continu-ously varying quantities. For example, consider a moving car. It is possible to create a functiondescribing thedisplacementof the car (where it is located in relation to a reference point) at anypoint in time as well as a function describing thevelocity(speed and direction of movement) ofthe car at any point in time. If the car were traveling at a constant velocity, then algebra wouldbe sufficient to determine the position of the car at any time; if the velocity is unknown but stillconstant, the position of the car could be used (along with the time) to find the velocity.
However, the velocity of a car cannot jump from zero to 35 miles per hour at the beginningof a trip, stay constant throughout, and then jump back to zero at the end. As the accelerator ispressed down, the velocity rises gradually, and usually not at a constantrate (i.e., the driver maypush on the gas pedal harder at the beginning, in order to speed up). Describing such motion andfinding velocities and distances at particular times cannot be done using methods taught in pre-calculus, but it is not only possible but straightforward with calculus.
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Calculus has two basic applications:differential calculusandintegral calculus. The simplestintroduction to differential calculus involves an explicit series of numbers. Given the series (42,43, 3, 18, 34), the differential of this series would be (1, -40, 15, 16). The new series is derivedfrom the difference of successive numbers which gives rise to its name "differential". Rarely, ifever, are differentials used on an explicit series of numbers as done here. Instead, they are derivedfrom a series of numbers defined by a continuous function which are described later.
Integral calculus, like differential calculus, can also be introduced via series of numbers.Notice that in the previous example, the original series can almost be derived solely from itsdifferential. Instead of taking the difference, however, integration involves taking the sum. Giventhe first number of the original series, 42 in this case, the rest of the original series can be derivedby adding each successive number in its differential (42+1, 43-40, 3+15, 18+16). Note thatknowledge of the first number in the original series is crucial in deriving the integral. As withdifferentials, integration is performed on continuous functions rather than explicit series ofnumbers, but the concept is still the same. Integral calculus allows us to calculate the area undera curve of almost any shape; in the car example, this enables you to find the displacement of thecar based on the velocity curve. This is because the area under the curve is the total distancemoved, as we will soon see.
Why learn calculus?
Calculus is essential for many areas of science and engineering. Both make heavy use ofmathematical functions to describe and predict physical phenomena that are subject to continualchange, and this requires the use of calculus. Take our car example: if you want to design cars,you need to know how to calculate forces, velocities, accelerations, and positions. All requirecalculus. Calculus is also necessary to study the motion of gases and particles, the interaction offorces, and the transfer of energy. It is also useful in business whenever rates are involved. Forexample, equations involving interest or supply and demand curves are grounded in the languageof calculus.
Calculus also provides important tools in understanding functions and has led to the develop-ment of new areas of mathematics including real and complex analysis, topology, and non-eu-clidean geometry.
Notwithstanding calculus'functionalutility (pun intended), many non-scientists and non-engineers have chosen to study calculus just for the challenge of doing so. A smaller number ofpersons undertake such a challenge and then discover that calculus is beautiful in and of itself.Is calculus then perhaps a manifestation of some divine ordering of the universe?
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What is involved in learning calculus?
Learning calculus, like much of mathematics, involves two parts:
• Understanding the concepts: You must be able to explain what it means when you takeaderivativerather than merely apply the formulas for finding a derivative. Otherwise,you will have no idea whether or not your solution is correct. Drawing diagrams, forexample, can help clarify abstract concepts.
• Symbolic manipulation: Like other branches of mathematics, calculus is written insymbols that represent concepts. You will learn what these symbols mean and how touse them. A good working knowledge oftrigonometryandalgebrais a must, especiallyin integral calculus. Sometimes you will need to manipulate expressions into a usableform before it is possible to perform operations in calculus.
What you should know before using this text
There are some basic skills that you need before you can use this text. Continuing with ourexample of a moving car:
• You will need to describe the motion of the car in symbols. This involves understandingfunctions.
• You need to manipulate these functions. This involves algebra.• You need to translate symbols into graphs and vice verse. This involves understanding
the graphing of functions.• It also helps (although it isn't necessarily essential) if you understand the functions used
in trigonometry since these functions appear frequently in science.
Scope
The first four chapters of this textbook cover the topics taught in a typical high school orfirst year college course. The first chapter,Precalculus, reviews those aspects of functions mostessential to the mastery of Calculus, the secondLimits, introduces the concept of the limit pro-cess. It also discusses some applications of limits and proposes using limits to examine slope andarea of functions. The next two chapters,DifferentiationandIntegration, apply limits to calculate
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derivatives and integrals. The Fundamental Theorem of Calculus is used, as are the essentialformulae for computation of derivatives and integrals without resorting to the limit process. Thethird and fourth chapters include articles that apply the concepts previously learned to calculatingvolumes, and so on as well as other important formulae.
The remainder of the central Calculus chapters cover topics taught in higher level Calculustopics: multivariable calculus, vectors, and series (Taylor, convergent, divergent).
Finally, the other chapters cover the same material, using formal notation. They introducethe material at a much faster pace, and cover many more theorems than the other two sections.They assume knowledge of some set theory and set notation.
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Precalculus
Algebra→Calculus← Contributing
Precalculus
• Algebra
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• Functions
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• Graphing linear functions
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• Exercises
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Exercises
Limits/An Introduction toLimits →
Calculus← Graphing linear functions
Precalculus/Exercises
Algebra
Convert to interval notation
1.2.3.4.5.6.7.8.9.10.
State the following intervals using inequalities
1.2.3.4.5.6.
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Which one of the following is a true statement?
Hint: the true statement is often referred to as thetriangle inequality.Give examples wherethe other two are false.
1.2.3.
Evaluate the following expressions
1.2.
3.
4.
5.
6.
7.
8.
9.
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Simplify the following
1.2.
3.4.
5.6.
7.
8.
Factor the following expressions
For 1-8, determine what values ofx make the expression 0 (i.e. determine the roots).
1.2.3.4.5.6.7.8.9.10.11.
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Simplify the following
1.
2.
3.
4.
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Functions
1. Let f(x) = x2.1. Computef(0) andf(2).2. What are the domain and range off?3. Doesf have an inverse? If so, find a formula for it.
2. Let f(x) = x + 2, g(x) = 1 / x.Give formulae for1.
f + g,1.2. f − g,3. g − f,4.
,5. f / g,6. g / f,7.
and8.
.2. Computef(g(2)) andg(f(2)).3. Do f andg have inverses? If so, find formulae for them.
3. Does this graph represent a function?
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4. Consider the following function
1. What is the domain?2. What is the range?
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3. Where isf continuous?5. Consider the following function
1. What is the domain?2. What is the range?3. Where isf continuous?
6. Consider the following function
1. What is the domain?2. What is the range?3. Where isf continuous?
7. Consider the following function
1. What is the domain?2. What is the range?3. Where isf continuous?
8. Consider the following function
1. What is the domain?2. What is the range?3. Where isf continuous?
Decomposition of functions
For each of the following functions,h, find functionsf andg such that (f(g(x)) = h(x)
1.2.
3.
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Graphing
1. Find the equation of the line that passes through the point (1,-1) and has slope 3.2. Find the equation of the line that passes through the origin and the point (2,3).
Solutions
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Solutions
Functions
1. 1. f(0) = 0, f(2) = 42. The domain is
; the range is
,3. No, sincef isn't one-to-one; for example,f( − 1) = f(1) = 1.
2. 1. 1. (f + g)(x) = x + 2 + 1 /x = (x2 + 2x + 1) / x.2. (f − g)(x) = x + 2 − 1 /x = (x2 + 2x − 1) / x.3. (g − f)(x) = 1 / x − x − 2 = (1 −x2 − 2x) / x.4.
.5. (f / g)(x) = x(x + 2) provided
. Note that 0 is not in the domain off / g, since it's not in the domain ofg,and you can't divide by something that doesn't exist!
6. (g / f)(x) = 1 / [x(x + 2)]. Although 0 is still not in the domain, we don'tneed to state it now, since 0 isn't in the domain of the expression 1 / [x(x +2)] either.
7..
8..
2. f(g(2)) = 5 / 2;g(f(2)) = 1 / 4.3. Yes;f − 1(x) = x − 2 andg − 1(x) = 1 / x. Note thatg and its inverse are the same.
3. As pictured, by the Vertical Line test, this graph represents a function.
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Algebra
Functions→Calculus← Precalculus
Algebra
This section is intended to review algebraic manipulation. It is important to understand alge-bra in order to do calculus. If you have a good knowledge of algebra, you should probably justskim this section to be sure you are familiar with the ideas.
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Rules of arithmetic and algebra
The following rules are always true.
• AdditionCommutative Law:•
.• Associative Law:
.• Additive Identity:
.• Additive Inverse:
.• Subtraction
Definition:•
.• Multiplication
Commutative law:•
.• Associative law:
.• Multiplicative Identity:
.• Multiplicative Inverse:
, whenever
• Distributive law:
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.• Division
• Definition:
, whenever
.
The above laws are true for alla, b, andc, whethera, b, andc are numbers, variables, func-tions, or other expressions. For instance,
=
=
=
=
Of course, the above is much longer than simply cancellingx + 3 out in both the numeratorand denominator. But, when you are cancelling, you are really just doing the above steps, so itis important to know what the rules are so as to know when you are allowed to cancel. Occasional-ly people do the following, for instance, which is incorrect:
.
The correct simplification is
,
where the number 2 cancels out in both the numerator and the denominator.
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Interval notation
There are a few different ways that one can express with symbols a specific interval (all thenumbers between two numbers). One way is with inequalities. If we wanted to denote the set ofall numbers between, say, 2 and 4, we could write "allx satisfying 2<x<4." This excludes theendpoints 2 and 4 because we use < instead of
. If we wanted in include the endpoints, we would write "allx satisfying
." This includes the endpoints.
Another way to write these intervals would be with interval notation. If we wished to convey"all x satisfying 2<x<4" we would write (2,4). This doesnot include the endpoints 2 and 4. If wewanted to include the endpoints we would write [2,4]. If we wanted to include 2 and not 4 wewould write [2,4); if we wanted to exclude 2 and include 4, we would write (2,4].
Thus, we have the following table:
Interval notationInequality notationEndpoint conditions
all x satisfyingNot including 2 nor 4
all x satisfyingIncluding 2 not 4
all x satisfyingIncluding 4 not 2
all x satisfyingIncluding both 2 and 4
In general, we have the following table:
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Set NotationInterval Nota-tion
Meaning
All values greater than or equal toa and less than or equal tob
All values greater thana and less thanb
All values greater than or equal toa and less thanb
All values greater thana and less than or equal tob
All values greater than or equal toa.
All values greater thana.
All values less than or equal toa.
All values less thana.
All values.
Note that
and
must always have an exclusive parenthesis rather than an inclusive bracket. This is because
is not a number, and therefore cannot be in our set.
is really just a symbol that makes things easier to write, like the intervals above.
Exponents and radicals
There are a few rules and properties involving exponents and radicals that you'd do well to
remember. As a definition we have that ifn is a positive integer thenan denotesn factors ofa.That is,
If
then we say that
. If n is a positive integer we say that
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If we have an exponent that's a fraction we say that
These definitions yield the following table of properties:
ExampleRule
Factoring and roots
Given the expressionx2 + 3x+ 2, one may ask "what are the values ofx that make this expres-sion 0?" If we factor we obtain
If x=-1 or -2, then one of the factors on the right becomes zero. Therefore, the whole mustbe zero. So, by factoring we have discovered the values ofx that render the expression zero.
These values are termed "roots." In general, given a quadratic polynomialpx2 + qx+ r that factorsas
then we have thatx = -c/a andx = -d/b are roots of the original polynomial.
A special case to be on the look out for is the difference of two squares,a2 − b2. In this case,we are always able to factor as
For example, consider 4x2 − 9. On initial inspection we would see that both 4x2 and 9 are
squares ((2x)2 = 4x2 and 32 = 9). Applying the previous rule we have
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Simplifying rational expressions
Consider the two polynomials
and
When we take the quotient of the two we obtain
The ratio of two polynomials is called arational expression. Many times we would like tosimplify such a beast. For example, say we are given
We may simplify this in the following way:
This is nice because we have obtained something we understand quite well,x − 1, fromsomething we didn't.
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Functions
Graphing linear functions→Calculus← Algebra
Functions
Classical understanding of functions
To provide the classical understanding of functions, think of afunctionas a kind of machine.You feed the machine raw materials, and the machine changes the raw materials into a finishedproduct based on a specific set of instructions. The kinds of functions we consider here, for themost part, take in a real number, change it in a formulaic way, and give out a real number (possi-bly the same as the one it took in). Think of this as aninput-output machine; you give the functionan input, and it gives you an output. For example, the squaring function takes the input 4 andgives the output value 16. The same squaring function takes the input − 1 and gives the outputvalue 1.
A function is usually written asf, g, or something similar - although it doesn't have to be. Afunction is always defined as "of a variable" which tells us what to replace in the formula for thefunction.
For example,
tells us:
• The functionf is a function ofx.• To evaluate the function at a certain number, replace thex with that number.• Replacingxwith that number in the right side of the function will produce the function's
output for that certain input.• In English, the definition of
is interpreted, "Given a number,f will return two more than the triple of that number."
Thus, if we want to know the value (or output) of the function at 3:
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We evaluate the function atx = 3.
The value of
at 3 is 11.
See? It's easy!
Note that
means the value of the dependent variable when
takes on the value of 3. So we see that the number11 is the output of the function when wegive the number3 as the input. We refer to the input as theargumentof the function (or theinde-pendent variable), and to the output as thevalue of the function at the given argument (or thedependent variable). A good way to think of it is the dependent variable
'depends' on the value of the independent variable
. This is read as "the value off at three is eleven", or simply "f of three equals eleven".
Notation
Functions are used so much that there is a special notation for them. The notation is some-what ambiguous, so familiarity with it is important in order to understand the intention of anequation or formula.
Though there are no strict rules for naming a function, it is standard practice to use the lettersf, g, andh to denote functions, and the variablex to denote an independent variable.y is used forboth dependent and independent variables.
When discussing or working with a functionf, it's important to know not only the function,but also its independent variablex. Thus, when referring to a functionf, you usually do not writef, but insteadf(x). The function is now referred to as "f of x". The name of the function is adjacent
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to the independent variable (in parentheses). This is useful for indicating the value of the functionat a particular value of the independent variable. For instance, if
,
and if we want to use the value off for x equal to 2, then we would substitute 2 forx on bothsides of the definition above and write
This notation is more informative than leaving off the independent variable and writingsimply 'f', but can be ambiguous since the parentheses can be misinterpreted as multiplication.
Modern understanding of functions
The formal definition of a function states that a function is actually arule that associates ele-ments of one set called thedomainof the function, with the elements of another set called therangeof the function. For each value we select from the domain of the function, there exists exact-ly one corresponding element in the range of the function. The definition of the function tells uswhich element in the range corresponds to the element we picked from the domain. Classically,the element picked from the domain is pictured as something that is fed into the function and thecorresponding element in the range is pictured as the output. Since we "pick" the element in thedomain whose corresponding element in the range we want to find, we have control over whatelement we pick and hence this element is also known as the "independent variable". The elementmapped in the range is beyond our control and is "mapped to" by the function. This element ishence also known as the "dependent variable", for it depends on which independent variable wepick. Since the elementary idea of functions is better understood from the classical viewpoint,we shall use it hereafter. However, it is still important to remember the correct definition offunctions at all times.
To make it simple, for the functionf(x), all of the possiblex values constitute the domain,and all of the valuesf(x) (y on the x-y plane) constitute the range.
Remarks
The following arise as a direct consequence of the definition of functions:
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1. By definition, for each "input" a function returns only one "output", corresponding tothat input. While the same output may correspond to more than one input, one inputcannot correspond to more than one output. This is expressed graphically as theverticalline test: a line drawn parallel to the axis of the dependent variable (normally vertical)will intersect the graph of a function only once. However, a line drawn parallel to theaxis of the independent variable (normally horizontal) may intersect the graph of afunction as many times as it likes. Equivalently, this has an algebraic (or formula-based) interpretation. We can always say ifa = b, thenf(a) = f(b), but if we only knowthatf(a) = f(b) then we can't be sure thata = b.
2. Each function has a set of values, the function'sdomain, which it can accept as input.Perhaps this set is all positive real numbers; perhaps it is the set {pork, mutton, beef}.This set must be implicitly/explicitly defined in the definition of the function. Youcannot feed the function an element that isn't in the domain, as the function is not de-fined for that input element.
3. Each function has a set of values, the function'srange, which it can output. This maybe the set of real numbers. It may be the set of positive integers or even the set {0,1}.This set, too, must be implicitly/explicitly defined in the definition of the function.
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This is an example of an expression which fails the vertical line test.
The vertical line test
The vertical line test, mentioned in the preceding paragraph, is a systematic test to find outif an equation involvingx andy can serve as a function (withx the independent variable andythe dependent variable). Simply graph the equation and draw a vertical line through each pointof thex-axis. If any vertical line ever touches the graph at more than one point, then the equationis not a function; if the line always touches at most one point of the graph, then the equation isa function.
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(There are a lot of useful curves, like circles, that aren't functions (see picture). Some peoplecall these graphs with multiple intercepts, like our circle, "multi-valued functions"; they wouldrefer to our "functions" as "single-valued functions".)
Important functions
Constant functionIt disregards the input and always outputs the constantc, and is a polyno-
mial of thezerothdegree wheref(x) = cx0= c(1) = c. Its graph is a hori-zontal line.
Linear functionTakes an input, multiplies bymand addsc. It is a polynomial of thefirstdegree. Its graph is a line (slanted, exceptm = 0).
Identity functionTakes an input and outputs it unchanged. A polynomial of thefirst de-
gree,f(x) = x1 = x. Special case of a linear function.
Quadratic functionA polynomial of theseconddegree. Its graph is a parabola, unlessa =0. (Don't worry if you don't know what this is.)
Polynomial functionThe numbern is called thedegree.
Signum function
Determines the sign of the argumentx.
Example functions
Some more simple examples of functions have been listed below.
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Gives 1 if input is positive, -1 if input is negative. Note that the function only acceptsnegative and positive numbers, not 0. Mathematics describes this condition by saying 0is not in the domain of the function.
Takes an input and squares it.
Exactly the same function, rewritten with a different independent variable. This is perfect-ly legal and sometimes done to prevent confusion (e.g. when there are already too manyuses ofx or y in the same paragraph.)
Note that we can define a function by a totally arbitrary rule.
It is possible to replace the independent variable with any mathematical expression, not justa number. For instance, if the independent variable is itself a function of another variable, thenit could be replaced with that function. This is called composition, and is discussed later.
Manipulating functions
Functions can be manipulated in the same ways as variables; they can be added, multiplied,raised to powers, etc. For instance, let
and
.
Then
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,
,
,
.
Composition of functions
However, there is one particular way to combine functions which cannot be done with vari-ables. The value of a functionf depends upon the value of another variablex; however, thatvariable could be equal to another functiong, so its value depends on the value of a third variable.If this is the case, then the first variable is a functionh of the third variable; this function (h) iscalled thecompositionof the other two functions (f andg). Composition is denoted by
.
This can be read as either "f composed with g" or "f of g of x."
For instance, let
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and
.
Then
.
Here,h is the composition off andg and we write
. Note that composition is not commutative:
, and
so
.
Composition of functions is very common, mainly because functions themselves are com-mon. For instance, squaring and sine are both functions:
,
Thus, the expression sin2x is a composition of functions:
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=sin2x
=
.
(Note that this isnot the same as
.) Since the function sine equals 1 / 2 ifx = π / 6,
.
Since the function square equals 1 / 4 ifx = 1 / 2,
.
Transformations
Transformations are a type of function manipulation that are very common. They consist ofmultiplying, dividing, adding or subtracting constants to either the input or the output. Multiply-ing by a constant is calleddilation and adding a constant is calledtranslation. Here are a fewexamples:
Dilation
Translation
Dilation
Translation
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See Picture License InformationHere
Examples of horizontal and vertical translations
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Examples of horizontal and vertical dilationsTranslations and dilations can be either horizontal or vertical. Examples of both vertical and
horizontal translations can be seen at right. The red graphs represent functions in their 'original'state, the solid blue graphs have been translated (shifted) horizontally, and the dashed graphshave been translated vertically.
Dilations are demonstrated in a similar fashion. The function
has had its input doubled. One way to think about this is that now any change in the inputwill be doubled. If I add one tox, I add two to the input off, so it will now change twice asquickly. Thus, this is a horizontal dilation by
because the distance to they-axis has beenhalved. A vertical dilation, such as
is slightly more straightforward. In this case, you double the output of the function. Theoutput represents the distance from thex-axis, so in effect, you have made the graph of thefunction 'taller'. Here are a few basic examples wherea is any positive constant:
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Rotation about originOriginal graph
Horizontal translation bya unitsrightHorizontal translation bya units left
Vertical dilation by a factor ofaHorizontal dilation by a factor ofa
Vertical translation bya unitsupVertical translation bya unitsdown
Reflection abouty-axisReflection aboutx-axis
Domain and Range
Domain
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The domain of the function is the interval from -1 to 1Thedomain of a function is the set of all points over which it is defined. More simply, it
represents the set of x-values which the function can accept as input. For instance, if
thenf(x) is only defined for values ofx between − 1 and 1, because the square root functionis not defined (in real numbers) for negative values. Thus, the domain, in interval notation, is
. In other words,
.
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The range of the function is the interval from 0 to 1
Range
The range of a function is the set of all values which it attains (i.e. the y-values). For in-stance, if:
,
thenf(x) can only equal values in the interval from 0 to 1. Thus, the range off is
.
One-to-one Functions
A function f(x) is one-to-one(or less commonlyinjective) if, for every value off, there isonly one value ofx that corresponds to that value off. For instance, the function
is not one-to-one, because bothx = 1 andx = - 1 result inf(x) = 0. However, the functionf(x) = x + 2 is one-to-one, because, for every possible value off(x), there is exactly one corre-
sponding value ofx. Other examples of one-to-one functions aref(x) = x3 + ax, where
. Note that if you have a one-to-one function and translate or dilate it, it remains one-to-one.(Of course you can't multiplyx or f by a zero factor).
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Horizontal Line Test
If you know what the graph of a function looks like, it is easy to determine whether or notthe function is one-to-one. If every horizontal line intersects the graph in at most one point, thenthe function is one-to-one. This is known as the Horizontal Line Test.
Algebraic 1-1 Test
If you dont know what the graph of the function looks like, it is also easy to determinewhether or not the function is one-to-one. The rule
applies.eg. Is
a 1-1 function?
Therefore by the algebraic 1-1 test, the function
is 1-1
Inverse functions
We callg(x) the inverse function off(x) if, for all x:
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.
A function f(x) has an inverse function if and only iff(x) is one-to-one. For example, the in-verse off(x) = x + 2 isg(x) = x - 2. The function
has no inverse.
Notation
The inverse function off is denoted asf - 1(x). Thus,f - 1(x) is defined as the function thatfollows this rule
f(f − 1(x)) = f − 1(f(x)) = x:
To determinef - 1(x) when given a functionf, substitutef - 1(x) for x and substitutex for f(x).
Then solve forf - 1(x), provided that it is also a function.
Example: Givenf(x) = 2x − 7, find f - 1(x).
Substitutef - 1(x) for x and substitutex for f(x). Then solve forf - 1(x):
To check your work, confirm thatf − 1(f(x)) = x:
f − 1(f(x)) =
f − 1(2x − 7) =
If f isn't one-to-one, then, as we said before, it doesn't have an inverse. Then this methodwill fail.
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Example: Givenf(x) = x2, find f - 1(x).
Substitutef - 1(x) for x and substitutex for f(x). Then solve forf - 1(x):
Since there are two possibilities forf - 1(x), it's not a function. Thusf(x) = x2 doesn't have aninverse. Of course, we could also have found this out from the graph by applying the HorizontalLine Test. It's useful, though, to have lots of ways to solve a problem, since in a specific casesome of them might be very difficult while others might be easy. For example, we might onlyknow an algebraic expression forf(x) but not a graph.
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Graphing linear functions
Precalculus/Exercises→Calculus← Functions
Graphing linear functions
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Graph of y=2xIt is sometimes difficult to understand the behavior of a function given only its definition; a
visual representation or graph can be very helpful. Agraph is a set of points in the Cartesianplane, where each point (x,y) indicates thatf(x) = y. In other words, a graph uses the position ofa point in one direction (thevertical-axisor y-axis) to indicate the value off for a position of thepoint in the other direction (thehorizontal-axisor x-axis).
Functions may be graphed by finding the value off for variousx and plotting the points (x,f(x)) in a Cartesian plane. For the functions that you will deal with, the parts of the function be-tween the points can generally be approximated by drawing a line or curve between the points.Extending the function beyond the set of points is also possible, but becomes increasingly inaccu-rate.
Example
Plotting points like this is laborious. Fortunately, many functions' graphs fall into generalpatterns. For a simple case, consider functions of the form
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The graph off is a single line, passing through the point (0,2) with slope 3. Thus, after plot-ting the point, a straightedge may be used to draw the graph. This type of function is calledlinearand there are a few different ways to present a function of this type.
Slope-intercept form
When we see a function presented as
we call this presentation theslope-intercept form. This is because, not surprisingly, thisway of writing a linear function involves the slope,m, and they-intercept,b.
x+y=7 y-x=7
Point-slope form
If someone walks up to you and gives you one point and a slope, you can draw one line andonly one line that goes through that point and has that slope. Said differently, a point and a slopeuniquely determine a line. So, if given a point (x0,y0) and a slopem, we present the graph as
We call this presentation thepoint-slope form. The point-slope and slope-intercept formare essentially the same. In the point-slope form we can use any point the graph passes through.Where as, in the slope-intercept form, we use they-intercept, that is the point (0,b).
Calculating slope
If given two points, (x1,y1) and (x2,y2), we may then compute the slope of the line that passes
through these two points. Remember, the slope is determined as "rise over run." That is, the slopeis the change iny-values divided by the change inx-values. In symbols,
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So now the question is, "what'sΔy andΔx?" We have thatΔy = y2 − y1 andΔx = x2 - x1.
Thus,
Two-point form
Two points also uniquely determine a line. Given points (x1,y1) and (x2,y2), we have the
equation
This presentation is in thetwo-point form . It is essentially the same as the point-slope formexcept we substitute the expression
for m.
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Limits
Limits/An Introduction toLimits →
Calculus← Precalculus/Exercises
Limits
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• Calculus/Limits/An Introduction to Limits
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• Finite Limits
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• Infinite Limits
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• Continuity
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• Formal Definition of the Limit
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• Calculus/Limits/Exercises
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An Introduction to Limits
Finite Limits→Calculus← Limits/Contents
Limits/An Introduction toLimits
Intuitive Look
A limit looks at what happens to a function when the input approaches a certain value. Thegeneral notation for a limit is as follows:
This is read as "The limit off(x) asx approachesa". We'll take up later the question of howwe can determine whether a limit exists forf(x) at a and, if so, what it is. For now, we'll look atit from an intuitive standpoint.
Let's say that the function that we're interested in isf(x) = x2, and that we're interested in itslimit asx approaches 2. Using the above notation, we can write the limit that we're interested inas follows:
One way to try to evaluate what this limit is would be to choose values near 2, computef(x)for each, and see what happens as they get closer to 2. This is implemented as follows:
1.9991.991.951.91.81.7x
3.9960013.96013.80253.613.242.89f(x) = x2
Here we chose numbers smaller than 2, and approached 2 from below. We can also choosenumbers larger than 2, and approach 2 from above:
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2.0012.012.052.12.22.3x
4.0040014.04014.20254.414.845.29f(x) = x2
We can see from the tables that asx grows closer and closer to 2,f(x) seems to get closerand closer to 4, regardless of whetherx approaches 2 from above or from below. For this reason,
we feel reasonably confident that the limit ofx2 asx approaches 2 is 4, or, written in limit nota-tion,
Now let's look at another example. Suppose we're interested in the behavior of the function
asx approaches 2. Here's the limit in limit notation:
Just as before, we can compute function values asx approaches 2 from below and fromabove. Here's a table, approaching from below:
1.9991.991.951.91.81.7x
-1000-100-20-10-5-3.333
And here from above:
2.0012.012.052.12.22.3x
1000100201053.333
In this case, the function doesn't seem to be approaching any value asx approaches 2. In thiscase we would say that the limit doesn't exist.
Both of these examples may seem trivial, but consider the following function:
This function is the same as
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Note that these functions are really completely identical; not just "almost the same," but actu-ally, in terms of the definition of a function, completely the same; they give exactly the sameoutput for every input.
In algebra, we would simply say that we can cancel the term (x − 2), and then we have the
functionf(x) = x2. This, however, would be a bit dishonest; the function that we have now is notreally the same as the one we started with, because it is defined whenx = 2, and our originalfunction was specifically not defined whenx = 2. In algebra we were willing to ignore this diffi-culty because we had no better way of dealing with this type of function. Now, however, in calcu-lus, we can introduce a better, more correct way of looking at this type of function. What wewant is to be able to say that, although the function doesn't exist whenx = 2, it works almost asthough it does. It may not get there, but it gets really, really close. That is,f(1.99999) = 3.99996.The only question that we have is: what do we mean by "close"?
Informal definition of a limit
As the precise definition of a limit is a bit technical, it is easier to start with an informaldefinition; we'll explain the formal definition later.
We suppose that a functionf is defined forx nearc (but we do not require that it be definedwhenx = c).
Definition: (Informal definition of a limit)We callL the limit of f(x) asx approachesc if f(x) becomes close toL whenx is close (but
not equal) toc.
When this holds we write
or
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Notice that the definition of a limit is not concerned with the value off(x) whenx = c (whichmay exist or may not). All we care about are the values off(x) whenx is close toc, on either theleft or the right (i.e. less or greater).
Limit rules
Now that we have defined, informally, what a limit is, we will list some rules that are usefulfor working with and computing limits. You will be able to prove all these once we formallydefine the fundamental concept of the limit of a function.
First, theconstant rulestates that iff(x) = b (that is,f is constant for allx) then the limit asx approachesc must be equal tob. In other words
Constant Rule for Limits
If b andc are constants then
.
Second, theidentity rule states that iff(x) = x (that is,f just gives back whatever numberyou put in) then the limit off asx approachesc is equal toc. That is,
Identity Rule for Limits
If c is a constant then
.
The next few rules tell us how, given the values of some limits, to compute others.
Operational Identities for LimitsSuppose that
and
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and thatk is constant. Then
•
•
•
•
•
Notice that in the last rule we need to require thatM is not equal to zero (otherwise we wouldbe dividing by zero which is an undefined operation).
These rules are known asidentities; they are the scalar product, sum, difference, product,and quotient rules for limits. (A scalar is a constant, and, when you multiply a function by aconstant, we say that you are performingscalar multiplication .)
Using these rules we can deduce another. Namely, using the rule for products many timeswe get that
for a positive integern.
This is called thepower rule.
Examples
Example 1
Find the limit
.
We need to simplify the problem, since we have no rules about this expression by itself. Weknow from the identity rule above that
. By the power rule,
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. Lastly, by the scalar multiplication rule, we get
.
Example 2
Find the limit
.
To do this informally, we split up the expression, once again, into its components. As above,
.
Also
and
. Adding these together gives
.
Example 3
Find the limit
.
From the previous example the limit of the numerator is
. The limit of the denominator is
As the limit of the denominator is not equal to zero we can divide. This gives
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.
Example 4
Find the limit
.
We apply the same process here as we did in the previous set of examples;
.
We can evaluate each of these;
and
Thus, the answer is
.
Example 5
Find the limit
.
To evaluate this seemingly complex limit, we will need to recall some sine and cosineidentities. We will also have to use two new facts. First, iff(x) is a trigonometric function (thatis, one of sine, cosine, tangent, cotangent, secant or cosecant) and is defined ata, then
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. Second,
.
To evaluate the limit, recognize that 1 − cosx can be multiplied by 1 + cosx to obtain (1 −
cos2x) which, by our trig identities, is sin2x. So, multiply the top and bottom by 1 + cosx. (Thisis allowed because it is identical to multiplying by one.) This is a standard trick for evaluatinglimits of fractions; multiply the numerator and the denominator by a carefully chosen expressionwhich will make the expression simplify somehow. In this case, we should end up with:
.
Our next step should be to break this up into
by the product rule. As mentioned above,
.
Next,
.
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Thus, by multiplying these two results, we obtain 0.
We will now present an amazingly useful result, even though we cannot prove it yet. Wecan find the limit atc of any polynomial or rational function, as long as that rational function isdefined atc (so we are not dividing by zero). That is,c must be in the domain of the function.
Limits of Polynomials and Rational functionsIf f is a polynomial or rational function that is defined atc then
We already learned this for trigonometric functions, so we see that it is easy to find limitsof polynomial, rational or trigonometric functions wherever they are defined. In fact, this is trueeven for combinations of these functions; thus, for example,
.
The Squeeze Theorem
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Graph showingf being squeezed betweeng andhThe Squeeze Theorem is very important in calculus, where it is typically used to find the
limit of a function by comparison with two other functions whose limits are known.
It is called the Squeeze Theorem because it refers to a functionf whose values are squeezedbetween the values of two other functionsg andh, both of which have the same limitL. If thevalue off is trapped between the values of the two functionsg andh, the values off must alsoapproachL.
Expressed more precisely:
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Theorem: (Squeeze Theorem)Suppose that
holds for allx in some open interval containinga, except possibly atx = a itself. Suppose alsothat
. Then
also.
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Plot of x*sin(1/x) for -0.5 < x <0.5Example: Compute
. Note that the sine of anything is in the interval [ − 1,1]. That is,
for all x. If x is positive, we can multiply these inequalities byx and get
. If x is negative, we can similarly multiply the inequalities by the positive number -x andget
. Putting these together, we can see that, for all nonzerox,
. But it's easy to see that
. So, by the Squeeze Theorem,
.
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Finding limits
Now, we will discuss how, in practice, to find limits. First, if the function can be built outof rational, trigonometric, logarithmic and exponential functions, then if a numberc is in thedomain of the function, then the limit atc is simply the value of the function atc.
If c is not in the domain of the function, then in many cases (as with rational functions) thedomain of the function includes all the points nearc, but notc itself. An example would be if wewanted to find
, where the domain includes all numbers besides 0.
In that case, in order to find
we want to find a functiong(x) similar to f(x), except with the hole atc filled in. The limitsof f andg will be the same, as can be seen from the definition of a limit. By definition, the limitdepends onf(x) only at the points wherex is close toc but not equal to it, so the limit atc doesnot depend on the value of the function atc. Therefore, if
,
also. And since the domain of our new functiong includesc, we can now (assumingg is stillbuilt out of rational, trigonometric, logarithmic and exponential functions) just evaluate it atc asbefore. Thus we have
.
In our example, this is easy; canceling thex's givesg(x) = 1, which equalsf(x) = x / x at allpoints except 0. Thus, we have
. In general, when computing limits of rational functions, it's a good idea to look for commonfactors in the numerator and denominator.
Lastly, note that the limit might not exist at all. There are a number of ways in which thiscan occur:
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"Gap"There is a gap (not just a single point) where the function is not defined. As an example,in
does not exist when
. There is no way to "approach" the middle of the graph. Note that the function also has nolimit at the endpoints of the two curves generated (atc = − 4 andc = 4). For the limit toexist, the point must be approachable fromboththe left and the right. Note also that thereis no limit at a totally isolated point on the graph.
"Jump"If the graph suddenly jumps to a different level, there is no limit. For example, letf(x) bethe greatest integer
. Then, ifc is an integer, whenx approachesc from the rightf(x) = c, while whenx ap-proachesc from the leftf(x) = c − 1. Thus
will not exist.
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A graph of 1/(x2) on the interval [-2,2].Vertical asymptote
In
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the graph gets arbitrarily high as it approaches 0, so there is no limit. (In this case wesometimes say the limit is infinite; see the next section.)
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A graph of sin(1/x) on the interval (0,1/π].Infinite oscillation
These next two can be tricky to visualize. In this one, we mean that a graph continuallyrises above and falls below a horizontal line. In fact, it does this infinitely often as you ap-proach a certainx-value. This often means that there is no limit, as the graph never ap-proaches a particular value. However, if the height (and depth) of each oscillation diminish-es as the graph approaches thex-value, so that the oscillations get arbitrarily smaller, thenthere might actually be a limit.
The use of oscillation naturally calls to mind the trigonometric functions. An example ofa trigonometric function that does not have a limit asx approaches 0 is
As x gets closer to 0 the function keeps oscillating between − 1 and 1. In fact, sin(1 /x)oscillates an infinite number of times on the interval between 0 and any positive value ofx. The sine function is equal to zero wheneverx = kπ, wherek is a positive integer. Betweenevery two integersk, sinx goes back and forth between 0 and − 1 or 0 and 1. Hence, sin(1/ x) = 0 for everyx = 1 / (kπ). In between consecutive pairs of these values, 1 / (kπ) and 1/ [(k + 1)π], sin(1 /x) goes back and forth from 0, to either − 1 or 1 and back to 0. We mayalso observe that there are an infinite number of such pairs, and they are all between 0 and1 / π. There are a finite number of such pairs between any positive value ofx and 1 /π, sothere must be infinitely many between any positive value ofx and 0. From our reasoning
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we may conclude that, asx approaches 0 from the right, the function sin(1 /x) does notapproach any specific value. Thus,
does not exist.
Using limit notation to describe asymptotes
Now consider the function
What is the limit asx approaches zero? The value ofg(0) does not exist; it is not defined.
Notice, also, that we can makeg(x) as large as we like, by choosing a smallx, as long as
. For example, to makeg(x) equal to one trillion, we choosex to be 10- 6. Thus,
does not exist.
However, wedoknow something about what happens tog(x) whenx gets close to 0 withoutreaching it. We want to say we can makeg(x) arbitrarily large (as large as we like) by takingxto be sufficiently close to zero, but not equal to zero. We express this symbolically as follows:
Note that the limit does not exist at 0; for a limit, being
is a special kind of not existing. In general, we make the following definition.
Definition: Informal definition of a limit being
We say thelimit of f(x) asx approachesc is infinity if f(x) becomes very big (as big as welike) whenx is close (but not equal) toc.
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In this case we write
or
.
Similarly, we say thelimit of f(x) as x approachesc is negative infinity if f(x) becomesvery negative whenx is close (but not equal) toc.
In this case we write
or
.
An example of the second half of the definition would be that
.
Key application of limits
To see the power of the concept of the limit, let's consider a moving car. Suppose we havea car whose position is linear with respect to time (that is, a graph plotting the position with re-spect to time will show a straight line). We want to find the velocity. This is easy to do from alge-bra; we just take the slope, and that's our velocity.
But unfortunately, things in the real world don't always travel in nice straight lines. Carsspeed up, slow down, and generally behave in ways that make it difficult to calculate their veloci-ties.
Now what we really want to do is to find the velocity at a given moment (the instantaneousvelocity). The trouble is that in order to find the velocity we need two points, while at any given
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time, we only have one point. We can, of course, always find the average speed of the car, giventwo points in time, but we want to find the speed of the car at one precise moment.
This is the basic trick of differential calculus, the first of the two main subjects of this book.We take the average speed at two moments in time, and then make those two moments in timecloser and closer together. We then see what the limit of the slope is as these two moments intime are closer and closer, and say that this limit is the slope at a single instant.
We will study this process in much greater depth later in the book. First, however, we willneed to study limits more carefully.
External links
• Online interactive exercises on limits• Tutorials for the calculus phobe
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Exercises
Differentiation→Calculus← Formal Definition of theLimit
Limits/Exercises
Limits with Graphs
Given the following graph, evaluate the succeeding limits
Basic Limit Exercises
1.
2.
Solutions
One Sided Limits
Evaluate the following limits or state that the limit does not exist.
1.
2.
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Two Sided Limits
Evaluate the following limits or state that the limit does not exist.
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1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
Solutions
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Limits to Infinity
Evaluate the following limits or state that the limit does not exist.
1.
2.
3.
4.
5.
6.
7.
Limits of Piece Functions
Evaluate the following limits or state that the limit does not exist.
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1. Consider the function
1.
2.
3.
2. Consider the function
1.
2.
3.
4.
5.
6.
3. Consider the function
1.
2.
3.
4.
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Contents
Limits/An Introduction toLimits →
Calculus← Precalculus/Exercises
Limits
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• Calculus/Limits/An Introduction to Limits
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• Finite Limits
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• Infinite Limits
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• Continuity
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• Formal Definition of the Limit
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• Calculus/Limits/Exercises
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Solutions
Basic Limit Exercises
(1)
Since this is a polynomial, two can simply be plugged in. This results in4(4)-2(3)+1=16-6+1=11
11
(2)
25
Harder Limit Exercises
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This page or section ofCalculusis astub.You can help Wikibooks byexpanding it.
(3)
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(4)D.N.E.(5)-6(6)6(7)3(8)13/8(9)10(10)D.N.E.(11)+infinity
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Finite Limits
Infinite Limits →Calculus← Limits/An Introduction toLimits
Finite Limits
Informal Finite Limits
Now, we will try to more carefully restate the ideas of the last chapter. We said then that theequation
meant that, whenx gets close to 2,f(x) gets close to 4. What exactly does this mean? Howclose is "close"? The first way we can approach the problem is to say that, atx = 1.99,f(x) =3.9601, which is pretty close to 4.
Sometimes however, the function might do something completely different. For instance,
supposef(x) = x4 − 2x2 − 3.77, sof(1.99) = 3.99219201. Next, if you take a value even closer to2, f(1.999) = 4.20602, in this case you actually move further from 4. As you can see here, theproblem with some functions is that, no matter how close we get, we can never be sure what theydo.
The solution is to find out what happensarbitrarily close to the point. In particular, we wantto say that, no matter how close we want the function to get to 4, if we makex close enough to2 then it will get there. In this case, we will write
and say "The limit off(x), asx approaches 2, equals 4" or "Asx approaches 2,f(x) approach-es 4." In general:
Definition: (New definition of a limit)
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We callL thelimit of f(x) asx approachesc if f(x) becomesarbitrarily close to L whenev-er x is sufficiently close(and not equal) toc.
When this holds we write
or
One-Sided Limits
Sometimes, it is necessary to consider what happens when we approach anx value from oneparticular direction. To account for this, we have one-sided limits. In a left-handed limit,x ap-proachesa from the left-hand side. Likewise, in a right-handed limit,x approachesa from theright-hand side.
For example, if we consider
, there is a problem because there is no way forx to approach 2 from the left hand side (thefunction is undefined here). But, ifx approaches 2 only from the right-hand side, we want to saythat
approaches 0.
Definition: (Informal definition of a one-sided limit)We callL the limit of f(x) as x approachesc from the right if f(x) becomesarbitrarily
closeto L wheneverx is sufficiently closeto andgreater than c.
When this holds we write
Similarly, we callL the limit of f(x) asx approachesc from the left if f(x) becomesarbi-trarily close to L wheneverx is sufficiently closeto andless thanc.
When this holds we write
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In our example, the left-handed limit
does not exist.
The right-handed limit, however,
.
It is a fact that
exists if and only if
and
exist and are equal to each other. In this case,
will be equal to the same number.
In our example, one limit does not even exist. Thus
does not exist either.
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Infinite Limits
Continuity→Calculus← Finite Limits
Infinite Limits
Informal infinite limits
Another kind of limit involves looking at what happens tof(x) asx gets very big. For exam-ple, consider the functionf(x) = 1 / x. As x gets very big, 1 /x gets very small. In fact, 1 /x getscloser and closer to zero the biggerx gets. Without limits it is very difficult to talk about this fact,becausex can keep getting bigger and bigger and 1 /x never actually gets to zero; but the lan-guage of limits exists precisely to let us talk about the behavior of a function as it approachessomething - without caring about the fact that it will never get there. In this case, however, wehave the same problem as before: how big doesx have to be to be sure thatf(x) is really goingtowards 0?
In this case, we want to say that, however close we wantf(x) to get to 0, forx big enoughf(x) is guaranteed to get that close. So we have yet another definition.
Definition: (Definition of a limit at infinity)We callL the limit of f(x) asx approaches infinity if f(x) becomesarbitrarily close to L
wheneverx is sufficiently large.
When this holds we write
or
Similarly, we callL thelimit of f(x) asx approaches negative infinityif f(x) becomesarbi-trarily close to L wheneverx is sufficiently negative.
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When this holds we write
or
So, in this case, we write:
and say "The limit, as x approaches infinity, equals 0," or "as x approaches infinity, thefunction approaches 0".
We can also write:
because makingx very negative also forces 1 /x to be close to 0.
Notice, however, thatinfinity is not a number; it's just shorthand for saying "no matter howbig." Thus, this is not the same as the regular limits we learned about in the last two chapters.
Limits at infinity of rational functions
One special case that comes up frequently is when we want to find the limit at
(or
) of a rational function. A rational function is just one made by dividing two polynomials by
each other. For example,f(x) = (x3 + x − 6) / (x2 − 4x + 3) is a rational function. Also, any polyno-mial is a rational function, since 1 is just a (very simple) polynomial, so we can write the function
f(x) = x2 − 3 asf(x) = (x2 − 3) / 1, the quotient of two polynomials.
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There is a simple rule for determining a limit of a rational function as the variable approachesinfinity. Look for the term with the highest exponent in the numerator. Look for the same in thedenominator. This rule is based on that information.
• If the exponent of the highest term in the numerator matches the exponent of the highestterm in the denominator, the limit (at both
and
) is the ratio of the coefficients of the highest terms.
• If thenumeratorhas the highest term, then the fraction is called "top-heavy" and neitherlimit (at
or at
) exists.
• If the denominatorhas the highest term, then the fraction is called "bottom-heavy" andthe limit (at both
and
) is zero.
Note that, if the numerator or denominator is a constant (including 1, as above), then this is
the same asx0. Also, a straight power ofx, like x3, has coefficient 1, since it is the same as 1x3.
Example
Find
.
The functionf(x) = (x − 5) / (x − 3) is the quotient of two polynomials,x − 5 andx − 3. Byour rule we look for the term with highest exponent in the numerator; it'sx. The term with highestexponent in the denominator is alsox. So, the limit is the ratio of their coefficients. Sincex = 1x,both coefficients are 1, so
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.
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Infinity is not a number
Most people seem to struggle with this fact when first introduced to calculus, and in particu-lar limits.
But
is different.
is not a number.
Mathematics is based on formal rules that govern the subject. When a list of formal rulesapplies to a type of object (e.g., "a number") those rules mustalwaysapply — no exceptions!
What makes
different is this: "there is no number greater than infinity". You can write down the formulain a lot of different ways, but here's one way:
. If you add one to infinity, you still have infinity; you don't have a bigger number. If youbelieve that, then infinity is not a number.
Since
does not follow the rules laid down for numbers, it cannot be a number. Every time you usethe symbol
in a formula where you would normally use a number, you have to interpret the formuladifferently. Let's look at how
does not follow the rules that every actual number does:
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Addition breaks
Every number has a negative, and addition is associative. For
we could write
and note that
. This is a good thing, since it means we can prove if you take one away from infinity, youstill have infinity:
But it also means we can prove 1 = 0, which is not so good.
.
Reinterpret formulas that use
We started off with a formula that does "mean" something, even though it used
and
is not a number.
What does this mean, compared to what it means when we have a regular number insteadof an infinity symbol:
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This formula says that I can make sure the values of
don't differ very much from
, so long I can control how muchx varies away from 2. I don't have to make
exactly equal to
, but I also can't controlx too tightly. I have to give you a range to varyx within. It's justgoing to be very, very small (probably) if you want to make
very very close to
. And by the way, it doesn't matter at all what happens whenx = 2.
If we could use the same paragraph as a template for my original formula, we'll see someproblems. Let's substitute 0 for 2, and
for
.
This formula says that I can make sure the values of
don't differ very much from
, so long I can control how muchx varies away from 0. I don't have to make
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exactly equal to
, but I also can't controlx too tightly. I have to give you a range to varyx within. It's just goingto be very, very small (probably) if you want to see that
gets very, veryclose to
. And by the way, it doesn't matter at all what happens whenx = 0.
It's close to making sense, but it isn't quite there. It doesn't make sense to say that some realnumber is really "close" to
. For example, whenx = .001 and
does it really makes sense to say 1000 is closer to
than 1 is? Solve the following equations forδ:
No real number is very close to
; that's what makes
so special! So we have to rephrase the paragraph:
This formula says that I can make sure the values of
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get as big as any number you pick, so long I can control how muchx varies away from 0. Idon't have to make
bigger than every number, but I also can't controlx too tightly. I have to give you a range tovaryx within. It's just going to be very, very small (probably) if you want to see that
gets very, verylarge. And by the way, it doesn't matter at all what happens whenx = 0.
You can see that the essential nature of the formula hasn't changed, but the exact details re-quire some human interpretation. While rigorous definitions and clear distinctions are essentialto the study of mathematics, sometimes a bit of casual rewording is okay. You just have to makesure you understand what a formula really means so you can draw conclusions correctly.
Exercises
Write out an explanatory paragraph for the following limits that include
. Remember that you will have to change any comparison of magnitude between a realnumber and
to a different phrase. In the second case, you will have to work out for yourself what theformula means.
1.
2.
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Continuity
Formal Definition of the Limit→
Calculus← Infinite Limits
Continuity
Defining Continuity
We are now ready to define the concept of a function beingcontinuous. The idea is that wewant to say that a function is continuous if you can draw its graph without taking your pencil offthe page. But sometimes this will be true for some parts of a graph but not for others. Therefore,we want to start by defining what it means for a function to be continuous atone point. The defini-tion is simple, now that we have the concept of limits:
Definition: (continuity at a point)If f(x) is defined on an open interval containingc, thenf(x) is said to becontinuous atc if
and only if
.Note that forf to be continuous atc, the definition in effect requires three conditions:
1. thatf is defined atc, sof(c) exists,2. the limit asx approachesc exists, and3. the limit andf(c) are equal.
If any of these do not hold thenf is not continuous atc.
The idea of the definition is that the point of the graph corresponding toc will be close tothe points of the graph corresponding to nearbyx-values. Now we can define what it means fora function to be continuous in general, not just at one point.
Definition: (continuity)A function is said to becontinuous on (a,b) if it is continuous at every point of the interval (a,b).
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We often use the phrase "the function is continuous" to mean that the function is continuousat every real number. This would be the same as saying the function was continuous on (−∞, ∞),but it is a bit more convenient to simply say "continuous".
Note that, by what we already know, the limit of a rational, exponential, trigonometric orlogarithmic function at a point is just its value at that point, so long as it's defined there. So, allsuch functions are continuous wherever they're defined. (Of course, they can't be continuouswhere they'renotdefined!)
Discontinuities
A discontinuity is a point where a function is not continuous. There are lots of possibleways this could happen, of course. Here we'll just discuss two simple ways.
Removable Discontinuities
The function
is not continuous atx = 3. It is discontinuous at that point because the fraction then becomes
, which is undefined. Therefore the function fails the first of our three conditions for continu-ity at the point 3; 3 is just not in its domain.
However, we say that this discontinuity isremovable. This is because, if we modify thefunction at that point, we can eliminate the discontinuity and make the function continuous. Tosee how to make the functionf(x) continuous, we have to simplifyf(x), getting
. We can define a new functiong(x) whereg(x) = x + 3. Note that the functiong(x) is notthe same as the original functionf(x), becauseg(x) is defined atx = 3, while f(x) is not. Thus,g(x) is continuous atx = 3, since
. However, whenever
, f(x) = g(x); all we did tof to getg was to make it defined atx = 3.
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In fact, this kind of simplification is often possible with a discontinuity in a rational function.We can divide the numerator and the denominator by a common factor (in our examplex − 3)to get a function which is the same except where that common factor was 0 (in our example atx= 3). This new function will be identical to the old except for being defined at new points wherepreviously we had division by 0.
Unfortunately this is not possible in every case. For example, the function
has a common factor ofx − 3 in both the numerator and denominator, but when you simplifyyou are left with
. Which is still not defined atx = 3. In this case the domain off(x) andg(x) are the same, andthey are equal everywhere they are defined, so they are in fact the same function. The reason thatg(x) differed fromf(x) in the first example was because we could take it to have a larger domainand not simply that the formulas definingf(x) andg(x) were different.
Jump Discontinuities
Unfortunately, not all discontinuities can be removed from a function. Consider this function:
Since
does not exist, there is no way to redefinek at one point so that it will be continuous at 0.These sorts of discontinuities are callednonremovablediscontinuities.
Note, however, that both one-sided limits exist;
and
. The problem is that they are not equal, so the graph "jumps" from one side of 0 to the other.In such a case, we say the function has ajumpdiscontinuity. (Note that a jump discontinuity isa kind of nonremovable discontinuity.)
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One-Sided Continuity
Just as a function can have a one-sided limit, a function can be continuous from a particularside. For a function to be continuous at a point from a given side, we need the following threeconditions:
1. the function is defined at the point,2. the function has a limit from that side at that point and3. the one-sided limit equals the value of the function at the point.
A function will be continuous at a point if and only if it is continuous from both sides at thatpoint. Now we can define what it means for a function to be continuous on a closed interval.
Definition: (continuity on a closed interval)A function is said to becontinuous on [a,b] if and only if
1. it is continuous on (a,b),2. it is continuous from the right ata and3. it is continuous from the left atb.
Notice that, if a function is continuous, then it is continuous on every closed interval con-tained in its domain.
Intermediate Value Theorem
The definition of continuity we've given might not seem to have much to do with the intu-itive notion we started with of being able to draw the graph without lifting one's pencil. Fortunate-ly, there is a connection, given by the so-called intermediate value theorem, which says, informal-ly, that if a function is continuous then its graph can be drawn without ever picking up one'spencil. More precisely:
Intermediate Value TheoremIf a functionf is continuous on a closed interval [a,b], then for every valuek betweenf(a) andf(b) there is a valuec betweena andb such thatf(c) = k.
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Application: bisection method
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A few steps of the bisection method applied over the starting range [a1;b1]. The bigger red dotis the root of the function.
The bisection method is the simplest and most reliable algorithm to find zeros of a continu-ous function.
Suppose we want to solve the equationf(x) = 0. Given two pointsa andb such thatf(a) andf(b) have opposite signs, the intermediate value theorem tells us thatf must have at least one rootbetweena andb as long asf is continuous on the interval [a,b]. If we know f is continuous ingeneral (say, because it's made out of rational, trigonometric, exponential and logarithmic func-tions), then this will work so long asf is defined at all points betweena andb. So, let's dividethe interval [a,b] in two by computingc = (a + b) / 2. There are now three possibilities:
1. f(c) = 0,2. f(a) andf(c) have opposite signs, or3. f(c) andf(b) have opposite signs.
In the first case, we're done. In the second and third cases, we can repeat the process on thesub-interval where the sign change occurs. In this way we home in to a small sub-interval contain-ing the zero. The midpoint of that small sub-interval is usually taken as a good approximation tothe zero.
Note that, unlike the methods you may have learned in algebra, this works foranycontinuousfunction that you (or your calculator) know how to compute.
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Formal Definition of the Limit
Limits/Exercises→Calculus← Continuity
Formal Definition of theLimit
In preliminary calculus, the concept of a limit is probably the most difficult one to grasp (ifnothing else, it took some of the most brilliant mathematicians 150 years to arrive at it); it is alsothe most important and most useful.
The intuitive definition of a limit is inadequate to prove anything rigorously about it. Theproblem lies in the vague term "arbitrarily close". We discussed earlier that the meaning of thisterm is that the closerx gets to the specified value, the closer the function must get to the limit,so that however close we want the function to the limit, we can accomplish this by makingxsufficiently close to our value. We can express this requirement technically as follows:
Definition: (Formal definition of a limit)Let f(x) be a function defined on an open intervalD that containsc, except possibly atx =
c. Let L be a number. Then we say that
if, for every
, there exists aδ > 0 such that for all
with
we have
.
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To further explain, earlier we said that "however close we want the function to the limit, wecan find a correspondingx close to our value." Using our new notation of epsilon (
) and delta (δ), we mean that if we want to makef(x) within
of L, the limit, then we know that makingx within δ of c puts it there.
Again, since this is tricky, let's resume our example from before:f(x) = x2, atx = 2. To start,let's say we wantf(x) to be within .01 of the limit. We know by now that the limit should be 4,so we say: for
, there is someδ so that as long as
, then
To show this, we can pickanyδ that is bigger than 0, so long as it works. For example, youmight pick .00000000000001, because you are absolutely sure that ifx is within.00000000000001 of 2, thenf(x) will be within .01 of 4. Thisδ works for
. But we can't just pick a specific value for
, like .01, because we said in our definition "forevery
." This means that we need to be able to show an infinite number ofδs, one for each
. We can't list an infinite number ofδs!
Of course, we know of a very good way to do this; we simply create a function, so that forevery
, it can give us aδ. In this case, it's a rather simple function; all we need is
.
So, in general, how do you show thatf(x) tends toL asx tends toc? Well imagine somebodygave you a small number
(e.g., say
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). Then you have to find aδ > 0 and show that whenever
we have |f(x) − L | < 0.03. Now if that person gave you a smaller
(say
) then you would have to find anotherδ, but this time with 0.03 replaced by 0.002. If you can dothis foranychoice of
then you have shown thatf(x) tends toL asx tends toc. Of course, the way you would do this ingeneral would be to create a function giving you aδ for every
, just as in the example above.
Formal Definition of the Limit at Infinity
Definition: (Limit of a function at infinity)We callL the limit of f(x) asx approaches
if for every number
there exists aδ such that wheneverx > δ we have
When this holds we write
or
as
Similarly, we callL the limit of f(x) asx approaches
if for every number
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, there exists a numberδ such that wheneverx < δ we have
When this holds we write
or
as
Notice the difference in these two definitions. For the limit off(x) asx approaches
we are interested in thosex such thatx > δ. For the limit off(x) asx approaches
we are interested in thosex such thatx < δ.
Examples
Here are some examples of the formal definition.
Example 1
We know from earlier in the chapter that
.
What isδ when
for this limit?
We start with the desired conclusion and substitute the given values forf(x) and
:
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.
Then we solve the inequality forx:
7.96 <x < 8.04
This is the same as saying
- 0.04 <x - 8 < 0.04.
(We want the thing in the middle of the inequality to bex − 8 because that's where we'retaking the limit.) We normally choose the smaller of
and 0.04 forδ, soδ = 0.04, but any smaller number will also work.
Example 2
What is the limit off(x) = x + 7 asx approaches 4?
There are two steps to answering such a question; first we must determine the answer —this is where intuition and guessing is useful, as well as the informal definition of a limit — andthen we must prove that the answer is right.
In this case, 11 is the limit because we knowf(x) = x + 7 is a continuous function whosedomain is all real numbers. Thus, we can find the limit by just substituting 4 in forx, so the an-swer is 4 + 7 = 11.
We're not done, though, because we never proved any of the limit laws rigorously; we juststated them. In fact, we couldn't have proved them, because we didn't have the formal definitionof the limit yet, Therefore, in order to be sure that 11 is the right answer, we need to prove thatno matter what value of
is given to us, we can find a value ofδ such that
whenever
For this particular problem, letting
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works (seechoosing deltafor help in determining the value ofδ to use in other problems). Now,we have to prove
given that
.
Since
, we know
which is what we wished to prove.
Example 3
What is the limit off(x) = x2 asx approaches 4?
As before, we use what we learned earlier in this chapter to guess that the limit is 42 = 16.Also as before, we pull out of thin air that
.
Note that, since
is always positive, so isδ, as required. Now, we have to prove
given that
.
We know that
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(because of the triangle inequality), so
Example 4
Show that the limit of sin(1 /x) asx approaches 0 does not exist.
We will proceed by contradiction. Suppose the limit exists; call itL. For simplicity, we'llassume that
; the case forL = 1 is similar. Choose
. Then if the limit wereL there would be someδ > 0 such that
for everyx with
. But, for everyδ > 0, there exists some (possibly very large)n such that
, but | sin(1 /x0) − L | = | 1 −L | , a contradiction.
Example 5
What is the limit ofxsin(1 /x) asx approaches 0?
By the Squeeze Theorem, we know the answer should be 0. To prove this, we let
. Then for allx, if 0 < | x | < δ, then
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as required.
Example 6
Suppose that
and
. What is
?
Of course, we know the answer should beL + M, but now we can prove this rigorously.Given some
, we know there's aδ1 such that, for anyx with
,
(since the definition of limit says "for any
", so it must be true for
as well). Similarly, there's aδ2 such that, for anyx with
,
. We can setδ to be the lesser ofδ1 andδ2. Then, for anyx with
,
, as required.
If you like, you can prove the other limit rules too using the new definition. Mathematicianshave already done this, which is how we know the rules work. Therefore, when computing alimit from now on, we can go back to just using the rules and still be confident that our limit iscorrect according to the rigorous definition.
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Formal Definition of a Limit Being Infinity
Definition: (Formal definition of a limit being infinity)Let f(x) be a function defined on an open intervalD that containsc, except possibly atx =
c. Then we say that
if, for every
, there exists aδ > 0 such that for all
with
we have
.
When this holds we write
or
as
.
Similarly, we say that
if, for every
, there exists aδ > 0 such that for all
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with
we have
.
When this holds we write
or
as
.
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Differentiation
Differentiation/DifferentiationDefined→
Calculus← Limits/Exercises
Differentiation
Basics of Differentiation
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• Differentiation Defined• Product and Quotient Rules• Derivatives of Trigonometric Functions• Chain Rule• More differentiation rules- More rules for differentiation• Higher Order derivatives- An introduction to second power derivatives• Implicit Differentiation• Derivatives of Exponential and Logarithm Functions• Exercises
Applications of Derivatives
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• Extrema and Points of Inflection• Newton's Method• Related Rates• Kinematics• Optimization• Euler's Method• Exercises
Important Theorems
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• Extreme Value Theorem• Rolle's Theorem• Mean Value Theorem
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Differentiation Defined
Product and Quotient Rules→Calculus← Limits/Exercises
Differentiation/Differentia-tion Defined
What is differentiation?
Differentiation is an operation that allows us to find a function that outputs therate ofchangeof one variable with respect to another variable when given that second variable.
Informally, we may suppose that we're tracking the position of a car on a two-lane road withno passing lanes. Assuming the car never pulls off the road, we can abstractly study the car'sposition by assigning it a variable,x. Since the car's position changes as the time changes, wesay thatx is dependent on time, orx = x(t). This tells where the car is at each specific time. Differ-entiation gives us a functiondx / dt which represents the car's speed, that is the rate of change ofits position with respect to time.
Equivalently, differentiation gives us the slope at any point of the graph of a non-linearfunction. For a linear function, of formf(x) = ax+ b, a is the slope. For non-linear functions, such
asf(x) = 3x2, the slope can depend onx; differentiation gives us a function which represents thisslope.
The Definition of Slope
Historically, the primary motivation for the study ofdifferentiation was the tangent lineproblem: for a given curve, find the slope of the straight line that is tangent to the curve at agiven point. The word tangent comes from the Latin wordtangens, which means touching. Thus,to solve the tangent line problem, we need to find the slope of a line that is "touching" a given
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curve at a given point, or, in modern language, that has the same slope. But what exactly do wemean by "slope" for a curve?
The solution is obvious in some cases: for example, a liney = mx+ c is its own tangent; the
slope at any point ism. For the parabolay = x2, the slope at the point (0,0) is 0; the tangent lineis horizontal.
But how can you find the slope of, say,y = sinx + x2 atx = 1.5? This is in general a nontrivialquestion, but first we will deal carefully with the slope of lines.
Of a Line
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Three lines with different slopesTheslopeof a line, also called thegradient of the line, is a measure of its inclination. A line
that is horizontal has slope 0, a line from the bottom left to the top right has a positive slope anda line from the top left to the bottom right has a negative slope.
The slope can be defined in two (equivalent) ways. The first way is to express it as howmuch the line climbs for a given motion horizontally. We denote a change in a quantity usingthe symbolΔ (pronounced "delta"). Thus, a change inx is written asΔx. We can therefore writethis definition of slope as:
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An example may make this definition clearer. If we have two points on a line,
and
, the change inx from P to Q is given by:
Likewise, the change iny from P to Q is given by:
This leads to the very important result below.
The slope of the line between the points (x1,y1) and (x2,y2) is
.
Alternatively, we can define slope trigonometrically, using the tangent function:
whereα is the angle from the line to the rightward-pointing horizontal (measured clockwise).If you recall that the tangent of an angle in a right triangle is defined as the length of the sideopposite the angle over the length of the leg adjacent to the angle, you should be able to spot theequivalence here.
Of a Graph of a Function
The graphs of most functions we are interested in are not straight lines (although they canbe) but rather curves. We cannot define the slope of a curve in the same way as we can for a line.
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In order for us to understand how to find the slope of a curve at a point, we will first have tocover the idea oftangency. Intuitively, atangent is a line whichjust touches a curve at a point,such that the angle between them at that point is zero. Consider the following four curves andlines:
(ii)(i)
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(iv)(iii)
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1. The lineL crosses, but is not tangent toC atP.2. The lineL crosses, and is tangent toC atP.3. The lineL crossesC at two points, but is tangent toC only atP.4. There are many lines that crossC atP, but none are tangent. In fact, this curve has no
tangent atP.
A secantis a line drawn through two points on a curve. We can construct a definition of atangent as the limit of a secant of the curve taken as the separation between the points tends tozero. Consider the diagram below.
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As the distanceh tends to zero, the secant line becomes the tangent at the pointx0. The two
points we draw our line through are:
and
As a secant line is simply a line and we know two points on it, we can find its slope,mh,
using the formula from before:
(We will refer to the slope asmh because it may, and generally will, depend onh.) Substitut-
ing in the points on the line,
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This simplifies to
This expression is called thedifference quotient. Note thath can be positive or negative —it is perfectly valid to take a secant through any two points on the curve — but cannot be 0.
The definition of the tangent line we gave was not rigorous, since we've only defined limitsof numbers— or, more precisely, of functions that output numbers — not oflines. But wecandefine theslopeof the tangent line at a point rigorously, by taking the limit of the slopes of thesecant lines from the last paragraph. Having done so, we canthendefine the tangent line as well.Note that we cannot simply seth to zero as this would imply division of zero by zero whichwould yield an undefined result. Instead we must find thelimit of the above expression ashtendsto zero:
Definition: (Slope of the graph of a function)Theslopeof the graph off(x) at the point (x0,f(x0)) is
If this limit does not exist, then we say the slope isundefined.
If the slope is defined, saym, then thetangent line to the graph off(x) at the point (x0,f(x0))
is the line with equation
This last equation is just the point-slope form for the line through (x0,f(x0)) with slopem.
Exercises
1. Find the slope of the tangent to the curvey = x2 at (1,1).
Answer: The definition of the slop off atx0 is
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Substituting inf(x) = x2 andx0 = 1 gives:
The Rate of Change of a Function at a Point
Consider the formula for average velocity in thex-direction,
, whereΔx is the change inx over the time intervalΔt. This formula gives the average veloci-ty over a period of time, but suppose we want to define the instantaneous velocity. To this endwe look at thechange in position as the change in time approaches 0. Mathematically this iswritten as:
, which we abbreviate by the symbol
. (The idea of this notation is that the letterd denotes change.) Compare the symbold withΔ. The (entirely non-rigorous) idea is that both indicate a difference between two numbers, butΔ denotes a finite difference whiled denotes an infinitesimal difference. Please note that thesymbolsdxanddt have no rigorous meaning on their own, since
, and we can't divide by 0.
(Note that the letters is often used to denote distance, which would yield
. The letterd is often avoided in denoting distance due to the potential confusion resultingfrom the expression
.)
The Definition of the Derivative
You may have noticed that the two operations we've discussed — computing the slope ofthe tangent to the graph of a function and computing the instantaneous rate of change of the
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function — involved exactly the same limit. That is, the slope of the tangent to the graph ofy =f(x) is
. Of course,
can, and generally will, depend onx, so we should really think of it as afunctionof x. We callthis process (of computing
) differentiation . Differentiation results in another function whose value for any valuex is theslope of the original function atx. This function is known as thederivative of the original func-tion.
Since lots of different sorts of people use derivatives, there are lots of different mathematicalnotations for them. Here are some:
•(read "f prime of x") for the derivative off(x),
• Dx[f(x)],• Df(x),•
for the derivative ofy as a function ofx or•
, which is more useful in some cases.
Most of the time the brackets are not needed, but are useful for clarity if we are dealing withsomething likeD(fg), where we want to differentiate the product of two functions,f andg.
The first notation has the advantage that it makes clear that the derivative is a function. Thatis, if we want to talk about the derivative off(x) atx = 2, we can just writef'(2).
In any event, here is the formal definition:
Definition: (derivative)Let f(x) be a function. Then
wherever this limit exists. In this case we say thatf is differentiable atx and itsderivative atxis f'(x).
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Examples
Example 1
The derivative off(x) = x / 2 is
,
no matter whatx is. This is consistent with the definition of the derivative as the slope of afunction.
Example 2
What is the slope of the graph ofy = 3x2 at (4,48)? We can do it "the hard (and imprecise)way", without using differentiation, as follows, using a calculator and using small differencesbelow and above the given point:
Whenx = 3.999,y = 47.976003.
Whenx = 4.001,y = 48.024003.
Then the difference between the two values ofx is Δx = 0.002.
Then the difference between the two values ofy is Δy = 0.048.
Thus, the slope
at the point of the graph at whichx = 4.
But, to solve the problem precisely, we compute
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=
=
=
=
=3(8)
=24.
We were lucky this time; the approximation we got above turned out to be exactly right. Butthis won't always be so, and, anyway, this way we didn't need a calculator.
In general, the derivative off(x) = 3x2 is
=
=
=
=
=
=3(2x)
=6x.
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Example 3
If
(the absolute value function) then
Here,f(x) is not smooth (though it is continuous) atx = 0 and so the limits
and
(the limits as 0 is approached from the right and left respectively) are not equal. From thedefinition,
, which does not exist. Thus,f'(0)is undefined, and sof'(x) has a discontinuity at 0. This sortof point of non-differentiability is called a cusp. Functions may also not be differentiable becausethey go to infinity at a point, or oscillate infinitely frequently.
Understanding the Derivative Notation
Wikipediahas related information at
SeePic-
Notation for differentiation
tureLi-censeInfor-ma-tionHere
The derivative notation is special and unique in mathematics. The most common notationfor derivatives you'll run into when first starting out with differentiating is the Leibniz notation,expressed as
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. You may think of this as "rate of change iny with respect tox". You may also think of it as"infinitesimal value ofy divided by infinitesimal value ofx". Either way is a good way of think-ing, although you should remember that the precise definition is the one we gave above. Often,in an equation, you will see just
, which literally means "derivative with respect to x". This means we should take the derivativeof whatever is written to the right; that is,
means
wherey = x + 2.
As you advance through your studies, you will see that we sometimes pretend thatdy anddxare separate entities that can be multiplied and divided, by writing things like
. Eventually you will see derivatives such as
, which just means that the input variable of our function is calledy and our output variableis calledx; sometimes, we will write
, to mean the derivative with respect toy of whatever is written on the right. In general, thevariables could be anything, say
.
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All of the following are equivalent for expressing the derivative ofy = x2
•
•
•••
Exercises
1. Using the definition of the derivative find the derivative of the functionf(x) = 2x + 3.
2. Using the definition of the derivative find the derivative of the functionf(x) = x3. Now try
f(x) = x4. Can you see a pattern? In the next section we will find the derivative off(x) = xn for alln.
3. The text states that the derivative of
is not defined atx = 0. Use the definition of the derivative to show this.
4. Graph the derivative toy = 4x2 on a piece of graph paper without solving fordy / dx. Then,solve fordy / dxand graph that; compare the two graphs.
5. Use the definition of the derivative to show that the derivative of sinx is cosx. Hint: Usea suitable sum to product formula and the fact that
.
Differentiation rules
The process of differentiation is tedious for complicated functions. Therefore, rules for differ-entiating general functions have been developed, and can be proved with a little effort. Oncesufficient rules have been proved, it will be fairly easy to differentiate a wide variety of functions.Some of the simplest rules involve the derivative of linear functions.
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Derivative of a Constant Function
For any fixed real numberc,
Intuition
The graph of the functionf(x) = c is a horizontal line, which has a constant slope of zero.Therefore, it should be expected that the derivative of this function is zero, regardless of thevalues ofx andc.
Proof
The definition of a derivative is
Let f(x) = c for all x. (That is,f is a constant function.) Thenf(x + Δx) = c. Therefore
Examples
1.
2.
Note that, in the second example,z is just a constant.
Derivative of a Linear Function
For any fixed real numbersm andc,
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The special case
shows the advantage of the
notation -- rules are intuitive by basic algebra, though this does not constitute a proof, andcan lead to misconceptions to what exactlydxanddyactually are.
Intuition
The graph ofy = mx+ c is a line with constant slopem.
Proof
If f(x) = mx+ c, thenf(x + Δx) = m(x + Δx) + c. So,
==
==
=m.
Constant multiple and addition rules
Since we already know the rules for some very basic functions, we would like to be able totake the derivative of more complex functions by breaking them up into simpler functions. Twotools that let us do this are the constant multiple rule and the addition rule.
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The Constant Rule
For any fixed real numberc,
The reason, of course, is that one can factorc out of the numerator, and then of the entirelimit, in the definition.
Example
We already know that
.
Suppose we want to find the derivative of 3x2
=
=
=
Another simple rule for breaking up functions is the addition rule.
The Addition and Subtraction Rules
Proof
From the definition:
By definition then, this last term is
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Example
What is:
=
=
=
=
The fact that both of these rules work is extremely significant mathematically because itmeans that differentiation islinear. You can take an equation, break it up into terms, figure outthe derivative individually and build the answer back up, and nothing odd will happen.
We now need only one more piece of information before we can take the derivatives of anypolynomial.
The Power Rule
For example, in the case ofx2 the derivative is 2x1 = 2x as was established earlier. A special
case of this rule is thatdx / dx= dx1 / dx= 1x0 = 1.
Since polynomials are sums of monomials, using this rule and the addition rule lets youdifferentiate any polynomial. A relatively simple proof for this can be derived from the binomialexpansion theorem.
This rule also applies to fractional and negative powers. Therefore
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=
=
=
Derivatives of polynomials
With these rules in hand, you can now find the derivative of any polynomial you comeacross. Rather than write the general formula, let's go step by step through the process.
The first thing we can do is to use the addition rule to split the equation up into terms:
We can immediately use the linear and constant rules to get rid of some terms:
Now you may use the constant multiplier rule to move the constants outside the derivatives:
Then use the power rule to work with the individual monomials:
And then do some algebra to get the final answer:
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These are not the only differentiation rules. There are other, more advanced,differentiationrules, which will be described in a later chapter.
Exercises
• Find the derivatives of the following equations:
f(x) = 42
f(x) = 6x + 10
f(x) = 2x2 + 12x + 3
• Use the definition of a derivative to prove the derivative of a constant function, of alinear function, and the constant rule and addition or subtraction rules.
• Answers:
f'(x) = 0
f'(x) = 6
f'(x) = 4x + 12
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Exercises
Extrema and Points of Inflec-tion →
Calculus← Implicit differentiation
Differentiation/Exercises
Find The Derivative By Definition
Find the derivative of the following functions using the limit definition of the derivative.
1.2.3.
4.5.6.
7.
8.
9.
Solutions
Prove Differentiation Rules
Use the definition of the derivative to prove the following general rules:
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1. For any fixed real numberc,
2. For any fixed real numbersm andc,
3. For any fixed real numberc,
4.
Solutions
Find The Derivative By Rules
Find the derivative of the following functions:
Power Rule
1.2.3.4.5.
6.
7.
8.
9.
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Product Rule
1.2.3.4.
Quotient Rule
1.
2.
3.
4.
5.
6.
7.
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Chain Rule
1.2.3.4.
5.6.
7.8.
9.10.11.
Exponentials
1.2.
3.
4.
Logarithms
1.2.3.4.5.
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Trig Functions
1.
2.
Solutions
More Differentiation
Given the above rules, practice differentiation on the following.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Answers
Click arrow to show answers
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1. 30x2(x3 + 5)9
2. 3x2 + 33. (x − 3)(x + 2) + (x + 4)(x + 2) + (x − 3)(x + 4)4.
5. 9x2
6.7.8.9.10.
Implicit Differentiation
Use implicit differentiation to find y'
1.2.3.
Solutions
Logarithmic Differentiation
Use logarithmic differentiation to find
1.2.
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Equation of Tangent Line
For each function,f, (a) determine for what values ofx the tangent line tof is horizontal and(b) find an equation of the tangent line tof at the given point.
1.
2.3.
4.
5.6.
7. Find an equation of the tangent line to the graph defined by
at the point (1,-1).8. Find an equation of the tangent line to the graph defined by
at the point (1,0).
Higher Order Derivatives
1. What is the second derivative of 3x4 + 3x2 + 2x?
Solutions
Relative Extrema
Find the relative maximum(s) and minimum(s), if any, of the following functions.
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1.
2.
3.
4.
5.6.
Range of Function
1. Show that the expressionx + 1 / x cannot take on any value strictly between 2 and -2.
Absolute Extrema
Determine the absolute maximum and minimum of the following functions on the givendomain
1.
on [0,3]2.
on [-1,1]3.
on [-1/2,2]
Determine Intervals of Change
Note: There are currently no answers given for these exercises.
Find the intervals where the following functions are increasing or decreasing
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1. 10-6x-2x2
2. 2x3-12x2+18x+153. 5+36x+3x2-2x3
4. 8+36x+3x2-2x3
5. 5x3-15x2-120x+36. x3-6x2-36x+2
Determine Intervals of Concavity
Find the intervals where the following functions are concave up or concave down
1. 10-6x-2x2
2. 2x3-12x2+18x+153. 5+36x+3x2-2x3
4. 8+36x+3x2-2x3
5. 5x3-15x2-120x+36. x3-6x2-36x+2
Word Problems
1. You peer around a corner. A velociraptor 64 meters away spots you. You run away ata speed of 6 meters per second. The raptor chases, running towards the corner you justleft at a speed of 4t meters per second (timet measured in seconds after spotting). Afteryou have run 4 seconds the raptor is 32 meters from the corner. At this time, how fastis death approaching your soon to be mangled flesh? That is, what is the rate of changein the distance between you and the raptor?
2. Two goldcarts leave an intersection at the same time. One heads north going 12 mphand the other heads east going 5 mph. How fast are the cars getting away from eachother after one hour?
3. You're making a can of volume 200 m3 with a gold side and silver top/bottom. Say
gold costs 10 dollars per m2 and silver costs 1 dollar per m2. What's the minimum costsof such a can?
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Graphing Functions
For each of the following, graph a function that abides by the provided characteristics
1.
2.
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Contents
Differentiation/DifferentiationDefined→
Calculus← Limits/Exercises
Differentiation
Basics of Differentiation
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• Differentiation Defined• Product and Quotient Rules• Derivatives of Trigonometric Functions• Chain Rule• More differentiation rules- More rules for differentiation• Higher Order derivatives- An introduction to second power derivatives• Implicit Differentiation• Derivatives of Exponential and Logarithm Functions• Exercises
Applications of Derivatives
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• Extrema and Points of Inflection• Newton's Method• Related Rates• Kinematics• Optimization• Euler's Method• Exercises
Important Theorems
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• Extreme Value Theorem• Rolle's Theorem• Mean Value Theorem
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Solutions
Find The Derivative By Definition
1. 2x
=
=
=
=
=
=
2. 2
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=
=
=
=
=
3. x
=
=
=
=
=
=
=
4. 4x + 4
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=
=
=
=
=
=
=
Prove Differentiation Rules
Proof of the Derivative of a Constant Function
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Proof of the Derivative of a Linear Function
Proof of the Constant Multiple Rule
=
=
=
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Proof of the Addition and Subtraction Rules
=
=
=
=
=
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Find The Derivative By Rules
1.2.
3.
4.
5.6.7.
8.
9.10.
11.
12.
13. 10x4 + 16x + 114. 49x6 + 40x4 + 3x2 + 2x − 1
Implicit Differentiation
Recall that
is the same asy'.
1.
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solve for
2.
solve for
Higher Order Derivatives
1. 36x2 + 6
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Product and Quotient Rules
Derivatives of TrigonometricFunctions→
Calculus← Differentiation/Differentia-tion Defined
Product and Quotient Rules
Product Rule
When we wish to differentiate a more complicated expression such as:
our only way (up to this point) to differentiate the expression is to expand it and get a polyno-mial, and then differentiate that polynomial. This method becomes very complicated and is partic-ularly error prone when doing calculations by hand. A beginner might guess that the derivativeof a product is the product of the derivatives, similar to the sum and difference rules, but this isnot true. To take the derivative of a product, we use the product rule.
Derivatives of products (Product rule)
Proof
Proving this rule is relatively straightforward, first let us state the equation for the derivative:
We will then apply one of the oldest tricks in the book—adding a term that cancels itself outto the middle:
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Notice that those terms sum to zero, and so all we have done is add 0 to the equation. Nowwe can split the equation up into forms that we already know how to solve:
Looking at this, we see that we can separate the common terms out of the numerators to get:
Which, when we take the limit, becomes:
, or the mnemonic "one D-two plus two D-one"
This can be extended to 3 functions:
For any number of functions, the derivative of their product is the sum, for each function,of its derivative times each other function.
Back to our original example of a product,
, we find the derivative by the product rule is
Note, its derivative wouldnot be
which is what you would get if you assumed the derivative of a product is the product of thederivatives.
To apply the product rule we multiply the first function by the derivative of the second andadd to that the derivative of first function multiply by the second function. Sometimes it helpsto remember the memorize the phrase "First times the derivative of the second plus the secondtimes the derivative of the first."
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Application, proof of the power rule
The product rule can be used to give a proof of the power rule for whole numbers. The proofproceeds bymathematical induction. We begin with the base casen = 1. If f1(x) = x then from
the definition is easy to see that
Next we suppose that for fixed value ofN, we know that forfN(x) = xN, fN'(x) = NxN − 1.
Consider the derivative offN + 1(x) = xN + 1,
We have shown that the statement
is true forn = 1 and that if this statement holds forn = N, then it also holds forn = N + 1.Thus by the principle of mathematical induction, the statement must hold for
.
Quotient rule
There is a similar rule for quotients. To prove it, we go to the definition of the derivative:
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This leads us to the so-called "quotient rule":
Derivatives of quotients (Quotient Rule)
Which some people remember with the mnemonic "low D-high minus high D-low (over)square the low and away we go!"
Examples
The derivative of (4x − 2) / (x2 + 1) is:
Remember: the derivative of a product/quotientis not the product/quotient of the derivatives.(That is, differentiation does not distribute over multiplication or division.) However one candistribute before taking the derivative. That is
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Derivatives of TrigonometricFunctions
Chain Rule→Calculus← Product and Quotient Rules
Derivatives of TrigonometricFunctions
Sine, Cosine, Tangent, Cosecant, Secant, Cotangent. These are functions that crop up continu-ously in mathematics and engineering and have a lot of practical applications. They also appearin more advanced mathematics, particularly when dealing with things such as line integrals withcomplex numbers and alternate representations of space like spherical and cylindrical coordinatesystems.
We use the definition of the derivative, i.e.,
,
to work these first two out.
Let us find the derivative of sinx, using the above definition.
Definition of derivative
trigonometric identity
factoring
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separation of terms
application of limit
solution
Now for the case ofcos x
Definition of derivative
trigonometric identity
factoring
separation of terms
application of limit
solution
Therefore we have established
Derivative of Sine and Cosine
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To find the derivative of the tangent, we just remember that:
which is a quotient. Applying the quotient rule, we get:
Then, remembering that cos2(x) + sin2(x) = 1, we simplify:
Derivative of the Tangent
For secants, we just need to apply the chain rule to the derivations we have already deter-mined.
So for the secant, we state the equation as:
u(x) = cos(x)
Take the derivative of both equations, we find:
Leaving us with:
Simplifying, we get:
Derivative of the Secant
Using the same procedure on cosecants:
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We get:
Derivative of the Cosecant
Using the same procedure for the cotangent that we used for the tangent, we get:
Derivative of the Cotangent
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Chain Rule
Higher Order Derivatives→Calculus← Derivatives of Trigonomet-ric Functions
Chain Rule
We know how to differentiate regular polynomial functions. For example:
However, there is a useful rule known as thechain method rule. The function above (f(x)
= (x2 + 5)2) can be consolidated into two nested partsf(x) = u2, whereu = m(x) = (x2 + 5).Therefore:
if
and
Then:
Then
Thechain rule states that if we have a function of the formy(u(x)) (i.e. y can be written asa function ofu andu can be written as a function ofx) then:
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Chain RuleIf a function F(x) is composed to two differentiable functions g(x)
and m(x), so that F(x)=g(m(x)), then F(x) is differentiable and,
We can now investigate the original function:
Therefore
This can be performed for more complicated equations. If we consider:
and let
andu=1+x2, so that
and
, then, by applying the chain rule, we find that
So, in just plain words, for the chain rule you take the normal derivative of thewhole thing(make the exponent the coefficient, then multiply by original function but decrease the exponentby 1) then multiply by the derivative of the inside.
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More Differentiation Rules
Higher Order Derivatives→Calculus← Chain Rule
More Differentiation Rules
External links
• Online interactive exercises on derivatives• Visual Calculus - Interactive Tutorial on Derivatives, Differentiation, and Integration
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Higher Order Derivatives
Implicit differentiation→Calculus← Chain Rule
Higher Order Derivatives
The second derivative, or second order derivative, is the derivative of the derivative of afunction. The derivative of the functionf(x) may be denoted by
, and its double (or "second") derivative is denoted by
. This is read as "f double prime of x," or "The second derivative of f(x)." Because thederivative of functionf is defined as a function representing the slope of functionf, the doublederivative is the function representing the slope of the first derivative function.
Furthermore, thethird derivative is the derivative of the derivative of the derivative of afunction, which can be represented by
. This is read as "f triple prime ofx", or "The third derivative off(x)". This can continue aslong as the resulting derivative is itself differentiable, with the fourth derivative, the fifthderivative, and so on. Any derivative beyond thefirst derivative can be referred to as ahigherorder derivative.
Notation
Let f(x) be a function in terms ofx. The following are notations for higher order derivatives.
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Notesnth Deriva-tive
4thDerivative3rd Deriva-tive
2nd Deriva-tive
Probably the most common nota-tion.
f(n)(x)f(4)(x)
Leibniz notation.
Another form of Leibniz notation.
Euler's notation.DnfD4fD3fD2f
Warning : You should not writefn(x) to indicate the nth derivative, as this is easily confusedwith the quantityf(x) all raised to the nth power.
The Leibniz notation, which is useful because of its precision, follows from
.
Newton's dot notation extends to the second derivative,
, but typically no further in the applications where this notation is common.
Examples
Example 1:
Find the third derivative of
with respect to x.
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Repeatedly apply thePower Ruleto find the derivatives.
•••
Example 2:
Find the third derivative of
with respect to x.
Use the differentiation rules forexponential expressions, logarithmic expressionsandpolyno-mials.
•
•
•
Applications:
For applications of the second derivative in finding a curve's concavity and points of inflec-tion, see "Extrema and Points of Inflection" and "Extreme Value Theorem". For applications ofhigher order derivatives in physics, see the "Kinematics" section.
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Implicit differentiation
Derivatives of Exponential andLogarithm Functions→
Calculus← Higher Order Derivatives
Implicit differentiation
Implicit differentiation takes a relation and turns it into a rectangular regular equation.
Explicit differentiation
For example, to differentiate a function explicitly,
First we can separate variables to get
Taking the square root of both sides we get a function ofy:
We can rewrite this as a fractional power as
Using the chain rule and simplifying we get,
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Implicit differentiation
Using the same equation
First, differentiate the individual terms of the equation:
Separate the variables:
Divide both sides by
, and simplify to get the same result as above:
Uses
Implicit differentiation is useful when differentiating an equation that cannot be explicitlydifferentiated because it is impossible to isolate variables.
For example, consider the equation,
Differentiate both sides of the equation (remember to use the product rule on the term xy) :
Isolate terms with y':
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Factor out a y' and divide both sides by the other term:
Implicit Differentiation
Generally, one will encounter functions expressed in explicit form, that is,y = f(x) form.You might encounter a function that contains a mixture of different variables. Many times it isinconvenient or even impossible to solve fory. A good example is the function
. It is too cumbersome to isolatey in this function. One can utilize implicit differentiation tofind the derivative. To do so, considery to be a nested function that is defined implicitly byx.You need to employ the chain rule whenever you take the derivative of a variable with respectto a different variable: i.e.,
(the derivative with respect tox) of x is 1;
of y is
.
Remember:
Therefore:
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Examples
Example 1
can be solved as:
then differentiated:
However, using implicit differentiation it can also be differentiated like this:
use the product rule:
solve for
:
Note that, if we substitute
into
, we end up with
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again.
Example 2
Find the derivative ofy2 + x2 = 25 with respect tox.
You are seeking
.
Take the derivative of each side of the equation with respect tox.
Inverse Trigonometric Functions
Arcsine, arccosine, arctangent. These are the functions that allow you to determine the anglegiven the sine, cosine, or tangent of that angle.
First, let us start with the arcsine such that:
y = arcsin(x)
To find dy/dxwe first need to break this down into a form we can work with:
x = sin(y)
Then we can take the derivative of that:
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...and solve fordy / dx:
At this point we need to go back to the unit triangle. Sincey is the angle and the oppositeside is sin(y) (which is equal tox), the adjacent side is cos(y) (which is equal to the square root
of 1 minusx2, based on the Pythagorean theorem), and the hypotenuse is 1. Since we have deter-mined the value of cos(y) based on the unit triangle, we can substitute it back in to the aboveequation and get:
Derivative of the Arcsine
We can use an identical procedure for the arccosine and arctangent:
Derivative of the Arccosine
Derivative of the Arctangent
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Derivatives of Exponential andLogarithm Functions
Extrema and Points of Inflec-tion →
Calculus← Implicit differentiation
Derivatives of Exponentialand Logarithm Functions
Exponential Function
First, we determine the derivative ofex using the definition of the derivative:
Then we apply some basic algebra with powers (specifically thatab + c = ab ac):
Sinceex does not depend onh, it is constant ash goes to 0. Thus, we can use the limit rulesto move it to the outside, leaving us with:
Now, the limit can be calculated by techniques we will learn later, for exampleCalculus/Im-proper_Integrals#Definition L'Hopital's rule, and we will see that
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so that we have proved the following rule:
Derivative of the exponential function
Now that we have derived a specific case, let us extend things to the general case. Assumingthata is a positive real constant, we wish to calculate:
One of the oldest tricks in mathematics is to break a problem down into a form that we al-
ready know we can handle. Since we have already determined the derivative ofex, we will at-
tempt to rewriteax in that form.
Using thateln(c) = c and that ln(ab) = b · ln(a), we find that:
Thus, we simply apply the chain rule:
Derivative of the exponential function
Logarithm Function
Closely related to the exponentiation is the logarithm. Just as with exponents, we will derivethe equation for a specific case first (the natural log, where the base ise), and then work to gener-alize it for any logarithm.
First let us create a variabley such that:
It should be noted that what we want to find is the derivative ofy or
.
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Next we will put both sides to the power ofe in an attempt to remove the logarithm fromthe right hand side:
ey = x
Now, applying the chain rule and the property of exponents we derived earlier, we take thederivative of both sides:
This leaves us with the derivative:
Substituting back our original equation ofx = ey, we find that:
Derivative of the Natural Logarithm
If we wanted, we could go through that same process again for a generalized base, but it iseasier just to use properties of logs and realize that:
Since 1 / ln(b) is a constant, we can just take it outside of the derivative:
Which leaves us with the generalized form of:
Derivative of the Logarithm
Logarithmic differentiation
We can use the properties of the logarithm, particularly the natural log, to differentiate moredifficult functions, such a products with many terms, quotients of composed functions, or func-
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tions with variable or function exponents. We do this by taking the natural logarithm of bothsides, re-arranging terms using the logarithm laws below, and then differentiating both sidesimplicitly, before multiplying through by y.
See the examples below.
Example 1
Suppose we wished to differentiate
We take the natural logarithm of both sides
Differentiating implicitly
Multiplying by y
Example 2
Let us differentiate a function
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Taking the natural logarithm of left and right
We then differentiate both sides, recalling the product rule
Multiplying by the original functiony
Example 3
Take a function
Then
We then differentiate
And finally multiply by y
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Extrema and Points of Inflection
Newton's Method→Calculus← Differentiation/Exercises
Extrema and Points of Inflec-tion
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The four types of extrema.Maxima andminima are points where a function reaches a highest or lowest value, respec-
tively. There are two kinds ofextrema (a word meaning maximumor minimum): global andlocal, sometimes referred to as "absolute" and "relative", respectively. A global maximum is apoint that takes the largest value on the entire range of the function, while a global minimum isthe point that takes the smallest value on the range of the function. On the other hand, local ex-trema are the largest or smallest values of the function in the immediate vicinity.
All extrema look like the crest of a hill or the bottom of a bowl on a graph of the function.A global extremum is always a local extremum too, because it is the largest or smallest value onthe entire range of the function, and therefore also its vicinity. It is also possible to have a func-tion with no extrema, global or local:y=x is a simple example.
At any extremum, the slope of the graph is necessarily zero, as the graph must stop risingor falling at an extremum, and begin to head in the opposite direction. Because of this, extremaare also commonly calledstationary points or turning points . Therefore, the first derivative ofa function is equal to zero at extrema. If the graph has one or more of thesestationary points,these may be found by setting the first derivative equal to zero and finding the roots of the result-ing equation.
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The functionf(x)=x3, which contains a point of inflexion at the point (0,0).However, a slope of zero does not guarantee a maximum or minimum: there is a third class
of stationary point called apoint of inflexion (also spelled point of inflection). Consider thefunction
.
The derivative is
The slope at x=0 is 0. We have a slope of zero, but while this makes it a stationary point,this doesn't mean that it is a maximum or minimum. Looking at the graph of the function youwill see that x=0 is neither, it's just a spot at which the function flattens out. True extrema requirethe a sign change in the first derivative. This makes sense - you have to rise (positive slope) toand fall (negative slope) from a maximum. In between rising and falling, on a smooth curve,there will be a point of zero slope - the maximum. A minimum would exhibit similar properties,just in reverse.
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Good (B andC, green) and bad (D andE, blue) points to check in order to classify the extremum(A, black). The bad points lead to an incorrect classification ofA as a minimum.
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This leads to a simple method to classify a stationary point - plug x values slightly left andright into the derivative of the function. If the results have opposite signs then it is a true maxi-mum/minimum. You can also use these slopes to figure out if it is a maximum or a minimum:the left side slope will be positive for a maximum and negative for a minimum. However, youmust exercise caution with this method, as, if you pick a point too far from the extremum, youcould take it on the far side ofanotherextremum and incorrectly classify the point.
The Extremum Test
A more rigorous method to classify a stationary point is called theextremum test, or 2ndDerivative Test. As we mentioned before, the sign of the first derivative must change for a sta-tionary point to be a true extremum. Now, thesecondderivative of the function tells us the rateof change of the first derivative. It therefore follows that if the second derivative is positive atthe stationary point, then the gradient is increasing. The fact that it is a stationary point in thefirst place means that this can only be a minimum. Conversely, if the second derivative is nega-tive at that point, then it is a maximum.
Now, if the second derivative is zero, we have a problem. It could be a point of inflexion,or it could still be an extremum. Examples of each of these cases are below - all have a secondderivative equal to zero at the stationary point in question:
• y = x3 has a point of inflexion atx = 0• y = x4 has a minimum atx = 0• y = − x4 has a maximum atx = 0
However, this is not an insoluble problem. What we must do is continue to differentiate untilwe get, at the(n+1)th derivative, a non-zero result at the stationary point:
If n is odd, then the stationary point is a true extremum. If the(n+1)th derivative is positive,it is a minimum; if the(n+1)th derivative is negative, it is a maximum. Ifn is even, then the sta-tionary point is a point of inflexion.
As an example, let us consider the function
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We now differentiate until we get a non-zero result at the stationary point atx=0 (assumewe have already found this point as usual):
Therefore,(n+1) is 4, son is 3. This is odd, and the fourth derivative is negative, so we havea maximum. Note that none of the methods given can tell you if this is a global extremum or justa local one. To do this, you would have to set the function equal to the height of the extremumand look for other roots.
See "Optimization" for a common application of these principles.
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Newton's Method
Related Rates→Calculus← Extrema and Points of In-flection
Newton's Method
Newton's Method (also called the Newton-Raphson method) is a recursive algorithm forapproximating the root of a differentiable function. We know simple formulas for finding theroots of linear and quadratic equations, and there are also more complicated formulae for cubicand quartic equations. At one time it was hoped that there would be formulas found for equationsof quintic and higher-degree, though it was later shown byNeils Henrik Abel that no suchequations exist. The Newton-Raphson method is a method for approximating the roots of polyno-mial equations of any order. In fact the method works for any equation, polynomial or not, aslong as the function is differentiable in a desired interval.
Newton's MethodLet f(x) be a differentiable function. Select a pointx1 based on a firstapproximation to the root, arbitrarily close to the function's root. Toapproximate the root you then recursively calculate using:
As you recursively calculate, thexn's become increasingly better approx-imations of the function's root.For n number of approximations,
In order to explain Newton's method, imagine thatx0 is already very close to a zero off(x).
We know that if we only look at points very close tox0 thenf(x) looks like it's tangent line. Ifx0
was already close to the place wheref(x) was zero, and nearx0 we know thatf(x) looks like its
tangent line, then we hope the zero of the tangent line atx0 is a better approximation thenx0 itself.
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The equation for the tangent line tof(x) atx0 is given by
y = f'(x0)(x − x0) + f(x0).
Now we sety = 0 and solve forx.
0 = f'(x0)(x − x0) + f(x0)
− f(x0) = f'(x0)(x − x0)
This value ofx we feel should be a better guess for the value ofx wheref(x) = 0. We chooseto call this value ofx1, and a little algebra we have
If our intuition was correct andx1 is in fact a better approximation for the root off(x), then
our logic should apply equally well atx1. We could look to the place where the tangent line at
x1 is zero. We callx2, following the algebra above we arrive at the formula
And we can continue in this way as long as we wish. At each step, if your current approxima-tion isxn our new approximation will be
Examples
Find the root of the function
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.
As you can seexn is gradually approaching zero (which we know is the root off(x)). One
can approach the function's root with arbitrary accuracy.
Answer:
has a root at
.
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Notes
This method fails whenf'(x) = 0. In that case, one should choose a new starting place. Occa-sionally it may happen thatf(x) = 0 andf'(x) = 0 have a common root. To detect whether this istrue, we should first find the solutions off'(x) = 0, and then check the value off(x) at these places.
Newton's method also may not converge for every function, take as an example:
For this function choosing anyx1 = r − h thenx2 = r + h would cause successive approxima-
tions to alternate back and forth, so no amount of iteration would get us any closer to the rootthan our first guess.
See Also
• Wikipedia:Newton's method• Wikipedia:Abel–Ruffini theorem
Wikimedia Commonshas media related to:category:Newton Method
SeePic-tureLi-censeIn-for-ma-tionHere
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Related Rates
Kinematics→Calculus← Newton's Method
Related Rates
Introduction
Process for solving related rates problems:
• Write out any relevant formulas and information.• Take the derivative of the primary equation with respect to time.• Solve for the desired variable.• Plug-in known information and simplify.
Related Rates
As stated in the introduction, when doing related rates, you generate a function which com-pares the rate of change of one value with respect to change in time. For example, velocity is therate of change of distance over time. Likewise, acceleration is the rate of change of velocity overtime. Therefore, for the variables for distance, velocity, and acceleration, respectively x, v, anda, and time, t:
Using derivatives, you can find the functions for velocity and acceleration from the distancefunction. This is the basic idea behind related rates: the rate of change of a function is thederivative of that function with respect to time.
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Common Applications
Filling Tank
This is the easiest variant of the most common textbook related rates problem: the fillingwater tank.
• The tank is a cube, with volume 1000L.• You have to fill the tank in ten minutes or you die.• You want to escape with your life and as much money as possible, so you want to find
the smallest pump that can finish the task.
We need a pump that will fill the tank 1000L in ten minutes. So, for pump rate p, volume ofwater pumped v, and minutes t:
Examples
Related rates can get complicated very easily.
Example 1:
• Write out any relevant formulas or pieces of information.
• Take the derivative of the equation above with respect to time. Remember to use theChain Ruleand theProduct Rule.
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Example 2:
• Write out any relevant formulas and pieces of information.
• Take the derivative of both sides of the volume equation with respect to time.
=
=
• Solve for
• Plug-in known information.
Example 3:
Note: Because the vertical distance is downward in nature, the rate of change of y is nega-tive. Similarly, the horizontal distance is decreasing, therefore it is negative (it is getting closerand closer).
The easiest way to describe the horizontal and vertical relationships of the plane's motion isthe Pythagorean Theorem.
• Write out any relevant formulas and pieces of information.
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(wheres is the distance between the plane and the house)
• Take the derivative of both sides of the distance formula with respect to time.
• Solve for
.
• Plug-in known information
=
=
=ft/s
Example 4:
• Write down any relevant formulas and information.
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Substitute
into the volume equation.
=
=
=
• Take the derivative of the volume equation with respect to time.
• Solve for
.
• Plug-in known information and simplify.
=
=ft/min
Example 5:
• Write out any relevant formulas and information.
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Use the Pythagorean Theorem to describe the motion of the ladder.
(wherel is the length of the ladder)
• Take the derivative of the equation with respect to time.
(
so
.)
• Solve for
.
• Plug-in known information and simplify.
=
=ft/sec
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Exercises
Problem Set
Here's a few problems for you to try:
1. A spherical balloon is inflated at a rate of100ft3/min. Assuming the rate of inflationremains constant, how fast is the radius of the balloon increasing at the instant the ra-dius is4 ft?
2. Water is pumped from a cone shaped reservoir (the vertex is pointed down)10 ft in
diameter and10 ft deep at a constant rate of3 ft3/min. How fast is the water levelfalling when the depth of the water is6 ft?
3. A boat is pulled into a dock via a rope with one end attached to the bow of a boat andthe other end held by a man standing6 ft above the bow of the boat. If the man pullsthe rope at a constant rate of2 ft/sec, how fast is the boat moving toward the dockwhen10 ft of rope is out?
Solution Set
Click arrow to show answers1.
2.
3.
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Kinematics
Optimization→Calculus← Related Rates
Kinematics
Introduction
Kinematics or the study of motion is a very relevant topic in calculus.
If x is the position of some moving object, andt is time, this section uses the followingconventions:
•is its position function
•is its velocity function
•is its accelerationfunction
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Differentiation
Average Velocity and Acceleration
Average velocity and acceleration problems use the algebraic definitions of velocity andacceleration.
•
•
Examples
Example 1:
A particle's position is defined by the equation
. Find theaverage velocity over the interval [2,7].
• Find the average velocity over the interval[2,7]:
=
=
=
Answer:
.
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Instantaneous Velocity and Acceleration
Instantaneous velocity and acceleration problems use the derivative definitions of velocityand acceleration.
•
•
Examples
Example 2:
A particle moves along a path with a position that can be determined by thefunction
.Determine the acceleration when t = 3.
• Find
• Find
• Find
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=
=
=
Answer:
Integration
•
•
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Optimization
Euler's Method→Calculus← Kinematics
Optimization
Introduction
Optimization is one of the uses of Calculus in the real world. Let us assume we are a pizzaparlor and wish to maximize profit. Perhaps we have a flat piece of cardboard and we need tomake a box with the greatest volume. How does one go about this process?
Obviously, this requires the use of maximums and minimums. We know that we find maxi-mums and minimums via derivatives. Therefore, one can conclude that Calculus will be a usefultool for maximizing or minimizing (also known as "Optimizing") a situation.
Examples
Volume Example
A box manufacturer desires to create a box with a surface area of 100 inches squared. Whatis the maximum size volume that can be formed by bending this material into a box? The box isto be closed. The box is to have a square base, square top, and rectangular sides.
• Write out known formulas and information
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• Eliminate the variableh in the volume equation
=
=
• Find the derivative of the volume equation in order to maximize the volume
• Set
and solve for
• Plug-in thex value into the volume equation and simplify
=
=
Answer:
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Sales Example
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A small retailer can selln units of a product for a revenue ofr(n) = 8.1n and at a cost ofc(n)
= n3 − 7n2 + 18n, with all amounts in thousands. How many units does it sell to maximize itsprofit?
The retailer's profit is defined by the equationp(n) = r(n) − c(n), which is the revenue generat-ed less the cost. The question asks for the maximum amount of profit which is the maximum ofthe above equation. As previously discussed, the maxima and minima of a graph are found whenthe slope of said graph is equal to zero. To find the slope one finds the derivative ofp(n). By us-ing the subtraction rulep'(n) = r'(n) − c'(n):
=
=
=
Therefore, when
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the profit will be maximized or minimized. Use thequadratic formulato find the roots, giving{3.798,0.869}. To find which of these is the maximum and minimum the function can be tested:
p(0.869) = − 3.97321,p(3.798) = 8.58802
Because we only consider the functions for all
(i.e., you can't haven = − 5 units), the only points that can be minima or maxima are thosetwo listed above. To show that 3.798 is in fact a maximum (and that the function doesn't remainconstant past this point) check if the sign ofp'(n) changes at this point. It does, and forn greaterthan 3.798P'(n) the value will remain decreasing. Finally, this shows that for this retailer selling3,798 units would return a profit of $8,588.02.
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Euler's Method
Extreme Value Theorem→Calculus← Optimization
Euler's Method
Euler's Method is a method for estimating the value of a function based upon the values ofthat function's first derivative.
The general algorithm for finding a value of
is:
Examples
The easiest way to keep track of the successive values generated by the algorithm is to drawa table with columns for
.
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Extreme Value Theorem
Rolle's Theorem→Calculus← Euler's Method
Extreme Value Theorem
Extreme Value TheoremIf f is a continuous function and closed on the interval [a,b], thenf has both a minimum and amaximum.
This introduces us to the aspect of global extrema and local extrema. (Also known as abso-lute extrema or relative extrema respectively.)
How is this so? Let us use an example.
f(x) = x2 and is closed on the interval [-1,2]. Find all extrema.
A critical point exists at (0,0). Just for practice, let us use the second derivative test to evalu-ate whether or not it is a minimum or maximum. (You should know it is a minimum from lookingat the graph.)
f''(c) > 0, thus it must be a minimum.
As mentioned before, one can find global extrema on a closed interval. How? Evaluate they coordinate at the endpoints of the interval and compare it to they coordinates of the criticalpoint. When you are finding extrema on a closed interval it is called a local extremum and whenit's for the whole graph it's called a global extremum.
1: Critical Point: (0,0) This is the lowest value in the interval. Therefore, it is a local mini-mum which also happens to be the global minimum.
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2: Left Endpoint (-1, 1) This point is not a critical point nor is it the highest/lowest value,therefore it qualifies as nothing.
3: Right Endpoint (2, 4) This is the highest value in the interval, and thus it is a local maxi-mum.
This example was to show you theextreme value theorem. The quintessential point is this:on a closed interval, the function will have both minima and maxima. However, if that intervalwas an open interval of all real numbers, (0,0) would have been a local minimum. On a closedinterval, always remember to evaluate endpoints to obtain global extrema.
First Derivative Test
Recall that the first derivative of a function describes the slope of the graph of the functionat every point along the graph for which the function is defined and differentiable.
Increasing/Decreasing:
• If
, then
is decreasing.
• If
, then
is increasing.
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Local Extrema:
• If
and
changes signs at
, then there exists a local extremum at
.
• If
for
and
for
, then
is a local minimum.
• If
for
and
, then
is a local maximum.
Example 1:
Let
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. Find all local extrema.
• Find
• Set
to find local extrema.
• Determine whether there is a local minimum or maximum at
.
Choose anx value smaller than
:
Choose anx value larger than
:
Therefore, there is a local minimum at
because
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and
changes signs at
.
Answer: local minimum:
.
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Second Derivative Test
Recall that the second derivative of a function describes the concavity of the graph of thatfunction.
• If
and
changes signs at
, then there is a point of inflection (change in concavity) at
.
• If
, then the graph of
is concave down.
• If
, then the graph of
is concave up.
Example 2:
Let
. Find any points of inflection on the graph of
.
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• Find
.
• Set
.
• Determine whether
changes signs at
.
Choose anx value that is smaller than 0:
Choose anx value that is larger than 0:
Therefore, there exists a point of inflection at
because
and
changes signs at
.
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Answer: point of inflection:
.
Wikipediahas related information at
SeePic-
Extreme value theorem
tureLi-censeInfor-ma-tionHere
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Rolle's Theorem
Mean Value Theorem forFunctions→
Calculus← Extreme Value Theorem
Rolle's Theorem
Rolle's TheoremIf a function,
, is continuous on the closed interval
, is differentiable on the open interval
, and
, then there exists at least one number c, in the interval
such that
Rolle's Theorem is important in proving theMean Value Theorem.
Examples
Example:
f(x) = x2 − 3x. Show that Rolle's Theorem holds true somewhere within this function. To doso, evaluate thex-intercepts and use those points as your interval.
Solution:
1: The question wishes for us to use thex-intercepts as the endpoints of our interval.
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Factor the expression to obtainx(x − 3) = 0.x = 0 andx = 3 are our two endpoints. We knowthatf(0) andf(3) are the same, thus that satisfies the first part of Rolle's theorem (f(a) = f(b)).
2: Now by Rolle's Theorem, we know that somewhere between these points, the slope willbe zero. Where? Easy: Take the derivative.
= 2x − 3
Thus, atx = 3 / 2, we have a spot with a slope of zero. We know that 3 / 2 (or 1.5) is between0 and 3. Thus, Rolle's Theorem is true for this (as it is for all cases). This was merely a demonstra-tion.
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Mean Value Theorem forFunctions
Integration/Contents→Calculus← Rolle's Theorem
Mean Value Theorem
Mean Value TheoremIf
is continuous on the closed interval
and differentiable on the open interval
, there exists a number,
, in the open interval
such that
.
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Examples
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What does this mean? As usual, let us utilize an example to grasp the concept. Visualize (or
graph) the functionf(x) = x3. Choose an interval (anything will work), but for the sake of simplici-ty, [0,2]. Draw a line going from point (0,0) to (2,8). Between the pointsx = 0 andx = 2 exists anumberx = c, where the derivative off at pointc is equal to the slope of the line you drew.
Solution:
1: Using the definition of the mean value theorem
insert values. Our chosen interval is [0,2]. So, we have
2: By the definition of the mean value theorem, we know that somewhere in the intervalexists a point that has the same slope as that point. Thus, let us take the derivative to find thispointx = c.
Now, we know that the slope of the point is 4. So, the derivative at this pointc is 4. Thus, 4
= 3x2. The square root of 4/3 is the point.
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Example 2:Find the point that satisifes the mean value theorem on the functionf(x) = sin(x)and the interval [0,π].
Solution:
1: Always start with the definition:
so,
(Remember, sin(π) and sin(0) are both 0.)
2: Now that we have the slope of the line, we must find the pointx = c that has the sameslope. We must now get the derivative!
The cosine function is 0 atπ / 2 + πn (wheren is an integer). Remember, we are bound bythe interval [0,π], soπ / 2 is the pointc that satisfies the Mean Value Theorem.
Differentials
Assume a functiony = f(x) that is differentiable in the open interval (a,b) that contains x.Δy=
The "Differential of x" is theΔx. This is an approximate change in x and can be considered"equivalent" todx. The same holds true for y. What is this saying? One can approximate a changein y by knowing a change in x and a change in x at a point very nearby. Let us view an example.
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Example: A schoolteacher has asked her students to discover what 4.12 is. The students,bereft of their calculators, are too lazy to multiply this out by hand or in their head and desire toutilize calculus. How can they approximate this?
1: Set up a function that mimics the procedure. What are they doing? They are taking anumber (Call it x) and they are squaring it to get a new number (call it y). Thus, y = x^2 Writeyourself a small chart. Make notes of values for x, y,Δx, Δy, and
. We are seeking what y really is, but we need the change in y first.
2: Choose a number close by that is easy to work with. Four is very close to 4.1, so writethat down as x. Yourδx is .1 (This is the "change" in x from the approximation point to the pointyou chose.)
3: Take the derivative of your function.
= 2x. Now, "split" this up (This is not really what is happening, but to keep things simple,assume you are "multiplying"dxover.)
3b. Now you havedy
. We are assumingdyanddxare approximately the same as the change in x, thus we can useΔx and y.
3c. Insert values:dy
. Thus,dy= .8.
4: To findF(4.1), takeF(4) +dy to get an approximation. 16 + .8 = 16.8; This approximationis nearly exact (The real answer is 16.81. This is only one hundredth off!)
Definition of Derivative
The exact value of the derivative at a point is the rate of change over an infinitely smalldistance, approaching zero. Therefore, if h approaches 0 and the function is f(x):
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If h approaches 0, then:
Cauchy's Mean Value Theorem
Cauchy's Mean Value TheoremIf
,
are continuous on the closed interval
and differentiable on the open interval
,
and
, then there exists a number,
, in the open interval
such that
.
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Integration
Indefinite integral→Calculus← Mean Value Theorem
Integration
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Basics of Integration
• Indefinite integral
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• Definite integral
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• Fundamental Theorem of Calculus
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Integration techniques
• Infinite Sums• Derivative Rules and the Substitution Rule• Integration by Parts• Complexifying• Rational Functions by Partial Fraction Decomposition• Trigonometric Substitutions• Tangent Half Angle Substitution• Trigonometric Integrals• Reduction Formula• Irrational Functions• Numerical Approximations• Integration techniques
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• Improper integrals
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• Exercises
Applications of Integration
• Area• Volume• Volume of solids of revolution• Arc length• Surface area• Work• Centre of mass• Pressure and force• Probability
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Exercises
Area→Calculus← Improper integrals
Integration/Exercises
Set One: Sums
[Insert Numbered Problems Here]
Solutions to Set One
Set Two: Integration of Polynomials
Given the above rules, practice indefinite integration on the following:
1.
2.
3.
4.
5.
Solutions to Set Two
Indefinite Integration
Antiderivatives
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1.
2.
3.
4.
5.
6.
7.
8.
Integration by parts
1. Consider the integral
. Find the integral in two different ways. (a) Integrate by parts withu = sin(x) andv' =cos(x). (b) Integrate by parts withu = cos(x) andv' = sin(x).
Compare your answers. Are they the same?
Solutions to Set Three
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Contents
Indefinite integral→Calculus← Mean Value Theorem
Integration
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Basics of Integration
• Indefinite integral
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• Definite integral
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• Fundamental Theorem of Calculus
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Integration techniques
• Infinite Sums• Derivative Rules and the Substitution Rule• Integration by Parts• Complexifying• Rational Functions by Partial Fraction Decomposition• Trigonometric Substitutions• Tangent Half Angle Substitution• Trigonometric Integrals• Reduction Formula• Irrational Functions• Numerical Approximations• Integration techniques
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• Improper integrals
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• Exercises
Applications of Integration
• Area• Volume• Volume of solids of revolution• Arc length• Surface area• Work• Centre of mass• Pressure and force• Probability
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Solutions
Solutions to Set One
Solutions to Set Two
1.
2.3.
4.
5.
Solutions to Set Three
1.
2.
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3.
= tanx + C
4.
5.
6.
7.
8. with the substitutionx = atanθ, we have
, andx2 + a2 = a2sec2θ, so that
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Indefinite integral
Definite integral→Calculus← Integration/Contents
Indefinite integral
Definition
Given functionsF andf such that
for everyx in some intervalI we say thatF is the antiderivative off on I. However,F is notthe only antiderivative. We can add any constant toF without changing the derivative. With this,we define the indefinite integral as follows:
whereF satisfies
andC is any constant.Note that the indefinite integral yields afamilyof functions.
Example
Since the derivative ofx4 is 4x3, the general antiderivative of 4x3 is x4 plus a constant. Thus,
Example: Finding antiderivatives
Let's take a look at 6x2. How would we go about finding the integral of this function? Recallthe rule from differentiation that
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In our circumstance, we have:
This is a start! We now know that the function we seek will have a power of 3 in it. Howwould we get the constant of 6? Well,
Thus, we say that 2x3 is an antiderivative of 6x2.
Indefinite integral identities
Basic Properties of Indefinite Integrals
Constant Rule for indefinite integrals
If c is a constant then
Sum/Difference Rule for indefinite integrals
Indefinite integrals of Polynomials
Say we are given a function of the form,f(x) = xn, and would like to determine the antideriva-tive of f. Considering that
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we have the following rule for indefinite integrals:
Power rule for indefinite integrals
for all
Example
Integral of the Inverse function
To integrate
, we should first remember
Therefore, since
is the derivative of lnx we can conclude that
Note that the polynomial integration rule does not apply when the exponent is -1. Thistechnique of integration must be used instead. Since the argument of the natural logarithm func-tion must be positive (on the real line), the absolute value signs are added around its argumentto ensure that the argument is positive.
Integral of the Exponential function
Since
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we see thatex is its own antiderivative. This allows us to find the integral of an exponentialfunction:
Integral of Sine and Cosine
Recall that
Sosin x is an antiderivative ofcos xand-cos xis an antiderivative ofsin x. Hence we getthe following rules for integratingsin xandcos x
We will find how to integrate more complicated trigonometric functions in the chapter onintegration techniques.
Example
Suppose we want to integrate the functionf(x) = x4 + 1 + 2sinx. An application of the sumrule from above allows us to use the power rule and our rule for integrating sinx as follows,
Example
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The Substitution Rule
The substitution rule is a valuable asset in the toolbox of any integration greasemonkey. Itis essentially the chain rule (a differentiation technique you should be familiar with) in reverse.First, let's take a look at an example:
Preliminary Example
Suppose we want to find
. That is, we want to find a function such that its derivative equals
. Stated yet another way, we want to find an antiderivative of
. Since sin(x) differentiates to cos(x), as a first guess we might try the function sin(x2). Butby the Chain Rule,
Which is almost what we want apart from the fact that there is an extra factor of 2 in front.But this is easily dealt with because we can divide by a constant (in this case 2). So,
Thus, we have discovered a function,
, whose derivative is
. That is,F is an antiderivative of
. This gives us
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Generalization
In fact, this technique will work for more general integrands. Supposeu is a differentiablefunction. Then to evaluate
we just have to notice that by the Chain Rule
As long asu' is continuous we have that
Now the right hand side of this equation is just the integral of cos(u) but with respect tou.If we write u instead ofu(x) this becomes
So, for instance, ifu(x) = x3 we have worked out that
General Substitution Rule
Now there was nothing special about using the cosine function in the discussion above, andit could be replaced by any other function. Doing this gives us the substitution rule for indefiniteintegrals:
Substitution rule for indefinite integralsAssumeu is differentiable with continuous derivative and thatf is continuous on the range
of u. Then
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Notice that it looks like you can "cancel" in the expression
to leave just adu. This does not really make any sense because
is not a fraction. But it's a good way to remember the substitution rule.
Examples
The following example shows how powerful a technique substitution can be. At first glancethe following integral seems intractable, but after a little simplification, it's possible to tackleusing substitution.
Example
We will show that
First, we re-write the integral:
.
Now we preform the following substitution:
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Which yields:
.
Integration by Parts
Integration by parts is another powerful tool for integration. It was mentioned above thatone could consider integration by substitution as an application of the chain rule in reverse. In asimilar manner, one may consider integration by parts as the product rule in reverse.
Preliminary Example
General Integration by Parts
Integration by parts for indefinite integrals
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Supposef andg are differentiable and their derivatives are continuous. Then
If we write u=f(x) andv=g(x), then by using the Leibnitz notationdu=f'(x) dxanddv=g'(x)dx the integration by parts rule becomes
Examples
Example
Find
Here we let:
u = x, so thatdu= dx,
dv= cos(x)dx , so thatv = sin(x).
Then:
Example
Find
In this example we will have to use integration by parts twice.
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Here we let
u = x2, so thatdu= 2xdx,
dv= exdx, so thatv = ex.
Then:
Now to calculate the last integral we use integration by parts again. Let
u = ''x, so thatdu= dx,
dv= exdx, so thatv = ex
and integrating by parts gives
So, finally we obtain
Example
Find
The trick here is to write this integral as
Now let
u = ln(x) sodu= 1 / xdx,
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v = x sodv= 1dx.
Then using integration by parts,
Example
Find
Again the trick here is to write the integrand as
. Then let
u = arctan(x); du = 1/(1+x2) dx
v = x; dv = 1·dx
so using integration by parts,
Example
Find
This example uses integration by parts twice. First let,
u = ex; thus du = exdx
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dv = cos(x)dx; thusv = sin(x)
so
Now, to evaluate the remaining integral, we use integration by parts again, with
u = ex; du = exdx
v = -cos(x); dv = sin(x)dx
Then
Putting these together, we have
Notice that the same integral shows up on both sides of this equation. We can simply addthe integral to both sides to get
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Definite integral
Fundamental Theorem of Cal-culus→
Calculus← Indefinite integral
Definite integral
Suppose we are given a function and would like to determine the area underneath its graphover an interval. We could guess, but how could we figure out the exact area? Below, using afew clever ideas, we actuallydefinesuch an area and show that by using what is called theDefi-nite integral we can indeed determine the exact area underneath a curve.
Definition of the Definite Integral
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Figure 1
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Figure 2
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The rough idea of defining the area under the graph off is to approximate this area with afinite number of rectangles. Since we can easily work out the area of the rectangles, we get anestimate of the area under the graph. If we use a larger number of rectangles we expect a betterapproximation. Somehow, it seems that we could use our old friend, the limit, and "approach"an infinite number of rectangles to get the exact area. Let's look at such an idea more closely.
Suppose we have a functionf that is positive and two numbersa,bsuch thata<b. Let's pickan integern and divide the interval [a,b] into n subintervals of equal width (see Figure 1). As theinterval [a,b] has widthb-a, each subinterval has width
We denote the endpoints of the subintervals by
. This gives us
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Figure 3Now for each
pick asample point
in the interval
and consider the rectangle of height
and widthΔx (see Figure 2). The area of this rectangle is
. By adding up the area of all the rectangles for
we get that the areaS is approximated by
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A more convenient way to write this is withsummation notation:
For each numbern we get a different approximation. Asn gets larger the width of the rectan-gles gets smaller which yields a better approximation (see Figure 3). In the limit ofAn asn tends
to infinity we get the area ofS.
Definition of the Definite IntegralSupposef is a continuous function on [a,b] and
. Then thedefinite integralof f betweena andb is
where
are any sample points in the interval [xi − 1,xi].It is a fact that iff is continuous on[a,b] then this limit always exists and does not depend
on the choice of the points
. For instance they may be evenly spaced, or distributed ambiguously throughout the interval.The proof of this is technical and is beyond the scope of this section.
Notation
When considering the expression,
, the functionf is called theintegrandand the interval[a,b] is the interval of integration. Alsoais called thelower limit andb theupper limitof integration.
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Figure 6One important feature of this definition is that we also allow functions which take negative
values. Iff(x)<0 for all x then
so
. So the definite integral off will be strictly negative. More generally iff takes on both posi-tive an negative values then
will be the area under the positive part of the graph off minus the area under the graph ofthe negative part of the graph (see Figure 6). For this reason we say that
is thesigned areaunder the graph.
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The area spanned byf(x)
A geometrical proof that anti-derivative gives the area
Suppose we have a function F(x) which returns the area between x and some unknown pointu. (Actually, u is the first number before x which satisfies F(u) = 0, but our solution is indepen-
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dent from u, so we won't bother ourselves with it.) We don't even know if something like F existsor not, but we're going to investigate what clue do we have if itdoesexist.
We can use F to calculate the area between a and b, for instance, which is obviously F(b)-F(a); F is something general. Now, consider a rather peculiar situation, the area bounded atx andx + Δx, in the limit of
. Of course it can be calculated by using F, but we're looking for another solution this time.As the right border approaches the left one, the shape seems to be an infinitesimal rectangle, withthe height of f(x) and width ofΔx. So, the area reads:
Of course, we could use F to calculate this area as well:
By combining these equations, we have
If we divide both sides byΔx, we get
which is an interesting result, because the left-hand side is the derivative of F with respectto x. This remarkable result doesn't tell us what F itself is, however it tells us what thederivativeof F is, and it isf.
Independence of Variable
It is important to notice that the variablex did not play an important role in the definition ofthe integral. In fact we can replace it with any other letter, so the following are all equal:
Each of these is the signed area under the graph off betweena andb.
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Left and Right Handed Riemann Sums
These methods are sometimes referred to as L-RAM and R-RAM, RAM standing for"Rectangular Approximation Method."
We could have decided to choose all our sample points
to be on the right hand side of the interval [xi − 1,xi] (see Figure 7). Then
for all i and the approximation that we calledAn for the area becomes
This is called theright-handed Riemann sum, and the integral is the limit
Alternatively we could have taken each sample point on the left hand side of the interval. Inthis case
(see Figure 8) and the approximation becomes
Then the integral off is
The key point is that, as long asf is continuous, these two definitions give the same answerfor the integral.
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Figure 7
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Figure 8
Example 1
In this example we will calculate the area under the curve given by the graph off(x) = x forx between 0 and 1. First we fix an integern and divide the interval [0,1] inton subintervals ofequal width. So each subinterval has width
To calculate the integral we will use the right-handed Riemann Sum. (We could have usedthe left-handed sum instead, and this would give the same answer in the end). For the right-handed sum the sample points are
Notice that
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. Putting this into the formula for the approximation,
Now we use theformula
to get
To calculate the integral off between0 and1 we take the limit asn tends to infinity,
Example 2
Next we show how to find the integral of the functionf(x) = x2 betweenx=a andx=b. Thistime the interval[a,b] has widthb-aso
Once again we will use the right-handed Riemann Sum. So the sample points we choose are
Thus
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We have to calculate each piece on the right hand side of this equation. For the first two,
For the third sum we have to use aformula
to get
Putting this together
Taking the limit asn tend to infinity gives
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Basic Properties of the Integral
From the definition of the integral we can deduce some basic properties. For all the followingrules, suppose thatf andg are continuous on[a,b].
The Constant Rule
Constant Rule
Whenf is positive, the height of the functioncf at a pointx is c times the height of the func-tion f. So the area undercf betweena andb is c times the area underf. We can also give a proofusing the definition of the integral, using the constant rule for limits,
Example
We saw in the previous section that
.
Using the constant rule we can use this to calculate that
Example
We saw in the previous section that
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We can use this and the constant rule to calculate that
There is a special case of this rule used for integrating constants:
Integrating Constants
If c is constant then
Whenc > 0 anda < b this integral is the area of a rectangle of heightc and widthb-awhichequalsc(b-a).
Example
The addition and subtraction rule
Addition and Subtraction Rules of Integration
As with the constant rule, the addition rule follows from the addition rule for limits:
=
=
=
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The subtraction rule can be proved in a similar way.
Example
From above
and
so
Example
The Comparison Rule
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Figure 9Comparison Rule
• Suppose
for all x in [a,b]. Then
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• Suppose
for all x in [a,b]. Then
• Suppose
for all x in [a,b]. Then
If
then each of the rectangles in the Riemann sum to calculate the integral off will be abovethey axis, so the area will be non-negative. If
then
and by linearity of the integral we get the second property. Finally if
then the area under the graph off will be greater than the area of rectangle with heightmandless than the area of the rectangle with heightM (see Figure 9). So
Example
Linearity with respect to endpoints
Additivity with respect to endpoints Supposea < c < b. Then
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Again suppose thatf is positive. Then this property should be interpreted as saying that thearea under the graph off betweena andb is the area betweena andc plus the area betweenc andb (see Figure 8)
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Figure 8Extension of Additivity with respect to limits of integrationWhena = b we have that
so
Also in defining the integral we assumed thata<b. But the definition makes sense even whenb<a in which case
so has changed sign. This gives
With these definitions,
whatever the order ofa,b,c.Example
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Even and odd functions
Recall that a functionf is called odd if it satisfiesf( − x) = − f(x) and is called even iff( − x)= f(x).
Supposef is a continuous odd function then for anya,
If f is a continuous even function then for anya,
Caution: For improper integrals (e.g. ifa is infinity, or if the function approaches infinityat 0 ora, etc.), the first equation above is only true if
exists. Otherwise the integral is undefined, and only the Cauchy principal value is 0.
Supposef is an odd function and consider first just the integral from-a to 0. We make thesubstitutionu=-x sodu=-dx. Notice that ifx=-a thenu=a and ifx=0 thenu=0. Hence
Now asf is odd,f( − u) = − f(u) so the integral becomes
Now we can replace the dummy variableu with any other variable. So we can replace it withthe letterx to give
Now we split the integral into two pieces
The proof of the formula for even functions is similar, and is left as an exercise.
Example
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Fundamental Theorem ofCalculus
Integration techniques→Calculus← Definite integral
Fundamental Theorem ofCalculus
Fundamental Theorem of Calculus
Wikipediahas related information at
SeePic-
Fundamental theorem of calculus
tureLi-censeInfor-ma-tionHere
Statement of the Fundamental Theorem
Suppose thatf is continuous on[a,b]. We can define a functionF by
Fundamental Theorem of Calculus Part ISupposef is continuous on[a,b] andF is de-fined by
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ThenF is differentiable on(a,b)and for all
,
Now recall thatF is said to be an antiderivative off if
.
Fundamental Theorem of Calculus Part II Suppose thatf is continuous on[a,b] and thatF is any antiderivative off. Then
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Figure 1Note: a minority of mathematicians refer to part one as two and part two as one. All mathe-
maticians refer to what is stated here as part 2 as The Fundamental Theorem of Calculus.
Proofs
Proof of Fundamental Theorem of Calculus Part I
Supposex is in (a,b). PickΔx so thatx + Δx is also in (a, b). Then
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and
.
Subtracting the two equations gives
Now
so rearranging this we have
According to themean value theoremfor integration, there exists ac in [x, x + Δx] such that
.
Notice thatc depends onΔx. Anyway what we have shown is that
,
and dividing both sides byΔx gives
.
Take the limit as
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we get the definition of the derivative ofF atx so we have
.
To find the other limit, we will use the squeeze theorem. The numberc is in the interval [x,x + Δx], sox≤ c ≤ x + Δx. Also,
and
. Therefore, according to the squeeze theorem,
.
As f is continuous we have
which completes the proof.
Proof of Fundamental Theorem of Calculus Part II
Define
Then by the Fundamental Theorem of Calculus part I we know thatP is differentiable on(a,b)and for all
SoP is an antiderivative off. Now we were assuming thatF was also an antiderivative sofor all
,
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A consequence of the Mean Value Theorem is that this implies there is a constantC suchthat for all
,
,
and asP andF are continuous we see this holds whenx=a and whenx=b as well. Since weknow thatP(a)=0 we can putx=a into the equation to get0=F(a) +C soC=-F(a). And puttingx=b gives
Integration of Polynomials
Using the power rule for differentiation we can find a formula for the integral of a power
using the Fundamental Theorem of Calculus. Letf(x) = xn. We want to find an antiderivative forf. Since the differentiation rule for powers lowers the power by 1 we have that
As long as
we can divide byn+1 to get
So the function
is an antiderivative off. If a,b>0 thenF is continuous on[a,b] and we can apply the Funda-mental Theorem of Calculus we can calculate the integral off to get the following rule.
Power Rule of Integration I
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as long as
anda,b > 0.Notice that we allow all values ofn, even negative or fractional. Ifn>0 then this works even
if a or b are negative.
Power Rule of Integration II
as long asn > 0.Examples
• To find
we raise the power by 1 and have to divide by 4. So
• The power rule also works for negative powers. For instance
• We can also use the power rule for fractional powers. For instance
• Using linearity the power rule can also be thought of as applying to constants. For exam-ple,
.
• Using the linearity rule we can now integrate any polynomial. For example
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Infinite Sums
Integration techniques/Recog-nizing Derivatives and the
Substitution Rule→
Calculus← Integration techniques
Integration techniques/Infi-nite Sums
The most basic, and arguably the most difficult, type of evaluation is to use the formal defini-tion of a Riemann integral.
Exact Integrals as Limits of Sums
Using the definition of an integral, we can evaluate the limit as n goes to infinity. Thistechnique requires a fairly high degree of familiarity with summationidentities. This techniqueis often referred to as evaluation "by definition," and can be used to find definite integrals, aslong as the integrands are fairly simple. We start with definition of the integral:
Thenpicking
to be
we get,
In some simple cases, this expression can be reduced to a real number, which can be interpret-ed as the area under the curve if f(x) is positive on [a,b].
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Example 1
Find
by writing the integral as a limit of Riemann sums.
In other cases, it is even possible to evaluate indefinite integrals using the formal definition.We can define the indefinite integral as follows:
Example 2
Supposef(x) = x2, then we can evaluate the indefinite integral as follows.
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Recognizing Derivatives and theSubstitution Rule
Integration techniques/Integra-tion by Parts→
Calculus← Integration techniques/Infi-nite Sums
Integrationtechniques/Recog-nizing Derivatives and the
Substitution Rule
After learning a simple list of antiderivatives, it is time to move on to more complex inte-grands, which are not at first readily integrable. In these first steps, we notice certain special caseintegrands which can be easily integrated in a few steps.
Recognizing Derivatives and ReversingDerivative Rules
If we recognize a functiong(x) as being the derivative of a functionf(x), then we can easilyexpress the antiderivative ofg(x):
For example, since
we can conclude that
Similarly, since we knowex is its own derivative,
The power rule for derivatives can be reversed to give us a way to handle integrals of powersof x. Since
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,
we can conclude that
or, a little more usefully,
.
Integration by Substitution
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Integration by SubstitutionFor many integrals, a substitution can be used to transform the integrand and make possible
the finding of an antiderivative. There are a variety of such substitutions, each depending on theform of the integrand.
Integrating with the derivative present
If a component of the integrand can be viewed as the derivative of another component ofthe integrand, a substitution can be made to simplify the integrand.
For example, in the integral
we see that 3x2 is the derivative ofx3 + 1. Letting
u = x3 + 1
we have
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or, in order to apply it to the integral,
du= 3x2dx.
With this we may write
Note that it was not necessary that we hadexactlythe derivative ofu in our integrand. Itwould have been sufficient to have any constant multiple of the derivative.
For instance, to treat the integral
we may letu = x5. Then
and so
the right-hand side of which is a factor of our integrand. Thus,
In general, the integral of a power of a function times that function's derivative may be inte-grated in this way. Since
,
we have
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Therefore,
There is a similar rule for definite integrals, but we have to change the endpoints.
Substitution rule for definite integrals
Assumeu is differentiable with continuous derivative and thatf is continuous on the range ofu.Supposec = u(a) andd = u(b). Then
Examples
Consider the integral
By using the substitutionu = x2 + 1, we obtaindu= 2x dxand
Note how the lower limitx = 0 was transformed intou = 02 + 1 = 1 and the upper limitx =
2 intou = 22 + 1 = 5.
Proof of the substitution rule
We will now prove the substitution rule for definite integrals. LetF be an anti derivative off so
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F'(x) = f(x). By the Fundamental Theorem of Calculus
Next we define a functionG by the rule
Then by the Chain ruleG is differentiable with derivative
Integrating both sides with respect tox and using the Fundamental Theorem of Calculus weget
But by the definition ofF this equals
Hence
which is the substitution rule for definite integrals.
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Integration by Parts
Integration techniques/Integra-tion by Complexifying→
Calculus← Integration techniques/Rec-ognizing Derivatives and the
Substitution Rule
Integration techniques/Inte-gration by Parts
Continuing on the path of reversing derivative rules in order to make them useful for integra-tion, we reverse the product rule.
Integration by Parts
If y = uvwhereu andv are functions ofx,
Then
Rearranging,
Therefore,
Therefore,
, or
.
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This is the integration by parts formula. It is very useful in many integrals involving productsof functions, as well as others.
For instance, to treat
we chooseu = x and
. With these choices, we havedu= dxandv = − cosx, and we have
Note that the choice ofu anddvwas critical. Had we chosen the reverse, so thatu = sinx and
, the result would have been
The resulting integral is no easier to work with than the original; we might say that this appli-cation of integration by parts took us in the wrong direction.
So the choice is important. One general guideline to help us make that choice is, if possible,to chooseu to be the factor of the integrand whichbecomes simplerwhen we differentiate it. Inthe last example, we see that sinx does not become simpler when we differentiate it: cosx is nosimpler than sinx.
An important feature of the integration by parts method is that we often need to apply it morethan once. For instance, to integrate
,
we start by choosingu = x2 anddv= ex to get
Note that we still have an integral to take care of, and we do this by applying integration byparts again, withu = x and
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, which gives us
So, two applications of integration by parts were necessary, owing to the power ofx2 in theintegrand.
Note thatany power of xdoes become simpler when we differentiate it, so when we see anintegral of the form
one of our first thoughts ought to be to consider using integration by parts withu = xn. Ofcourse, in order for it to work, we need to be able to write down an antiderivative fordv.
Example
Use integration by parts to evaluate the integral
Solution: If we letu = sin(x) andv' = ex, then we haveu' = cos(x) andv = ex. Using our rulefor integration by parts gives
We do not seem to have made much progress. But if we integrate by parts again withu =
cos(x) andv' = ex and henceu' = − sin(x) andv = ex, we obtain
We may solve this identity to find the anti-derivative ofexsin(x) and obtain
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With definite integral
For definite integrals the rule is essentially the same, as long as we keep the endpoints.
Integration by parts for definite integrals Supposef andg are differen-tiable and their derivatives are continuous. Then
.
This can also be expressed in Leibniz notation.
Wikipediahas related information at
SeePic-
Integration by parts
tureLi-censeInfor-ma-tionHere
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Integration by Complexifying
Integration techniques/PartialFraction Decomposition→
Calculus← Integration techniques/Inte-gration by Parts
Integration techniques/Inte-gration by Complexifying
This technique requires an understanding and recognition of complex numbers. SpecificallyEuler's formula:
Recognize, for example, that the real portion:
Given an integral of the general form:
We can complexify it:
With basic rules of exponents:
It can be proven that the "real portion" operator can be moved outside the integral:
The integral easily evaluates:
Multiplying and dividing by (1-2i):
Which can be rewritten as:
Applying Euler's forumula:
Expanding:
Taking the Real part of this expression:
So:
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Partial Fraction Decomposition
Integrationtechniques/Trigono-metric Substitution→
Calculus← Integration techniques/Inte-gration by Complexifying
Integration techniques/Par-tial Fraction Decomposition
Suppose we want to find
. One way to do this is to simplify the integrand by finding constantsA andB so that
This can be done by cross multiplying the fraction which gives
As both sides have the same denominator we must have 3x + 1 = A(x + 1) + Bx. This is anequation forx so must hold whatever valuex is. If we put inx = 0 we get 1 =A and puttingx =- 1 gives − 2 = −B soB = 2. So we see that
Returning to the original integral
=
=
Rewriting the integrand as a sum of simpler fractions has allowed us to reduce the initialintegral to a sum of simpler integrals. In fact this method works to integrate any rational function.
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Method of Partial Fractions:
• Step 1Use long division to ensure that the degree ofP(x) less than the degreeof Q(x).
• Step 2Factor Q(x) as far as possible.• Step 3Write down the correct form for the partial fraction decomposition (see
below) and solve for the constants.
To factor Q(x) we have to write it as a product of linear factors (of the formax+ b) and irre-
ducible quadratic factors (of the formax2 + bx+ c with b2 − 4ac< 0).
Some of the factors could be repeated. For instance ifQ(x) = x3 − 6x2 + 9x we factorQ(x)as
Q(x) = x(x2 − 6x + 9) = x(x − 3)(x − 3) = x(x − 3)2.
It is important that in each quadratic factor we haveb2 − 4ac< 0, otherwise it is possible to
factor that quadratic piece further. For example ifQ(x) = x3 − 3x2 − 2x then we can write
Q(x) = x(x2 − 3x + 2) = x(x − 1)(x + 2)
We will now show how to writeP(x) / Q(x) as a sum of terms of the form
and
Exactly how to do this depends on the factorization ofQ(x) and we now give four cases thatcan occur.
Case (a)Q(x) is a product of linear factors with no repeats.
This means thatQ(x) = (a1x + b1)(a2x + b2)...(anx + bn) where no factor is repeated and no
factor is a multiple of another.
For each linear term we write down something of the form
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, so in total we write
Example 1
Find
Here we haveP(x) = 1 + x2,Q(x) = (x + 3)(x + 5)(x + 7) andQ(x) is a product of linear fac-tors. So we write
Multiply both sides by the denominator
1 + x2 = A(x + 5)(x + 7) + B(x + 3)(x + 7) + C(x + 3)(x + 5)
Substitute in three values ofx to get three equations for the unknown constants,
soA = 5 / 4,B = − 13 / 2,C = 25 / 4, and
We can now integrate the left hand side.
Case (b)Q(x) is a product of linear factors some of which are repeated.
If (ax+ b) appears in the factorisation ofQ(x) k-times. Then instead of writing the piece
we use the more complicated expression
Example 2
Find
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HereP(x)=1" and "Q(x)=(x+1)(x+2)2 We write
Multiply both sides by the denominator 1 =A(x + 2)2 + B(x + 1)(x + 2) + C(x + 1)
Substitute in three values ofx to get 3 equations for the unknown constants,
soA=1, B=-1, C=-1, and
We can now integrate the left hand side.
Case (c)Q(x) contains some quadratic pieces which are not repeated.
If (ax2 + bx+ c) appears we use
Case (d)Q(x) contains some repeated quadratic factors.
If (ax2 + bx+ c) appears k-times then use
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Trigonometric Substitution
A Wikibookian suggests thatSolving Integrals by Trigonometric substitutionbemergedinto this book or chapter.Discuss whether or not this merger should happen on thediscussion page.
Integration techniques/TangentHalf Angle→
Calculus← Integration techniques/Par-tial Fraction Decomposition
Integration tech-niques/TrigonometricSubsti-
tution
If the integrand contains a single factor of one of the forms
we can try a trigonometric substitution.
• If the integrand contains
let x = asinθ and use theidentity1 − sin2θ = cos2θ.• If the integrand contains
let x = atanθ and use the identity 1 + tan2θ = sec2θ.• If the integrand contains
let x = asecθ and use the identity sec2θ − 1 = tan2θ.
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Trigonometric Substitutions
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Trigonometric Substitutions
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Trigonometric Substitutions
Sine substitution
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This substitution is easily derived from a triangle, using thePythagorean Theorem.If the integrand contains a piece of the form
we use the substitution
This will transform the integrand to a trigonometic function. If the new integrand can't beintegrated on sight then the tan-half-angle substitution described below will generally transformit into a more tractable algebraic integrand.
Eg, if the integrand is√(1-x2),
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If the integrand is√(1+x)/√(1-x), we can rewrite it as
Then we can make the substitution
Tangent substitution
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This substitution is easily derived from a triangle, using thePythagorean Theorem.When the integrand contains a piece of the form
we use the substitution
E.g, if the integrand is (x2+a2)-3/2 then on making this substitution we find
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If the integral is
then on making this substitution we find
After integrating by parts, and using trigonometric identities, we've ended up with an expres-sion involving the original integral. In cases like this we must now rearrange the equation so thatthe original integral is on one side only
As we would expect from the integrand, this is approximatelyz2/2 for largez.
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Secant substitution
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This substitution is easily derived from a triangle, using thePythagorean Theorem.If the integrand contains a factor of the form
we use the substitution
Example 1
Find
Example 2
Find
We can now integrate by parts
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Tangent Half Angle
Integrationtechniques/Trigono-metric Integrals→
Calculus← Integration tech-niques/TrigonometricSubstitu-
tion
Integration techniques/Tan-gent Half Angle
Another useful change of variables is
With this transformation, using the double-angle trig identities,
This transforms a trigonometric integral into a algebraic integral, which may be easier tointegrate.
For example, if the integrand is 1/(1 + sinx ) then
This method can be used to further simplify trigonometric integrals produced by the changesof variables described earlier.
For example, if we are considering the integral
we can first use the substitutionx= sinθ, which gives
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then use the tan-half-angle substition to obtain
In effect, we've removed the square root from the original integrand. We could do this witha single change of variables, but doing it in two steps gives us the opportunity of doing thetrigonometric integral another way.
Having done this, we can split the new integrand into partial fractions, and integrate.
This result can be further simplified by use of the identities
ultimately leading to
In principle, this approach will work with any integrand which is the square root of aquadratic multiplied by the ratio of two polynomials. However, it should not be applied automati-cally.
E.g, in this last example, once we deduced
we could have used the double angle formula, since this contains only even powers of cosand sin. Doing that gives
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Using tan-half-angle on this new, simpler, integrand gives
This can be integrated on sight to give
This is the same result as before, but obtained with less algebra, which shows why it is bestto look for the most straightforward methods at every stage.
A more direct way of evaluating the integralI is to substitutet = tanθ right from the start,which will directly bring us to the line
above. More generally, the substitutiont = tanx gives us
so this substitution is the preferable one to use if the integrand is such that all the squareroots would disappear after substitution, as is the case in the above integral.
Example
Using the trigonometric substitutiont = atanx, then
and
(when −π / 2 < x < π / 2). So,
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Alternate Method
In general, to evaluate integrals of the form
,
it is extremely tedious to use the aforementioned "tan half angle" substitution directly, asone easily ends up with a rational function with a 4th degree denominator. Instead, we may firstwrite the numerator as
.
Then the integral can be written as
which can be evaluated much more easily.
Example
Evaluate
.
Let
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.
Then
.
Comparing coefficients of cosx, sinx and the constants on both sides, we obtain
yieldingp = q = 1/2, r = 2. Substituting back into the integrand,
.
The last integral can now be evaluated using the "tan half angle" substitution describedabove, and we obtain
.
The original integral is thus
.
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Trigonometric Integrals
Integration techniques/Reduc-tion Formula→
Calculus← Integration techniques/Tan-gent Half Angle
Integration tech-niques/Trigonometric Inte-
grals
When the integrand is primarily or exclusively based on trigonometric functions, the follow-ing techniques are useful.
Powers of Sine and Cosine
We will give a general method to solve generally integrands of the form cosm (x)sinn(x).First let us work through an example.
Notice that the integrand contains an odd power of cos. So rewrite it as
We can solve this by making the substitutionu = sin(x) so du = cos(x) dx. Then we can writethe whole integrand in terms of u by using the identity
cos(x)2 = 1 - sin2(x)=1-u2.
So
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This method works whenever there is an odd power of sine or cosine.
To evaluate
wheneither m or n isodd.
• If m is odd substituteu=sinx and use the identity cos2x = 1 - sin2x=1-u2.• If n is odd substituteu=cosx and use the identity sin2x = 1 - cos2x=1-u2.
Example
Find
.
As there is an odd power of sin we letu = cosx so du = - sin(x)dx. Notice that whenx=0 wehaveu=cos(0)=1and whenx = π / 2 we haveu = cos(π / 2) = 0.
When bothm andn are even things get a little more complicated.
To evaluate
when bothm andn areeven.
Use theidentitiessin2x = 1/2 (1- cos 2x) and cos2x = 1/2 (1+ cos 2x).
Example
Find
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As sin2x = 1/2 (1- cos 2x) and cos2x = 1/2 (1+ cos 2x) we have
and expanding, the integrand becomes
Using the multiple angle identities
then we obtain on evaluating
Powers of Tan and Secant
To evaluate
.
1. If n is even and
then substituteu=tanx and use theidentity sec2x = 1 + tan2x.2. If n andm are both odd then substituteu=secx and use theidentity tan2x =
sec2x-1.3. If n is odd andm is even then use theidentity tan2x = sec2x-1 and apply a re-
duction formula to integrate
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Example 1
Find
.
There is an even power of secx. Substitutingu = tanx givesdu= sec2xdxso
Example 2
Find
.
Let u = cosx sodu= − sinxdx. Then
Example 3
Find
.
The trick to do this is to multiply and divide by the same thing like this:
Making the substitutionu = secx + tanx sodu= secxtanx + sec2xdx,
More trigonometric combinations
For the integrals
or
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or
use theidentities
•
•
•
Example 1
Find
We can use the fact that sina cosb=(1/2)(sin(a+b)+sin(a-b)), so
Now use the oddness property of sin(x) to simplify
And now we can integrate
Example 2
Find:
.
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Using the identities
Then
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Reduction Formula
Integration techniques/Irra-tional Functions→
Calculus← Integration tech-niques/Trigonometric Integrals
Integration techniques/Re-duction Formula
A reduction formulais one that enables us to solve an integral problem byreducingit to aproblem of solving an easier integral problem, and then reducing that to the problem of solvingan easier problem, and so on.
For example, if we let
Integration by parts allows us to simplify this to
which is our desired reduction formula. Note that we stop at
.
Similarly, if we let
then integration by parts lets us simplify this to
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Using the trigonometric identity, tan2=sec2-1, we can now write
Rearranging, we get
Note that we stop atn=1 or 2 if n is odd or even respectively.
As in these two examples, integrating by parts when the integrand contains a power oftenresults in a reduction formula.
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Irrational Functions
Integrationtechniques/Numeri-cal Approximations→
Calculus← Integration techniques/Re-duction Formula
Integration techniques/Irra-tional Functions
Integration of irrational functions is more difficult than rational functions, and many cannotbe done. However, there are some particular types that can be reduced to rational forms by suit-able substitutions.
Type 1
Integrand contains
Use the substitution
.
Example
Find
.
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Type 2
Integral is of the form
Write Px+ Q as
.
Example
Find
.
Type 3
Integrand contains
,
or
This was discussed in "trigonometric substitutions above". Here is a summary:
1. For
, usex = asinθ.2. For
, usex = atanθ.3. For
, usex = asecθ.
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Type 4
Integral is of the form
Use the substitution
.
Example
Find
.
Type 5
Other rational expressions with the irrational function
1. If a > 0, we can use
.2. If c > 0, we can use
.3. If ax2 + bx+ c can be factored asa(x − α)(x − β), we can use
.4. If a < 0 andax2 + bx+ c can be factored as −a(α − x)(x − β), we can usex = αcos2θ
+ βsin2θ, / theta+ β.
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Numerical Approximations
Improper integrals→Calculus← Integration techniques/Irra-tional Functions
Integration techniques/Nu-merical Approximations
It is often the case, when evaluating definite integrals, that an antiderivative for the integrandcannot be found, or is extremely difficult to find. In some instances, a numerical approximationto the value of the definite value will suffice. The following techniques can be used, and arelisted in rough order of ascending complexity.
Riemann Sum
This comes from the definition of an integral. If we pick n to be finite, then we have:
where
is any point in the i-th sub-interval [xi − 1,xi] on [a,b].
Right Rectangle
A special case of the Riemann sum, where we let
, in other words the point on the far right-side of each sub-interval on, [a,b]. Again if we pickn to be finite, then we have:
Left Rectangle
Another special case of the Riemann sum, this time we let
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, which is the the point on the far left side of each sub-interval on [a,b]. As always, this is an ap-proximation when n is finite. Thus, we have:
Trapezoidal Rule
Simpson's Rule
Remember, n must be even,
Further reading
• Numerical Methods/Numerical Integration
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Improper integrals
Integration/Exercises→Calculus← Integration techniques/Nu-merical Approximations
Improper Integrals
The definition of a definite integral:
requires the interval [a,b] be finite. The Fundamental Theorem of Calculus requires thatf becontinuous on [a,b]. In this section, you will be studying a method of evaluating integrals thatfail these requirements—either because their limits of integration are infinite, or because a finitenumber of discontinuities exist on the interval [a,b]. Integrals that fail either of these requirementsareimproper integrals.
L'Hopital's Rule
L'Hopital's Rule is included in this section because limits involving infinity often appear inimproper integration. L'Hopital's Rule describes how to evaluate limits involving infinity and or0 if the limit evaluates to an indeterminate form.
All of the following expressions are indeterminate forms.
These expressions are calledindeterminatebecause you cannot determine their exact valuein the indeterminate form. Depending on the situation, each indeterminate form could evaluateto a variety of values.
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Theorem
If
is indeterminate of type
or
,
then
Note:
can approach a finite valuec,
or
.
Example:
One might think the value
Consider
Plugging the value ofx into the limit yields
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(indeterminate form).
Let
=
=
=
(indeterminateform)
We now apply L'Hopital's Rule by taking the derivative of the top and bottom with respectto x.
Returning to the expression above
=
=
(indeterminateform)
We apply L'Hopital's Rule once again
Therefore
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And
Careful: this does not prove that
because
Improper Integrals with Infinite Limits ofIntegration
Consider the integral
Assigning a finite upper boundb in place of infinity gives
This improper integral can be interpreted as the area of the unbounded region betweenf(x)=1/x^2, y=0 (thex-axis), andx=1.
Definition
1. Suppose
exists for all
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. Then we define
=
as long as this limit exists and is finite.
If it does exist we say the integral isconvergentand otherwise we say itis divergent.
2. Similarly if
exists for all
we define
=
3. Finally supposec is a fixed real number and that
and
are both convergent. Then we define
=
Example: Convergent Improper Integral
We claim that
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To do this we calculate
=
=
=
=
Example: Divergent Improper Integral
We claim that the integral
diverges.
This follows as
=
=
=
=
Therefore
diverges.
Example: Improper Integral
Find
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To calculate the integral use integration by parts twice to get
Now
and because exponentials overpower polynomials, we see that
and
as well. Hence,
Example: Powers
Show
}}
If
then
=
=
=
=
Notice that we had to assume that
do avoid dividing by zero. However thep = 1 case was done in a previous example.
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Improper Integrals with a Finite NumberDiscontinuities
First we give a definition for the integral of functions which have a discontinuity at onepoint.
Definition of improper integrals with a singlediscontinuity
If f is continuous on the interval [a,b) and is discontinuous atb, we define :
=
If the limit in question exists we say the integralconvergesand otherwisewe say itdiverges.
Similarly if f is continuous on the interval(a,b] and is discontinuous ata,we define
=
Finally supposef has an discontinuity at a point c in (a,b) and is continu-ous at all other points in [a,b]. If
and
converge we define
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=
Example 1
Show
}}
If
then
=
=
=
=
Notice that we had to assume that
do avoid dividing by zero. So instead we do thep = 1 case separately,
which diverges.
We can also give a definition of the integral of a function with a finite number of discontinu-ities
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Definition: Improper integrals with finitenumber of discontinuities
Supposef is continuous on [a,b] except at points
in [a,b]. We define
as long as each integral on the right converges.
Notice that by combining this definition with the definition for improper integrals with infi-nite endpoints, we can define the integral of a function with a finite number of discontinuitieswith one or more infinite endpoints.
Example 2
The integral
is improper because the integrand is not continuous at x=2. However had we not notice thatwe might have been tempted to apply the fundamental theorem of calculus and conclude that itequals
which is not correct. In fact the integral diverges.
Comparison Test
There are integrals which cannot easily be evaluated. However it may still be possible toshow they are convergent by comparing them to an integral we already know converges.
Theorem (Comparison Test)Let f andg be continuous functions definedfor all
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.
1. Suppose
for all
. Then if
converges so does
2. Suppose
for all
. Then if
diverges so does
A similar theorem holds for improper integrals of the form
and for improper integrals with discontinuities.
Example: Use of comparsion test to show convergence
Show that
converges.
For allx we know that
so
. This implies that
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.
We have seen that
converges. So putting
and
into the comparison test we get that the integral
converges as well.
Example: Use of Comparsion Test to show divergence
Show that
diverges.
Just as in the previous example we know that
for all x. Thus
We have seen that
diverges. So putting
and
into the comparison test we get that
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diverges as well.
An extension of the comparison theorem
To apply the comparison theorem you do not really need
for all
. What we actually need is this inequality holds for sufficiently largex (i.e. there is a numberc such that
for all
). For then
so the first integral converges if and only if third does, and we can apply the comparisontheorem to the
piece.
Example
Show that
converges.
The reason that this integral converges is because for largex thee − x factor in the integrand
is dominant. We could try comparingx7 / 2e − x with e − x, but as
, the inequality
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is the wrong way around to show convergence.
Instead we rewrite the integrand as
Since the limit
we know that forx sufficiently large we have
. So for largex,
Since the integral
converges the comparison test tells us that
converges as well.
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Area
Volume→Calculus← Integration/Exercises
Area
Introduction
Finding the area between two curves, usually given by two explicit functions, is often usefulin calculus.
In general the rule for finding the area between two curves is
or
If f(x) is the upper function and g(x) is the lower function
This is true whether the functions are in the first quadrant or not.
Area between two curves
Suppose we are given two functionsy1=f(x) andy2=g(x) and we want to find the area be-
tween them on the interval[a,b]. Also assume thatf(x)≥ g(x) for all x on the interval[a,b]. Beginby partitioning the interval[a,b] into n equal subintervals each having a length ofΔx=(b-a)/n.Next choose any point in each subinterval,xi* . Now we can 'create' rectangles on each interval.
At the pointxi* , the height of each rectangle isf(xi*)-g(xi*) and the width isΔx. Thus the area
of each rectangle is[f(x i*)-g(xi*)] Δx. An approximationof the area,A, between the two curves
is
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.
Now we take the limit asn approaches infinity and get
which gives the exact area. Recalling the definition of the definite integral we notice that
.
This formula of finding the area between two curves is sometimes known as applying integra-tion with respect to thex-axis since the rectangles used to approximate the area have their baseslying parallel to thex-axis. It will be most useful when the two functions are of the formy1=f(x)
andy2=g(x). Sometimes however, one may find it simpler to integrate with respect to they-axis.
This occurs when integrating with respect to thex-axis would result in more than one integral tobe evaluated. These functions take the formx1=f(y) andx2=g(y) on the interval[c,d]. Note that
[c,d] are values ofy. The derivation of this case is completely identical. Similar to before, wewill assume thatf(y)≥ g(y) for all y on [c,d]. Now, as before we can divide the interval intonsubintervals and create rectangles to approximate the area betweenf(y) andg(y). It may be usefulto picture each rectangle having their 'width',Δy, parallel to they-axis and 'height',f(yi*)-g(yi*)
at the pointyi* , parallel to thex-axis. Following from the work above we may reason that an
approximationof the area,A, between the two curves is
.
As before, we take the limit asn approaches infinity to arrive at
,
which is nothing more than a definite integral, so
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.
Regardless of the form of the functions, we basically use the same formula.
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Volume
Volume of solids of revolution→
Calculus← Area
Volume
When we think about volume from an intuitive point of view, we typically think of it as theamount of "space" an item occupies. Unfortunately assigning a number that measures this amountof space can prove difficult for all but the simplest geometric shapes. Calculus provides a newtool for calculating volume that can greatly extend our ability to calculate volume. In order tounderstand the ideas involved it helps to think about the volume of a cylinder. The volume of a
cylinder is calculated using the formulaV = πr2h. The base of the cylinder is a circle whose area
is given byA = πr2. Notice that the volume of a cylinder is derived by taking the area of its baseand multiplying by the heighth. For more complicated shapes, we could think of approximatingthe volume by taking the area of some cross section at some heightx and multiplying by somesmall change in heightΔx them adding up the heights of all of these approximations from thebottom to the top of the object. This would appear to be a Riemann sum. Keeping this in mind,we can develop a more general formula for the volume of solids in
(3 dimensional space).
Formal Definition
Formally the ideas above suggest that we can calculate the volume of a solid by calculatingthe integral of the cross-sectional area along some dimension. In the above example of a cylinder,the every cross section was given by the same circle, so the cross-sectional area is therefore aconstant function, and the dimension of integration was vertical (although it could have been anyone we desired). Generally, ifS is a solid that lies in
betweenx = a andx = b, letA(x) denote the area of a cross section taken in the plane perpen-dicular to thex direction, and passing through the pointx. If the functionA(x) is continuous on[a,b], then the volumeVS of the solidS is given by:
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Examples
Example 1: A right cylinder
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Now we will calculate the volume of a right cylinder using our new ideas about how to calcu-late volume. Since we already know the formula for the volume of a cylinder this will give us a"sanity check" that our formulas make sense. First, we choose a dimension along which to inte-grate. In this case, it will greatly simplify the calculations to integrate along the height of thecylinder, so this is the direction we will choose. Thus we will call the vertical direction (see dia-gram)x. Now we find the function,A(x), which will describe the cross-sectional area of ourcylinder at a height ofx. The cross-sectional area of a cylinder is simply a circle. Now simply
recall that the area of a circle isπr2, and soA(x) = πr2. Before performing the computation, wemust choose our bounds of integration. In this case, we simply definex = 0 to be the base of thecylinder, and so we will integrate fromx = 0 tox = h, whereh is the height of the cylinder. Final-ly, we integrate:
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Vcylin-
der
=
πr2(h− 0)
=
πr2h.
This is exactly the familiar formula for the volume of a cylinder.
Example 2: A right circular cone
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For our next example we will look at an example where the cross sectional area is not con-stant. Consider a right circular cone. Once again the cross sections are simply circles. But nowthe radius varies from the base of the cone to the tip. Once again we choosex to be the verticaldirection, with the base atx = 0 and the tip atx = h, and we will letR denote the radius of thebase. While we know the cross sections are just circles we cannot calculate the area of the crosssections unless we find some way to determine the radius of the circle at heightx.
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Luckily in this case it is possible to use some of what we know from geometry. We canimagine cutting the cone perpendicular to the base through some diameter of the circle all theway to the tip of the cone. If we then look at the flat side we just created, we will see simply atriangle, whose geometry we understand well. The right triangle from the tip the base at heightx is similar to the right triangle with from the tip with heighth triangle. This tells us that
. So that we see that the radius of the circle at heightx is
. Now using the familiar formula for the area of a circle we see that
.
Now we are ready to integrate.
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Vcone
By u-substitution we may letu = h − x, thendu= − dxand our integralbecomes
Example 3: A sphere
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In a similar fashion, we can use our definition to prove the well known formula for the vol-ume of a sphere. First, we must find our cross-sectional area function,A(x). Consider a sphereof radiusRwhich is centered at the origin in
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. If we again integrate vertically thenx will vary from − R to R. In order to find the area of aparticular cross section it helps to draw a right triangle whose between the center of the sphere,the center of the circular cross section, and a point along the circumference of the cross section.As shown in the diagram the side lengths of this triangle will beR, | x | , andr. Wherer is theradius of the circular cross section. Then by the Pythagorean theorem
and find thatA(x) = π(R2 − | x | 2). It is slightly helpful to notice that |x | 2 = x2 so we do not needto keep the absolute value.
So we have that
Vsphere
Extension to Non-trivial Solids
Now that we have shown our definition agrees with our prior knowledge, we will see howit can help us extend our horizons to solids whose volumes are not possible to calculate usingelementary geometry.
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Volume of solids of revolution
Arc length→Calculus← Volume
Volume of solids of revolu-tion
Revolution solids
A solid is said to be of revolution when it is formed by rotating a given curve against an axis.For example, rotating a circle positioned at(0,0) against they-axiswould create a revolutionsolid, namely, a sphere.
Calculating the volume
Calculating the volume of this kind of solid is very similar to calculating anyvolume: wecalculate the basal area, and then we integrate through the height of the volume.
Say we want to calculate the volume of the shape formed rotating over thex-axisthe areacontained between the curvesf(x) andg(x) in the range [a,b]. First calculate the basal area:
| πf(x)2 − πg(x)2 |
And then integrate in the range [a,b]:
Alternatively, if we want to rotate in they-axisinstead,f andg must be invertible in the range[a,b], and, following the same logic as before:
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Arc length
Surface area→Calculus← Volume of solids of revolu-tion
Arc length
Suppose that we are given a functionf and we want to calculate the length of the curve drawnout by the graph off. If the graph were a straight line this would be easy — the formula for thelength of the line is given by Pythagoras' theorem. And if the graph were a polygon we can calcu-late the length by adding up the length of each piece.
The problem is that most graphs are not polygons. Nevertheless we can estimate the lengthof the curve by approximating it with straight lines. Suppose the curveC is given by the formulay = f(x) for
. We divide the interval [a,b] into n subintervals with equal widthΔx and endpoints
. Now letyi = f(xi) soPi = (xi,yi) is the point on the curve abovexi. The length of the straight
line betweenPi andPi + 1 is
So an estimate of the length of the curveC is the sum
As we divide the interval [a,b] into more pieces this gives a better estimate for the length ofC. In fact we make that a definition.
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Definition (Length of a Curve)
The length of the curvey = f(x) for
is defined to be
The Arclength Formula
Suppose thatf' is continuous on [a,b]. Then the length of the curve given byy = f(x) betweena andb is given by
And in Leibniz notation
Proof: Consideryi + 1 − yi = f(xi + 1) − f(xi). By theMean Value Theoremthere is a pointzi
in (xi + 1,xi) such that
.
So
=
=
=
=
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Putting this into the definiton of the length ofC gives
Now this is the definition of the integral of the function
betweena andb (notice thatg is continuous because we are assuming thatf' is continuous).Hence
as claimed.
Arclength of a parametric curve
For a parametric curve, that is, a curve defined byx = f(t) andy = g(t), the formula is slightlydifferent:
Proof: The proof is analogous to the previous one: Consideryi + 1 − yi = g(ti + 1) − g(ti) and
xi + 1 − xi = f(ti + 1) − f(ti). By the Mean Value Theorem there are pointsci anddi in (ti + 1,ti) such
that
and
.
So
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=
=
=
=
Putting this into the definiton of the length of the curve gives
This is equivalent to:
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Surface area
Work →Calculus← Arc length
Surface area
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Suppose we are given a functionf and we want to calculate the surface area of the functionf rotated around a given line. The calculation of surface area of revolution is related to the arclength calculation.
If the functionf is a straight line, other methods such as surface area formulas for cylindersand conical frustra can be used. However, iff is not linear, an integration technique must be used.
Recall the formula for the lateral surface area of a conical frustrum:
wherer is the average radius andl is the slant height of the frustrum.
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For y=f(x) and
, we divide[a,b] into subintervals with equal widthΔx and endpoints
. We map each point
to a conical frustrum of widthΔx and lateral surface area
.
We can estimate the surface area of revolution with the sum
As we divide[a,b] into smaller and smaller pieces, the estimate gives a better value for thesurface area.
Definition (Surface of Revolution)
The surface area of revolution of the curvey=f(x) about a line for
is defined to be
The Surface Area Formula
Supposef is a continuous function on the interval[a,b] andr(x) represents the distance fromf(x) to the axis of rotation. Then the lateral surface area of revolution about a line is given by
And in Leibniz notation
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Proof:
=
=
=
As
and
, we know two things:
1. the average radius of each conical frustrumr i approaches a single value
2. the slant height of each conical frustruml i equals an infitesmal segment of arc length
From the arc length formula discussed in the previous section, we know that
Therefore
=
=
Because of the definition of an integral
, we can simplify the sigma operation to an integral.
Or if f is in terms ofy on the interval[c,d]
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Work
Centre of mass→Calculus← Surface area
Work
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Centre of mass
Parametric Introduction→Calculus← Work
Centre of mass
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Parametric and Polar Equations
Parametric Introduction→Calculus← Probability
Parametric and Polar Equa-tions
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Parametric Equations
• Introduction to Parametric Equations• Differentiation and Parametric Equations• Integration and Parametric Equations
Polar Equations
• Introduction to Polar Equations• Differentiation and Polar Equations• Integration and Polar Equations
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Parametric Introduction
Parametric Differentiation→Calculus← Parametric and Polar Equa-tions
Parametric Introduction
Introduction
Parametric equations are typically definied by two equations that specify both the x and ycoordinates of a graph using a parameter. They are graphed using the parameter (usually t) tofigure out both the x and y coordinates.
Example 1:
Note: This parametric equation is equivalent to the rectangular equation
.
Example 2:
Note: This parametric equation is equivalent to the rectangular equation
and the polar equation
.
Parametric equations can be plotted by using a t-table to show values of x and y for eachvalue of t. They can also be plotted by eliminating the parameter though this method removesthe parameter's importance.
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Forms of Parametric Equations
Parametric equations can be described in three ways:
• Parametric form• Vector form• An equality
The first two forms are used far more often, as they allow us to find the value of the compo-nent at the given value of the parameter. The final form is used less often; it allows us to verifya solution to the equation, or find the parameter (or some constant multiple thereof).
Parametric Form
A parametric equation can be shown inparametric formby describing it with a system ofequations. For instance:
Vector Form
Vector formcan be used to describe a parametric equation in a similar manner toparametricform. In this case, a position vector is given:
Equalities
A parametric equation can also be described with a set of equalities. This is done by solvingfor the parameter, and equating the components. For example:
From here, we can solve for t:
And hence equate the two right-hand sides:
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Converting Parametric Equations
There are a few common place methods used to change a parametric equation to rectangularform. The first involves solving for t in one of the two equations and then replacing the new ex-pression for t with the variable found in the second equation.
Example 1:
becomes
Example 2:
Given
Isolate the trigonometric functions
Use the "Beloved Identity"
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Parametric Differentiation
Taking Derivatives of Parametric Systems
Just as we are able to differentiate functions of x, we are able to differentiate x and y, whichare functions of t. Consider:
We would find the derivative of x with respect to t, and the derivative of y with respect tot:
In general, we say that if
and
then:
and
It's that simple.
This process works for any amount of variables.
Slope of Parametric Equations
In the above process, x' has told us only the rate at which x is changing, not the rate for y,and vice versa. Neither is the slope.
In order to find the slope, we need something of the form
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.
We can discover a way to do this by simple algebraic manipulation:
So, for the example in section 1, the slope at any time t:
In order to find a vertical tangent line, set the horizontal change, or x', equal to 0 and solve.
In order to find a horizontal tangent line, set the vertical change, or y', equal to 0 and solve.
If there is a time when both x' and y' are 0, that point is called a singular point.
Concavity of Parametric Equations
Solving for the second derivative of a parametric equation can be more complex that it mayseem at first glance. When you have take the derivative of
in terms of t, you are left with
:
.
By multiplying this expression by
, we are able to solve for the second derivative of the parametric equation:
.
Thus, the concavity of a parametric equation can be described as:
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So for the example in sections 1 and 2, the concavity at any time t:
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Parametric Integration
Introduction
Because most parametric equations are given in explicit form, they can be integrated likemany other equations. Integration has a variety of applications with respect to parametric equa-tions, especially in kinematics and vector calculus.
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Polar Introduction
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A polar grid with several angles labeled in degreesThepolar coordinate systemis a two-dimensional coordinate system in which each point
on a plane is determined by an angle and a distance. The polar coordinate system is especiallyuseful in situations where the relationship between two points is most easily expressed in termsof angles and distance; in the more familiar Cartesian coordinate system or rectangular coordinatesystem, such a relationship can only be found through trigonometric formulae.
As the coordinate system is two-dimensional, each point is determined by two polar coordi-nates: the radial coordinate and the angular coordinate. The radial coordinate (usually denotedasr) denotes the point's distance from a central point known as thepole(equivalent to theoriginin the Cartesian system). The angular coordinate (also known as the polar angle or the azimuthangle, and usually denoted byθ or t) denotes the positive or anticlockwise (counterclockwise)angle required to reach the point from the 0° ray orpolar axis(which is equivalent to the positivex-axis in the Cartesian coordinate plane).
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Plotting points with polar coordinates
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The points (3,60°) and (4,210°) on a polar coordinate systemEach point in the polar coordinate system can be described with the two polar coordinates,
which are usually calledr (the radial coordinate) andθ (the angular coordinate, polar angle, orazimuth angle, sometimes represented asφ or t). Ther coordinate represents the radial distancefrom the pole, and theθ coordinate represents the anticlockwise (counterclockwise) angle fromthe 0° ray (sometimes called the polar axis), known as the positive x-axis on the Cartesian coordi-nate plane.
For example, the polar coordinates (3, 60°) would be plotted as a point 3 units from the poleon the 60° ray. The coordinates (−3, 240°) would also be plotted at this point because a negativeradial distance is measured as a positive distance on the opposite ray (the ray reflected about theorigin, which differs from the original ray by 180°).
One important aspect of the polar coordinate system, not present in the Cartesian coordinatesystem, is that a single point can be expressed with an infinite number of different coordinates.This is because any number of multiple revolutions can be made around the central pole withoutaffecting the actual location of the point plotted. In general, the point (r, θ) can be representedas (r, θ ± n×360°) or (−r, θ ± (2n + 1)180°), wheren is any integer.
The arbitrary coordinates (0,θ) are conventionally used to represent the pole, as regardlessof theθ coordinate, a point with radius 0 will always be on the pole. To get a unique representa-tion of a point, it is usual to limitr to negative and non-negative numbersr ≥ 0 andθ to the inter-val [0, 360°) or (−180°, 180°] (or, in radian measure, [0, 2π) or (−π, π]).
Angles in polar notation are generally expressed in either degrees or radians, using the con-version 2π rad = 360°. The choice depends largely on the context. Navigation applications use
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degree measure, while some physics applications (specifically rotational mechanics) and almostall mathematical literature on calculus use radian measure.
Converting between polar and Cartesian coordinates
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A diagram illustrating the conversion formulaeThe two polar coordinatesr andθ can be converted to the Cartesian coordinatesx andy by
using the trigonometric functions sine and cosine:
while the two Cartesian coordinatesx andy can be converted to polar coordinater by
(by a simple application of the Pythagorean theorem).
To determine the angular coordinateθ, the following two ideas must be considered:
• For r = 0,θ can be set to any real value.• For r ≠ 0, to get a unique representation forθ, it must be limited to an interval of size
2π. Conventional choices for such an interval are [0, 2π) and (−π, π].
To obtainθ in the interval [0, 2π), the following may be used (arctan denotes the inverse ofthe tangent function):
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To obtainθ in the interval (−π, π], the following may be used:
One may avoid having to keep track of the numerator and denominator signs by use of theatan2 function, which has separate arguments for the numerator and the denominator.
Polar equations
The equation defining an algebraic curve expressed in polar coordinates is known as apolarequation. In many cases, such an equation can simply be specified by definingr as a function ofθ. The resulting curve then consists of points of the form (r(θ), θ) and can be regarded as thegraph of the polar functionr.
Different forms of symmetry can be deduced from the equation of a polar functionr. Ifr(−θ) = r(θ) the curve will be symmetrical about the horizontal (0°/180°) ray, ifr(π−θ) = r(θ) itwill be symmetric about the vertical (90°/270°) ray, and ifr(θ−α°) = r(θ) it will be rotationallysymmetricα° counterclockwise about the pole.
Because of the circular nature of the polar coordinate system, many curves can be describedby a rather simple polar equation, whereas their Cartesian form is much more intricate. Amongthe best known of these curves are the polar rose, Archimedean spiral, lemniscate, limaçon, andcardioid.
For the circle, line, and polar rose below, it is understood that there are no restrictions onthe domain and range of the curve.
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Circle
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A circle with equationr(θ) = 1The general equation for a circle with a center at (r0, φ) and radiusa is
This can be simplified in various ways, to conform to more specific cases, such as theequation
for a circle with a center at the pole and radiusa.
Line
Radial lines (those running through the pole) are represented by the equation
,
whereφ is the angle of elevation of the line; that is,φ = arctanm wherem is the slope ofthe line in the Cartesian coordinate system. The non-radial line that crosses the radial lineθ = φperpendicularly at the point (r0, φ) has the equation
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Polar rose
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A polar rose with equationr(θ) = 2 sin 4θA polar rose is a famous mathematical curve that looks like a petaled flower, and that can
be expressed as a simple polar equation,
for any constantφ0 (including 0). Ifk is an integer, these equations will produce ak-petaled
rose if k is odd, or a 2k-petaled rose ifk is even. Ifk is rational but not an integer, a rose-likeshape may form but with overlapping petals. Note that these equations never define a rose with2, 6, 10, 14, etc. petals. The variablea represents the length of the petals of the rose.
Archimedean spiral
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One arm of an Archimedean spiral with equationr(θ) = θ for 0 < θ < 6πThe Archimedean spiral is a famous spiral that was discovered by Archimedes, which also
can be expressed as a simple polar equation. It is represented by the equation
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Changing the parameterawill turn the spiral, whilebcontrols the distance between the arms,which for a given spiral is always constant. The Archimedean spiral has two arms, one forθ > 0and one forθ < 0. The two arms are smoothly connected at the pole. Taking the mirror image ofone arm across the 90°/270° line will yield the other arm. This curve is notable as one of the firstcurves, after theConic Sections, to be described in a mathematical treatise, and as being a primeexample of a curve that is best defined by a polar equation.
Conic sections
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Ellipse, showing semi-latus rectumA conic section with one focus on the pole and the other somewhere on the 0° ray (so that
the conic's semi-major axis lies along the polar axis) is given by:
wheree is the eccentricity and
is the semi-latus rectum (the perpendicular distance at a focus from the major axis to thecurve). Ife > 1, this equation defines a hyperbola; ife = 1, it defines a parabola; and ife < 1, itdefines an ellipse. The special casee= 0 of the latter results in a circle of radius
.
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Polar Differentiation
Differential calculus
We have the following formulas:
To find the Cartesian slope of the tangent line to a polar curver(θ) at any given point, thecurve is first expressed as a system of parametric equations.
Differentiating both equations with respect toθ yields
Dividing the second equation by the first yields the Cartesian slope of the tangent line to thecurve at the point (r, r(θ)):
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Polar Integration
Introduction
Integrating a polar equation requires a different approach than integration under the Cartesiansystem, hence yielding a different formula, which is not as straightforward as integrating thefunctionf(x).
Proof
In creating the concept of integration, we used Riemann sums on rectangles to approximatethe area under the curve. However, with polar graphs, one must use triangles that start from theorigin, and have a radius ending on the curve. If you don't mind skipping the proof, this is theform to use to integrate a polar expression of the formr = f(θ):
,
where (a,f(a)) and (b,f(b)) are the ends of the curve that you wish to integrate.
Integral calculus
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The integration regionR is bounded by the curver = f(θ) and the raysθ = a andθ = b.
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Let R denote the region enclosed by a curver = f(θ) and the raysθ = a andθ = b, where 0< b − a < 2π. Then, the area ofR is
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The regionR is approximated byn sectors (here,n = 5).This result can be found as follows. First, the interval [a,b] is divided inton subintervals,
wheren is an arbitrary positive integer. Thusθ, the length of each subinterval, is equal tob − a(the total length of the interval), divided byn, the number of subintervals. For each subinterval
, let θi be the midpoint of the subinterval, and construct a circular sector with the center at
the origin, radiusr i = f(θi), central angleδθ, and arc lengthr iδθ. The area of each constructed
sector is therefore equal to
. Hence, the total area of all of the sectors is
As the number of subintervalsn is increased, the approximation of the area continues toimprove. In the limit as
, the sum becomes the Riemann integral.
Generalization
Using Cartesian coordinates, an infinitesimal area element can be calculated asdA=
. The substitution rule for multiple integrals states that, when using other coordinates, theJacobian determinant of the coordinate conversion formula has to be considered:
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Hence, an area element in polar coordinates can be written as
Now, a function that is given in polar coordinates can be integrated as follows:
Here,R is the same region as above, namely, the region enclosed by a curver = f(θ) and theraysθ = a andθ = b.
The formula for the area ofR mentioned above is retrieved by takingg identically equal to1.
Applications
Polar integration is often useful when the corresponding integral is either difficult or impossi-ble to do with the Cartesian coordinates. For example, let's try to find the area of the closed unit
circle. That is, the area of the region enclosed byx2 + y2 = 1.
In Cartesian
In order to evaluate this, one usually uses trigonometric substitution. By setting sinθ = x, weget both
and
.
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Putting this back into the equation, we get
In Polar
To integrate in polar coordinates, we first realize
and in order to include the whole circle,a = 0 andb = 2π.
An interesting example
A less intuitive application of polar integration yields the Gaussian integral
Try it! (Hint: multiply
and
.)
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Sequences and Series
Sequences→Calculus← Polar Integration
Sequences and Series
Basics
• Sequences• Series
Series and calculus
• Taylor series• Power series• Exercises
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Exercises
1. Assume that thenth partial sum of aseriesis given by
.a) Does the series converge? If so, to what value?
b) What is the formula for thenth term of the series?
2. Find the value to which each the following series converges:a)
b)
c)
d)
3. Determine whether each the following series converges or diverges:a)
b)
c)
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d)
e)
f)
g)
4. Determine whether each the following series converges conditionally, converges abso-lutely, or diverges:
a)
b)
c)
d)
e)
f)
g)
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Hints
1.a) take a limit
b) sn = sn − 1 + an
2.a) sum of an infinite geometric series
b) sum of an infinite geometric series
c) telescoping series
d) rewrite so that all exponents aren
3.a)p-series
b) geometric series
c) limit comparison test
d) direct comparison test
e) divergence test
f) alternating series test
g) alternating series test
4.a) direct comparison test
b) alternating series test; integral test or direct comparison test
c) divergence test
d) alternating series test; limit comparison test
e) divergence test
f) ratio test
g) divergence test
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Answers only
1.a) The series converges to 2.
b)
2.a) 4
b)
c) 1
d) −1/5
3.a) converges
b) converges
c) diverges
d) diverges
e) diverges
f) converges
g) diverges
4.a) converges conditionally
b) converges conditionally
c) diverges
d) converges absolutely
e) diverges
f) converges absolutely
g) diverges
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Full solutions
1.a) The series converges to 2 since:
b)
2.a) The series is
and so is geometric with first terma = 3 and common ratior = 1/4. So
b)
c) Note that
by partial fractions. So
All but the first and last terms cancel out, so
d) The series simplifies to
and so is geometric. Thus
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3.a) This is ap-series withp = 2. Sincep > 1, the series converges.
b) This is a geometric series with common ratior = 1/2, and so converges since |r | < 1.
c) This series can be compared to ap-series:
The
symbol means the two series are "asymptotically equivalent"—that is, they either bothconverge or both diverge because their terms behave so similarly when summed asn getsvery large. This can be shown by thelimit comparison test:
Since the limit is positive and finite, the two series either both converge or both diverge.The simpler series diverges because it is ap-series withp = 1 (harmonic series), and so theoriginal series diverges by the limit comparison test.
d) This series can be compared to a smallerp-series:
Thep-series diverges sincep = 1 (harmonic series), so the larger series diverges by theappropriatedirect comparison test.
e)solution to come
f) solution to come
g) solution to come
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4.a)solution to come
b) solution to come
c) solution to come
d) solution to come
e)solution to come
f) solution to come
g) solution to come
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Sequences
A sequenceis an ordered list of objects (or events). Like a set, it contains members (alsocalledelementsor terms), and the number of terms (possibly infinite) is called thelengthof thesequence. Unlike a set, order matters, and the exact same elements can appear multiple times atdifferent positions in the sequence.
For example, (C, R, Y) is a sequence of letters that differs from (Y, C, R), as the orderingmatters. Sequences can befinite, as in this example, orinfinite, such as the sequence of all evenpositive integers (2, 4, 6,...).
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An infinite sequence of real numbers (in blue). This sequence is neither increasing, nor decreas-ing, nor convergent. It is however bounded.
Examples and notation
There are various and quite different notions of sequences in mathematics, some of which(e.g., exact sequence) are not covered by the notations introduced below.
A sequence may be denoted (a1, a2, ...). For shortness, the notation (an) is also used.
A more formal definition of afinite sequencewith terms in a setS is a function from {1, 2,...,n} to Sfor somen ≥ 0. An infinite sequencein Sis a function from {1, 2, ...} (the set of natu-ral numbers without 0) toS.
Sequences may also start from 0, so the first term in the sequence is thena0.
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A finite sequence is also called an n-tuple. Finite sequences include theempty sequence( )that has no elements.
A function from all integers into a set is sometimes called abi-infinite sequence, since itmay be thought of as a sequence indexed by negative integers grafted onto a sequence indexedby positive integers.
Types and properties of sequences
A subsequence of a given sequence is a sequence formed from the given sequence by delet-ing some of the elements without disturbing the relative positions of the remaining elements.
If the terms of the sequence are a subset of an ordered set, then amonotonically increasingsequence is one for which each term is greater than or equal to the term before it; if each term isstrictly greater than the one preceding it, the sequence is calledstrictly monotonically increasing.A monotonically decreasing sequence is defined similarly. Any sequence fulfilling the monotonic-ity property is called monotonic ormonotone. This is a special case of the more general notionof monotonic function.
The termsnon-decreasingandnon-increasingare used in order to avoid any possible confu-sion with strictly increasing and strictly decreasing, respectively. If the terms of a sequence areintegers, then the sequence is an integer sequence. If the terms of a sequence are polynomials,then the sequence is a polynomial sequence.
If Sis endowed with a topology, then it becomes possible to considerconvergenceof an infi-nite sequence inS. Such considerations involve the concept of the limit of a sequence.
Sequences in analysis
In analysis, when talking about sequences, one will generally consider sequences of the form
or
which is to say, infinite sequences of elements indexed by natural numbers. (It may be conve-nient to have the sequence start with an index different from 1 or 0. For example, the sequence
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defined byxn = 1/log(n) would be defined only forn ≥ 2. When talking about such infinite se-
quences, it is usually sufficient (and does not change much for most considerations) to assumethat the members of the sequence are defined at least for all indices large enough, that is, greaterthan some givenN.)
The most elementary type of sequences are numerical ones, that is, sequences of real orcomplex numbers.
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Series
Introduction
An arithmetic series is the sum of a sequence of terms with a common difference. A geomet-ric series is the sum of terms with a common ratio. For example, an interesting series which ap-pears in many practical problems in science, engineering, and mathematics is the geometric series
r + r2 + r3 + r4 + ... where the ... indicates that the series continues indefinitely. A common wayto study a particular series (following Cauchy) is to define a sequence consisting of the sum ofthe firstn terms. For example, to study the geometric series we can consider the sequence whichadds together the first n terms:
Generally by studying the sequence of partial sums we can understand the behavior of theentire infinite series.
Two of the most important questions about a series are
• Does it converge?• If so, what does it converge to?
For example, it is fairly easy to see that forr > 1, the geometric seriesSn(r) will not converge
to a finite number (i.e., it will diverge to infinity). To see this, note that each time we increase
the number of terms in the series,Sn(r) increases byrn + 1, sincern + 1 > 1 for all r > 1 (as we de-
fined), Sn(r) must increase by a number greater than one every term. When increasing the sum
by more than one for every term, it will diverge.
Perhaps a more surprising and interesting fact is that for |r | < 1,Sn(r) will converge to a fi-
nite value. Specifically, it is possible to show that
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Indeed, consider the quantity
Since
as
for | r | < 1, this shows that
as
. The quantity 1 -r is non-zero and doesn't depend onn so we can divide by it and arrive atthe formula we want.
We'd like to be able to draw similar conclusions about any series.
Unfortunately, there is no simple way to sum a series. The most we will be able to do inmost cases is determine if it converges. The geometric and the telescoping series are the onlytypes of series in which we can easily find the sum of.
Convergence
It is obvious that for a series to converge, thean must tend to zero (because sum of any infi-
nite terms is infinity, except when the sequence approaches 0), but even if the limit of the se-quence is 0, is not sufficient to say it converges.
Consider the harmonic series, the sum of 1/n, and group terms
As m tends to infinity, so does this final sum, hence the series diverges.
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We can also deduce something about how quickly it diverges. Using the same grouping ofterms, we can get an upper limit on the sum of the first so many terms, thepartial sums.
or
and the partial sums increase like logm, very slowly.
Notice that to discover this, we compared the terms of the harmonic series with a series weknew diverged.
This is aconvergence test(also known as the direct comparison test) we can apply to anypair of series.
• If bn converges and |an|≤|bn| thenan converges.• If bn diverges and |an|≥|bn| thenan diverges.
There are many such tests, the most important of which we'll describe in this chapter.
Absolute convergence
Theorem: If the series ofabsolutevalues,
, converges, then so does the series
We say such a seriesconverges absolutely.
Proof:
Let ε > 0
According to the Cauchy criterion for series convergence, existsN so that for allN < m,n:
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We know that:
And then we get:
Now we get:
Which is exactly the Cauchy criterion for series convergence.
Q.E.D
The converse does not hold.The series 1-1/2+1/3-1/4 ... converges, even though the seriesof its absolute values diverges.
A series like this that converges, but not absolutely, is said toconverge conditionally.
If a series converges absolutely, we can add terms in any order we like. The limit will stillbe the same.
If a series converges conditionally, rearranging the terms changes the limit. In fact, we canmake the series converge to any limit we like by choosing a suitable rearrangement.
E.g, in the series 1-1/2+1/3-1/4 ..., we can add only positive terms until the partial sum ex-ceeds 100, subtract 1/2, add only positive terms until the partial sum exceeds 100, subtract 1/4,and so on, getting a sequence with the same terms that converges to 100.
This makes absolutely convergent series easier to work with. Thus, all but one of conver-gence tests in this chapter will be for series all of whose terms are positive, which must be abso-lutely convergent or divergent series. Other series will be studied by considering the correspond-ing series of absolute values.
Ratio test
For a series with termsan, if
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then
• the series converges (absolutely) ifr<1• the series diverges ifr>1• the series could do either ifr=1, so the test is not conclusive in this case.
E.g, suppose
then
so this series converges.
Integral test
If f(x) is a monotonically decreasing, always positive function, then the series
converges ifand only ifthe integral
converges.
E.g, considerf(x)=1/xp, for a fixedp.
• If p=1 this is the harmonic series, which diverges.• If p<1 each term is larger than the harmonic series, so it diverges.• If p>1 then
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The integral converges, forp>1, so the series converges.
We can prove this test works by writing the integral as
and comparing each of the integrals with rectangles, giving the inequalities
Applying these to the sum then shows convergence.
Limit comparison test
Given an infinite series
with positive terms only, if one can find another infinite series
with positive terms for which
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for a positive and finiteL (i.e., the limit exists and is not zero), then the two series eitherboth converge or both diverge. That is,
•converges if
converges, and•
diverges if
diverges.
Example:
For largen, the terms of this series are similar to, but smaller than, those of the harmonicseries. We compare the limits.
so this series diverges.
Alternating series
If the signs of thean alternate,
then we call this analternatingseries.The series sum converges provided that
and
.
The error in a partial sum of an alternating series is smaller than the first omitted term.
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Geometric series
The geometric series can take either of the following forms
or
As you have seen at the start, the sum of the geometric series is
.
Telescoping series
Expanding (or "telescoping") this type of series is informative. If we expand this series, weget:
Additive cancellation leaves:
Thus,
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and all that remains is to evaluate the limit.
There are other tests that can be used, but these tests are sufficient for all commonly encoun-tered series.
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Taylor series
Taylor Series
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sin(x)and Taylor approximations, polynomials of degree1, 3, 5, 7, 9, 11and13.
TheTaylor seriesof an infinitely oftendifferentiablereal (or complex)function f definedon anopen interval(a-r, a+r) is thepower series
Here,n! is thefactorialof n andf (n)(a) denotes thenth derivativeof f at the pointa. If thisseries converges for everyx in the interval (a-r, a+r) and the sum is equal tof(x), then the func-tion f(x) is calledanalytic. To check whether the series converges towardsf(x), one normallyuses estimates for the remainder term ofTaylor's theorem. A function is analytic if and only if apower seriesconverges to the function; the coefficients in that power series are then necessarilythe ones given in the above Taylor series formula.
If a = 0, the series is also called aMaclaurin series.
The importance of such a power series representation is threefold. First, differentiation andintegration of power series can be performed term by term and is hence particularly easy. Second,an analytic function can be uniquely extended to aholomorphic functiondefined on an open disk
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in thecomplex plane, which makes the whole machinery ofcomplex analysisavailable. Third,the (truncated) series can be used to approximate values of the function near the point of expan-sion.
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The functione-1/x²is not analytic: the Taylor series is 0, although the function is not.
Note that there are examples ofinfinitely often differentiable functionsf(x) whose Taylorseries converge, but arenot equal tof(x). For instance, for the function defined piecewise bysaying thatf(x) = exp(−1/x²) if x ≠ 0 andf(0) = 0, all the derivatives are zero atx = 0, so theTaylor series off(x) is zero, and itsradius of convergenceis infinite, even though the functionmost definitely is not zero. This particular pathology does not afflictcomplex-valued functionsof a complex variable. Notice that exp(−1/z²) does not approach 0 asz approaches 0 along theimaginary axis.
Some functions cannot be written as Taylor series because they have asingularity; in thesecases, one can often still achieve a series expansion if one allows also negative powers of thevariablex; seeLaurent series. For example,f(x) = exp(−1/x²) can be written as a Laurent series.
TheParker-Sockacki theoremis a recent advance in finding Taylor series which are solutionsto differential equations. This theorem is an expansion on thePicard iteration.
Derivation/why this works
If a function f(x) is written as a infinite power series, it will look like this:
f(x)=c0(x-a)0+c1(x-a)1+c2(x-a)2+c3(x-a)3+c4(x-a)4+c5(x-a)5+c6(x-a)6+c7(x-
a)7+...
where a is half the radius of convergence and c0,c1,c2,c3,c4... are coefficients. If we substitute
a for x:
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f(a)=c0
If we differentiate:
f´(x)=1c1(x-a)0+2c2(x-a)1+3c3(x-a)2+4c4(x-a)3+5c5(x-a)4+6c6(x-a)5+7c7(x-
a)6+...
If we substitute a for x:
f´(a)=1c1
If we differentiate:
f´´(x)=2c2+3*2*c3(x-a)1+4*3*c4(x-a)2+5c5*4*(x-a)3+6*5*c6(x-a)4+7c7*6*(x-
a)5+...
If we substitute a for x:
f´´(a)=2c2
Extrapolating:
n!cn=fn(a)
where f0(x)=f(x) and f1(x)=f´(x) and so on.We can actually go ahead and say that the powerapproximation of f(x) is:
f(x)=Sn=0¥((fn(a)/n!)*(x-a)n)
<this needs to be improved>
List of Taylor series
Several important Taylor series expansions follow. All these expansions are also valid forcomplex argumentsx.
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Exponential functionandnatural logarithm:
Geometric series:
Binomial series:
Trigonometric functions:
Hyperbolic functions:
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Lambert's W function:
The numbersBk appearing in the expansions of tan(x) and tanh(x) are theBernoulli numbers.
The C(α,n) in the binomial expansion are thebinomial coefficients. TheEk in the expansion of
sec(x) areEuler numbers.
Multiple dimensions
The Taylor series may be generalized to functions of more than one variable with
History
The Taylor series is named for mathematicianBrook Taylor, who first published the powerseries formula in 1715.
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Constructing a Taylor Series
Several methods exist for the calculation of Taylor series of a large number of functions.One can attempt to use the Taylor series as-is and generalize the form of the coefficients, or onecan use manipulations such as substitution, multiplication or division, addition or subtraction ofstandard Taylor series (such as those above) to construct the Taylor series of a function, by virtueof Taylor series being power series. In some cases, one can also derive the Taylor series by repeat-edly applyingintegration by parts. The use ofcomputer algebra systemsto calculate Taylor seriesis common, since it eliminates tedious substitution and manipulation.
Example 1
Consider the function
for which we want a Taylor series at 0.
We have for the natural logarithm
and for the cosine function
We can simply substitute the second series into the first. Doing so gives
Expanding by usingmultinomial coefficientsgives the required Taylor series. Note that co-sine and thereforef are even functions, meaning thatf(x) = f( − x), hence the coefficients of the
odd powersx, x3, x5, x7 and so on have to be zero and don't need to be calculated. The first fewterms of the series are
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The general coefficient can be represented usingFaà di Bruno's formula. However, thisrepresentation does not seem to be particularly illuminating and is therefore omitted here.
Example 2
Suppose we want the Taylor series at 0 of the function
We have for the exponential function
and, as in the first example,
Assume the power series is
Then multiplication with the denominator and substitution of the series of the cosine yields
Collecting the terms up to fourth order yields
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Comparing coefficients with the above series of the exponential function yields the desiredTaylor series
Convergence
Generalized Mean Value Theorem
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Power series
The study ofpower seriesis aimed at investigating series which can approximate somefunction over a certain interval.
Motivations
Elementary calculus (differentiation) is used to obtain information on a line which touchesa curve at one point (i.e. a tangent). This is done by calculating the gradient, or slope of the curve,at a single point. However, this does not provide us with reliable information on the curve's actualvalueat given points in a wider interval. This is where the concept of power series becomesuseful.
An example
Consider the curve ofy = cos(x), about the pointx = 0. A naïve approximation would be theline y = 1. However, for a more accurate approximation, observe that cos(x) looks like an invertedparabola aroundx = 0 - therefore, we might think about which parabola could approximate theshape of cos(x) near this point. This curve might well come to mind:
In fact, this is the best estimate for cos(x) which uses polynomials of degree 2 (i.e. a highest
term ofx2) - but how do we know this is true? This is the study of power series: finding optimalapproximations to functions using polynomials.
Definition
A power seriesis aseriesof the form
a0x0 + a1x
1 + ... +anxn
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or, equivalently,
Radius of convergence
When using a power series as an alternative method of calculating a function's value, theequation
can only be used to studyf(x) where the power series converges - this may happen for a finiterange, or for allreal numbers.
The size of the interval (around its center) in which the power series converges to the func-tion is known as theradius of convergence.
An example
(a geometric series)
this converges when |x | < 1, the range -1 <x < +1, so the radius of convergence - centeredat 0 - is1. It should also be observed that at theextremitiesof the radius, that is wherex = 1 andx = -1, the power series does not converge.
Another example
Using theratio test, this series converges when the ratio of successive terms is less than one:
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or
which is always true - therefore, this power series has an infinite radius of convergence. Ineffect, this means that the power series canalwaysbe used as a valid alternative to the original
function, ex.
Abstraction
If we use the ratio test on an arbitrary power series, we find it converges when
and diverges when
The radius of convergence is therefore
If this limit diverges to infinity, the series has an infinite radius of convergence.
Differentiation and Integration
Within its radius of convergence, a power series can be differentiated and integrated termby term.
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Both the differential and the integral have the same radius of convergence as the originalseries.
This allows us to sum exactly suitable power series. For example,
This is a geometric series, which converges for |x | < 1. Integrating both sides, we get
which will also converge for |x | < 1. Whenx = -1 this is the harmonic series, whichdi-verges'; whenx = 1 this is an alternating series with diminishing terms, whichconvergesto ln2 - this is testing the extremities.
It also lets us write power series for integrals we cannot do exactly such as the error function:
The left hand side can not be integrated exactly, but the right hand side can be.
This gives us a power series for the sum, which has an infinite radius of convergence, lettingus approximate the integral as closely as we like.
Further reading
• "Decoding the Rosetta Stone"article by Jack W. Crenshaw 2005-10-12
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Multivariable and differentialcalculus
Vectors→Calculus← Sequences and Series/Exer-cises
Multivariable and differen-tial calculus
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• Vectors
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• Lines and Planes in Space
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• Multivariable Calculus
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• Ordinary Differential Equations
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• Partial Differential Equations
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• Exercises
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Vectors
Lines and Planes in Space→Calculus← Multivariable and differen-tial calculus
Vectors
Two-Dimensional Vectors
Introduction
In most mathematics courses up until this point, we deal withscalars. These are quantitieswhich only need one number to express. For instance, the amount of gasoline used to drive tothe grocery store is a scalar quantity because it only needs one number: 2 gallons.
In this unit, we deal withvectors. A vector is adirected line segment-- that is, a line seg-ment that points one direction or the other. As such, it has aninitial point and aterminal point .The vector starts at the initial point and ends at the terminal point, and the vector points towardsthe terminal point. A vector is drawn as a line segment with an arrow at the terminal point:
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The same vector can be placed anywhere on the coordinate plane and still be the same vector-- the only two bits of information a vector represents are themagnitudeand thedirection. Themagnitude is simply the length of the vector, and the direction is the angle at which it points.Since neither of these specify a starting or endinglocation, the same vector can be placed any-where. To illustrate, all of the line segments below can be defined as the vector with magnitude
and angle 45 degrees:
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It is customary, however, to place the vector with the initial point at the origin as indicatedby the black vector. This is called thestandard position.
Component Form
In standard practice, we don't express vectors by listing the length and the direction. We in-stead usecomponent form, which lists the height (rise) and width (run) of the vectors. It iswritten as follows:
Other ways of denoting a vector in component form include:
and
From the diagram we can now see the benefits of the standard position: the two numbers forthe terminal point's coordinates are the same numbers for the vector's rise and run. Note that wenamed this vectoru. Just as you can assign numbers to variables in algebra (usually x, y, and z),you can assign vectors to variables in calculus. The letters u, v, and w are usually used, and eitherboldface or an arrow over the letter is used to identify it as a vector.
When expressing a vector in component form, it is no longer obvious what the magnitudeand direction are. Therefore, we have to perform some calculations to find the magnitude anddirection.
Magnitude
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whereux is the width, or run, of the vector;uy is the height, or rise, of the vector. You should
recognize this formula as the Pythagorean theorem. It is -- the magnitude is the distance betweenthe initial point and the terminal point.
The magnitude of a vector can also be called the norm.
Direction
whereθ is the direction of the vector. This formula is simply the tangent formula for righttriangles.
Vector Operations
For these definitions, assume:
Vector Addition
Vector Addition is often calledtip-to-tail addition, because this makes it easier to remember.
The sum of the vectors you are adding is called the resultant vector, and is the vector drawnfrom the initial point (tip) of the first vector to the terminal point (tail) of the second vector. Al-though they look like the arrows, the pointy bit is the tail, not the tip. (Imagine you were walkingthe direction the vector was pointing... you would start at the flat end (tip) and walk toward thepointy end.)
It looks like this:
(Notice, the black lined vector is the sum of the two dotted line vectors!)
Numerically:
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Or more generally:
Scalar Multiplication
Graphically, multiplying a vector by a scalar changes only the magnitude of the vector bythat same scalar. That is, multiplying a vector by 2 will "stretch" the vector to twice its originalmagnitude, keeping the direction the same.
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Numerically, you calculate the resultant vector with this formula:
, where c is a constant scalar.As previously stated, the magnitude is changed by the same constant:
Since multiplying a vector by a constant results in a vector in the same direction, we canreason that two vectors are parallel if one is a constant multiple of the other -- that is, that
if
for some constant c.
We can also divide by a non-zero scalar by instead multiplying by the reciprocal, as withdividing regular numbers:
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Dot Product
The dot product, sometimes confusingly called the scalarproduct, of two vectors is givenby:
or which is equivalent to:
whereθ is the angle difference between the two vectors. This provides a convenient way offinding the angle between two vectors:
Applications of Scalar Multiplication and Dot Product
Unit Vectors
A unit vector is a vector with a magnitude of 1. Theunit vector of u is a vector in the samedirection as
, but with a magnitude of 1:
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The process of finding the unit vector of u is callednormalization. As mentioned inscalarmultiplication , multiplying a vector by constant C will result in the magnitude being multipliedby C. We know how to calculate the magnitude of
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. We know that dividing a vector by a constant will divide the magnitude by that constant.Therefore, if that constant is the magnitude, dividing the vector by the magnitude will result ina unit vector in the same direction as
:
, where
is the unit vector of
Standard Unit Vectors
A special case ofUnit Vectorsare theStandard Unit Vectorsi andj : i points one unit directlyright in the x direction, andj points one unit directly up in the y direction:
Using the scalar multiplication and vector addition rules, we can then express vectors in adifferent way:
If we work that equation out, it makes sense. Multiplying x byi will result in the vector
. Multiplying y by j will result in the vector
. Adding these two together will give us our original vector,
. Expressing vectors using i and j is calledstandard form.
Projection and Decomposition of Vectors
Sometimes it is necessary to decompose a vector
into two components: one component parallel to a vector
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, which we will call
; and one component perpendicular to it,
.
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Since the length of
is (
), it is straightforward to write down the formulas for
and
:
and
Length of a vector
The length of a vector is given by the dot product of a vector with itself, andθ = 0deg:
Perpendicular vectors
If the angleθ between two vectors is 90 degrees or
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(if the two vectors are orthogonal to each other), that is the vectors are perpendicular, then thedot product is 0. This provides us with an easy way to find a perpendicular vector: if you have avector
, a perpendicular vector can easily be found by either
Polar coordinates
Polar coordinates are an alternative two-dimensional coordinate system, which is often usefulwhen rotations are important. Instead of specifying the position along thexandyaxes, we specifythe distance from the origin,r, and the direction, an angleθ.
Looking at this diagram, we can see that the values ofx andy are related to those ofr andθ by the equations
Because tan-1 is multivalued, care must be taken to select the right value.
Just as for Cartesian coordinates the unit vectors that point in thex andy directions are spe-cial, so in polar coordinates the unit vectors that point in ther andθ directions are also special.
We will call these vectors
and
, pronounced r-hat and theta-hat. Putting a circumflex over a vector this way is often usedto mean the unit vector in that direction.
Again, on looking at the diagram we see,
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Three-Dimensional Coordinates and Vectors
Basic definition
Two-dimensional Cartesian coordinates as we've discussed so far can be easily extended tothree-dimensions by adding one more value: 'z'. If the standard (x,y) coordinate axes are drawnon a sheet of paper, the 'z' axis would extend upwards off of the paper.
Similar to the two coordinate axes in two-dimensional coordinates, there are threecoordi-nate planesin space. These are thexy-plane, theyz-plane, and thexz-plane. Each plane is the"sheet of paper" that contains both axes the name mentions. For instance, the yz-plane containsboth the y and z axes and is perpendicular to the x axis.
Therefore, vectors can be extended to three dimensions by simply adding the 'z' value.
To facilitate standard form notation, we add another standard unit vector:
Again, both forms (component and standard) are equivalent.
Magnitude: Magnitude in three dimensions is the same as in two dimensions, with the addi-tion of a 'z' term in the radicand.
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Three dimensions
The polar coordinate system is extended into three dimensions with two different coordinatesystems, the cylindrical and spherical coordinate systems, both of which include two-dimensionalor planar polar coordinates as a subset. In essence, the cylindrical coordinate system extendspolar coordinates by adding an additional distance coordinate, while the spherical system insteadadds an additional angular coordinate.
Cylindrical coordinates
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a point plotted with cylindrical coordinatesThecylindrical coordinate systemis a coordinate system that essentially extends the two-
dimensional polar coordinate system by adding a third coordinate measuring the height of a pointabove the plane, similar to the way in which the Cartesian coordinate system is extended intothree dimensions. The third coordinate is usually denotedh, making the three cylindrical coordi-nates (r, θ, h).
The three cylindrical coordinates can be converted to Cartesian coordinates by
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Spherical coordinates
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A point plotted using spherical coordinatesPolar coordinates can also be extended into three dimensions using the coordinates (ρ, φ,
θ), whereρ is the distance from the origin,φ is the angle from the z-axis (called the colatitudeor zenith and measured from 0 to 180°) andθ is the angle from the x-axis (as in the polar coordi-nates). This coordinate system, called thespherical coordinate system, is similar to the latitudeand longitude system used for Earth, with the origin in the centre of Earth, the latitudeδ beingthe complement ofφ, determined byδ = 90° −φ, and the longitudel being measured byl = θ −180°.
The three spherical coordinates are converted to Cartesian coordinates by
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Cross Product
The cross product of two vectors is adeterminant:
and is also apseudovector.
The cross product of two vectors is orthogonal to both vectors. The magnitude of the crossproduct is the product of the magnitude of the vectors and the sin of the angle between them.
This magnitude is the area of the parallelogram defined by the two vectors.
The cross product islinear andanticommutative. For any numbersa andb,
If both vectors point in the same direction, their cross product is zero.
Triple Products
If we have three vectors we can combine them in two ways, a triple scalar product,
and a triple vector product
The triple scalar product is a determinant
If the three vectors are listed clockwise, looking from the origin, the sign of this product ispositive. If they are listed anticlockwise the sign is negative.
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The order of the cross and dot products doesn't matter.
Either way, the absolute value of this product is the volume of the parallelepiped defined bythe three vectors,u, v, andw
The triple vector product can be simplified
This form is easier to do calculations with.
The triple vector product isnot associative.
There are special cases where the two sides are equal, but in general the brackets matter.They must not be omitted.
Three-Dimensional Lines and Planes
We will user to denote the position of a point.
The multiples of a vector,a all lie on a line through the origin. Adding a constant vectorbwill shift the line, but leave it straight, so the equation of a line is,
This is aparametric equation. The position is specified in terms of the parameters.
Any linear combination of two vectors,a andb lies on a single plane through the origin,provided the two vectors are not colinear. We can shift this plane by a constant vector again andwrite
If we choosea andb to beorthonormalvectors in the plane (i.e unit vectors at right angles)thens andt are Cartesian coordinates for points in the plane.
These parametric equations can be extended to higher dimensions.
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Instead of giving parametric equations for the line and plane, we could use constraints. E.g,for any point in thexyplanez=0
For a plane through the origin, the single vector normal to the plane,n, is at right angle withevery vector in the plane, by definition, so
is a plane through the origin, normal ton.
For planes not through the origin we get
A line lies on the intersection of two planes, so it must obey the constraint for both planes,i.e
These constraint equations con also be extended to higher dimensions.
Vector-Valued Functions
Vector-Valued Functions are functions that instead of giving a resultant scalar value, givea resultant vector value. These aid in the create of direction and vector fields, and are thereforeused in physics to aid with visualizations of electric, magnetic, and many other fields. They areof the following form:
Introduction
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Limits, Derivatives, and Integrals
Put simply, the limit of a vector-valued function is the limit of its parts.
Proof:
Suppose
Therefore for anyε > 0 there is aφ > 0 such that
But by the triangle inequality
So
Therefore
A similar argument can be used through parts a_n(t)
Now let
again, and that for anyε>0 there is a correspondingφ>0 such 0<|t-c|<φ implies
Then
therefore!:
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From this we can then create an accurate definition of a derivative of a vector-valued func-tion:
The final step was accomplished by taking what we just did with limits.
By the Fundamental Theorem of Calculus integrals can be applied to the vector's compo-nents.
In other words: the limit of a vector function is the limit of its parts, the derivative of a vectorfunction is the derivative of its parts, and the integration of a vector function is the integrationof it parts.
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Velocity, Acceleration, Curvature, and a brief mentionof the Binormal
Assume we have a vector-valued function which starts at the origin and as its independentvariables changes the points that the vectors point at trace a path.
We will call this vector
, which is commonly known as theposition vector.
If
then represents a position and t represents time, then in model with Physics we know thefollowing:
is displacement.
where
is the velocity vector.
is the speed.
where
is the acceleration vector.
The only other vector that comes in use at times is known as the curvature vector.
The vector
used to find it is known as the unit tangent vector, which is defined as
or shorthand
.
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The vector normal
to this then is
.
We can verify this by taking the dot product
Also note that
and
and
Then we can actually verify:
Therefore
is perpendicular to
What this gives rise to is theUnit Normal Vector
of which the top-most vector is the Normal vector, but the bottom half
is known as the curvature. Since the Normal vector points toward the inside of a curve, thesharper a turn, the Normal vector has a large magnitude, therefore the curvature has a small value,and is used as an index in civil engineering to reflect the sharpness of a curve (clover-leaf high-ways, for instance).
The only other thing not mentioned is the Binormal that occurs in 3-d curves
, which is useful in creating planes parallel to the curve.
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Lines and Planes in Space
Multivariable calculus→Calculus← Vectors
Lines and Planes in Space
Introduction
For many practical applications, for example for describing forces in physics and mechanics,you have to work with the mathematical descriptions oflinesandplanesin 3-dimensional space.
Parametric Equations
Line in Space
A line in space is defined by two points in space, which I will callP1 andP2. Let
be the vector from the origin toP1, and
the vector from the origin toP2. Given these two points, every other pointP on the line can
be reached by
where
is the vector fromP1 andP2:
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Plane in Space
The same idea can be used to describe a plane in 3-dimensional space, which is uniquelydefined by three points (which do not lie on a line) in space (P1,P2,P3). Let
be the vectors from the origin toPi. Then
with:
Note that the starting point does not have to be
, but can be any point in the plane. Similarly, the only requirement on the vectors
and
is that they have to be two non-collinear vectors in our plane.
Vector Equation (of aPlane in Space, or of aLine in a Plane)
An alternative representation of a Plane in Space is obtained by observing that a plane isdefined by a pointP1 in that plane and a direction perpendicular to the plane, which we denote
with the vector
. As above, let
describe the vector from the origin toP1, and
the vector from the origin to another pointP in the plane. Since any vector that lies in theplane is perpendicular to
, thevector equationof the plane is given by
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In 2 dimensions, the same equation uniquely describes a Line.
Scalar Equation (of aPlane in Space, or of aLine in a Plane)
If we express
and
through their components
writing out the scalar product for
provides us with thescalar equationfor a plane in space:
ax+ by+ cz= dwhere
.
In 2d space, the equivalent steps lead to the scalar equation for a line in a plane:
ax+ by= c
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Multivariable calculus
Ordinary differential equations→
Calculus← Lines and Planes in Space
Multivariable calculus
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In your previous study of calculus, we have looked at functions and their behavior. Most ofthese functions we have examined have been all in the form
f(x) : R → R,
and only occasional examination of functions of two variables. However, the study of func-tions ofseveralvariables is quite rich in itself, and has applications in several fields.
We write functions of vectors - many variables - as follows:
f : Rm → Rn
andf(x) for the function that maps a vector inRm to a vector inRn.
Before we can do calculus inRn, we must familiarize ourselves with the structure ofRn. We
need to know which properties ofR can be extended toRn
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Topology inRn
We are already familiar with the nature of the regular real number line, which is the setR,
and the two-dimensional plane,R2. This examination oftopologyin Rn attempts to look at a
generalization of the nature ofn-dimensional spaces -R, or R23, or Rn.
Lengths and distances
If we have a vector inR2, we can calculate its length using the Pythagorean theorem. Forinstance, the length of the vector (2, 3) is
We can generalize this toRn. We define a vector's length, written |x|, as the square root ofthe sum of the squares of each of its components. That is, if we have a vectorx=(x1,...,xn),
Now that we have established some concept of length, we can establish the distance betweentwo vectors. We define this distance to be the length of the two vectors' difference. We write thisdistanced(x, y), and it is
This distance function is sometimes referred to as ametric. Other metrics arise in differentcircumstances. The metric we have just defined is known as theEuclideanmetric.
Open and closed balls
In R, we have the concept of aninterval, in that we choose a certain number of other pointsabout some central point. For example, the interval [-1, 1] is centered about the point 0, and in-cludes points to the left and right of zero.
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In R2 and up, the idea is a little more difficult to carry on. ForR2, we need to consider points
to the left, right, above, and below a certain point. This may be fine, but forR3 we need to includepoints in more directions.
We generalize the idea of the interval by considering all the points that are a given, fixed
distance from a certain point - now we know how to calculate distances inRn, we can make ourgeneralization as follows, by introducing the concept of anopen balland aclosed ballrespective-ly, which are analogous to the open and closed interval respectively.
anopen ball
is a set in the form {x ∈ Rn|d(x, a) < r}
aclosed ball
is a set in the form {x ∈ Rn|d(x, a) ≤ r}
In R, we have seen that the open ball is simply an open interval centered about the point
x=a. In R2 this is a circle with no boundary, and inR3 it is a sphere with no outer surface. (Whatwould the closed ball be?)
Boundary points
If we have some area, say a field, then the common sense notion of theboundaryis the points'next to' both the inside and outside of the field. For a set, S, we can define this rigorously bysaying the boundary of the set contains all those points such that we can find points both insideand outside the set. We call the set of such points∂S
Typically, when it exists the dimension of∂S is one lower than the dimension of S. e.g theboundary of a volume is a surface and the boundary of a surface is a curve.
This isn't always true; but it is true of all the sets we will be using.
A setS is boundedif there is some positive number such that we can encompass this set bya closed ball about0. --> if every point in it is within a finite distance of the origin, i.e there existssomer>0 such thatx is in S implies |x|<r.
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Curves and parameterizations
If we have a functionf : R → Rn, we say thatf's image (the set {f(t) | t ∈ R} - or some subset
of R) is acurvein Rn andf is its parametrization.
Parameterizations are not necessarily unique - for example,f(t) = (cost, sin t) such thatt ∈[0, 2π) is one parametrization of the unit circle, andg(t) = (cosat, sinat) such thatt ∈ [0, 2π/a)is a whole family of parameterizations of that circle.
Collision and intersection points
Say we have two different curves. It may be important to consider
• when the two curves cross each other - where theyintersect• when the two curves hit each other at the same time - where theycollide.
Intersection points
Firstly, we have two parameterizationsf(t) andg(t), and we want to find out when they inter-sect, this means that we want to know when the function values of each parametrization are thesame. This means that we need to solve
f(t) = g(s)
because we're seeking the function values independent of the times they intersect.
For example, if we havef(t) = (t, 3t) andg(t) = (t, t2), and we want to find intersection points:
f(t) = g(s)
(t, 3t) = (s, s2)
t = s and 3t = s2
with solutions (t, s) = (0, 0) and (3, 3)
So, the two curves intersect at the points (0, 0) and (3, 3).
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Collision points
However, if we want to know when the points "collide", withf(t) andg(t), we need to knowwhen both the function valuesandthe times are the same, so we need to solve instead
f(t) = g(t)
For example, using the same functions as before,f(t) = (t, 3t) andg(t) = (t, t2), and we wantto find collision points:
f(t) = g(t)
(t, 3t) = (t, t2)
t = t and 3t = t2
which gives solutionst = 0, 3 So the collision points are (0, 0) and (3, 3).
We may want to do this to actually model physical problems, such as in ballistics.
Continuity and differentiability
If we have a parametrizationf : R → Rn, which is built up out ofcomponent functionsin theform f(t) = (f1(t),...,fn(t)), f is continuous if and only if each component function is also.
In this case the derivative off(t) is
ai = (f1′(t),...,fn′(t)). This is actually a specific consequence of a more general fact we willsee later.
Tangent vectors
Recall in single-variable calculus that on a curve, at a certain point, we can draw a line thatis tangent to that curve at exactly at that point. This line is called atangent. In the several variablecase, we can do something similar.
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We can expect thetangent vectorto depend onf′(t) and we know that a line is its own tan-gent, so looking at a parametrised line will show us precisely how to define the tangent vectorfor a curve.
An arbitrary line isf(t)=at+b, with :fi(t)=ait+bi, so
fi′(t)=ai and
f′(t)=a, which is the direction of the line, its tangent vector.
Similarly, for any curve, the tangent vector isf′(t).
Angle between curves
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We can then formulate the concept of theanglebetween two curves by considering the anglebetween the two tangent vectors. If two curves, parametrized byf1 andf2 intersect at some point,
which means that
f1(s)=f2(t)=c,
the angle between these two curves atc is the angle between the tangent vectorsf1′(s) and
f2′(t) is given by
Tangent lines
With the concept of the tangent vector as being analogous to being the gradient of the linein the one variable case, we can form the idea of thetangent line. Recall that we need a point onthe line and its direction.
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If we want to form the tangent line to a point on the curve, sayp, we have the direction ofthe linef′(p), so we can form the tangent line
x(t)=p+t f′(p)
Different parameterizations
One such parametrization of a curve is not necessarily unique. Curves can have several differ-ent parametrizations. For example, we already saw that the unit circle can be parametrized byg(t) = (cosat, sinat) such thatt ∈ [0, 2π/a).
Generally, iff is one parametrization of a curve, andg is another, with
f(t0) = g(s0)
there is a functionu(t) such thatu(t0)=s0, andg'(u(t)) = f(t) neart0.
This means, in a sense, the functionu(t) "speeds up" the curve, but keeps the curve's shape.
Surfaces
A surface in space can be described by the image of a functionf : R2→ Rn. f is said to be
the parametrization of that surface.
For example, consider the function
f(α, β) = α(2,1,3)+β(-1,2,0)
This describes an infinite plane inR3. If we restrictα andβ to some domain, we get a paral-
lelogram-shaped surface inR3.
Surfaces can also be described explicitly, as the graph of a functionz = f(x, y) which has astandard parametrization asf(x,y)=(x, y, f(x,y)), or implictly, in the formf(x, y, z)=c.
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Level sets
The concept of thelevel set(or contour) is an important one. If you have a functionf(x, y,
z), a level set inR3 is a set of the form {(x,y,z)|f(x,y,z)=c}. Each of these level sets is a surface.
Level sets can be similarly defined in anyRn
Level sets in two dimensions may be familiar from maps, or weather charts. Each line repre-sents a level set. For example, on a map, each contour represents all the points where the heightis the same. On a weather chart, the contours represent all the points where the air pressure is thesame.
Limits and continuity
Before we can look at derivatives of multivariate functions, we need to look at how limitswork with functions of several variables first, just like in the single variable case.
If we have a functionf : Rm → Rn, we say thatf(x) approachesb (in Rn) asx approachesa
(in Rm) if, for all positiveε, there is a corresponding positive numberδ, |f(x)-b| < ε whenever |x-a| < δ, with x ≠ a.
This means that by making the difference betweenx anda smaller, we can make the differ-ence betweenf(x) andb as small as we want.
If the above is true, we say
• f(x) haslimit b ata•
• f(x) approachesb asx approachesa• f(x) → b asx → a
These four statements are all equivalent.
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Rules
Since this is an almost identical formulation of limits in the single variable case, many ofthe limit rules in the one variable case are the same as in the multivariate case.
For f andg, mappingRm to Rn, andh(x) a scalar function mappingRm to R, with
• f(x) → b asx → a• g(x) → c asx → a• h(x) → H asx → a
then:
•
•
and consequently
•
•
when H≠0
•
Continuity
Again, we can use a similar definition to the one variable case to formulate a definition ofcontinuity for multiple variables.
If f : Rm → Rn, f is continuous at a pointa in Rm if f(a) is defined and
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Just as for functions of one dimension, iff, g are both continuous atp, f+g, λf (for a scalar
λ), f·g, andf×g are continuous also. Ifφ : Rm → R is continus atp, φf, f/φ are too ifφ is neverzero.
From these facts we also have that ifA is some matrix which isn×m in size, withx in Rm,a functionf(x)=A x is continuous in that the function can be expanded in the formx1a1+...+xmam,
which can be easily verified from the points above.
If f : Rm → Rn which is in the formf(x) = (f1(x),...,fn(x) is continuous if and only if each of
its component functions are a polynomial or rational function, whenever they are defined.
Finally, if f is continuous atp, g is continuous atf(p), g(f(x)) is continuous atp.
Special note about limits
It is important to note that we can approach a pointin more than one direction, and thus, thedirection that we approach that point counts in our evaluation of the limit. It may be the case thata limit may exist moving in one direction, but not in another.
Differentiable functions
We will start from the one-variable definition of the derivative at a pointp, namely
Let's change above to equivalent form of
which achieved after pulling f'(p) inside and putting it over a common denominator.
We can't divide by vectors, so this definition can't be immediately extended to the multiplevariable case. Nonetheless, we don't have to: the thing we took interest in was the quotient oftwo small distances (magnitudes), not their other properties (like sign). It's worth noting that
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'other' property of vector neglected is its direction. Now we can divide by the absolute value ofa vector, so lets rewrite this definition in terms of absolute values
Another form of formula above is obtained by letingh = x − p we havex = p + h and if
, the
, so
,
whereh can be thought of as a 'small change'.
So, how can we use this for the several-variable case?
If we switch all the variables over to vectors and replace the constant (which performs alinear map in one dimension) with a matrix (which denotes also a linear map), we have
or
If this limit exists for somef : Rm → Rn, and there is a linear mapA : Rm → Rn (denoted bymatrixA which ism×n), we refer to this map as being the derivative and we write it as Dp f.
A point on terminology - in referring to the action of taking the derivative (giving the linearmapA), we write Dp f, but in referring to the matrixA itself, it is known as theJacobianmatrix
and is also writtenJp f. More on the Jacobian later.
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Properties
There are a number of important properties of this formulation of the derivative.
Affine approximations
If f is differentiable atp for x close top, |f(x)-(f(p)+A(x-p))| is small compared to |x-p|, whichmeans thatf(x) is approximately equal tof(p)+A(x-p).
We call an expression of the formg(x)+c affine, wheng(x) is linear andc is a constant.f(p)+A(x-p) is an affine approximation tof(x).
Jacobian matrix and partial derivatives
The Jacobian matrix of a function is in the form
for a f : Rm → Rn, Jp f' is am×n matrix.
The consequence of this is that iff is differentiable atp, all the partial derivatives off existatp.
However, it is possible that all the partial derivatives of a function exist at some point yetthat function is not differentiable there, so it's very important not to mix derivative (linear map)with the Jacobian (matrix) especially when cited situation arised.
Continuity and differentiability
Furthermore, if all the partial derivatives exist, and are continuous in some neighbourhoodof a pointp, thenf is differentiable atp. This has the consequence that for a functionf which hasits component functions built from continuous functions (such as rational functions, differentiablefunctions or otherwise),f is differentiable everywheref is defined.
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We use the terminologycontinuously differentiablefor a function differentiable atp whichhas all its partial derivatives existing and are continuous in some neighbourhood atp.
Rules of taking Jacobians
If f : Rm → Rn, andh(x) : Rm → R are differentiable at 'p' :
•••
Important: make sure the order is right - matrix multiplication is not commutative!
Chain rule
The chain rule for functions of several variables is as follows. Forf : Rm → Rn andg : Rn
→ Rp, andg o f differentiable atp, then the Jacobian is given by
Again, we have matrix multiplication, so one must preserve this exact order. Compositionsin one order may be defined, but not necessarily in the other way.
Alternate notations
For simplicity, we will often use various standard abbreviations, so we can write most of theformulae on one line. This can make it easier to see the important details.
We can abbreviate partial differentials with a subscript, e.g,
When we are using a subscript this way we will generally use the HeavisideD rather than∂,
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Mostly, to make the formulae even more compact, we will put the subscript on the functionitself.
If we are using subscripts to label the axes,x1, x2 …, then, rather than having two layers of
subscripts, we will use the number as the subscript.
We can also use subscripts for the components of a vector function,u=(ux, uy, uy) or
u=(u1,u2…un)
If we are using subscripts for both the components of a vector and for partial derivatives wewill separate them with a comma.
The most widely used notation is hx. Both h1 and∂1h are also quite widely used whenever
the axes are numbered. The notation∂xh is used least frequently.
We will use whichever notation best suits the equation we are working with.
Directional derivatives
Normally, a partial derivative of a function with respect to one of its variables, say,xj, takes
the derivative of that "slice" of that function parallel to thexj'th axis.
More precisely, we can think of cutting a functionf(x1,...,xn) in space along thexj'th axis,
with keeping everything but thexj variable constant.
From the definition, we have the partial derivative at a pointp of the function along this sliceas
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provided this limit exists.
Instead of the basis vector, which corresponds to taking the derivative along that axis, wecan pick a vector in any direction (which we usually take as being a unit vector), and we take thedirectional derivativeof a function as
whered is the direction vector.
If we want to calculate directional derivatives, calculating them from the limit definition is
rather painful, but, we have the following: iff : Rn → R is differentiable at a pointp, |p|=1,
There is a closely related formulation which we'll look at in the next section.
Gradient vectors
The partial derivatives of a scalar tell us how much it changes if we move along one of theaxes. What if we move in a different direction?
We'll call the scalarf, and consider what happens if we move an infintesimal directiondr=(dx,dy,dz), using the chain rule.
This is the dot product ofdr with a vector whose components are the partial derivatives off, called the gradient off
We can form directional derivatives at a pointp, in the directiond then by taking the dotproduct of the gradient withd
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.
Notice that gradf looks like a vector multiplied by a scalar. This particular combination ofpartial derivatives is commonplace, so we abbreviate it to
We can write the action of taking the gradient vector by writing this as anoperator. Recallthat in the one-variable case we can writed/dx for the action of taking the derivative with respectto x. This case is similar, but∇ acts like a vector.
We can also write the action of taking the gradient vector as:
Properties of the gradient vector
Geometry
• Gradf(p) is a vector pointing in the direction of steepest slope off. |gradf(p)| is the rateof change of that slope at that point.
For example, if we consider h(x, y)=x2+y2. The level sets ofh are concentric circles, centredon the origin, and
gradh points directly away from the origin, at right angles to the contours.
• Along a level set, (∇f)(p) is perpendicular to the level set {x|f(x)=f(p) atx=p}.
If dr points along the contours off, where the function is constant, thendf will be zero. Sincedf is a dot product, that means that the two vectors,df and gradf, must be at right angles, i.e thegradient is at right angles to the contours.
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Algebraicproperties
Like d/dx, ∇ is linear. For any pair of constants,a andb, and any pair of scalar functions,fandg
Since it's a vector, we can try taking its dot and cross product with other vectors, and withitself.
Divergence
If the vector functionu mapsRn to itself, then we can take the dot product ofu and∇. Thisdot product is called the divergence.
If we look at a vector function likev=(1+x2,xy) we can see that to the left of the origin allthev vectors are converging towards the origin, but on the right they are diverging away fromit.
Div u tells us how muchu is converging or diverging. It is positive when the vector is diverg-ing from some point, and negative when the vector is converging on that point.
Example:
Forv=(1+x2, xy), div v=3x, which is positive to the right of the origin, wherev is diverging,and negative to the left of the origin, wherev is converging.
Like grad, div is linear.
Later in this chapter we will see how the divergence of a vector function can be integratedto tell us more about the behaviour of that function.
To find the divergence we took the dot product of∇ and a vector with∇ on the left. If wereverse the order we get
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To see what this means consideri·∇ This is Dx, the partial differential in thei direction.
Similarly, u·∇ is the the partial differential in theu direction, multiplied by |u|
Curl
If u is a three-dimensional vector function onR3 then we can take its cross product with∇.This cross product is called thecurl.
Curl u tells us if the vectoru is rotating round a point. The direction of curlu is the axis ofrotation.
We can treat vectors in two dimensions as a special case of three dimensions, withuz=0 and
Dzu=0. We can then extend the definition of curlu to two-dimensional vectors
This two dimensional curl is a scalar. In four, or more, dimensions there is no vector equiva-lent to the curl.
Example:Consideru=(-y, x). These vectors are tangent to circles centred on the origin, so appear to
be rotating around it anticlockwise.
ExampleConsideru=(-y, x-z, y), which is similar to the previous example.
Thisu is rotating round the axisi+k
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Later in this chapter we will see how the curl of a vector function can be integrated to tellus more about the behaviour of that function.
Product and chain rules
Just as with ordinary differentiation, there are product rules for grad, div and curl.
• If g is a scalar andv is a vector, then
the divergence ofgv is
the curl ofgv is
• If u andv are both vectors then
the gradient of their dot product is
the divergence of their cross product is
the curl of their cross product is
We can also write chain rules. In the general case, when both functions are vectors and thecomposition is defined, we can use the Jacobian defined earlier.
whereJu is the Jacobian ofu at the pointv.
Normally J is a matrix but if either the range or the domain ofu is R1 then it becomes avector. In these special cases we can compactly write the chain rule using only vector notation.
• If g is a scalar function of a vector andh is a scalar function ofg then
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• If g is a scalar function of a vector then
This substitution can be made in any of the equations containing∇
Second order differentials
We can also consider dot and cross products of∇ with itself, whenever they can be defined.Once we know how to simplify products of two∇'s we'll know out to simplify products withthree or more.
The divergence of the gradient of a scalarf is
This combination of derivatives is theLaplacian of f. It is commmonplace in physics andmultidimensional calculus because of its simplicity and symmetry.
We can also take the Laplacian of a vector,
The Laplacian of a vector is not the same as the divergence of its gradient
Both the curl of the gradient and the divergence of the curl are always zero.
This pair of rules will prove useful.
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Integration
We have already considered differentiation of functions of more than one variable, whichleads us to consider how we can meaningfully look at integration.
In the single variable case, we interpret the definite integral of a function to mean the areaunder the function. There is a similar interpretation in the multiple variable case: for example, if
we have a paraboloid inR3, we may want to look at the integral of that paraboloid over someregion of thexyplane, which will be thevolumeunder that curve and inside that region.
Riemann sums
When looking at these forms of integrals, we look at the Riemann sum. Recall in the one-variable case we divide the interval we are integrating over into rectangles and summing the areasof these rectangles as their widths get smaller and smaller. For the multiple-variable case, we
need to do something similar, but the problem arises how to split upR2, or R3, for instance.
To do this, we extend the concept of the interval, and consider what we call an-interval. Ann-interval is a set of points in some rectangular region with sides of some fixed width in each
dimension, that is, a set in the form {x∈Rn|ai ≤ xi ≤ bi with i = 0,...,n}, and its area/size/volume
(which we simply call itsmeasureto avoid confusion) is the product of the lengths of all its sides.
So, ann-interval inR2 could be some rectangular partition of the plane, such as {(x,y) | x ∈[0,1] andy ∈ [0, 2]|}. Its measure is 2.
If we are to consider the Riemann sum now in terms of sub-n-intervals of a regionΩ, it is
wherem(Si) is the measure of the division ofΩ into k sub-n-intervalsSi, andx*i is a point
in Si. The index is important - we only perform the sum whereSi falls completely withinΩ - any
Si that is not completely contained inΩ we ignore.
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As we take the limit ask goes to infinity, that is, we divide upΩ into finer and finer sub-n-intervals, and this sum is the same no matter how we divide upΩ, we get theintegral of f overΩ which we write
f∫
Ω
For two dimensions, we may write
and likewise forn dimensions.
Iterated integrals
Thankfully, we need not always work with Riemann sums every time we want to calculatean integral in more than one variable. There are some results that make life a bit easier for us.
For R2, if we have some region bounded between two functions of the other variable (sotwo functions in the formf(x) = y, or f(y) = x), between a constant boundary (so, betweenx = aandx =b or y = a andy = b), we have
An important theorem (calledFubini's theorem) assures us that this integral is the same as
.
Order of integration
In some cases the first integral of the entire iterated integral is difficult or impossible tosolve, therefore, it can be to our advantage to change the order of integration.
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As of the writing of this, there is no set method to change an order of integration from dxdyto dydx or some other variable. Although, it is possible to change the order of integration in anx and y simple integration by simply switching the limits of integration around also, in non-simple x and y integrations the best method as of yet is to recreate the limits of the integrationfrom the graph of the limits of integration.
In higher order integration that can't be graphed, the process can be very tedious. For exam-ple, dxdydz can be written into dzdydx, but first dxdydz must be switched to dydxdz and then todydzdx and then to dzdydx (but since 3-dimensional cases can be graphed, doing this would beseemingly idiotic).
Parametric integrals
If we have a vector function,u, of a scalar parameter,s, we can integrate with respect tossimply by integrating each component ofu seperately.
Similarly, if u is given a function of vector of parameters,s, lying in Rn, integration withrespect to the parameters reduces to a multiple integral of each component.
Line integrals
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In one dimension, saying we are integrating froma to b uniquely specifies the integral.
In higher dimensions, saying we are integrating froma to b is not sufficient. In general, wemust also specify the path taken betweena andb.
We can then write the integrand as a function of the arclength along the curve, and integrateby components.
E.g, given a scalar functionh(r ) we write
whereC is the curve being integrated along, andt is the unit vector tangent to the curve.
There are some particularly natural ways to integrate a vector function,u, along a curve,
where the third possibility only applies in 3 dimensions.
Again, these integrals can all be written as integrals with respect to the arclength,s.
If the curve is planar andu a vector lieing in the same plane, the second integral can beusefully rewritten. Say,
wheret, n, andb are the tangent, normal, and binormal vectors uniquely defined by thecurve.
Then
For the 2-d curves specifiedb is the constant unit vector normal to their plane, andub is al-
ways zero.
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Therefore, for such curves,
Green's Theorem
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Let C be a piecewise smooth, simple closed curve that bounds a region S on the Cartesianplane. If two function M(x,y) and N(x,y) are continuous and their partial derivatives are continu-ous, then
In order for Green's theorem to work there must be no singularities in the vector field withinthe boundaries of the curve.
Green's theorem works by summing the circulation in each infinitesimal segment of areaenclosed within the curve.
Inverting differentials
We can use line integrals to calculate functions with specified divergence, gradient, or curl.
• If gradV = u
whereh is any function of zero gradient and curlu must be zero.
• If div u = V
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wherew is any function of zero divergence.
• If curl u = v
wherew is any function of zero curl.
For example, if V=r2 then
and
so this line integral of the gradient gives the original function.
Similarly, if v=k then
Consider any curve from0 to p=(x, y', z), given byr=r (s) with r (0)=0 andr (S)=p for someS, and do the above integral along that curve.
and curlu is
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as expected.
We will soon see that these three integrals do not depend on the path, apart from a constant.
Surface and Volume Integrals
Just as with curves, it is possible to parameterise surfaces then integrate over those parame-ters without regard to geometry of the surface.
That is, to integrate a scalar functionV over a surfaceA parameterised byr ands we calcu-late
whereJ is the Jacobian of the tranformation to the parameters.
To integrate a vector this way, we integrate each component seperately.
However, in three dimensions, every surface has an associated normal vectorn, which canbe used in integration. We writedS=ndS.
For a scalar function,V, and a vector function,v, this gives us the integrals
These integrals can be reduced to parametric integrals but, written this way, it is clear thatthey reflect more of the geometry of the surface.
When working in three dimensions,dV is a scalar, so there is only one option for integralsover volumes.
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Gauss's divergence theorem
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We know that, in one dimension,
Integration is the inverse of differentiation, so integrating the differential of a function returnsthe original function.
This can be extended to two or more dimensions in a natural way, drawing on the analogiesbetween single variable and multivariable calculus.
The analog ofD is ∇, so we should consider cases where the integrand is a divergence.
Instead of integrating over a one-dimensional interval, we need to integrate over an-dimen-sional volume.
In one dimension, the integral depends on the values at the edges of the interval, so we ex-pect the result to be connected with values on the boundary.
This suggests a theorem of the form,
This is indeed true, for vector fields in any number of dimensions.
This is calledGauss's theorem.
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There are two other, closely related, theorems for grad and curl:
•
,
•
,
with the last theorem only being valid where curl is defined.
Stokes' curl theorem
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These theorems also hold in two dimensions, where they relate surface and line integrals.Gauss's divergence theorem becomes
wheres is arclength along the boundary curve and the vectorn is the unit normal to the curvethat lies in the surfaceS, i.e in the tangent plane of the surface at its boundary, which is not neces-sarily the same as the unit normal associated with the boundary curve itself.
Similarly, we get
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,
whereC is the boundary ofS
In this case the integral does not depend on the surfaceS.
To see this, suppose we have different surfaces,S1 andS2, spanning the same curveC, then
by switching the direction of the normal on one of the surfaces we can write
The left hand side is an integral over a closed surface bounding some volumeV so we canuse Gauss's divergence theorem.
but we know this integrand is always zero so the right hand side of (2) must always be zero,i.e the integral is independent of the surface.
This means we can choose the surface so that the normal to the curve lying in the surface isthe same as the curves intrinsic normal.
Then, ifu itself lies in the surface, we can write
just as we did for line integrals in the plane earlier, and substitute this into (1) to get
This isStokes' curl theorem
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Ordinary differential equations
Partial differential equations→Calculus← Multivariable calculus
Ordinary differential equa-tions
Ordinary differential equations involve equations containing:
• variables• functions• their derivatives
and their solutions.
In studying integration, youalreadyhave considered solutions to very simple differentialequations. For example, when you look to solving
for g(x), you are really solving the differential equation
Notations and terminology
The notations we use for solving differential equations will be crucial in the ease of solubilityfor these equations.
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This document will be usingthree notations primarily:
• f' to denote the derivative of f• D f to denote the derivative off•
to denote the derivative off (for separable equations).
Terminology
Consider the differential equation
Since the equation's highest derivative is 2, we say that the differential equation is oforder2.
Some simple differential equations
A key idea in solving differential equations will be that ofintegration.
Let us consider the second order differential equation (remember that a function acts on avalue).
How would we go about solving this? It tells us that on differentiating twice, we obtain theconstant 2 so, if we integrate twice, we should obtain our result.
Integrating once first of all:
We have transformed the apparently difficult second order differential equation into a rathersimpler one, viz.
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This equation tells us that if we differentiate a function once, we get 2x + C1. If we integrate
once more, we should find the solution.
This is thesolutionto the differential equation. We will get
for all values ofC1 andC2.
The valuesC1 andC2 are related to quantities known asinitial conditions.
Why are initial conditions useful? ODEs (ordinary differential equations) are useful inmodeling physical conditions. We may wish to model a certain physical system which is initiallyat rest (so one initial condition may be zero), or wound up to some point (so an initial conditionmay be nonzero, say 5 for instance) and we may wish to see how the system reacts under suchan initial condition.
When we solve a system with given initial conditions, we substitute them after our processof integration.
Example
When we solved
say we had the initial conditions
and
. (Note, initial conditions need not occur at f(0)).
After we integrate we make substitutions:
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Without initial conditions, the answer we obtain is known as thegeneral solutionor the solu-tion to thefamily of equations. With them, our solution is known as aspecific solution.
Basic first order DEs
In this section we will considerfour main types of differential equations:
• separable• homogeneous• linear• exact
There are many other forms of differential equation, however, and these will be dealt within the next section
Separable equations
A separableequation is in the form (using dy/dx notation which will serve us greatly here)
Previously we have only dealt with simple differential equations with g(y)=1. How do wesolve such a separable equation as above?
We groupx anddx terms together, andy anddy terms together as well.
Integrating both sides with respect to y on the left hand side and x on the right hand side:
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we will obtain the solution.
Worked example
Here is a worked example illustrating the process.
We are asked to solve
Separating
Integrating
Lettingk = eC where k is a constant we obtain
which is the general solution.
Verification
This step does not need to be part of your work, but if you want to check your solution, youcan verify your answer by differentiation.
We obtained
as the solution to
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Differentiating our solution with respect to x,
And since
, we can write
We see that we obtain our original differential equation, thus our work must be correct.
Homogeneous equations
A homogeneousequation is in the form
This looks difficult as it stands, however we can utilize the substitution
so that we are now dealing with F(v) rather than F(y/x).
Now we can express y in terms of v, asy=xvand use the product rule.
The equation above then becomes, using the product rule
Then
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which is a separable equation and can be solved as above.
However let's look at a worked equation to see how homogeneous equations are solved.
Worked example
We have the equation
This does not appear to be immediately separable, but let us expand to get
Substitutingy=xvwhich is the same as substitutingv=y/x:
Now
Cancelingv from both sides
Separating
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Integrating both sides
which is our desired solution.
Linear equations
A linear first order differential equation is a differential equation in the form
Multiplying or dividing this equation by any non-zero function ofx makes no difference toits solutions so we could always divide bya(x) to make the coefficient of the differential 1, butwriting the equation in this more general form may offer insights.
At first glance, it is not possible to integrate the left hand side, but there is one special case.If b happens to be the differential ofa then we can write
and integration is now straightforward.
Since we can freely multiply by any function, lets see if we can use this freedom to writethe left hand side in this special form.
We multiply the entire equation by an arbitrary,I(x), getting
then impose the condition
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If this is satisfied the new left hand side will have the special form. Note that multiplyingIby any constant will leave this condition still satisfied.
Rearranging this condition gives
We can integrate this to get
We can set the constantk to be 1, since this makes no difference.
Next we useI on the original differential equation, getting
Because we've chosenI to put the left hand side in the special form we can rewrite this as
Integrating both sides and dividing byI we obtain the final result
We call I an integrating factor. Similar techniques can be used on some other calclulusproblems.
Example
Consider
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First we calculate the integrating factor.
Multiplying the equation by this gives
or
We can now integrate
Exact equations
An exact equation is in the form
f(x, y) dx + g(x, y) dy = 0
and, has the property that
Dx f = Dy g
(If the differential equation does not have this property then we can't proceed any further).
As a result of this, if we have an exact equation then there exists a function h(x, y) such that
Dy h = f and Dx h = g
So then the solutions are in the form
h(x, y) = c
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by using the fact of the total differential. We can find then h(x, y) by integration
Basic second and higher order ODE's
The generic solution of anth order ODE will containn constants of integration. To calculatethem we needn more equations. Most often, we have either
boundary conditions, the values ofy and its derivatives take for two different values ofx
or
initial conditions, the values ofy and its firstn-1derivatives take for one particular valueof x.
Reducible ODE's
1. If the independent variable,x, does not occur in the differential equation then its ordercan be lowered by one. This will reduce a second order ODE to first order.
Consider the equation:
Define
Then
Substitute these two expression into the equation and we get
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=0
which is a first order ODE
Example
Solve
if at x=0, y=Dy=1
First, we make the substitution, getting
This is a first order ODE. By rearranging terms we can separate the variables
Integrating this gives
u2 / 2 = c + 1 / 2y
We know the values ofy andu whenx=0 so we can findc
Next, we reverse the substitution
and take the square root
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To find out which sign of the square root to keep, we use the initial condition, Dy=1 atx=0,again, and rule out the negative square root. We now have another separable first order ODE,
Its solution is
Sincey=1 whenx=0, d=2/3, and
2. If the dependent variable,y, does not occur in the differential equation then it may alsobe reduced to a first order equation.
Consider the equation:
Define
Then
Substitute these two expressions into the first equation and we get
=0
which is a first order ODE
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Linear ODEs
An ODE of the form
is calledlinear. Such equations are much simpler to solve than typical non-linear ODEs.Though only a few special cases can be solved exactly in terms of elementary functions, there ismuch that can be said about the solution of a generic linear ODE. A full account would be beyondthe scope of this book
If F(x)=0 for all x the ODE is calledhomogeneous
Two useful properties of generic linear equations are
1. Any linear combination of solutions of an homogeneous linear equation is also a solu-tion.
2. If we have a solution of a nonhomogeneous linear equation and we add any solutionof the corresponding homogenous linear equation we get another solution of the nonho-mogeneous linear equation
Variation of constants
Suppose we have a linear ODE,
and we know one solution,y=w(x)
The other solutions can always be written asy=wz. This substitution in the ODE will give
us terms involving every differential ofzupto thenth, no higher, so we'll end up with annth orderlinear ODE forz.
We know thatz is constant is one solution, so the ODE forzmust not contain az term, which
means it will effectively be ann-1th order linear ODE. We will have reduced the order by one.
Lets see how this works in practice.
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Example
Consider
One solution of this isy=x2, so substitutey=zx2 into this equation.
Rearrange and simplify.
x2D2z + 6xDz= 0
This is first order for Dz. We can solve it to get
Since the equation is linear we can add this to any multiple of the other solution to get thegeneral solution,
y = Ax - 3 + Bx2
Linear homogeneous ODE's with constant coefficients
Suppose we have a ODE
(Dn + a1Dn - 1 + ... +an - 1D + a0)y = 0
we can take an inspired guess at a solution (motivate this)
y = epx
For this function Dny=pny so the ODE becomes
(pn + a1pn - 1 + ... +an - 1p + a0)y = 0
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y=0 is a trivial solution of the ODE so we can discard it. We are then left with the equation
pn + a1pn - 1 + ... +an - 1p + a0) = 0
This is called thecharacteristicequation of the ODE.
It can have up ton roots, p1, p2 … pn, each root giving us a different solution of the ODE.
Because the ODE is linear, we can add all those solution together in any linear combinationto get a general solution
To see how this works in practice we will look at the second order case. Solving equationslike this of higher order uses the exact same principles; only the algebra is more complex.
Secondorder
If the ODE is second order,
D2y + bDy+ cy= 0
then the characteristic equation is a quadratic,
p2 + bp+ c = 0
with roots
What these roots are like depends on the sign ofb2-4c, so we have three cases to consider.
1) b2 > 4c
In this case we have two different real roots, so we can write down the solution straightaway.
2) b2 < 4c
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In this case, both roots are imaginary. We could just put them directly in the formula, but ifwe are interested in real solutions it is more useful to write them another way.
Defining k2=4c-b2, then the solution is
For this to be real, theA's must be complex conjugates
Make this substitution and we can write,
y = Ae - bx / 2cos(kx+ a)
If b is positive, this is a damped oscillation.
3) b2 = 4c
In this case the characteristic equation only gives us one root,p=-b/2. We must use anothermethod to find the other solution.
We'll use the method of variation of constants. The ODE we need to solve is,
D2y - 2pDy+ p2y = 0
rewritingb andc in terms of the root. From the characteristic equation we know one solution
is y = epx so we make the substitutiony = zepx, giving
(epxD2z + 2pepxDz+ p2epxz) - 2p(epxDz+ pepxz) + p2epxz = 0
This simplifies to D2z=0, which is easily solved. We get
so the second solution is the first multiplied byx.
Higher order linear constant coefficient ODE's behave similarly: an exponential for everyreal root of the characteristic and a exponent multiplied by a trig factor for every complex conju-gate pair, both being multiplied by a polynomial if the root is repeated.
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E.g, if the characteristic equation factors to
(p - 1)4(p - 3)(p2 + 1)2 = 0
the general solution of the ODE will be
y = (A + Bx+ Cx2 + Dx3)ex + Ee3x + Fcos(x + a) + Gxcos(x + b)
The most difficult part is finding the roots of the characteristic equation.
Linear nonhomogeneous ODEs with constant coefficients
First, let's consider the ODE
Dy - y = x
a nonhomogeneous first order ODE which we know how to solve.
Using the integrating factore-x we find
y = ce - x + 1 - x
This is the sum of a solution of the corresponding homogeneous equation, and a polynomial.
Nonhomogeneous ODE's of higher order behave similarly.
If we have a single solution,yp of the nonhomogeneous ODE, called aparticular solution,
then the general solution isy=yp+yh, whereyh is the general solution of the homogeneous
ODE.
Findyp for an arbitraryF(x) requires methods beyond the scope of this chapter, but there are
some special cases where findingyp is straightforward.
Remember that in the first order problemyp for a polynomialF(x) was itself a polynomial
of the same order. We can extend this to higher orders.
Example:
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D2y + y = x3 - x + 1
Consider a particular solution
yp = b0 + b1x + b2x2 + x3
Substitute fory and collect coefficients
x3 + b2x2 + (6 + b1)x + (2b2 + b0) = x3 - x + 1
Sob2=0, b1=-7, b0=1, and the general solution is
y = asinx + bcosx + 1 - 7x + x3
This works because all the derivatives of a polynomial are themselves polynomials.
Two other special cases are
wherePn,Qn,An, andBn are all polynomials of degreen.
Making these substitutions will give a set of simultaneous linear equations for the coeffi-cients of the polynomials.
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Partial differential equations
Multivariable and differentialcalculus:Exercises→
Calculus← Ordinary differential equa-tions
Partial differential equations
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Introduction
First order
Any partial differential equation of the form
whereh1, h2 … hn, andb are all functions of bothu andRn can be reduced to a set of ordi-
nary differential equations.
To see how to do this, we will first consider some simpler problems.
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Special cases
We will start with the simple PDE
Becauseu is only differentiated with respect toz, for any fixedx andy we can treat this like
the ODE,du/dz=u. The solution of that ODE iscez, wherec is the value ofu whenz=0, for thefixed x and y
Therefore, the solution of the PDE is
u(x,y,z) = u(x,y,0)ez
Instead of just having a constant of integration, we have an arbitary function. This will betrue for any PDE.
Notice the shape of the solution, an arbitary function of points in thexy, plane, which isnormal to the 'z' axis, and the solution of an ODE in the 'z' direction.
Now consider the slightly more complex PDE
where h can be any function, and eacha is a real constant.
We recognize the left hand side as beinga·∇, so this equation says that the differential ofuin thea direction ish(u). Comparing this with the first equation suggests that the solution can bewritten as an arbitary function on the plane normal toa combined with the solution of an ODE.
Remembering fromCalculus/Vectorsthat any vectorr can be split up into components paral-lel and perpendicular toa,
we will use this to split the components ofr in a way suggested by the analogy with (1).
Let's write
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and substitute this into (2), using the chain rule. Because we are only differentiating in thea direction, adding any function of the perpendicular vector tos will make no difference.
First we calculate grads, for use in the chain rule,
On making the substitution into (2), we get,
which is an ordinary differential equation with the solution
The constantccan depend on the perpendicular components, but not upon the parallel coordi-nate. Replacings with a monotonic scalar function ofs multiplies the ODE by a function ofs,which doesn't affect the solution.
Example:
u(x,t)x = u(x,t)t
For this equation,a is (1, -1),s=x-t, and the perpendicular vector is (x+t)(1, 1). The reducedODE isdu/ds=0so the solution is
u=f(x+t)
To find f we need initial conditions onu. Are there any constraints on what initial conditionsare suitable?
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Consider, if we are given
• u(x,0), this is exactlyf(x),• u(3t,t), this isf(4t) andf(t) follows immediately• u(t3+2t,t), this isf(t3+3t) andf(t) follows, on solving the cubic.• u(-t,t), then this isf(0), so if the given function isn't constant we have a inconsistency,
and if it is the solution isn't specified off the initial line.
Similarly, if we are givenu on any curve which the linesx+t=c intersect only once, and towhich they are not tangent, we can deducef.
For any first order PDE with constant coefficients, the same will be true. We will have a setof lines, parallel tor=at, along which the solution is gained by integrating an ODE with initialconditions specified on some surface to which the lines aren't tangent.
If we look at how this works, we'll see we haven't actually used the constancy ofa, so let'sdrop that assumption and look for a similar solution.
The important point was that the solution was of the formu=f(x(s),y(s)), where (x(s),y(s)) isthe curve we integrated along -- a straight line in the previous case. We can add constant func-tions of integration tos without changing this form.
Consider a PDE,
a(x,y)ux + b(x,y)uy = c(x,y,u)
For the suggested solution,u=f(x(s),y(s)), the chain rule gives
Comparing coefficients then gives
so we've reduced our original PDE to a set of simultaneous ODE's. This procedure can bereversed.
The curves (x(s),y(s)) are calledcharacteristicsof the equation.
Example:Solveyux = xuy givenu=f(x) for x≥0 The ODE's are
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subject to the initial conditions ats=0,
This ODE is easily solved, giving
so the characteristics are concentric circles round the origin, and in polar coordinatesu(r,θ)=f(r)
Considering the logic of this method, we see that the independence ofa andb from u hasnot been used either, so that assumption too can be dropped, giving the general method forequations of thisquasilinearform.
Quasilinear
Summarising the conclusions of the last section, to solve a PDE
subject to the initial condition that on the surface, (x1(r1,…,rn-1, …xn(r1,…,rn-1), u=f(r1,…,rn-
1) --this being an arbitary paremetrisation of the initial surface--
• we transform the equation to the equivalent set of ODEs,
subject to the initial conditions
• Solve the ODE's, givingxi as a function ofs and ther i.• Invert this to gets and ther i as functions of thexi.• Substitute these inverse functions into the expression foru as a function ofsand ther i
obtained in the second step.
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Both the second and third steps may be troublesome.
The set of ODEs is generally non-linear and without analytical solution. It may even beeasier to work with the PDE than with the ODEs.
In the third step, ther i together withs form a coordinate system adapted for the PDE. We
can only make the inversion at all if the Jacobian of the transformation to Cartesian coordinatesis not zero,
This is equivalent to saying that the vector (a1, &hellip:, an) is never in the tangent plane to
a surface of constants.
If this condition is not false whens=0 it may become so as the equations are integrated. Wewill soon consider ways of dealing with the problems this can cause.
Even when it is technically possible to invert the algebraic equations it is obviously inconve-nient to do so.
Example
To see how this works in practice, we willa/ consider the PDE,
uux + uy + ut = 0
with generic initial condition,
u = f(x,y) on t = 0
Naming variables for future convenience, the corresponding ODE's are
subject to the initial conditions atτ=0
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These ODE's are easily solved to give
These are the parametric equations of a set of straight lines, the characteristics.
The determinant of the Jacobian of this coordinate transformation is
This determinant is 1 whent=0, but if fr is anywhere negative this determinant will eventual-
ly be zero, and this solution fails.
In this case, the failure is because the surfacesfr = - 1 is an envelope of the characteristics.
For arbitaryf we can invert the transformation and obtain an implicit expression foru
u = f(x - tu,y - x)
If f is given this can be solved foru.
1/ f(x,y) = ax, The implicit solution is
This is a line in theu-x plane, rotating clockwise ast increases. Ifa is negative, this lineeventually become vertical. Ifa is positive, this line tends towardsu=0, and the solution is validfor all t.
2/ f(x,y)=x2, The implicit solution is
This solution clearly fails when 1 + 4tx < 0, which is just whensfr = - 1 . For anyt>0 this
happens somewhere. Ast increases this point of failure moves toward the origin.
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Notice that the point whereu=0 stays fixed. This is true for any solution of this equation,whateverf is.
We will see later that we can find a solution after this time, if we consider discontinuoussolutions. We can think of this as a shockwave.
3/ f(x,y) = sin(xy)The implicit solution is
and we can not solve this explitly foru. The best we can manage is a numerical solution ofthis equation.
b/We can also consider the closely related PDE
uux + uy + ut = y
The corresponding ODE's are
subject to the initial conditions atτ=0
These ODE's are easily solved to give
Writing f in terms ofu, s, andτ, then substituting into the equation forx gives an implicitsolution
It is possible to solve this foru in some special cases, but in general we can only solve thisequation numerically. However, we can learn much about the global properties of the solutionfrom further analysis
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Characteristic initial value problems
What if initial conditions are given on a characteristic, on an envelope of characteristics, ona surface with characteristic tangents at isolated points?
Discontinuous solutions
So far, we've only considered smooth solutions of the PDE, but this is too restrictive. Wemay encounter initial conditions which aren't smooth, e.g.
If we were to simply use the general solution of this equation for smooth initial conditions,
we would get
which appears to be a solution to the original equation. However, since the partial differen-tials are undefined on the characteristicx+ct=0, so it becomes unclear what it means to say thatthe equation is true at that point.
We need to investigate further, starting by considering the possible types of discontinuities.
If we look at the derivations above, we see we've never use any second or higher orderderivatives so it doesn't matter if they aren't continuous, the results above will still apply.
The next simplest case is when the function is continuous, but the first derivative is not, e.g|x|. We'll initially restrict ourselves to the two-dimensional case,u(x, t) for the generic equation.
Typically, the discontinuity is not confined to a single point, but is shared by all points onsome curve, (x0(s), t0(s) )
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Then we have
We can then compareu and its derivatives on both sides of this curve.
It will prove useful to name thejumpsacross the discontinuity. We say
Now, since the equation (1) is true on both sides of the discontinuity, we can see that bothu+ andu-, being the limits of solutions, must themselves satisfy the equation. That is,
Subtracting then gives us an equation for the jumps in the differentials
We are considering the case whereu itself is continuous so we know that [u]=0. Differentiat-ing this with respect tos will give us a second equation in the differential jumps.
The last two equations can only be both true if one is a multiple of the other, but multiplyingsby a constant also multiplies the second equation by that same constant while leaving the curveof discontinuity unchanged, hence we can without loss of generality defines to be such that
But these are the equations for a characteristic, i.ediscontinuities propagate along charac-teristics. We could use this property as an alternative definition of characteristics.
We can deal similarly with discontinuous functions by first writing the equation inconserva-tion form, so called because conservation laws can always be written this way.
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Notice that the left hand side can be regarded as the divergence of (au, bu). Writing theequation this way allows us to use the theorems of vector calculus.
Consider a narrow strip with sides parallel to the discontinuity and widthh
We can integrate both sides of (1) over R, giving
Next we use Green's theorem to convert the left hand side into a line integral.
Now we let the width of the strip fall to zero. The right hand side also tends to zero but theleft hand side reduces to the difference between two integrals along the part of the boundary ofRparallel to the curve.
The integrals along the opposite sides ofRhave different signs because they are in oppositedirections.
For the last equation to always be true, the integrand must always be zero, i.e
Since, by assumption [u] isn't zero, the other factor must be, which immediately implies thecurve of discontinuity is a characteristic.
Once again,discontinuities propagate along characteristics.
Above, we only considered functions of two variables, but it is straightforward to extendthis to functions ofn variables.
The initial condition is given on ann-1 dimensional surface, which evolves along the charac-teristics. Typical discontinuities in the initial condition will lie on an-2 dimensional surfaceembedded within the initial surface. This surface of discontinuity will propagate along the charac-teristics that pass through the initial discontinuity.
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The jumps themselves obey ordinary differential equations, much asu itself does on a charac-teristic. In the two dimensional case, foru continuous but not smooth, a little algebra shows that
while u obeys the same equation as before,
We can integrate these equations to see how the discontinuity evolves as we move along thecharacteristic.
We may find that, for some futures, [ux] passes through zero. At such points, the discontinu-
ity has vanished, and we can treat the function as smooth at that characteristic from then on.
Conversely, we can expect that smooth functions may, under the righr circumstances, be-come discontinuous.
To see how all this works in practice we'll consider the solutions of the equation
for three different initial conditions.
The general solution, using the techniques outlined earlier, is
u is constant on the characteristics, which are straight lines with slope dependent onu.
First considerf such that
While u is continuous its derivative is discontinuous atx=0, whereu=0, and atx=a, whereu=1. The characteristics through these points divide the solution into three regions.
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All the characteristics to the right of the characteristic throughx=a, t=0 intersect thex-axisto the right ofx=1, whereu=1 sou is 1 on all those characteristics, i.e wheneverx-t>a.
Similarly the characteristic through the origin is the linex=0, to the left of whichu remainszero.
We could find the value ofu at a point in between those two characteristics either by findingwhich intermediate characteristic it lies on and tracing it back to the initial line, or via the generalsolution.
Either way, we get
At largert the solutionu is more spread out than att=0 but still the same shape.
We can also consider what happens whena tends to 0, so thatu itself is discontinuous atx=0.
If we write the PDE in conservation form then use Green's theorem, as we did above for thelinear case, we get
[u²] is the difference of two squares, so if we takes=t we get
In this case the discontinuity behaves as if the value ofu on it were the average of the limit-ing values on either side.
However, there is a caveat.
Since the limiting value to the left isu- the discontinuity must lie on that characteristic, and
similarly for u+; i.e the jump discontinuity must be on an intersection of characteristics, at a point
whereu would otherwise be multivalued.
For this PDE the characteristic can only intersect on the discontinuity if
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If this is not true the discontinuity can not propagate. Something else must happen.
The limit a=0 is an example of a jump discontinuity for which this condition is false, so wecan see what happens in such cases by studying it.
Taking the limit of the solution derived above gives
If we had taken the limit of any other sequence of initials conditions tending to the samelimit we would have obtained a trivially equivalent result.
Looking at the characteristics of this solution, we see that at the jump discontinuity character-istics on whichu takes every value betweeen 0 and 1 all intersect.
At later times, there are two slope discontinuities, atx=0 andx=t, but no jump discontinuity.
This behaviour is typical in such cases. The jump discontinuity becomes a pair of slope dis-continuities between which the solution takes all appropriate values.
Now, lets consider the same equation with the initial condition
This has slope discontinuities atx=0 andx=a, dividing the solution into three regions.
The boundaries between these regions are given by the characteristics through these initialpoints, namely the two lines
These characteristics intersect att=a, so the nature of the solution must change then.
In between these two discontinuities, the characteristic throughx=b at t=0 is clearly
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All these characteristics intersect at the same point, (x,t)=(a,a).
We can use these characteristics, or the general solution, to writeu for t<a
As t tends toa, this becomes a step function. Sinceu is greater to the left than the right ofthe discontinuity, it meets the condition for propagation deduced above, so fort>a u is a stepfunction moving at the average speed of the two sides.
This is the reverse of what we saw for the initial condition previously considered, two slopediscontinuities merging into a step discontinuity rather than vice versa. Which actually happensdepends entirely on the initial conditions. Indeed, examples could be given for which both process-es happen.
In the two examples above, we started with a discontinuity and investigated how it evolved.It is also possible for solutions which are initially smooth to become discontinuous.
For example, we saw earlier for this particular PDE that the solution with the initial conditionu=x² breaks down when 2xt+1=0. At these points the solution becomes discontinuous.
Typically, discontinuities in the solution of any partial differential equation, not merely onesof first order, arise when solutions break down in this way and propagate similarly, merging andsplitting in the same fashion.
Fully non-linear PDEs
It is possible to extend the approach of the previous sections to reduce any equation of theform
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to a set of ODE's, foranyfunction,F.
We will not prove this here, but the corresponding ODE's are
If u is given on a surface parameterized byr1…rn then we have, as before,n initial conditions
on then, xi
given by the parameterization andoneinitial condition onu itself,
but, because we have an extran ODEs for theui's, we need an extran initial conditions.
These are,n-1 consistency conditions,
which state that theui's are the partial derivatives ofu on the initial surface, andoneinitial
condition
stating that the PDE itself holds on the initial surface.
Thesen initial conditions for theui will be a set of algebraic equations, which may have
multiple solutions. Each solution will give a different solution of the PDE.
Example
Consider
The initial conditions atτ=0 are
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and the ODE's are
Note that the partial derivatives are constant on the characteristics. This always happen whenthe PDE contains only partial derivatives, simplifying the procedure.
These equations are readily solved to give
On eliminating the parameters we get the solution,
which can easily be checked. abc
Second order
Suppose we are given a second order linear PDE to solve
The natural approach, after our experience with ordinary differential equations and withsimple algebraic equations, is attempt a factorisation. Let's see how for this takes us.
We would expect factoring the left hand of (1) to give us an equivalent equation of the form
and we can immediately divide through bya. This suggests that those particular combina-tions of first order derivatives will play a special role.
Now, when studying first order PDE's we saw that such combinations were equivalent tothe derivatives along characteristic curves. Effectively, we changed to a coordinate system de-fined by the characteristic curve and the initial curve.
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Here, we have two combinations of first order derivatives each of which may define a differ-ent characteristic curve. If so, the two sets of characteristics will define a natural coordinate sys-tem for the problem, much as in the first order case.
In the new coordinates we will have
with each of the factors having become a differentiation along its respective characteristiccurve, and the left hand side will become simplyurs giving us an equation of the form
If A, B, andC all happen to be zero, the solution is obvious. If not, we can hope that thesimpler form of the left hand side will enable us to make progress.
However, before we can do all this, we must see if (1) can actually be factored.
Multiplying out the factors gives
On comparing coefficients, and solving for theα's we see that they are the roots of
Since we are discussing real functions, we are only interested in real roots, so the existenceof the desired factorization will depend on the discriminant of this quadratic equation.
• If b(x,y)2 > 4a(x,y)c(x,y)
then we have two factors, and can follow the procedure outlined above. Equations like thisare calledhyperbolic
• If b(x,y)2 = 4a(x,y)c(x,y)
then we have only factor, giving us a single characteristic curve. It will be natural to usedistance along these curves as one coordinate, but the second must be determined by otherconsiderations.
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The same line of argument as before shows that use the characteristic curve this way givesa second order term of the formurr , where we've only taken the second derivative with re-spect to one of the two coordinates. Equations like this are calledparabolic
• If b(x,y)2 < 4a(x,y)c(x,y)
then we have no real factors. In this case the best we can do is reduce the second orderterms to the simplest possible form satisfying this inequality, i.eurr+uss
It can be shown that this reduction is always possible. Equations like this are calledelliptic
It can be shown that, just as for first order PDEs, discontinuities propagate along characteris-tics. Since elliptic equations have no real characteristics, this implies that any discontinuities theymay have will be restricted to isolated points; i.e, that the solution is almost everywhere smooth.
This is not true for hyperbolic equations. Their behavior is largely controlled by the shapeof their characteristic curves.
These differences mean different methods are required to study the three types of secondequation. Fortunately, changing variables as indicated by the factorisation above lets us reduceany second order PDE to one in which the coefficients of the second order terms are constant,which means it is sufficient to consider only three standard equations.
We could also consider the cases where the right hand side of these equations is a givenfunction, or proportional tou or to one of its first order derivatives, but all the essential propertiesof hyperbolic, parabolic, and elliptic equations are demonstrated by these three standard forms.
While we've only demonstrated the reduction in two dimensions, a similar reduction appliesin higher dimensions, leading to a similar classification. We get, as the reduced form of the sec-ond order terms,
where each of theais is equal to either 0, +1, or -1.
If all theais have thesame signthe equation iselliptic
If anyof theais arezerothe equation isparabolic
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If exactly oneof theais has theopposite signto the rest the equation ishyperbolic
In 2 or 3 dimensions these are the only possibilities, but in 4 or more dimensions there is afourth possibility:at least twoof theais arepositive, andat least twoof theais arenegative.
Such equations are calledultrahyperbolic. They are less commonly encountered than theother three types, so will not be studied here.
When the coefficients are not constant, an equation can be hyperbolic in some regions ofthexy plane, and elliptic in others. If so, different methods must be used for the solutions in thetwo regions.
Elliptic
Standard form, Laplace's equation:
Quote equation in spherical and cylindrical coordinates, and give full solution for cartesianand cylindrical coordinates. Note averaging property Comment on physical significance, rotationinvariance of laplacian.
Hyperbolic
Standard form, wave equation:
Solution, any sum of functions of the form
These are waves. Compare with solution from separating variables. Domain of dependance,etc.
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Parabolic
The canonical parabolic equation is the diffusion equation:
Here, we will consider some simple solutions of the one-dimensional case.
The properties of this equation are in many respects intermediate between those of hyperbol-ic and elliptic equation.
As with hyperbolic equations but not elliptic, the solution is well behaved if the value isgiven on the initial surfacet=0.
However, the characteristic surfaces of this equation are the surfaces of constantt, thus thereis no way for discontinuities to propagate to positive t.
Therefore, as with elliptic equations but not hyberbolic, the solutions are typically smooth,even when the initial conditions aren't.
Furthermore, at a local maximum ofh, its Laplacian is negative, soh is decreasing witht,while at local minima, where the Laplacian will be positive,h will increase witht. Thus, initialvariations inh will be smoothed out ast increases.
In one dimension, we can learn more by integrating both sides,
Provided thathx tends to zero for largex, we can take the limit asa andb tend to infinity,
deducing
so the integral ofh over all space is constant.
This means this PDE can be thought of as describing some conserved quantity, initiallyconcentrated but spreading out, or diffusing, over time.
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This last result can be extended to two or more dimensions, using the theorems of vectorcalculus.
We can also differentiate any solution with respect to any coordinate to obtain another solu-tion. E.g ifh is a solution then
sohx also satisfies the diffusion equation.
Similarity solution
Looking at this equation, we might notice that if we make the change of variables
then the equation retains the same form. This suggests that the combination of variablesx²/t,which is unaffected by this variable change, may be significant.
We therefore assume this equation to have a solution of the special form
then
and substituting into the diffusion equation eventually gives
which is an ordinary differential equation.
Integrating once gives
Reverting toh, we find
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This last integral can not be written in terms of elementary functions, but its values are wellknown.
In particular the limiting values ofh at infinity are
taking the limit ast tends to zero gives
We see that the initial discontinuity is immediately smoothed out. The solution at later timesretains the same shape, but is more stretched out.
The derivative of this solution with respect tox
is itself a solution, withh spreading out from its initial peak, and plays a significant role inthe further analysis of this equation.
The same similiarity method can also be applied to some non-linear equations.
Separating variables
We can also obtain some solutions of this equation by separating variables.
giving us the two ordinary differential equations
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and solutions of the general form
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Multivariable and differentialcalculus:Exercises
Extensions→Calculus←Partial differential equations
Multivariable and differen-tial calculus:Exercises
Exercise
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Extensions
Systems of ordinary differen-tial equations→
Calculus←Partial differential equations
Extensions
Further Analysis
• Systems of Ordinary Differential Equations
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Formal Theory of Calculus
• Real numbers
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• Complex numbers
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Systems of ordinary differentialequations
We have already examined cases where we have a single differential equation and foundseveral methods to aid us in finding solutions to these equations. But what happens if we havetwo or more differential equations that depend on each other? For example, consider the casewhere
Dtx(t) = 3y(t)2 + x(t)t
and
Dty(t) = x(t) + y(t)
Such a set of differential equations is said to becoupled. Systems of ordinary differentialequations such as these are what we will look into in this section.
First order systems
A general system of differential equations can be written in the form
Instead of writing the set of equations in a vector, we can write out each equation explicitly,in the form:
If we have the system at the very beginning, we can write it as:
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where
and
or write each equation out as shown above.
Why are these forms important? Often, this arises as a single, higher order differentialequation that is changed into a simpler form in a system. For example, with the same example,
Dtx(t) = 3y(t)2 + x(t)t
Dty(t) = x(t) + y(t)
we can write this as a higher order differential equation by simple substitution.
Dty(t) - y(t) = x(t)
then
Dtx(t) = 3y(t)2 + (Dty(t) - y(t))t
Dtx(t) = 3y(t)2 + tDty(t) - ty(t)
Notice now that the vector form of the system is dependent ont since
the first component is dependent ont. However, if instead we had
notice the vector field is no longer dependent ont. We call such systemsautonomous. Theyappear in the form
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We can convert between an autonomous system and a non-autonomous one by simplymaking a substitution that involvest, such asy=(x, t), to get a system:
In vector form, we may be able to separateF in a linear fashion to get something that lookslike:
whereA(t) is a matrix andb is a vector. The matrix could contain functions or constants,clearly, depending on whether the matrix depends ont or not.
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Real numbers
Fields
You are probably already familiar with many different sets of numbers from your past experi-ence. Some of the commonly used sets of numbers are
• Natural numbers, usually denoted with anN, are the numbers 0,1,2,3,...• Integers, usually denoted with aZ, are the positive and negative natural numbers: ...-
3,-2,-1,0,1,2,3...• Rational numbers, denoted with aQ, are fractions of integers (excluding division by
zero): -1/3, 5/1, 0, 2/7. etc.• Real numbers, denoted with aR, are constructed and discussed below.
Note that different sets of numbers have different properties. In the set integers for example,any number always has anadditive inverse: for any integerx, there is another integert such thatx + t = 0 This should not be terribly surprising: from basic arithmetic we know thatt = − x. Tryto prove to yourself that not all natural numbers have an additive inverse.
In mathematics, it is useful to note the important properties of each of these sets of numbers.The rational numbers, which will be of primary concern in constructing the real numbers, havethe following properties:
There exists a number 0 such that for any other numbera, 0+a=a+0=a
For any two numbersa andb, a+b is another number
For any three numbersa,b, andc, a+(b+c)=(a+b)+c
For any numbera there is another number-a such thata+(-a)=0
For any two numbersa andb, a+b=b+a
For any two numbersa andb,a*b is another number
There is a number 1 such that for any numbera, a*1=1*a=a
For any two numbersa andb, a*b=b*a
For any three numbersa,bandc, a(bc)=(ab)c
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For any three numbersa,bandc, a(b+c)=ab+ac
For every numbera there is another numbera-1 such that aa-1=1
As presented above, these may seem quite intimidating. However, these properties arenothing more than basic facts from arithmetic. Any collection of numbers (and operations + and* on those numbers) which satisfies the above properties is called afield. The properties aboveare usually calledfield axioms.As an exercise, determine if the integers form a field, and if not,which field axiom(s) they violate.
Even though the list of field axioms is quite extensive, it does not fully explore the propertiesof rational numbers. Rational numbers also have anordering.' A total ordering must satisfyseveral properties: for any numbersa, b,andc
if a ≤ b andb ≤ a thena = b (antisymmetry)
if a ≤ b andb ≤ c thena ≤ c (transitivity)
a ≤ b or b ≤ a (totality)
To familiarize yourself with these properties, try to show that (a) natural numbers, integersand rational numbers are all totally ordered and more generally (b) convince yourself that anycollection of rational numbers are totally ordered (note that the integers and natural numbers areboth collections of rational numbers).
Finally, it is useful to recognize one more characterization of the rational numbers: everyrational number has a decimal expansion which is either repeating or terminating. The proof ofthis fact is omitted, however it follows from the definition of each rational number as a fraction.When performing long division, the remainder at any stage can only take on positive integervalues smaller than the denominator, of which there are finitely many.
Constructing the Real Numbers
There are two additional tools which are needed for the construction of the real numbers:the upper bound and the least upper bound.Definition A collection of numbersE is boundedabove if there exists a numbermsuch that for allx in E x≤m. Any numbermwhich satisfies thiscondition is called an upper bound of the setE.
Definition If a collection of numbersE is bounded above withm as an upper bound ofE,and all other upper bounds ofE are bigger thanm, we callm theleast upper boundor supremumof E, denoted by supE.
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Many collections of rational numbers do not have a least upper bound which is also rational,although some do. Suppose the the numbers 5 and 10/3 are, together, taken to beE. The number10/3 is not only an upper bound ofE, it is a least upper bound. In general, there are many upperbounds (12, for instance, is an upper bound of the collection above), but there can be at most oneleast upper bound.
Consider the collection of numbers
: You may recognize these decimals as the first few digits of pi. Since each decimal termi-nates, each number in this collection is a rational number. This collection has infinitely manyupper bounds. The number 4, for instance, is an upper bound. There is no least upper bound, atleast not in the rational numbers. Try to convince yourself of this fact by attempting to constructsuch a least upper bound: (a) why does pi not work as a least upper bound (hint: pi does not havea repeating or terminating decimal expansion), (b) what happens if the proposed supremum isequal to pi up to some decimal place, and zeros after (c) if the proposed supremum is bigger thanpi, can you find a smaller upper bound which will work?
In fact, there are infinitely many collections of rational numbers which do not have a rationalleast upper bound. We define a real number to be any number that is the least upper bound ofsome collection of rational numbers.
Properties of Real Numbers
The reals are well ordered.
For all reals;a, b, c
Eitherb>a, b=a, or b<a.
If a<b andb<c thena<c
Also
b>a impliesb+c>a+c
b>a andc>0 impliesbc>ac
b>a implies-a>-b
Upper bound axiom
Every non-empty set of real numbers which is bounded above has a supremum.
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The upper bound axiom is necessary for calculus. It is not true for rational numbers.
We can also define lower bounds in the same way.
Definition A setE is bounded below if there exists a real M such that for allx∈E x≥M AnyM which satisfies this condition is called an lower bound of the setE
Definition If a set,E, is bounded below,M is an lower bound ofE, and all other lowerbounds ofE are less thanM, we callM thegreatest lower boundor inifimumof E, denoted byinf E
The supremum and infimum of finite sets are the same as their maximum and minimum.
Theorem
Every non-empty set of real numbers which is bounded below has an infimum.
Proof:
Let E be a non-empty set of of real numbers, bounded below
Let L be the set of all lower bounds of E
L is not empty, by definition of bounded below
Every element of E is an upper bound to the set L, by definition
Therefore, L is a non empty set which is bounded above
L has a supremum, by the upper bound axiom
1/ Every lower bound of E is≤sup L, by definition of supremum
Suppose there were ane∈E such that e<sup L
Every element of L is≤e, by definition
Therefore e is an upper bound of L and e<sup L
This contradicts the definition of supremum, so there can be no such e.
If e∈E then e≥sup L, proved by contradiction
2/ Therefore, sup L is a lower bound of E
inf E exists, and is equal to sup L, on comparing definition of infinum to lines 1 & 2
Bounds and inequalities, theorems:
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Theorem: (The triangle inequality)
Proofby considering cases
If a≤b≤c then|a-c|+|c-b| = (c-a)+(c-b)= 2(c-b)+(b-a)>b-a= |b-a|
Exercise:Prove the other five cases.
This theorem is a special case of the triangle inequality theorem from geometry: The sumof two sides of a triangle is greater than or equal to the third side. It is useful whenever we needto manipulate inequalities and absolute values.
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Complex numbers
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Complex NumbersIn mathematics, acomplex number is a number of the form
wherea andb are real numbers, andi is the imaginary unit, with the propertyi 2 = −1. Thereal numbera is called thereal partof the complex number, and the real numberb is theimagi-nary part. Real numbers may be considered to be complex numbers with an imaginary part ofzero; that is, the real numbera is equivalent to the complex numbera+0i.
For example, 3 + 2i is acomplex number, with real part 3 and imaginary part 2. Ifz = a +bi, the real part (a) is denoted Re(z), orℜ(z), and the imaginary part (b) is denoted Im(z), orℑ(z).
Complex numbers can be added, subtracted, multiplied, and divided like real numbers andhave other elegant properties. For example, real numbers alone do not provide a solution for ev-ery polynomial algebraic equation with real coefficients, while complex numbers do (this is thefundamental theorem of algebra).
Equality
Two complex numbers are equal if and only if their real parts are equalandtheir imaginaryparts are equal. That is,a + bi = c + di if and only if a = c andb = d.
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Notation and operations
The set of all complex numbers is usually denoted byC, or in blackboard bold by
(Unicodeℂ). The real numbers,R, may be regarded as "lying in"C by considering everyreal number as a complex:a = a + 0i.
Complex numbers are added, subtracted, and multiplied by formally applying the associative,
commutative and distributive laws of algebra, together with the equationi2 = −1:
Division of complex numbers can also be defined (see below). Thus, the set of complexnumbers forms a field which, in contrast to the real numbers, is algebraically closed.
In mathematics, the adjective "complex" means that the field of complex numbers is theunderlying number field considered, for example complex analysis, complex matrix, complexpolynomial and complex Lie algebra.
The field of complex numbers
Formally, the complex numbers can be defined as ordered pairs of real numbers (a, b) togeth-er with the operations:
So defined, the complex numbers form a field, the complex number field, denoted byC (afield is an algebraic structure in which addition, subtraction, multiplication, and division are de-fined and satisfy certain algebraic laws. For example, the real numbers form a field).
The real numbera is identified with the complex number (a, 0), and in this way the field ofreal numbersR becomes a subfield ofC. The imaginary uniti can then be defined as the complexnumber (0, 1), which verifies
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In C, we have:
• additive identity ("zero"): (0, 0)• multiplicative identity ("one"): (1, 0)• additive inverse of (a,b): (−a, −b)• multiplicative inverse (reciprocal) of non-zero (a, b):
Since a complex numbera + bi is uniquely specified by an ordered pair (a, b) of real num-bers, the complex numbers are in one-to-one correspondence with points on a plane, called thecomplex plane.
The complex plane
A complex numberz can be viewed as a point or a position vector in a two-dimensionalCartesian coordinate system called thecomplex planeorArgand diagram . The point and hencethe complex numberz can be specified by Cartesian (rectangular) coordinates. The Cartesiancoordinates of the complex number are the real partx = Re(z) and the imaginary party = Im(z).The representation of a complex number by its Cartesian coordinates is called theCartesian formor rectangular formor algebraic formof that complex number.
Polar form
Alternatively, the complex numberzcan be specified by polar coordinates. The polar coordi-nates arer = |z| ≥ 0, called theabsolute valueor modulus, andφ = arg(z), called theargumentof z. For r = 0 any value ofφ describes the same number. To get a unique representation, a con-ventional choice is to set arg(0) = 0. Forr > 0 the argumentφ is unique modulo 2π; that is, ifany two values of the complex argument differ by an exact integer multiple of 2π, they are consid-ered equivalent. To get a unique representation, a conventional choice is to limitφ to the interval(-π,π], i.e. −π < φ ≤ π. The representation of a complex number by its polar coordinates is calledthepolar formof the complex number.
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Conversion from the polar form to the Cartesian form
Conversion from the Cartesian form to the polar form
The previous formula requires rather laborious case differentiations. However, many pro-gramming languages provide a variant of the arctangent function. A formula that uses the arccosfunction requires fewer case differentiations:
Notation of the polar form
The notation of the polar form as
is calledtrigonometric form. The notation cisφ is sometimes used as an abbreviation forcosφ + i sinφ. UsingEuler's formulait can also be written as
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which is calledexponential form.
Multiplication, division, exponentiation, and rootextraction in the polar form
Multiplication, division, exponentiation, and root extraction are much easier in the polarform than in the Cartesian form.
Usingsum and difference identitiesits possible to obtain that
and that
Exponentiation with integer exponents; according tode Moivre's formula,
Exponentiation with arbitrary complex exponents is discussed in the article onexponentia-tion.
The addition of two complex numbers is just the addition of two vectors, and multiplicationby a fixed complex number can be seen as a simultaneous rotation and stretching.
Multiplication by i corresponds to a counter-clockwise rotation by 90° (π/2 radians). The
geometric content of the equationi2 = −1 is that a sequence of two 90 degree rotations results ina 180 degree (π radians) rotation. Even the fact (−1) · (−1) = +1 from arithmetic can be under-stood geometrically as the combination of two 180 degree turns.
All the roots of any number, real or complex, may be found with a simple algorithm. Thenth roots are given by
for k = 0, 1, 2, …,n−1, where
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represents the principalnth root ofr.
Absolute value, conjugation and distance
Theabsolute value(or modulusor magnitude) of a complex numberz = r eiφ is defined as|z| = r. Algebraically, ifz = a + bi, then
One can check readily that the absolute value has three important properties:
if and only if
(triangle inequality)
for all complex numberszandw. It then follows, for example, that | 1 | = 1 and |z / w | = |z| / | w | . By defining thedistancefunctiond(z, w) = |z − w| we turn the set of complex numbersinto ametric spaceand we can therefore talk about limits and continuity.
Thecomplex conjugateof the complex numberz= a + bi is defined to bea − bi, written as
or
. As seen in the figure,
is the "reflection" ofz about the real axis. The following can be checked:
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if and only if z is real
if z is non-zero.
The latter formula is the method of choice to compute the inverse of a complex number if itis given in rectangular coordinates.
That conjugation commutes with all the algebraic operations (and many functions;e.g.
) is rooted in the ambiguity in choice ofi (−1 has two square roots). It is important to note,however, that the function
is not complex-differentiable.
Complex fractions
We can divide a complex number (a + bi) by another complex number (c + di) ≠ 0 in twoways. The first way has already been implied: to convert both complex numbers into exponentialform, from which their quotient is easily derived. The second way is to express the division as afraction, then to multiply both numerator and denominator by the complex conjugate of the de-nominator. The new denominator is a real number.
Matrix representation of complex numbers
While usually not useful, alternative representations of the complex field can give some in-sight into its nature. One particularly elegant representation interprets each complex number as
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a 2×2 matrix with real entries which stretches and rotates the points of the plane. Every suchmatrix has the form
wherea andb are real numbers. The sum and product of two such matrices is again of thisform. Every non-zero matrix of this form is invertible, and its inverse is again of this form.Therefore, the matrices of this form are a field. In fact, this is exactly the field of complex num-bers. Every such matrix can be written as
which suggests that we should identify the real number 1 with the identity matrix
and the imaginary uniti with
a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeedequal to the 2×2 matrix that represents −1.
The square of the absolute value of a complex number expressed as a matrix is equal to thedeterminant of that matrix.
If the matrix is viewed as a transformation of the plane, then the transformation rotates pointsthrough an angle equal to the argument of the complex number and scales by a factor equal tothe complex number's absolute value. The conjugate of the complex numberzcorresponds to thetransformation which rotates through the same angle aszbut in the opposite direction, and scalesin the same manner asz; this can be represented by the transpose of the matrix corresponding toz.
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If the matrix elements are themselves complex numbers, the resulting algebra is that of thequaternions. In other words, this matrix representation is one way of expressing the Cayley-Dickson construction of algebras.
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References
Table of Trigonometry→Calculus← Complex numbers
References
• Table of Trigonometry• Summation notation• Tables of Derivatives• Tables of Integrals
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Table of Trigonometry
Wikipediahas related information at
SeePic-
Trigonometric_identity
tureLi-censeInfor-ma-tionHere
Definitions
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Double Angle Identities
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Product-to-sum identities
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Summation notation
Summation notation allows an expression that contains a sum to be expressed in a simple,compact manner. The uppercase Greek letter sigma,Σ, is used to denote the sum of a set ofnumbers.
Example
Let f be a function andN,M are integers withN < M. Then
We sayN is the lower limit andM is the upper limit of the sum.
We can replace the letteri with any other variable. For this reasoni is referred to as adummyvariable. So
Conventionally we use the lettersi, j, k, m for dummy variables.
Example
Here, thedummy variableis i, thelower limit of summation is 1, and theupper limit is 5.
ExampleSometimes, you will see summation signs with no dummy variable specified, e.g.,
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In such cases the correct dummy variable should be clear from the context.
You may also see cases where the limits are unspecified. Here too, they must be deducedfrom the context.
Common summations
Wikipediahas related information at
SeePic-
Summation#Capital-sigma notation
tureLi-censeInfor-ma-tionHere
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Tables of Derivatives
Wikipediahas related information at
SeePic-
Table_of_derivatives
tureLi-censeInfor-ma-tionHere
General Rules
Powers and Polynomials
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Trigonometric Functions
Exponential and Logarithmic Functions
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Hyperbolic and Inverse Hyperbolic Functions
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Tables of Integrals
Wikipediahas related information at
SeePic-
Lists of integrals
tureLi-censeInfor-ma-tionHere
Rules
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Powers
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Basic Trigonometric Functions
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Reciprocal Trigonometric Functions
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Exponential and Logarithmic Functions
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Acknowledgements
Acknowledgements
Portions of this book have been copied from relevantWikipediaarticles.
Contributors
In alphabetical order (by surname or display name):
• Aaron Paul (AKAGrimm)• "Professor M." (no user page available)• Chaotic llama• User:Cronholm144• User:Fephisto• User:Juliusross• User:Stranger104• User:Whiteknight
Further Reading
The following books listCalculusas a prerequisite:
• Electrodynamics• Clock and data recovery• Signal Processing
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Other Calculus Textbooks
Other calculus textbooks available online:
• Calculus Refresherby Paul Garrett, notes on first-year calculus (PDF/TeX).• Difference Equations to Differential Equations: An Introduction to Calculusby Dan
Sloughter, available under a Creative Commons license (PDF).• The Calculus of Functions of Several Variablesby Dan Sloughter, available under a
Creative Commons license (PDF).• Lecture Notes for Applied Calculus(PDF) by Karl Heinz Dovermann, first-semester
calculus without using limits.• Elements of the Differential and Integral Calculusby William Granville (1911), a
classic calculus textbook now available online in various forms. (It is also partiallyavailable atWikisource.)
• Calculus(3rd Ed., 1994) byMichael Spivak, is a more rigorous introductory calculustextbook.
Using infinitesimals
• Elementary Calculus: An Approach Using Infinitesimals(2nd Ed., 1986) by H. JeromeKeisler, an out-of-print nonstandard calculus textbook now available online under aCreative Commons license (PDF).
• Yet Another Calculus Textby Dan Sloughter, an introduction to calculus using infinitesi-mals available under a Creative Commons license (PDF).
• Calculusby Benjamin Crowell, an introduction to calculus available under a CreativeCommons license (PDF). Also see Crowell's"Five Free Calculus Textbooks"(2004)review on Slashdot.
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Choosing delta
Recall the definition of a limit:
A numberL is the limit of a functionf(x) asx approachesc if and only if for all numbersε> 0 there exists a numberδ > 0 such that
whenever
.
In other words, given a numberε we must construct a numberδ such that assuming
we can prove
;moreover, this proof must work forall values ofε > 0.
Note: this definition is not constructive -- it does not tell you how tofind the limit L, onlyhow to check whether a particular value is indeed the limit. We use the informal definition of thelimit, experience with similar problems, or theorems (L'Hopital's rule, for example), to determinethe value, and then can prove the correctness of this value using the formal definition.
Example 1: Suppose we want to find the limit off(x) = x + 5 asx approachesc = 9. We knowthat the limitL is 9+5=14, and desire to prove this.
We chooseδ = ε (this will be explained later). Then, since we assume
we can show
,which is what we wanted to prove.
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We choseδ by working backwards from the formula we are trying to prove:
.In this case, we desire to prove
,given
,so the easiest way to prove it is by choosingδ = ε. This example, however, is too easy to
adequately explain how to chooseδ in general. Lets try something harder:
Example 2: Prove that the limit off(x) = x² - 9 asx approaches 2 isL = -5.
We want to prove that
given
.
We chooseδ by working backwards. First, we need to rewrite the equation we want to proveusingδ instead ofx:
Note: we used the fact that |x + 2| <δ + 4, which can be proven with the triangle inequality.
Word of caution: the above series of equations is not a logical series of steps, and is not partof any proof, but is an informal technique used to help write the proof. We will select a value ofδ so that the last equation is true, and then use the last equation to prove the equations above itin turn (which is what was meant earlier byworking backwards).
Note: in the equations above, whenδ was substituted forx, the sign < was replaced with =.This can be done (but is not necessary) because we are not told that |x-2| =δ, but rather |x-2| <δ. The justification for this becomes clear when the above equations are used in backwards orderin the proof.
We can solve this last equation forδ using the quadratic formula:
Note:δ is alwaysin terms ofε. A constant value ofδ (e.g.,δ = 0.5) will never work.
Now, we have a value ofδ, and we can do our proof:
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given
,
.
Here a few more examples of choosingδ; try to figure them out before reading the explana-tion.
Example 3: Prove that the limit off(x) = sin(x)/x asx approaches 0 isL = 1.
Explanation:
Example 4: Prove thatf(x) = 1/x has no limit asx approaches 0.
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More Differentation Rules
Higher Order Derivatives→Calculus← Chain Rule
More Differentiation Rules
External links
• Online interactive exercises on derivatives• Visual Calculus - Interactive Tutorial on Derivatives, Differentiation, and Integration
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Mean Value Theorem
Integration/Contents→Calculus← Rolle's Theorem
Mean Value Theorem
Mean Value TheoremIf
is continuous on the closed interval
and differentiable on the open interval
, there exists a number,
, in the open interval
such that
.
Examples
See Picture License InformationHere
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What does this mean? As usual, let us utilize an example to grasp the concept. Visualize (or
graph) the functionf(x) = x3. Choose an interval (anything will work), but for the sake of simplici-ty, [0,2]. Draw a line going from point (0,0) to (2,8). Between the pointsx = 0 andx = 2 exists anumberx = c, where the derivative off at pointc is equal to the slope of the line you drew.
Solution:
1: Using the definition of the mean value theorem
insert values. Our chosen interval is [0,2]. So, we have
2: By the definition of the mean value theorem, we know that somewhere in the intervalexists a point that has the same slope as that point. Thus, let us take the derivative to find thispointx = c.
Now, we know that the slope of the point is 4. So, the derivative at this pointc is 4. Thus, 4
= 3x2. The square root of 4/3 is the point.
Example 2:Find the point that satisifes the mean value theorem on the functionf(x) = sin(x)and the interval [0,π].
Solution:
1: Always start with the definition:
so,
(Remember, sin(π) and sin(0) are both 0.)
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2: Now that we have the slope of the line, we must find the pointx = c that has the sameslope. We must now get the derivative!
The cosine function is 0 atπ / 2 + πn (wheren is an integer). Remember, we are bound bythe interval [0,π], soπ / 2 is the pointc that satisfies the Mean Value Theorem.
Differentials
Assume a functiony = f(x) that is differentiable in the open interval (a,b) that contains x.Δy=
The "Differential of x" is theΔx. This is an approximate change in x and can be considered"equivalent" todx. The same holds true for y. What is this saying? One can approximate a changein y by knowing a change in x and a change in x at a point very nearby. Let us view an example.
Example: A schoolteacher has asked her students to discover what 4.12 is. The students,bereft of their calculators, are too lazy to multiply this out by hand or in their head and desire toutilize calculus. How can they approximate this?
1: Set up a function that mimics the procedure. What are they doing? They are taking anumber (Call it x) and they are squaring it to get a new number (call it y). Thus, y = x^2 Writeyourself a small chart. Make notes of values for x, y,Δx, Δy, and
. We are seeking what y really is, but we need the change in y first.
2: Choose a number close by that is easy to work with. Four is very close to 4.1, so writethat down as x. Yourδx is .1 (This is the "change" in x from the approximation point to the pointyou chose.)
3: Take the derivative of your function.
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= 2x. Now, "split" this up (This is not really what is happening, but to keep things simple, assumeyou are "multiplying"dxover.)
3b. Now you havedy
. We are assumingdyanddxare approximately the same as the change in x, thus we can useΔx and y.
3c. Insert values:dy
. Thus,dy= .8.
4: To findF(4.1), takeF(4) +dy to get an approximation. 16 + .8 = 16.8; This approximationis nearly exact (The real answer is 16.81. This is only one hundredth off!)
Definition of Derivative
The exact value of the derivative at a point is the rate of change over an infinitely smalldistance, approaching zero. Therefore, if h approaches 0 and the function is f(x):
If h approaches 0, then:
Cauchy's Mean Value Theorem
Cauchy's Mean Value TheoremIf
,
are continuous on the closed interval
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and differentiable on the open interval
,
and
, then there exists a number,
, in the open interval
such that
.
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Infinite series
Sequences→Calculus← Polar Integration
Sequences and Series
Basics
• Sequences• Series
Series and calculus
• Taylor series• Power series• Exercises
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List of Contributors
Cronholm144The Grimm RipperDysprosiaJuliusrossTiledWhiteknightSwiftMrwojoIamunknownAdrignolaRoadrunnerThenub314TechnochefCarandolXiaodaiRobert HorningPseudohanSBJohnnyGwilmKarl WickZginderZondorW3asalWnealSbroolsThomas.haslwanterNumberTheoristRam einsteinMike.lifeguardMknMetricMihoshiLarge and in chargeLeonusIntManJgukHerbythymeHerraoticGraemeb1967HagindazEtscrivnerGeoffreyDcljrEric119DavidMcKenzieDavidmanheim
Darklama
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