Calculus Handout PARTY!!! Week

Preparation for High School

Keep in mind that high school TMSCA is completely different from that of middle school. Although this may seem to be a disadvantage, there is actually a positive outcome to the test restructuralization. Those who did not excel in certain aspects (Number Sense, General Math, Calculator) are now given a chance to start fresh with everybody else.

As many may know from looking at some practice tests, the high school TMSCA contains many calculus problems! 30% of the general math, 60% of the calculator, and 10% of the number sense exams contain calculus problems.Lastly, if you want to have a starting advantage against your competitors, PRACTICE, PRACTICE, AND PRACTICE! With sufficient exercise (and looking up solutions online or asking Me/Tony), your scores can dramatically increase.

Below are some guidelines for scoring and a projected TMSCA state winning score for a 9th grader:

Number of Questions

Scoring Highest Possible Score

State-Winning Score

Number Sense

80 (questions completed)*5 – (questions incorrect or skipped)*9

400 260

Calculator 35 Number Crunchers14 Geometry21 Stated

(questions completed)*5 – (questions incorrect or skipped)*9

350 240

General Math

60 (questions completed)*6 – (questions incorrect)*7

No penalty for skipped

360 270

Science 20 Biology20 Chemistry20 Physics

(questions completed)*6 – (questions incorrect)*7

No penalty for skipped

360 270

Note that when competing in any UIL competitions (local, regional, or state) grade divisions are not taken into account. All grades compete with each other for each contest type. The top 3 at the local competition and the top team for each contest type advance to regionals. The top 3 at the state competition and the top team for each contest type advance to state.

For more info and resources (many tests and links to important online resources) go to the high school math club website: www.chhsmathclub.wikispaces.com

Overview:Calculus is the mathematical study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus (concerning rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under curves); these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Calculus has widespread uses in science, economics, and engineering and can solve many problems that algebra alone cannot.Calculus is a major part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis.Calculus has historically been called "the calculus of infinitesimals", or "infinitesimal calculus". The word "calculus" comes from Latin (calculus) and means a small stone used for counting. More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, calculus of variations, lambda calculus, and process calculus.

Derivatives:The derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative is basically a way to measure a point’s slope (the slope of the point’s tangent line).

The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.There are several methods to denote the derivative:

In Leibniz's notation (infinitesimal change in x is denoted by dx, and the derivative of y with respect to x)

Shorthand notation (derivative of a function)*the presence of multiple tick marks indicates the number of derivatives you perform the derivative of the function f(x) taken once is shown below

There are multiple ways to calculate a derivative (with respect to x): Common Way (using exponents)

o Functions of Addition and Subtraction1. Separate the function by terms2. Take the exponent and place it in front of the term3. Subtract the exponent by 1

o Examples1. f(x) = 5x2 + 30x - 25

f ’(x) = (2)5x2-(1) + (1)30x1-(1) – (0)250-(1)

f ’(x) = 10x + 302. f(x) = 3x3 + 5

f ’(x) = (3)3x3-(1) + (0)50-(1)

f ’(x) = 9x2

f ’’(x) = (2)9x2-(1)

f ”(x) = 18x

o Functions of Multiplication (Product Rule) Let h(x) = f(x) * g(x)

h ‘(x) = f ‘(x) * g(x) + f(x) * g ‘(x)o Examples

1. f(x) = (2x3 + 1)(5x2)f ‘(x) = (2x3 + 1)’ * (5x2) + (2x3 + 1) * (5x2)’f ‘(x) = (6x2)(5x2) + (2x3 + 1)(10x)f ‘(x) = 30x4 + 20x4 + 10x

f ‘(x) = 50x4 + 10x2. f(x) = sin(x)cos(x)

*Refer to the list of derivatives on the next pagef ‘(x) = (sin(x))’ * (cos(x)) + (sin(x)) * (cos(x))’f ‘(x) = cos(x) * cos(x) + sin(x) * -sin(x)f ‘(x) = cos2(x) – sin2(x)

o Functions of division (quotient rule)

Let h(x) = f (x )g (x)

h ‘(x) = f' ( x)∗g ( x )−f ( x )∗g' (x )

g ( x )2

o Examples

1. f(x) = x2+1x2−234

f ‘(x) = (x¿¿2+1) '∗(x¿¿2−234)−(x¿¿2+1)∗(x¿¿2−234)'(x¿¿2−234)2 ¿

¿¿¿¿

f ‘(x) = (2 x)∗(x¿¿2−234)−(x¿¿2+1)∗(2 x)(x¿¿2−234)2¿

¿¿

Formula Bashing

o h indicates the derivative you wish to calculate (explained ahead) with limits

o f() is the function you are differentiating

The application to finding the derivative function is that you can determine the slope of a curved line using this function.For example, you are given the function f(x) = x2 and wish to calculate the instantaneous slope of this quadratic curve at the x-coordinate 10.

1. Find f’(x)f(x) = x2

f ‘(x) = (2)x2-(1)

f ‘(x) = 2x2. Plug the value at x from which you want to solve the slope for

f ‘(10) = 2(10)f ‘(10) = 20

Note that derivatives are closely related to limits (mentioned later) which you might have saw under the bullet point above labeled “formula bashing.”Also note that the formulas listed above do not work for specific functions such as trigonometric or logarithmic. Several of these special function’s derivatives are located on the next page:

Simple Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Integrals:An integral (also called the antiderivative, primitive integral or indefinite integral) of a function f is a differentiable function F whose derivative is equal to f (F ′ = f). 

The process of solving for antiderivatives is called antidifferentiation (or indefinite integration) and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.

Indefinite Integral:

The indefinite integral is the exact opposite of the derivative. Its notation is as follows:

∫ f ( x )dxThe integral is calculated as exactly opposite of that of the integral (with respect to x):

1. Separate the function by terms2. Add one to the exponent3. Take the new exponent value, inverse it, then place it in front of

the term

Examples:1. ∫2 x3+x2−5 x+3¿∫( 14 )2 x

3+(1)

+( 13 )x2+(1)

−( 12 )5 x1+(1)+(11)30+(1)

¿∫( 12 ) x4¿

+(13 )x3

−( 52 ) x2+3¿

Definite Integral:

Here’s where things get exciting!!! Definite integrals are used to calculate 2-d areas under a curve. The notation is as follows:

a represents the lower bound of the curved area, b represents the upper bound of the curved areathe functions F() represent indefinite integrals

Indefinite Integral (no specific values)

Definite Integral (from a to b)

Just like derivatives, there are several integrals that can’t be integrated using traditional methods. These integrals are located below:

Limits:In mathematics, a limit is the value that a function or sequence "approaches" as the input or index approaches some value. Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals.The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.In formulas, limit is usually abbreviated as lim as in lim(an) = a, and the fact of approaching a limit is represented by the right arrow (→) as in an → a.

Limit of a Function 

Suppose f is a real-valued function and c is a real number. The expression

means that f(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, the above equation can be read as "the limit of f of x, as x approaches c, is L".Note that the above definition of a limit is true even if f(c) ≠ L. Indeed, the function f need not even be defined at c.For example, if

then f(1) is not defined (see division by zero), yet as x moves arbitrarily close to 1, f(x) correspondingly approaches 2:

f(0.9)

f(0.99)

f(0.999)

f(1.0)f(1.001)

f(1.01)

f(1.1)

1.900

1.990 1.999

⇒ undefined ⇐ 2.001 2.01

02.100

Thus, f(x) can be made arbitrarily close to the limit of 2 just by making x sufficiently close to 1.

In other words, 

This can also be calculated algebraically, as  

for all real numbers  .

Now since    is continuous in   at 1, we can now plug in 1 for  , thus In addition to limits at finite values, functions can also have limits at infinity. For example, consider

f(100) = 1.9900 f(1000) = 1.9990 f(10000) = 1.99990As x becomes extremely large, the value of f(x) approaches 2, and the value of f(x) can be made as close to 2 as one could wish just by picking x sufficiently large. In this case, the limit of f(x)as x approaches infinity is 2. In mathematical notation,

Limit of a Sequence 

Consider the following sequence: 1.79, 1.799, 1.7999,... It can be observed that the numbers are "approaching" 1.8, the limit of the sequence.Formally, suppose a1, a2, ... is a sequence of real numbers. It can be stated that the real number L is the limit of this sequence, namely:

to mean

For every real number ε > 0, there exists a natural number n0 such that for all n > n0, |an − L| < ε.Intuitively, this means that eventually all elements of the sequence get arbitrarily close to the limit, since the absolute value |an − L| is the distance between an and L. Not every sequence has a limit; if it does, it is

called convergent, and if it does not, it is divergent. One can show that a convergent sequence has only one limit.The limit of a sequence and the limit of a function are closely related. On one hand, the limit as n goes to infinity of a sequence a(n) is simply the limit at infinity of a function defined on the natural numbers n. On the other hand, a limit L of a function f(x) as x goes to infinity, if it exists, is the same as the limit of any arbitrary sequence an that approaches L, and where an is never equal to L. Note that one such sequence would be L + 1/n.

Other Notes:

Calculus Theorems

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The first fundamental theorem of calculus states that, if   is continuous on the closed interval   and   is the indefinite integral of   on  , then

This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral.

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The second fundamental theorem of calculus holds for   a continuous function on an open interval   and   any point in  , and states that if   is defined by

then

at each point in  .

______________________________________________________________________________The extreme value theorem states that if a real-valued function f is continuous in the closed and bounded interval [a,b], thenf must attain its maximum and minimum value, each at least once. That is, there exist numbers c and d in [a,b] such that:

A related theorem is the boundedness theorem which states that a continuous function f in the closed interval [a,b] is bounded on that interval. That is, there exist real numbers m and M such that:

______________________________________________________________________________The intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value.Version I. The intermediate value theorem states the following: If f is a real-valued continuous function on the interval [a, b], and u is a number between f(a) and f(b), then there is a c ∈ [a, b] such that f(c) = u.

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The mean value theorem states, roughly: given a planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints.The theorem is used to prove global statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.More precisely, if a function f is continuous on the closed interval [a, b], where a < b, and differentiable on the open interval (a, b), then there exists a point c in (a, b) such that

[1]

______________________________________________________________________________Rolle’s theorem states that: If a real-valued function f is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists a c in the open interval (a, b) such that

This version of Rolle's theorem is used to prove the mean value theorem, of which Rolle's theorem is indeed a special case. It is also the basis for the proof of Taylor's theorem.

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The squeeze/sandwich/pinching theorem is formally stated as follows.

Let I be an interval having the point a as a limit point. Let f, g, and h be functions defined on I, except possibly at a itself. Suppose that for every x in I not equal to a, we have:

and also suppose that:

Then  

The functions g and h are said to be lower and upper bounds (respectively) of f.

Here a is not required to lie in the interior of I. Indeed, if a is an endpoint of I, then the above limits are left- or right-hand limits.

A similar statement holds for infinite intervals: for example, if I = (0, ∞), then the conclusion holds, taking the limits as x → ∞.