Calculus Resume Parie.

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CALCULUS RESUME

Physics Department of Education

Faculty of Mathematic and Science

Ganesha University of Education

2011

By:

Gde Parie Perdana

Class A, Semester I

1113021059



CALCULUS SUMMARY

I. FUNCTION

a. Definition of Function

A function f is a matching rule that links each element x in a set, called the region of

origin (domain), with a unique value f(x) of the second set, the set obtained in this

way is called the outcome function (the codomain).

Example of Function

If f is a function from A to B we write:

f: A → B

Which means that f maps A to B. A is the area of origin (domain) of f and B is

called the results (codomain) of f . A is a collection of things, such as numbers.

Here are some examples:

Set of even numbers: {..., -4, -2, 0, 2, 4 ...}

Set of odd numbers: {..., -3, -1, 1, 3, ...}

Set of prime numbers: {2, 3, 5, 7, 11, 13, ...}

Positive multiples of 3 that are less than 10: {3, 6, 9}

b. Domain, Codomain and Range

In this illustration: The set "A" is the Domain,



The set "B" is the Codomain,

And the set of elements that get pointed to in B (the actual values produced by

the function) are the Range, also called the Image.

c. Sort of Function

Elementary Function

o Constant function

Is the simplest function with the general form is y = a. Identity function

is the function that the general form y = x.

o Linear functions

The general form is y = a + bx. Where a is a constant and b is

coefficients.

o Quadratic equation

The general form of the quadratic equation is y = ax2 + bx + c. With a≠0.

The letters of a, b and c are called coefficients: the quadratic coefficient

a is the coefficient of x2, the linear coefficient b is the coefficient of x,

and c is a constant coefficient also called interest-free.

o Trigonometry functions

Trigonometric functions are functions in the form of sine and cosine

(trigonometric parameters). A simple example such as y= sin ax.

o Polynomial function

Polynomial function is a function that contains a lot of interest in the

independent variables, has the form

. Where, n is a positive

integer called the power of the polynomial.

Rational Function

The definition of a rational function is a quotient polynomial function.

Implicit Function

Implicit Function is a function of independent and the nondependent variables

are placed on the same segment.



Explicit Function

Explicit Function is a function where the independent and the nondependent

variables are at a different segment.

Parametric Function Parametric function is functions of the independent variables are bounded to

other variables.

d. Two Special Functions

Absolute Function

The absolute function is even function. This function is defined by:

|| { } Function of the largest integer

The greatest integer is neither even function nor odd function. ⟦⟧ = largest

integer, smaller or equal to x.

The graph of absolute function and function of the largest integer:

e. Operation of Functions

A function is not same as a number but same as two numbers a and b can be add to

get a new number a + b likewise to function f and g can be add to find new function

f +g.

If you have two functions f and g by the formula , √ . We can

make a new function (f + g), (f – g), (f . g), and by giving value x to each of

function. Suppose that f and g have natural domain so that each operation we can be

defined:



Operation Domain

√ [0,)

√ [0,)

√ [0,)

√ (0,)

f. Translation

By observing how the function is formed from the simple, can help us to drawing

the graph.

If we have the basic function

|

|, then we can draw the graph of

| |, || , and | | by apply the concept of translation.

g. Trigonometry Function

Trigonometric functions are functions in the form of sine and cosine (trigonometric

parameters). A simple example such as .

The graph of sine and cosine



Base on the graph above, we can get some points.

1. Both function and have the value in interval -1 until 1.

2.

Both of the graphs repeated in contiguous interval as long as 2.3. The graph of is symmetry to origin point, (0.0), meanwhile

is symmetry to the y axis (so, sine function is odd function and

cosine function is even function).

4. The graph of is same as , but has translation units to

the right.

h. Period and amplitude of trigonometric function

A function is called to be periodic if there is number p so that:

For all the members of real number x in domain of f. the smallest p the number like

that called the period of f . the function of sine called be periodic because for all of x. Its true if:

If the periodic functions of f get the maximum and minimum value we defined that

the amplitude A is half of the distance between highest and lowest point.

i. Relation to Angle Trigonometry

Angles are commonly measured either in degrees or in radius. One radian is by

definition the angle corresponding to an arc of length 1 on the unit circle.



This leads to the results

j. Trigonometric Identity



II. LIMIT

a. Definition

When we say that it mean that x near c but not same as c.

Then f (x) closes to L.

b. Right and Left Limit

Saying that its means that x near from right to c then f (x) near to

L. Similarly, saying that its means that x near from left to c but

not same as c then f (x) near to L.

c. Rigorous Study of Limits

To say that means that for each given (no matter how small)

there is a corresponding such that | | provided that

| | .



d. Theorem of Limit

Let n be a positive integer, k be a constant, and f and g be functions that have

limits at c.

1.

2. 3. 4.

[ ]

5. [ ]

6. [ ]

7.

provided

8. [] [ ]

9. , provided when n is even.

Example, if and , find * + * + =

= [ ].

= [4]2. √

= 32



e. Trigonometry Limit

b

a

bx

ax

x

it bx

ax

x

it

b

a

bx

ax

x

it bx

ax

x

it

b

a

bx

ax

x

it bx

ax

x

it

b

a

bx

ax

x

it bx

ax

x

it

sin

tan

0

limtan

sin

0

lim.4

tan

tan

0

limsin

sin

0

lim.3

tan0

limtan

0

lim.2

sin0

limsin

0

lim.1

f. Limit at Infinity

Limit

For example f defined at [c,

) for some numbers of c. We say that if

for each there is number related to M so that

| |

Limit

For example f defined at [c,) for some numbers of c. We say that

if for each

there is number related to M so that:



g. Infinite Limit

We say that if for every positive number M , there exists a

corresponding such that

Example:

and

the graph is,

h. Asymptote

We say that the line of x = c is the vertical asymptote of if one or more

from the four formula is true.

1.

2.

3.

4.

Likewise the line of is the horizontal asymptote of the graph of if

atau



i. Continuity at one point

If f has a definition in an open interval which is contain c. We say that f continue at

c if . By this definition we want to give three conditions:

1. There is

2. There is (where c consisted in the domain of f ).

3.



j. Continuity on an interval

We say that f continue at an open interval (a,b) if f continue at every points in the

interval. f continue at an close interval [a,b] if f :

1. Continue at (a,b)

2.

3.



III. DERIVATIVE

a. Gradients

Tangent line is a line that touches a curve at only one point. The slope of the

tangent line is a derivative of the curve that is in contact with the tangent line.

Tangent line the curve y = f (x) at point P (c, f (c)) is the line through P with slope.

b. Derivative

The derivative of a function f (x) is another function f '(x) whose value on any

number c is or

c. Law of Derivative

The constant function rule, if

with k a constant, then for any x,

.

The identity function rule, if f (x), then .

The powers rule, if , where n is a positive integer, then

Constant Multiple Rule, if k is a constant and f is a function which in

differentiation, then

Sum rule, if f and g is functions which in differentiation, then



Difference rule, if f and g is functions which in differentiation, then

Product rule, if f and g is functions which in differentiation, then

Quotient rule, if f and g is functions which in differentiation with ,

then

d. Derivative of trigonometry function

There are several formulas derived in the sine and cosine functions.

In addition, there is also a more specializes.

e. Leibniz Notation for the Derivative

Suppose now that the independent variable change from x to x + x. the

corresponding change in the dependent variable, y, will be

And the ratio

Represents the slope of a secant line though (x,f(x)), as show in graph below.

y

f(x+

(x,

F(x) (x,f(x))

x x+ x



as 0, the slope of this second line approached that of the tangent line, and

for this letter slope of this secant line approaches that of the tangent line, and for

this letter slope we use the symbol. Thus,

f. The Chain Rule

Facilitate the functioning of the chain rule decrease the degree of the polynomial

function. Suppose if dan . If g differentiable in x and f

differentiable in , then composite function , defined by differentiable in x and ()

g. Higher Derivatives

Notation for derivatives

Derivative Notation f’ Notation y’ Notation D Leibniz Notation

First

Second

Third

Fourth

n-th

h. Implicit differentiation

To derivative the implicit function we have to use implicit differentiation, as an

example of a function . By using the chain rule, we obtain:

()

( )

Thus, the implicit function derivative is

.



IV. APPLICATION OF DERIVATIVE

a. Maxima and Minima

Suppose that S is the domain of f , which have within c point. We can say that:

1. is maximum value f on S, if for all x in S;

2.

is minimum value f on S, if

for all x in S;

3. is extreme value f on S, if is maximum or minimum value

4. The function we want maximize or minimize is objective function.

If f continue at close interval [], then f achieve maximum or minimum value in

there. Suppose that f is an interval I which contain c point. If is extreme value,

then c must be critical point; that is c must be one of:

1. Tip point of I ;

2. Stationery point of f, that is a point where ; or

3. Singular point of f, that is a point where does not exist.



b. The Monotony and Concavity

Suppose f is defined on interval I (open, close or neither). We say that:

1. f is increasing I if, for ever pair of number x1 and x2 in I .

2. f is decreasing I if, for ever pair of number x1 and x2 in I .

3. f is strictly monotonic on I if f is either increasing on I or decreasing on I .

c. Monotonicity Theorem

Suppose f continuous on an interval I and differentiable at every interior point of I .

1. If for all x interior to I , then f is increasing on I .

2. If for all x interior to I , then f decreasing on I .

d. The Second Law and Concavity

Suppose f differentiable on an open interval I . We say that f (as well as its graph) is

concave up on I if f’ is increasing on I, and we say that f is concave down on I if f’

is decreasing on I .



e.

Concavity TheoremSuppose that f be twice differentiable on the open interval I.

1. If for all x in I , then f is concave up on I .

2. If for all x in I , then f is concave down on I .

f. Inflection Points

Let f be continuous at c. we call (c,f (c)) an inflection point of the graph of f if f is

concave up on one side and concave down on the other side. Below is the graph.



g. Local Extreme

Global maximum value is the simply the largest of the local maximum values.

Similarly, the global minimum value is the smallest of the local minimum values.

Let S, the domain of f , contains the point c. We say that:

1. is a local maximum value of f if there is an interval (a,b) containing c

such that is the maximum value of f on ;

2.

is a local minimum value of f if there is an interval (a,b) containing csuch that is the minimum value of f on ;

3. is a local extreme value of f if it is either a local maximum or a local

minimum value.

h. First and Second Derivative Test

To proof the graph we can use the first derivative of function. Let f be continuous

on an open interval (a,b) that contains a critical point c.

1. If for all x in (a,c) and for all x in (c,b), then is a local

maximum value of f .

2. If for all x in (a,c) and for all x in (c,b), then is a local

minimum value of f .

3. If has the same sign on both sides of c, then is not a local extreme

value of f .

Besides the first derivative of function, there is another test for local maxima andminima that is sometimes easier to apply. It is the second derivative at the



stationary points. Let and

exist at every point in an open interval (a,b)

containing c, and suppose that .

1. If , then is a local maximum value of I .

2. If , then is a local minimum value of I .

i. Example sophisticated graph

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Calculus Resume Parie.

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