1

Notes on

Antiderivatives

So far our studies have concentrated on given a function how do we find its

derivative? Now we ask the reverse question: Given a derivative how do we find the

function?

The following definition provides some clarity for this last

question.

Definition

: Let

f

denote a function on an interval

I

. A function

F

is said to be an

antiderivative of

f

on the interval

I

provided

for all values of

.

Since

for

, the function

is an antiderivative of the

function

on the interval

. There are other antiderivatives of

such

as

,

and

. In fact, since

, where

C

is a constant, all

the functions

, where

C

is any constant, are antiderivatives of

.

So the are

infinitely many antiderivatives for the function

. Are there any others?

To answer this question the following theorems provide the basis

for the theory.

In addition these theorems have important consequences.

Rolle

’s Theorem

:

Suppose

is a function which is continuous on the interval

,

differentiable on the interval

and for which

. Then there is a number

,

,

for which

.

The Mean Value Theorem (for Derivatives)

:

Suppose

is a function which is

continuous on the interval

and differentiable

on the interval

.

Then there is a

number

,

, for which

.

Observations

:

The following are consequences of these theorems.

Rolle’s Theorem is a special

case of the Mean Value Theorem.

Rolle’s Theorem guarantees the existence of at least one critical point on the

interval

.

The geometrical interpretation of the Mean Value Theorem is that there is a

tangent line to the curve

which is parallel to the (secant) line joining the

points

and

.

When the function

is interpreted as the position of a particle moving along a

line, then the M

ean Value Theorem says that there is a point in time that the

instantaneous velocity is the same as the average velocity over the time

interval

.

2

The following theorems provide the answer to the questions posed earlier. The proofs

are based on the Mean Value Theorem and are included

at the end of these notes

for

completeness.

Theorem 1

: Suppose

f

is a differentiable function on an interval

I

for which

for

all

. Then

f

is a consta

nt function, that is to say,

for some constant

C

.

Theorem 2

: Suppose

f

is a function defined on an interval

I

and suppose also that

F

and

G

are both antiderivatives of

f

on the interval

I

. Then there is a constant

C

for

which

for all values of

x

in the interval

I

.

One can now answer

the ques

tion originally posed, namely, d

oes the family

of

functions

,

C

a constant, represent all the antiderivatives of the function

on the

interval

? According to Theorem 2, the answer is “YES”!

Indeed the function

is

a known antiderivative of

. If

is any other antideriv

ative of

, Theorem 2

states

for some constant

.

Definition

: Suppose a function

on an interval

I

, the family of all antiderivatives of

on

the interval

I

is called the

indefinite integral of

and this family is denoted by the

symbol

.

Observation

: Notice that Theorem 2 states that if

is a function for

wh

ich

, then

, where C is a constant called the

constant of integration

. Conversely, if

and

are functions for

which

, where C is

a constant, then

.

Example 1

: Verify that the function

satisfies the hypotheses of

Rolle’s Theorem on the interval

. Then find all the numbers on that interval that

satisfies t

he conclusion of Rolle’s Theorem.

Solution

: The given function is a polynomial and therefore continuous everywhere and in

particular on the interval

. Since polynomials are differentiable everywhere, the

given function is differen

tiable on the interval

.

Furthermore

and

. So the given function satisfies the

hypotheses of Rolle’s Theorem.

Find the numbers

so

that

:

(

U

se the Quadratic Formula

)

3

.

There are two possibilities, namely

and

. The second

possibility

is negative and therefore not in the interval

.

However the first

possibility

and therefore in the interval

.

Example 2

:

Verify that the function

satisfies the hypotheses of the Mean

Value Theorem on the interval

. Then find all the numbers

that satisfy the

conclusion of the Mean Value Theorem.

Solution

:

The given function is a combination of

functions that are continuous

everywhere and therefore continuous on the interval

. The given function is also

differentiable on the interval

since

everywhere. So the given funct

ion

satisfies the hypotheses of the Mean Value Theorem.

Find

so that

:

. So this number lies in

the interval

.

Example

3

:

Consider the function

. Show that there is no number

in the

interval

for which

. Why does this not

contradict the Mean

Value Theorem?

Solution

:

. Since this equation

which has no real solutions, there is no such number

. This does not contradict the Mean

Value Theorem because the given function has a discontinuity at

and is not

continuous on the interval

. (

It is not differentiable on

either

.

)

4

Example

4

:

Show that the equation

has exactly one real root.

Solution

:

The function

is a combination of continuous functions and

therefore continuous everywhere. Furthermore

and

. So

by the Intermediate Value Theorem there is at least one real root on the interval

.

Suppose there are two real roots,

.

(

We will reach a contradiction.

) For all values

o

f

x

,

. So the function

is continuous on the

interval

and differentiable on the interval

. Further

and

.

So by Rolle’s Theorem there must exist a number

,

, for which

.

So

. This is a contradiction.

So there cannot be two real roots of

the equation.

Ex

ample

5

:

Evaluate each of the following indefinite integrals.

a)

c)

b)

d)

Solution

:

a) Since

,

the function

is an antiderivative of

.

Therefore

.

b)

Since

, the function

is an antiderivative of

.

Therefore

.

c)

Since

, the function

is an antiderivative of

.

Therefore

d)

Since

, the function

is an anti

derivative of

.

Therefore

.

The proofs of the t

heorems are now presented. It is not important for you to

memorize these proofs, but

as

you read them

you can

see why

the theorems hold true

.

Rolle’s Theor

em

: Suppose

is a function which is continuous on the interval

,

differentiable on the interval

and for which

. Then there is a number

,

, for which

.

Proof

:

There are three cases to consider.

5

Case 1: Suppose

is a constant function. Then

for all values of

x

in the

interval

. Select a number

. Then

.

Case 2: Suppose

for some value of

. Since

is continuous on the

interval

, the function

has a maximum value, call it

, on the interval

.

Then the number

c

is a critical point for the function

and, since

is differentiable on the

interval

,

.

Case 3: Suppose

for some value of

. Since

is continuous on the

interval

, the function

has a

minimum

value, call it

, on the interval

.

Then the number

c

is a critical point for the function

and, since

is differentiable on the

interval

,

.

The Mean Value Theorem (for Derivatives)

: Suppose

is a function which is

continuous on the interval

and differentiable on the interval

.

Then there is a

number

,

, for which

.

Proof

: Define the function

which is a

combination of the given function and a polynomial. Since

is continuous

on the

interval

, so is

. Since

is

differentiable on the interval

, so is

.

Furthermore

and

.

So by Rolle’s Theorem, there is a

number

,

, for which

. Since

, it follows

that

and

.

Theorem 1

: Suppose

f

is a differentiable function on an interval

I

for which

for

all

. Then

f

is a constant function, that is to say,

for some constant

C

.

Proof

: Let

be a fixed point in the interval

I

and let x denote any other point in the

interval

I

. For demonstration purposes, suppo

se

and note that the interval

is

contained in the interval

I

. Since f is differentiable on the interval

I

, it is continuous on

the interval

I

. Therefore

f

is continuous on the interval

and differentiable on the

interval

. So by the Mean Value Theorem for Derivatives there is a point

in

6

the interval

for which

. Since

,

and it

follows that

. So

. Since the point

x

is

arbitrary, f is a constant function.

Theorem 2

: Suppose

f

is a function defined on an interval

I

and suppose a

lso that

F

and

G

are both antiderivatives of

f

on the interval

I

. Then there is a constant

C

for

which

for all values of

x

in the interval

I

.

Proof

: Define

for

. Since

F

and

G

ar

e antiderivatives of

f

on the

interval

I

, the definition states that

and

for all

.

Then

for all

. By Theorem 1,

for

all

for some constant

C

. Therefore

or

.

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