# Antiderivative

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## Contents

## Definition

Given a function , we say that a function is an *antiderivative* of if for all values of .

#### Example 1

If , the antiderivative of is where can be any real number.

#### Example 2

If , the antiderivative of is where can be any real number.

## Physical interpretation

The derivative of distance is velocity. The derivative of velocity is acceleration

#### Example

A boy throws a baseball up in the air from the edge of a cliff 400 ft above the ground. The initial speed of the ball as it leaves his hand is 40 ft/sec. When does the ball reach its maximum height? Note: g (acceleration due to gravity) is 32 ft/s^{2}

#### Solution

We know that the acceleration is -32ft/s^{2} a(t)=dv/dt=-32

So by taking the antiderivative: v(t)= -32(t)+C

Now we need to determine what the constant C is. To do this, we set t=0.

v(0)=40 so 40=0+C, so v(t)= -32t+40

To find the maximum height reached by the ball, set v(t)=0 to find the time at which it reaches this point

0= -32(t)+40

32t=40

t= 4/5 seconds

since s'(t)=v(t), we must antidifferentiate again

s(t)= -16t^{2} +40t+D

Now we must determine what the constant D is. To do this, we set t=0

s(0)= 400

So, s(t)= -16t^{2} +40t +400

When the ball hits the ground, s(t)=0

-16t^{2} +40t+400=0

-2t^{2}+5t+50=0

2t^{2}-5t-50=0

Solve for t.