1. Antiderivatives

2. We will also learn some applications.
Think About It
f '(x)
Suppose this is the graph of the derivative of a function
What do we know about the original function?
Critical numbers
Where it is increasing, decreasing
What do we not know?
In this chapter we will learn how to determine the original function when given the derivative.
We will also learn some applications.

3. Anti-Derivatives Derivatives give us the rate of change of a function
What if we know the rate of change …
Can we find the original function?
If f '(x) = f(x)
Then f(x) is an antiderivative of f’(x)
Example – let f(x) = 12x2
Then f '(x) = 24x
So f(x) = 12x2 is the antiderivative of f’(x) = 24x

4. Finding An Antiderivative
Given f(x) = 12x3
What is the antiderivative, f(x)?
Use the power rule backwards
Recall that for f(x) = xn … f '(x) = n • x n – 1
That is …
Multiply the expression by the exponent
Decrease exponent by 1
Now do opposite (in opposite order)
Increase exponent by 1
Divide expression by new exponent

5. Family of Antiderivatives
Consider a family of parabolas
f(x) = x2 + n which differ only by value of n
Note that f '(x) is the same for each version of f
Now go the other way …
The antiderivative of 2x must be different for each of the original functions
So when we take an antiderivative
We specify F(x) + C
Where C is an arbitrary constant
This indicates that multiple antiderivatives could exist from one derivative

6. Indefinite Integral
The family of antiderivatives of a function f indicated by
The symbol is a stylized S to indicate summation

7. Indefinite Integral The indefinite integral is a family of functions
The + C represents an arbitrary constant
The constant of integration

8. Properties of Indefinite Integrals
The power rule
The integral of a sum (difference) is the sum (difference) of the integrals

9. Properties of Indefinite Integrals
The derivative of the indefinite integral is the original function
A constant can be factored out of the integral

10. Try It Out
Determine the indefinite integrals as specified below