1. Antiderivatives Chapter 4.9

2. Antiderivatives
The antiderivative problem: Given the derivative of a function, f ‘(x), can you guess the original function, f(x)? I’m thinking of a function. Its derivative is . Which of the following could be my function? A. 12x2 – 2 B. x4 – x2 C. x4 – x2 + 1 D. x4 – x2 + x – 12

3. Definition
An antiderivative of f(x) is a function F(x) such that F’(x) = f(x). Question: Which of the following is an antiderivative of f(x) = 6x2 ? A. 2x3 B. 2x3 – 5 C. 2x D. all of the above

4. The family of antiderivatives of y = 2x

5. Terminology and Notation
The antiderivative of a function is also called the indefinite integral. The symbol for the antiderivative of f(x) is We often write F(x) as an antiderivative of f(x), and so

6. Antiderivative Rules Antiderivative of 0:
Antiderivative of a constant:
Antiderivative of a power of x:
p is a real number, p ≠ -1
The Power Rule

7. Voting Questions
1. A. 0 B. C C. 12x D. 12x + C 2. A. B. C. D.

8. Constant Multiples, Sums, Differences
Constant multiple rule:
Antiderivative of a sum:
Antiderivative of a difference:

9. Voting Question
A. B.
C. D.

10. Voting Question
A. 3x2 + C B. 3x2 + 5x + C
C. D.

11. Products, Quotients There is no product rule for antiderivatives!
There is no quotient rule for antiderivatives!
Consequently, antiderivatives of products and quotients cannot be directly computed by using a general rule.

12. A. B. C. D.

13. 1. A. B. C. D. 2. A. B. C. D.

14. A Trig Antiderivative
We know that , so working in reverse, we have More generally, Working in reverse, we get Dividing both sides by a gives

15. Antiderivatives Involving Trig Functions

16. Examples
1. 2.

17. Particular Antiderivatives
When the value of an antiderivative F(x) is given for a particular value of x, we can determine the value of C. Example: Find the antiderivative of f (x) = 8x3 – 2x-2 that satisfies F(1) = 5.

18. Position, velocity, acceleration
Position, y(t)
Velocity, y ’ (t)
Acceleration, y ’’ (t)

19. Example
A ball is thrown into the air from an initial height of 80 ft with initial velocity 20 ft/sec. Near the surface of the earth, acceleration due to gravity is about -32 ft/sec2. Find the position of the ball as a function of time (t), and determine the height of the ball after 2 seconds.