Antiderivatives (3/21/12) There are times when we would like to reverse the derivative process. Given the rate of change of a function, what does that.

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Antiderivatives (3/21/12) There are times when we would like to reverse the derivative process. Given the rate of change of a function, what does that tell us about the function? Computing antiderivatives is sometimes easy, sometimes hard, and sometimes impossible! Contrast this with computing derivatives.

Definition and examples An antiderivative F (x ) of a function f (x ) is a function whose derivative F '(x ) is equal to f (x ). For example, what is an antiderivative of f (x ) = x 2 + 4x – 5? Can you find others? How about f (x ) = e x ? Can you find others? How about f (x ) = 1 / x ? Others?

Clicker Question 1 What is an antiderivative of f (x ) = x 5 – 3x + 7 ? A. x 6 – 3x 2 B. x 6 – 3x 2 + 7x C. (1/5)x 6 – (3/2)x 2 + 7x D. (1/6)x 6 – (3/2)x 2 + 7x E. 5x 4 – 3

Most General Antiderivative If F (x ) is one particular antiderivative of f (x ), then the most general antiderivative is F (x ) + C where C is a constant number. For example, the most general antiderivative of f (x ) = sin(x) is ?? How about f (x ) = 1 / (1 + x 2 ) ?

Clicker Question 2 What is the most general antiderivative of f (x ) = x 3 – 6x /x ? A. x 4 – 6x 2 + 3x C B. (1/4)x 4 – 3x 2 + 3x + ln(x) + C C. (1/4)x 4 – 3x 2 + 3x + ln(x) D. (1/4)x 4 – 3x 2 + 3x C E. 3x 2 – 6

An Application Suppose the velocity v (t ) (in feet per second) of an object at time t is given. What does an antiderivative of v (t ) measure ? Call this function s (t ). If we choose C so that s (0) = 0, then it measures the distance from the starting point.

Clicker Question 3 Suppose the velocity of an object (in feet/sec) at t seconds is given by v (t ) = 5 + 4t. How far is it from its starting position after 3 seconds? A. 18 feetB. 41 feet C. 15 feetD. 25 feet E. 33 feet

Assignment for Friday Read Section 4.9. Do Exercises 1 – 17 odd and 21 on page 348.