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Antiderivatives (4/8/09) 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.

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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?

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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 ) ?

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

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Some Antiderivative Facts Function Simplest Antiderivative x n (n -1)(1/(n +1))x n+1 a x (1/ln(a)) a x sin(x)- cos(x), etc. 1/xln(|x|) 1 / (1 + x 2 )arctan(x) 1 / (1 – x 2 )arcsin(x)

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Evaluating C If we are given one additional piece of information about the antiderivative, we can then evaluate the constant C. For example, find the antiderivative of f (x ) = x 2 + 4x – 5 which has a value of 6 when x = 1?

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Clicker Question 2 Which antiderivative of g (x ) = e x has the property that its value is 5 when x = 0 ? A. e x + 4 B. e x + 5 C. 5e x D. e x + C E. 4e x + 1

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Assignment for Friday Read Section 4.9. Do exercises 1 – 15 odd, 21 – 31 odd, 57, and 59. Attendance will not be taken at Friday’s class. However, HW#3 will be returned, we will go over this assignment, and we will review for the test as needed. Test #2 is on Monday (4/13) 8:40-10 for Section 2, 12:20-1:40 for Section 1. Special Office Hours: Friday 1:30-3:30.

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