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Look Ahead Today – HW#4 out Monday – Regular class (not lab) Tuesday – HW#4 due at 4:45 Wednesday – Last class - return clickers Thursday – Regular office hours 3:15-4:45 Friday – Extra help here 2-3 pm; HW#4 back Monday 5/4 – Office Hours 1-3 pm; Exam 6-9 pm Emerson

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Clicker Question 1 What is ? A. (x 2 + 3) + C B. ½ (x 2 + 3) + C C. ¼ (x 2 + 3) + C D. 1/3 (x 2 + 3) 3/2 + C E. 2 (x 2 + 3) + C

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Clicker Question 2 What is tan(t ) sec 2 (t ) dt ? A. tan(t 2 ) + C B. ln(tan(t)) + C C. tan 2 (t ) + C D ½ tan 2 (t ) + C E. 2 tan 2 (t ) + C

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The Fundamental Theorem of Calculus (4/24/09) It tells us how the antiderivative and the (definite) integral are related. Part 1 says that we can find out the integral of a function f on an interval by evaluating an antiderivative F of f. Part 2 says we can manufacture an antiderivative of f by turning the integral into a function by “freeing up” the right hand endpoint.

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FTC Part 1 This is the one we have been using for a week now, and it’s what most people think of as the Fundamental Theorem. Theorem. If F (x) is any antiderivative of f (x), then

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The Idea Behind Part 1 Since F is an antiderivative of f, f is the rate of change of F. The integral of f on [a, b] is just an adding up of all its values there. If we add up all the rates of change of F over the interval [a, b], we will get the total change in F, which is just F (b) – F (a).

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FTC Part 2 This half of the theorem is lesser known and less often used. Theorem. Given f (x), if we define F (x) by then F is an antiderivative of f.

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Example of Part 2 There is no known antiderivative formula for f (x) = e x^2, but Part 2 of the Fundamental Theorem says that we can manufacture an antiderivative by turning the right hand endpoint of the integral into the input variable:

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Summary and Assignment In summary: FTC: Given f on [a, b], Part 1: antiderivative (a function) integral (a number) via evaluation at b and a. Part 2: integral (a number) antiderivative (a function) via turning b into the input variable x. Assignment: Work on HW#4 and on these ideas. See you Monday.

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