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32-FTC-last-problem

32-FTC-last-problem - Math 1a 1 One Last Fundamental...

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Math 1a One Last Fundamental Theorem Problem Fall, 2009 1 (a) Let f ( x ) = Z x 0 t n dt for some fixed n > 0. Find f 0 ( x ). (b) Let g ( x ) = Z x n 0 t 1 /n dt for the same fixed n > 0. Find g 0 ( x ). (c) Let F ( x ) = f ( x ) + g ( x ) = Z x 0 t n dt + Z x n 0 t 1 /n dt for this value of n > 0. Write down F 0 ( x ). (d) What is F (0)? (e) Use your answers to parts (d) and (e) to find a simple expression for F ( x ). Explain this answer geometrically.
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For your convenience, we reprint here a few questions from Wednesday’s worksheet. These questions (and answers) are from Maria Terrell’s GoodQuestions Project at the Cornell University Department of Mathematics: 2 True or False : If f is continuous on the interval [ a, b ], d dx Z b a f ( x ) dx = f ( x ). To the right is the graph of a function f . For the next two functions, we consider the function g ( x ) = Z x 0 f ( t ) dt . - 1 1 2 . . 1 2 3 4 . . x y . 3 For 0 < x < 2, g ( x ) is (a) increasing and concave up. (b) increasing and concave down. (c) decreasing and concave up. (d) decreasing and concave down.
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