solutions_11_25

# solutions_11_25 - MATH 1A THE FUNDAMENTAL THEOREM OF...

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MATH 1A: THE FUNDAMENTAL THEOREM OF CALCULUS (SOLUTIONS) 1. Let f ( t ) be the function deﬁned on the interval [ - 5 , 5] by the graph shown. Deﬁne the area function F by F ( x ) = Z x 0 f ( t ) dt. M 5 M 3 M 1 1 3 5 x M 2 2 4 f L x R By the fundamental theorem of calculus, we know that F 0 ( x ) = f ( x ). (1) Where is F increasing and decreasing? Solution. F is increasing whenever f is positive, i.e. for x [ - 5 , 3 . 5). F is decreasing whenever f is negative, i.e. x (3 . 5 , 5]. ± (2) Where is the absolute maximum and the absolute minimum of F ? Solution. Since F increases up to 3 . 5 and then decreases, the absolute maximum is at x = 3 . 5. Since the only critical point of F is 3 . 5, which is a maximum, the absolute minimum must be at one of the endpoints. Now, we have F (5) = F ( - 5) + Z 5 - 5 f ( t ) dt. The area under the graph of f ( t ) above the x axis is more than that below the x axis. Hence the integral must be positive. Therefore F (5) > F ( -

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## This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.

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solutions_11_25 - MATH 1A THE FUNDAMENTAL THEOREM OF...

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