t5(3) - the graph of y = f ( x ), indicating any extreme...

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Methods of Calculus, Final Exam v.V April 26, 2002 Instructions: Write your complete solutions in your exam book. Do not write on this paper. This is a closed-book test. No calculators are allowed to be used. To receive credit, you must show all work. Some numerical answers may involve the function ln or the number e . You are not required to simplify your answers. (1) Find the derivatives of the following functions. (a) (10 pts.) f ( x ) = 9 - 8 x + e x + ln x + 1 x 1 / 5 (b) (10 pts.) f ( x ) = 7 x 2 x 2 + 12 (c) (10 pts.) f ( x ) = x 5 ( 1 - x 2 ) 4 (d) (10 pts.) f ( x ) = 5 e 4 x 2 (e) (10 pts.) f ( x ) = ln ( 7 - x 2 ) (2) (10 pts.) Find the antiderivative Z ± x - 6 e 3 x + 8 x ² dx (3) (10 pts.) A company’s marginal cost function is M C ( x ) = 3 x 2 - 8 x + 15 where x is the production level. If the ±xed costs are 15, ±nd the cost function. (4) (20 pts.) For the function f ( x ) = 9 x + x - 1 ±nd the values of x where f ( x ) has a possible relative maximum or minimum point. Use the second derivative to determine the nature of the function at these points. Sketch
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Unformatted text preview: the graph of y = f ( x ), indicating any extreme points and asymptotes. (5) (20 pts.) Find the area of the region bounded by the curves y = x 2 +2 x-1 and y = 2 x . Sketch a graph showing the two curves and the region which they bound. (6) (20 pts.) When a drug is injected into a muscle it begins to diuse into the bloodstream. The concentration in the veins increases until it reaches its maximum value. The concentration then decays exponentially. Suppose the concentration in the blood is given by the function f ( t ) = 0 . 01 te-t/ 2 , where t is the time since the injection, in hours, and f ( t ) is given in terms of some suitable units. Find the value of t for which the concentration is maximized. Find the value of t for which the function f ( t ) has an inection point. Sketch the graph of f ( t ) for t 0....
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This note was uploaded on 07/13/2011 for the course MAC 2233 taught by Professor Staff during the Spring '08 term at FAU.

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