111S08_Final

# Question 9 15 points fundamental theorem of calculus

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Question 9. (15 points) (*Fundamental Theorem of Calculus. incr/decr functions*) Let f ( x ) be the function shown below. Define h ( x ) = integraltext x 1 f ( t ) dt . At what x values (if any) on the interval [0,6] does h ( x ) have a local maximum? Justify your answer. Question 10. (16 points, 4 points each part) (*Linear approx/Taylor polynomials*) Suppose a function f has f (4) = 7, f (4) = 4, and 0 f ′′ ( x ) < 1 x for x > 0. (a) (4 points) Find the best approximation you can for f (4 . 5) using a linear approximation.

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Math 111 (Spring 2008) Final Exam (5/9/2008) 5 (b) (4 points) Find the smallest interval you can be sure contains f (4 . 5). (c) (4 points) If you also know that f ′′ (4) = 1 3 and f ′′′ ( x ) < 0 for x > 0, give the order 2 Taylor polynomial for f centered at x = 4. (d) (4 points) Is this polynomial an over-approximation or under-approximation for f ( x ) if x > 4? Don’t forget to justify your answer.
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• Fall '06
• MARTIN,C.
• 5/9/2008

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