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Question 1 (6 points) + - + f(x) -4 + - - + + - f'(x) -5 -3 -1 1 4 - + - + + - + - f" (x) -4 -3 -2 -1 1 3 Given the sign charts of f.f' and f" above, answer the following questions. Note that each of the dotted dividing lines correspond to zeros of the corresponding function. (a) Identify the local maxima of f. (b) In words, explain how you got your answer to (a). (c) Identify any local minima of f. (d) In words, explain how you got your answer in part (c). (e) Identify any inflection points of f. (f) In words, explain how you got your answer in part (e). Question 2 (7 points) Find the antiderivative F(x) of 3 f(x) = - 5 sin(x) +x 0.5 2.c 2 with F(1) = Question 3 (6 points) Consider the following sketch for a function y = f(x): 3- 2 1. -2 -1 o -1- 24 -3 Use the graph to find the following values. If a value doesn't exist write "DNE". (a) f( - 1) (b) f(2) (c) lim f(x) I-100 (d) lim f(x) x->2 (e) lim f(x) x > - 1+ (f) lim f(x) 3 Question 4 (6 points) Find k so that the following function is continuous on any interval. Show your work. 2 cos (x - 3), x > 3 f(x) = - k.2² +5, x < 3 Question 5 (7 points) Let I t - 3 F(x) = dt. t² + 2 Using F'(x) to explain why your answer is correct, find the largest possible interval where F(x) is increasing. Question 6 (6 points) For the equation x 3 e 55 - 3y2 = 7. dy find the derivative da Question 7 (6 points) Let f(x) = In(5x - 4) (a) Find the local linearization of f near x = 1 (b) Use the linearization to approximate f(1.3) We want to determine the largest possible area for a rectangle that has its bottom edge on the x-axis, and its top left and right corners on the function y = x2 + 9. A sample rectangle (not necessarily the one with the largest area) is shown below. 9 8 7 6 5 4 3 2 1 3 - 2 - 1 1 2 3 (a) A student solving this problem models the area of a rectangle described above as A = - 2x³ + 18x Using a few sentences, explain how the student arrived at this formula. In particular, what did the student use as the width of the rectangle? What did they use for the height of the rectangle, and why? (b) Showing all your work and using calculus techniques, determine the value of x that maximizes the area function A from part (a). (c) What is the largest area possible for the rectangle?

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