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MATH 161 - EXAM II S’10 NAME __________________________ 1. (3 points each) Given the function, f (x) = 4x 3 – x 4 , find the following: a) critical numbers of f (x) b) intervals where f (x) is increasing and decreasing increasing _______________________ decreasing __________________________ c) relative maximums and minimums if they exist relative maximums ____________________ relative minimums __________________________ d) intervals where f (x) is concave up and concave down concave up ______________________ concave down _________________________ e) inflection points if they exist _________________________________________________ f) find the absolute maximum and minimum, if either exists, for f (x) on –1 < x < 2 absolute maximums ____________________ absolute minimums __________________________ 2. (10 points) Use the information below to graph the f (x) Domain: All real x, except x = –2 and x = 2 f (0) = 0 f (3) = 0 f ' (x) > 0 on (–∞, –2), (–2, 0) and (2, ∞) f ' (x) < 0 on (0, 2) f '' (x) < 0 on (–2, 2) and (2,∞) f '' (x) > 0 on (–∞, –2) vertical asymptote x = –2 and x = 2 horizontal asymptote y = 1 3. (10 points) Find the equation of the tangent line for the function 4. (10 points) Use logarithmic differentiation to find the derivative of : at x = 1 5. (10 points each) Differentiate each of the following. DO NOT SIMPLIFY. a) y = x 2 e 3x + 1 b) f (x) = (ln (x + 2)) 5 6. (10 points) Find the indefinite integral a) (4x 5 – 3 + ) dx b) 7. (12 points) A farmer wants to fence a corral next to a barn. He wants to separate the corral into equal sections. The total area for the corrals must equal 1200 square feet. What are the dimensions of the pasture if he wants to minimize the fencing? (check to make sure that this is a maximum!) Barn X X Y X