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Math 151 Week In Review 5.2, 5.3, 5.5 2 1. f ( x ) ( 2 x 2 4 x 30 ) 5 a) Find all critical values of f. b) Find all local extrema of f. c) Find the absolute max and min of f on the interval [-2, 4]. d) Find the absolute max and min of f on the interval [-2, 5]. 2. g ( x ) e x f ( x ) where f(0)=2, f'(0)=-2, and f"(0)=1. Find g'(0) and g"(0). What does the 2nd derivative test say about g at x=0? 3. f ( x ) ln | x 3 | 2 ( x 3) 2 Find any local extrema and inflection points of f. Where is f concave up? 4. Find the absolute extrema of f ( x ) x 3 3 x 2 9 x 3 on the interval a) [-6, 2] b) [-3, 4] Sketch a graph of f and f'. If f' has a local extremum at c then f has _____ at c. 5. Find c which satisfies the conclusion of the Mean Value Theorem for a) f ( x ) x 2 2 x on [-1, 2]. b) f ( x ) x 2 on [a, b] for any a<b. 6. Find all local extrema and inflection points of f ( x) e x . If F'(x)=f(x) and F(0)=0, sketch a graph of F. 2 7. Find the intervals where f is increasing, decreasing and find all local extrema. f(x) = xsin x + cos x on [0, 2π]. 8. f ( x ) x arctan x 1 2 ln( 1 x 2 ) Find f'(x) and graph it. Approximate a graph of f by looking at f'. 9. The table shows values of f' and f'' at some values of x. For each value of x, what does the 2nd derivative test say about the graph of f? x 1 2 3 4 5 f'(x) 1 0 0 2 0 f"(x) 1 0 1 -1 -2 10. The derivative of f is given by f ' ( x ) x4 x2 8 . Find the x values where f has local max, local min, inflection point(s) and find the intervals where f is concave up, concave down. 11. A right circular cylinder with top and bottom is to have a volume of 16π cubic cm. Find the dimensions that minimize the surface area. 12. Courtesy Art Belmonte Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 3 cm and 4 cm if two sides of the rectangle lie along the legs.