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Review Problems Midterm 2 December 8, 2007 1. Problem 1 - Find the indicated derivative of the following functions. (a) Find dy dx if y Answer:y 0 = (b) If dy dx = = p3 3xx 2x p (32 x x 1 1 1)4=3 4, nd . d3 y dx3 . 4 Answer:y 000 = (x2 4) 3= 2 t2 ). if y = ln( p (c) Find dy dt 5 2t Answer:y 0 = 2t (5 12t) 3t 2 (d) Find dy dt if y = [ln( t 1 )] . Answer:y 0 = t(t 3t 2 ln 1) (t 1) 2. Problem 2 - The function and its rst and second derivatives are given. Use these to nd any horizontal and vertical asymptotes, critical points, relative maxima, relative minima, and point of inection. Then sketch the graph of the function. 1 y y y x = 2 (x 3)2 6x (x 3)3 12x + 18 (x 3)4 0 = 00 = Answer: VA:x = 3; HA: y = 1; critical point: (0; 0); relative min: (0; 0); POI: ( 3=2; 1=9) 3. Problem 3 - Prot Suppose that in a monopolistic market, the demand function for a commodity is p = 7000 x 10x 2 3 where x is the number of units and p is the price (in dollars). If a company's average cost function for this commodity is ( )= C x 40; 000 x + 600 + 8x nd the maximum prot. Answer: Prot P (x) = 6400x the prot; P (64) = 208; 490:67; 18x2 2 x3 3 40; 000; x = 64 maximizes 4. Problem 4 - Problem 21 page 740 Answer: Dimension of the box: length = 8, width = 4, height = 8. 5. Problem 5 - Problem 9 page 729 Answer: 40 people will maximize the revenue. 6. Problem 6 - Use second derivative test to nd the relative maxima, relative minima, and point of inection, and sketch the graph of y = x4 8x3 + 16x2 Answer: relative min: (0; 0) and (4; 0); relative max: (2; 16)); POI: (2 p23 ; 7:111) and (2 + p23 ; 7:111) 3