Math 132 Fall 2007 Exam II Integral Formula 1 1 sec t 3 dt sec t tan t ln sec t tan t C 2 2 1 Suppose that f x 2 2 ln 2 3 2 f 3 ln 2 a 2 ln 2 9 2 g 9 ln 2 b x Calculate D f 9 the derivative of f x evaluated at x 9 4 ln 2 3 4 h 3 ln 2 c 4 ln 2 9 4 i 9 ln 20 d 4 ln 2 27 4 j 27 ln 2 e 3 2 Calculate log3 x dx 1 a 2 1 ln 3 f 2 ln 3 1 1 ln 3 2 g 3 ln 3 b 2 c 2 ln 3 h 3 2 ln 3 d 2 ln 3 i 1 2 ln 3 e 2 ln 3 1 j 1 2 ln 3 x 3 Suppose that f x x Calculate D f 4 D f 4 is the derivative of f x evaluated at x 4 a 1 ln 2 b 2 2 ln 2 c 4 2 ln 2 d 2 1 ln 2 e 8 ln 2 f 2 ln 2 g 4 ln 2 h 4 1 ln 2 i 2 4 ln 2 j 8 1 ln 2 4 A radioactive substance has mass 120g at time t 4 and mass 90g at time t 6 What is the mass at t 12 1979 1985 1991 1997 1203 a b c d e 32 32 32 32 21 f 1209 32 g 1215 32 h 1221 32 i 1227 32 j 1233 32 5 The mass of a microbe colony splashing about in a nutrient broth triples every 12 hours What is the colony s doubling time a 8 b 8 ln 2 3 f 12 ln 2 g 3 ln 12 2 8 ln 2 ln 3 c 8 ln 3 d h 3 ln 2 i 2 ln 3 12 ln 2 ln 3 2 ln 12 j 3 e 6 Suppose that u t is the unique solution of the initial value problem d u t M 5 u t u 0 1 where M is a constant dt If a 3 f 15 lim u t 12 then what is M t b 4 g 20 c 5 h 30 d 10 i 60 e 12 j 120 1 7 Suppose that f x arcsec 5 x Calculate D f The derivative of f x at 3 1 x 3 a f 9 4 9 4 5 4 5 g 4 b c h 4 3 4 3 d i 5 3 5 3 9 20 9 j 20 e 8 Suppose that f x 120 arctan x What is D f 4 The derivative of f x at x 4 a 2 f 8 b 3 g 10 c 4 h 12 d 5 i 15 e 6 j 20 1 2 2 x 9 Calculate dx xe 0 1 4 e f 4 a 1 2 e g 2 b 3 4 3e h 4 c d 1 e 2 i e j 2 e 2 10 Calculate x sin x dx 0 a 2 b 1 2 g 4 f 5 2 2 2 c 2 2 h 3 2 2 d 3 e 4 2 j 1 i 2 2 e 4 11 Calculate 25 x ln x dx 1 a e5 1 b 2 e5 1 c 3 e5 1 d 4 e5 1 e 5 e5 1 f e5 1 g 2 e5 1 h 3 e5 1 i 4 e5 1 j 5 e5 1 2 5x 2 12 Calculate dx x2 x 1 a ln 2 f ln 6 b ln 3 2 g ln 3 c 2 ln 2 3 h ln 2 d 2 ln 3 9 i ln 2 e 3 ln 2 27 j ln 2 13 Find an ordered triple of positive integers such that 2 2 1 x 3x 4 dx ln ln x x 2 2 1 a 2 3 4 f 3 2 3 b 3 2 4 g 2 3 2 c 4 3 2 h 4 2 2 d 4 2 3 e 3 4 2 i 4 3 3 j 3 2 2 3 2 14 Calculate sin x cos x dx 0 a 1 15 f 2 5 b 2 15 g 7 15 c 1 5 h 8 15 d 4 15 i 3 5 e 1 3 j 2 3 15 Use the reduction formula 1 1 2 xn e x dx n 2 x2 n 1 e dx x 0 2 1 2e 0 1 x2 and the approximation dx e 1 3 4 0 1 1 2 4e 9 5 f 16 4 e a 1 3 2 4e 5 5 g 8 4e b 3 1 4 2e 5 3 h 8 4e c 2 4 x to approximate x e dx 0 3 3 4 2e 3 1 i 4 e d 9 3 16 4 e 9 1 j 16 e e 1 16 Calculate 1 2 1 x dx 2 2 8 1 f 2 a 2 4 4 g 8 b 2 2 4 h 4 c 1 8 4 i 2 d 1 4 8 j 8 e 1 17 Calculate 0 x 1 x ln 1 2 2 ln 1 2 b 1 2 c 2 ln 1 2 a 1 d 2 2 ln 1 2 e 1 ln 1 2 f 1 ln 1 2 1 1 g ln 1 2 2 1 1 h ln 1 2 2 2 ln 1 2 i 2 2 2 ln 1 2 j 2 2 2 dx 1 4 x2 2 x 4 18 Evaluate dx 2 x2 1 0 a 2 1 b 2 1 f g 1 4 1 4 1 h 2 c d 1 2 i 1 e 1 j 2 19 The region in the first quadrant bounded above by y 2 x and below by y x is rotated about the x axis What is the volume of the resulting solid of revolution a 2 f 5 2 11 6 8 g 3 b 5 3 9 h 4 c 3 2 7 i 4 d 4 3 5 j 4 e 2 20 The region above the x axis and under the parabola y 1 x 2 1 x 3 is rotated about the y axis What is the volume of the resulting solid of revolution a 5 3 f 5 8 3 16 g 3 b 10 3 21 h 4 c d 4 i 6 9 2 20 j 3 e
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