Math 132 Fall 2007 Exam II 22 October 2007 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 x Calculate D f 9 the derivative of f x evaluated at x 9 2 ln 2 9 2 g 9 ln 2 4 ln 2 3 4 h 3 ln 2 b 4 ln 2 9 4 i 9 ln 20 c d 4 ln 2 27 4 j 27 ln 2 e Solution c f x 2 sqrt x f x 2 x D f x 12 x 2 ln 2 x D f 9 4 3 ln 2 3 2 Calculate log3 x dx 1 a 2 1 ln 3 b 2 1 ln 3 c 2 ln 3 d 2 ln 3 e 2 ln 3 1 f 2 ln 3 1 g 3 2 ln 3 h 3 2 ln 3 i 1 2 ln 3 j 1 2 ln 3 Solution g Int log 3 x x Int log 3 x x ln x dx log3 x dx ln 3 Int ln x ln 3 x student intparts Int ln x ln 3 x ln x Integration by parts with u ln x x ln x 1 ln x dx dx ln 3 ln 3 ln 3 Int ln x ln 3 x value student intparts Int ln x ln 3 x ln x x ln x x ln x d x ln 3 ln 3 ln 3 Answer Int log 3 x x 1 3 subs x 3 1 ln 3 x ln x 1 ln 3 x subs x 1 1 ln 3 x ln x 1 ln 3 x 3 2 ln 1 Answer log3 x dx 3 ln 3 ln 3 1 Answer value rhs Answer Answer 3 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 8 1 ln 2 g 4 ln 2 h 4 1 ln 2 i 2 4 ln 2 j Solution j f x x sqrt x x f x x D f x x x 1 ln x 1 2 x x D f 4 4 ln 4 8 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 b c d e a 32 32 32 32 21 f 1209 32 g 1215 32 h 1221 32 i 1227 32 j 1233 32 Solution g m t A exp lambda t m t A e t eqn1 m 4 120 4 eqn1 A e 120 eqn2 m 6 90 6 eqn2 A e 90 eqn3 lhs eqn1 lhs eqn2 rhs eqn1 rhs eqn2 4 eqn3 e 6 4 3 e eqn4 combine lhs eqn3 exp rhs eqn3 eqn4 e 2 4 3 eqn5 lambda solve eqn4 lambda 4 ln 2 3 Substitute this value of lambda eqn5 eqn6 subs eqn5 eqn1 1 into eqn1 to find A eqn6 A e eqn7 A solve eqn6 A 2 ln 4 3 120 640 eqn7 A 3 Answer m 12 m 12 subs eqn5 eqn7 t 12 m t Answer m 12 A e 12 640 3 6 ln 4 3 e answer simplify Answer answer 1215 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 Solution e Let T be the doubling time and let m t A 2 t T be the colony s population at time t m t A 2 t T m t A 2 t T Then eqn m 12 3 m 0 eqn A 2 solve eqn T 12 T 3A 12 ln 2 ln 3 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 lim u t 12 then what is M t a 3 f 15 b 4 g 20 c 5 h 30 d 10 i 60 e 12 j 120 Solution i eqn1 Int 1 M 5 u u Int 1 t C 1 eqn1 du 1 dt C M 5 u eqn2 map value eqn1 1 eqn2 ln M 5 u t C 5 eqn3 map z 5 z eqn2 eqn3 ln M 5 u 5 t 5 C eqn4 map z exp z eqn3 eqn4 M 5 u e eqn5 u solve eqn4 u eqn5 u M 1 5 t 5 C e 5 t 5 C 5 5 eqn6 limit rhs eqn5 t infinity 12 eqn6 M 5 12 M solve eqn6 M M 60 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 Solution f f x arcsec 5 x f x arcsec 5 x D f x 1 x2 25 1 x2 D f 1 3 9 16 16 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 Solution e f x 120 arctan sqrt x f x 120 arctan x D f x 60 x 1 x simplify D f 4 6 1 2 2 x 9 Calculate dx xe 0 1 4 e f 4 1 2 e g 2 b a 3 4 3e h 4 c d 1 e 2 i e j 2 e Solution a J Int x exp 2 x x 0 1 2 J 1 2 xe 2 x dx 0 K student intparts J x Integration by parts with u x 1 K e 4 1 2 1 2 2 x e dx 0 value K 1 4 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 Solution e J Int x 2 sin x x 0 Pi J x2 sin x dx 0 K student intparts J x 2 Integration by parts with u x 2 K 2 x cos x dx 2 …
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