Math 132 Fall 2007 Exam IIIntegral Formula: = d⌠⌡( )sec t3t + + 12( )sec t ( )tan t12( )ln + ( )sec t ( )tan t C 1. Suppose that = = = = ( )f x 2x. Calculate ( )( )D f 9 , the derivative of ( )f x evaluated at x = 9.) a) 23 ln(2) b) 29 ln(2) c) 43 ln(2) d) 49 ln(2) e) 427 ln(2) f) 23 ( )ln 2 g) 29 ( )ln 2 h) 43 ( )ln 2 i) 49 ( )ln 20 j) 427 ( )ln 22. Calculate d⌠⌠⌠⌠⌡⌡⌡⌡13( )log3x x.a) − − − − 21( )ln 3 b) + + + + 21( )ln 3 c) − − − − 2 ( )ln 3 d) + + + + 2 ( )ln 3 e) − − − − 2 ( )ln 3 1 f) + + + + 2 ( )ln 3 1 g) − − − − 32( )ln 3 h) + + + + 32( )ln 3 i) − − − − 12( )ln 3 j) + + + + 12( )ln 33. Suppose that = = = = ( )f x xx. 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 24. A radioactive substance has mass 120g at time = = = = t 4 and mass 90g at time = = = = t 6. What is the mass at = = = = t 12 ? a) 197932 b) 198532 c) 199132 d) 199732 e) 120321 f) 120932 g) 121532 h) 122132 i) 122732 j) 1233325. 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 c) 8 ( )ln 3 d) 8 ( )ln 2( )ln 3 e) 12 ( )ln 2( )ln 3 f) 12ln32 g) 3 ( )ln 122 h) 3 ( )ln 2 i) 2 ( )ln 3 j) 2 ( )ln 1236. Suppose that ( )u t is the unique solution of the initial value problem , = = = = ddt( )u t − − − − M 5 ( )u t = = = = ( )u 0 1 where M is a constant. If lim → → → → t ∞∞∞∞ = = = = ( )u t 12 then what is M? a) 3 b) 4 c) 5 d) 10 e) 12 f) 15 g) 20 h) 30 i) 60 j) 1207. Suppose that = = = = ( )fx( )arcsec 5x. Calculate ( )Df −−−−13. (The derivative of f(x) at = = = = x −−−−13 ). a) −−−−94 b) −−−−54 c) −−−−43 d) −−−−53 e) −−−−920 f) 94 g) 54 h) 43 i) 53 j) 9208. Suppose that = = = = ( )f x 120 ( )arctan x . What is ( )( )D f 4 ? (The derivative of f(x) at = = = = x 4 ). a) 2 b) 3 c) 4 d) 5 e) 6 f) 8 g) 10 h) 12 i) 15 j) 209. Calculate d⌠⌠⌠⌠⌡⌡⌡⌡012x e( )2 xx. a) 14 b) 12 c) 34 d) 1 e) 2 f) e4 g) e2 h) 3 e4 i) e j) 2 e10. Calculate d⌠⌠⌠⌠⌡⌡⌡⌡0ππππx2( )sin x x . a) ππππ2 b) − − − − ππππ21 c) − − − − ππππ22 d) − − − − ππππ23 e) − − − − ππππ24 f) + + + + ππππ25 g) + + + + ππππ24 h) + + + + ππππ23 i) + + + + ππππ22 j) + + + + ππππ2111. Calculate 25 d⌠⌠⌠⌠⌡⌡⌡⌡1ex4( )ln x x .a) − − − − e51 b) − − − − 2 e51 c) − − − − 3 e51 d) − − − − 4 e51 e) − − − − 5 e51 f) + + + + e51 g) + + + + 2 e51 h) + + + + 3 e51 i) + + + + 4 e51 j) + + + + 5 e5112. Calculate d⌠⌠⌠⌠⌡⌡⌡⌡12 + + + + 5 x 2 + + + + x2xx .a) ( )ln 2 b) ( )ln 3 c) 2 ( )ln 2 d) 2 ( )ln 3 e) 3 ( )ln 2 f) ( )ln 6 g) ln23 h) ln32 i) ln92 j) ln27213. Find an ordered triple ( , ,αααα ββββ γγγγ) of positive integers , ,αααα ββββ γγγγ such that = = = = d⌠⌠⌠⌠⌡⌡⌡⌡12 + + + + − − − − x23 x 4x ( ) + + + + x 22x − − − − + + + + αααα ( )ln ββββ ββββ ( )ln αααα1γγγγ . a) , ,2 3 4 b) ( , ,3 2 4 ) c) ( , ,4 3 2) d) ( , ,4 2 3) e) , ,3 4 2 f) ( , ,3 2 3) g) ( , ,2 3 2) h) ( , ,4 2 2) i) ( , ,4 3 3) j) ( , ,3 2 2)14. Calculate d⌠⌠⌠⌠⌡⌡⌡⌡0ππππ( )sin x3( )cos x2x.a) 1/15 b) 2/15 c) 1/5 d) 4/15 e) 1/3 f) 2/5 g) 7/15 h) 8/15 i) 3/5 j) 2/315. Use the reduction formula = = = = d⌠⌠⌠⌠⌡⌡⌡⌡01xne( )−−−−x2x − − − − ( ) − − − − n 1 d⌠⌠⌠⌠⌡⌡⌡⌡01x( ) − − − − n 2e( )−−−−x2x212 e and the approximation
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