Math 132 Fall 2007 Final Exam 1. Calculate d⌠⌠⌠⌠⌡⌡⌡⌡0ππππ2( )cos x ( )sin x3x.a) 1 b) 12 c) 13 d) 14 e) 15 f) 23 g) 34 h) 32 i) 43 j) 162. Let = = = = ( )Fx d⌠⌠⌠⌠⌡⌡⌡⌡x2 + + + + 5 t4 + + + + 1 t3t. Calculate the derivative D( F )( 2 ) of F at 2. a) 4 b) 5 c) 6 d) 7 e) 8 f) −−−−4 g) −−−−5 h) −−−−6 i) −−−−7 j) −−−−83. Calculate d⌠⌠⌠⌠⌡⌡⌡⌡01x( ) + + + + x 1 ( ) + + + + x 2x. a) ln98 b) ln76 c) ln54 d) ln43 e) ln32 f) ln95 g) ln83 h) ln94 i) ln163 j) ln1694. Calculate d⌠⌠⌠⌠⌡⌡⌡⌡01 + + + + + + + + 8 x22 x 6( ) + + + + 1 x ( ) + + + + 1 x2x. a) 14 ( )ln 2 b) 12 ( )ln 2 c) ( )ln 2 d) 2 ( )ln 2 e) 3 ( )ln 2 f) 4 ( )ln 2 g) 5 ( )ln 2 h) 6 ( )ln 2 i) 7 ( )ln 2 j) 8 ( )ln 25. Calculate d⌠⌠⌠⌠⌡⌡⌡⌡1ex2( )ln x x.a) 13 e3 b) 13 ( − 2 e31 ) c) 13 ( − e32 ) d) 23 ( − e31 ) e) 13 ( + 2 e31 ) f) 13 ( + e32 ) g) 23 ( + e31 ) h) 19 ( + 2e31 ) i) 19 ( + e32 ) j) 29 ( + e31 )6. What is the derivative of x1x with respect to x at = = = = x12 ? a) −−−− ( )ln 2 b) −−−−12 ( )ln 2 c) − − − − 1 ( )ln 2 d) − − − − 112 ( )ln 2 e) ( )ln 2 f) 12 ( )ln 2 g) + + + + 1 ( )ln 2 h) + + + + 112 ( )ln 2 i) 14 ( )ln 2 j) 147. If = = = = ( )y 0 0 and dydx = ( )cosx − − − − 1y2, then what is ( )yx?a) ( )sin ππππ ( )cos x b) ( )sin ( )sin x c) cosππππ ( )cos x2 d) ( )cos ( )sin x -1 e) ( )arcsin x2 f) ( )arcsin ( )arcsin x g) ( )sin ( )tan x h) ( )tan ( )sin x i) ( )arcsin ( )tan x j) ( )arcsin ( )arctan x8. Consider the following three statements about a series ∑∑∑∑ = = = = n 1∞∞∞∞an with positive terms: I: The series converges because = = = = lim → → → → n ∞∞∞∞an0. II: The series converges because = = = = lim → → → → n ∞∞∞∞a + + + + n 1bn1.1 and ∑∑∑∑ = = = = n 1∞∞∞∞bn converges. III: The series converges because = = = = lim → → → → n ∞∞∞∞a + + + + n 1an1 . For each statement, determine whether the reasoning is correct or incorrect. a) I: correct, II: correct, III: correct b) I: correct, II: correct, III: incorrect c) I: correct, II: incorrect, III: correct d) I: correct, II: incorrect, III: incorrect e) I: incorrect, II: correct, III: correct f) I: incorrect, II: correct, III: incorrect g) I: incorrect, II: incorrect, III: correct h) I: incorrect, II: incorrect, III: incorrect i) Wrong answer j) Bonus wrong answer9. Consider the following three statements about a series ∑∑∑∑ = = = = n 1∞∞∞∞an with positive terms: I: The series converges because < < < < an1 + + + + 10 n . II: The series diverges because < < < < 1n2an . III: The series converges because = = = = lim → → → → n ∞∞∞∞a + + + + n 1an0. For each statement, determine whether the reasoning is correct ( C ) or incorrect ( F ). a) I: C, II: C, III: C b) I: C, II: C, III: F c) I: C, II: F, III: C d) I: C, II: F, III: F e) I: F, II: C, III: C f) I: F, II: C, III: F g) I: F, II: F, III: C h) I: F, II: F, III: F i) Wrong answer j) Bonus wrong answer10. Consider the three series I: ∑∑∑∑ = = = = n 0∞∞∞∞n53n , II: ∑∑∑∑ = = = = n 0∞∞∞∞10n!n , and III: ∑∑∑∑ = = = = n 2∞∞∞∞1n ( )ln n and the statements ( C ) The series converges ( D ) The series diverges For each series, decide which of statements (C), (D) is correct. a) I: C, II: C, III: C b) I: C, II: C, III: D c) I: C, II: D, III: C d) I: C, II: D, III: D e) I: D, II: C, III: C f) I: D, II: C, III: D g) I: D, II: D, III: C h) I: D, II: D, III: D i) Wrong answer j) Bonus wrong answer11. Consider the two series I: ∑∑∑∑ = = = = n 1∞∞∞∞(
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