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Unformatted text preview: Kim, Jin Homework 3 Due: Sep 19 2007, 3:00 am Inst: Diane Radin 1 This printout should have 22 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points A function h has graph 3 6 3 6 3 6 9 3 6 9 If f ( x ) = Z x 5 h ( t ) dt, ( x  5) , which of the following is the graph of f ? 1. 3 6 3 6 3 6 9 3 6 9 2. 3 6 3 6 3 6 9 3 6 9 3. 3 6 3 6 3 6 9 3 6 9 correct 4. 3 6 3 6 3 6 9 3 6 9 5. 3 6 3 6 3 6 9 3 6 9 Explanation: When f is defined by f ( x ) = Z x 5 h ( t ) dt, ( x  5) , then, first note that f ( x ) is not defined for x < 5. This eliminates one of the graphs. On the other hand, by the Fundamental theorem of Calculus, f ( x ) = h ( x ) x > 5 , , x = 5; thus f ( 5) = 0 rules out two more of the five given graphs. But 5 < x < 3 = h ( x ) > , Kim, Jin Homework 3 Due: Sep 19 2007, 3:00 am Inst: Diane Radin 2 so the graph of f must be increasing on ( 5 , 3). This eliminates one of the two remaining graphs, leaving only 3 6 3 6 3 6 9 3 6 9 which must therefore be the graph of f . keywords: graph, integral, Fundamental The orem Calculus, FTC 002 (part 1 of 1) 10 points Find the value of F 00 ( / 6) when F ( x ) = Z x 2 e 4 cos 2 d . 1. F 00 6 = 2 e 2 2. F 00 6 = 2 3 e 2 3. F 00 6 = 4 e 2 4. F 00 6 = 4 3 e 3 correct 5. F 00 6 = 4 e 3 6. F 00 6 = 2 3 e 3 Explanation: By the Fundamental theorem of calculus, F ( x ) = 2 e 4 cos 2 x , so after a second differentiation we see that F 00 ( x ) = (4 e 4 cos 2 x ) 4 sin x cos x. At x = / 6, therefore, F 00 6 = 4 3 e 3 . keywords: 003 (part 1 of 1) 10 points Determine g ( x ) when g ( x ) = Z 2 x 4 t 2 tan t dt. 1. g ( x ) = 4 x 2 sec x 2. g ( x ) = 8 x sec x tan x 3. g ( x ) = 8 x sec 2 x 4. g ( x ) = 4 x 2 sec x 5. g ( x ) = 8 x sec x tan x 6. g ( x ) = 8 x sec 2 x 7. g ( x ) = 4 x 2 tan x correct 8. g ( x ) = 4 x 2 tan x Explanation: By Properties of integrals and the Funda mental Theorem of Calculus, d dx Z a x f ( t ) dt = d dx  Z x a f ( t ) dt = f ( x ) . When g ( x ) = Z 2 x f ( t ) dt, f ( t ) = 4 t 2 tan t, therefore, g ( x ) = 4 x 2 tan x . Kim, Jin Homework 3 Due: Sep 19 2007, 3:00 am Inst: Diane Radin 3 keywords: Stewart5e, FTC, derivative form, properties of integrals, 004 (part 1 of 1) 10 points Determine F ( x ) when F ( x ) = Z x 3 2 sin t t dt. 1. F ( x ) = 2 sin x x 2. F ( x ) = 2 cos( x ) x 3. F ( x ) = sin x x 4. F ( x ) = sin( x ) x correct 5. F ( x ) = cos( x ) x 6. F ( x ) = 2 cos x x 7. F ( x ) = 2 cos x x 8. F ( x ) = sin( x ) x Explanation: By the Fundamental Theorem of Calculus and the Chain Rule, d dx Z g ( x ) a f ( t ) dt = f ( g ( x )) g ( x ) ....
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 Spring '09
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