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Unformatted text preview: Version 010 Homework 01 Odell (58340) 1 This printout should have 14 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. Welcome to Quest Learning and As sessment. Best of luck this semester. 001 10.0 points Find the value of f (0) when f ( t ) = 6( t + 1) and f (1) = 6 , f (1) = 5 . 1. f (0) = 5 2. f (0) = 2 3. f (0) = 1 4. f (0) = 3 5. f (0) = 4 correct Explanation: The most general antiderivative of f has the form f ( t ) = 3 t 2 + 6 t + C where C is an arbitrary constant. But if f (1) = 6, then f (1) = 3 + 6 + C = 6 , i.e., C = 3 . From this it follows that f ( t ) = 3 t 2 + 6 t 3 , and the most general antiderivative of the latter is f ( t ) = t 3 + 3 t 2 3 t + D , where D is an arbitrary constant. But if f (1) = 5, then f (1) = 1 + 3 3 + D = 5 , i.e., D = 4 . Consequently, f ( t ) = t 3 + 3 t 2 3 t + 4 . At x = 0, therefore, f (0) = 4 . 002 10.0 points Consider the following functions: ( A ) F 1 ( x ) = cos 2 x 2 , ( B ) F 2 ( x ) = sin 2 x 2 , ( C ) F 3 ( x ) = cos 2 x 4 . Which are antiderivatives of f ( x ) = sin x cos x ? 1. none of them 2. F 3 only 3. F 1 and F 2 only 4. F 1 and F 3 only 5. F 1 only 6. all of them 7. F 2 and F 3 only correct 8. F 2 only Explanation: By trig identities, cos 2 x = 2 cos 2 x 1 = 1 2 sin 2 x , while sin 2 x = 2 sin x cos x . But d dx sin x = cos x, d dx cos x = sin x . Consequently, by the Chain Rule, ( A ) Not antiderivative. Version 010 Homework 01 Odell (58340) 2 ( B ) Antiderivative. ( C ) Antiderivative. 003 10.0 points Find f ( x ) on ( 2 , 2 ) when f ( x ) = 3 2 sin x + 2 sec 2 x and f ( 4 ) = 4. 1. f ( x ) = 3 2 tan x + 3 2 sin x 2. f ( x ) = 2 tan x + 3 2 sin x + 5 3. f ( x ) = 2 tan x 3 2 cos x + 5 correct 4. f ( x ) = 9 2 tan x 3 2 cos x 5. f ( x ) = 2 tan x + 3 2 cos x 1 Explanation: The most general antiderivative of f ( x ) = 3 2 sin x + 2 sec 2 x is f ( x ) = 3 2 cos x + 2 tan x + C with C an arbitrary constant. But if f parenleftBig 4 parenrightBig = 4, then f parenleftBig 4 parenrightBig = 3 + 2 + C = 4 , so C = 5 . Consequently, f ( x ) = 2 tan x 3 2 cos x + 5 . 004 10.0 points Find the unique antiderivative F of f ( x ) = e 4 x + 3 e 2 x + 3 e 2 x e 2 x for which F (0) = 0. 1. F ( x ) = 1 2 e 2 x 3 x + 3 4 e 4 x 5 4 2. F ( x ) = 1 2 e 2 x +3 x 3 4 e 4 x + 1 4 correct 3. F ( x ) = 1 4 e 4 x + 3 x 3 4 e 4 x 1 2 4. F ( x ) = 1 4 e 4 x 3 x + 1 2 e 2 x 1 2 5. F ( x ) = 1 2 e 2 x 3 x + 3 4 e 2 x 1 4 6. F ( x ) = 1 2 e 2 x + 3 x 1 2 e 2 x Explanation: After division, e 4 x + 3 e 2 x + 3 e 2 x e 2 x = e 2 x + 3 + 3 e 4 x ....
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 Spring '10
 ZHENG

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