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badruddin (ssb776) – HW02 – Radin – (57410) 1 This print-out should have 14 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Determine f ( t ) when f ′′ ( t ) = 2(3 t 4) and f (1) = 3 , f (1) = 2 . 1. f ( t ) = t 3 + 4 t 2 8 t + 5 2. f ( t ) = 3 t 3 4 t 2 + 8 t 5 3. f ( t ) = t 3 4 t 2 + 8 t 3 correct 4. f ( t ) = 3 t 3 + 8 t 2 8 t 1 5. f ( t ) = 3 t 3 8 t 2 + 8 t 1 6. f ( t ) = t 3 + 8 t 2 8 t + 1 Explanation: The most general anti-derivative of f ′′ has the form f ( t ) = 3 t 2 8 t + C where C is an arbitrary constant. But if f (1) = 3, then f (1) = 3 8 + C = 3 , i.e., C = 8 . From this it follows that f ( t ) = 3 t 2 8 t + 8 . The most general anti-derivative of f is thus f ( t ) = t 3 4 t 2 + 8 t + D , where D is an arbitrary constant. But if f (1) = 2, then f (1) = 1 4 + 8 + D = 2 , i.e., D = 3 . Consequently, f ( t ) = t 3 4 t 2 + 8 t 3 . 002 10.0 points Find all functions g such that g ( x ) = x 2 + 2 x + 3 x . 1. g ( x ) = 2 x parenleftbigg 1 5 x 2 + 2 3 x + 3 parenrightbigg + C cor- rect 2. g ( x ) = x ( x 2 + 2 x + 3 ) + C 3. g ( x ) = x parenleftbigg 1 5 x 2 + 2 3 x + 3 parenrightbigg + C 4. g ( x ) = 2 x parenleftbigg 1 5 x 2 + 2 3 x 3 parenrightbigg + C 5. g ( x ) = 2 x ( x 2 + 2 x + 3 ) + C 6. g ( x ) = 2 x ( x 2 + 2 x 3 ) + C Explanation: After division g ( x ) = x 3 / 2 + 2 x 1 / 2 + 3 x 1 / 2 , so we can now find an antiderivative of each term separately. But d dx parenleftbigg ax r r parenrightbigg = ax r 1 for all a and all r negationslash = 0. Thus 2 5 x 5 / 2 + 4 3 x 3 / 2 + 6 x 1 / 2 = 2 x parenleftbigg 1 5 x 2 + 2 3 x + 3 parenrightbigg is an antiderivative of g . Consequently, g ( x ) = 2 x parenleftbigg 1 5 x 2 + 2 3 x + 3 parenrightbigg + C

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badruddin (ssb776) – HW02 – Radin – (57410) 2 with C an arbitrary constant. 003 10.0 points Consider the following functions: ( A ) F 1 ( x ) = cos 2 x 4 , ( B ) F 2 ( x ) = cos 2 x 2 , ( C ) F 3 ( x ) = sin 2 x . Which are anti-derivatives of f ( x ) = sin x cos x ? 1. F 1 and F 3 only 2. F 2 only 3. F 2 and F 3 only 4. F 3 only 5. all of them 6. none of them correct 7. F 1 and F 2 only 8. F 1 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 anti-derivative. ( B ) Not anti-derivative. ( C ) Not anti-derivative. 004 10.0 points Find f ( π/ 2) when f ( t ) = cos 1 3 t 6 sin 2 3 t and f (0) = 1. 1. f ( π/ 2) = 0 2. f ( π/ 2) = 1 3. f ( π/ 2) = 1 4. f ( π/ 2) = 2 correct 5. f ( π/ 2) = 3 Explanation: The function f must have the form f ( t ) = 3 sin 1 3 t + 9 cos 2 3 t + C where the constant C is determined by the condition f (0) = 3 sin 0 + 9 cos 0 + C = 1 .
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