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HW 12 solutions-1

# HW 12 solutions-1 - mandel(tgm245 HW12 Radin(56470 This...

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mandel (tgm245) – HW12 – Radin – (56470) 1 This print-out should have 15 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Find all functions g such that g ( x ) = x 2 + 2 x + 4 x . 1. g ( x ) = 2 x ( x 2 + 2 x + 4 ) + C 2. g ( x ) = 2 x parenleftbigg 1 5 x 2 + 2 3 x + 4 parenrightbigg + C cor- rect 3. g ( x ) = x ( x 2 + 2 x + 4 ) + C 4. g ( x ) = 2 x ( x 2 + 2 x 4 ) + C 5. g ( x ) = 2 x parenleftbigg 1 5 x 2 + 2 3 x 4 parenrightbigg + C 6. g ( x ) = x parenleftbigg 1 5 x 2 + 2 3 x + 4 parenrightbigg + C Explanation: After division g ( x ) = x 3 / 2 + 2 x 1 / 2 + 4 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 + 8 x 1 / 2 = 2 x parenleftbigg 1 5 x 2 + 2 3 x + 4 parenrightbigg is an antiderivative of g . Consequently, g ( x ) = 2 x parenleftbigg 1 5 x 2 + 2 3 x + 4 parenrightbigg + C with C an arbitrary constant. 002 10.0 points Find the value of f (0) when f ′′ ( t ) = 2(3 t + 4) and f (1) = 3 , f (1) = 2 . 1. f (0) = 8 2. f (0) = 7 3. f (0) = 5 correct 4. f (0) = 9 5. f (0) = 6 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 , and the most general anti-derivative of the latter is 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 = 5 . Consequently, f ( t ) = t 3 + 4 t 2 8 t + 5 .

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mandel (tgm245) – HW12 – Radin – (56470) 2 At x = 0, therefore, f (0) = 5 . 003 10.0 points Find the value of f (0) when f ( t ) = 5 sin 2 t , f parenleftBig π 2 parenrightBig = 3 . 1. f (0) = 3 2. f (0) = 1 3. f (0) = 2 correct 4. f (0) = 0 5. f (0) = 1 Explanation: Since d dx cos mt = m sin mt , for all m negationslash = 0, we see that f ( t ) = 5 2 cos 2 t + C where the arbitrary constant C is determined by the condition f ( π/ 2) = 3. But cos 2 t vextendsingle vextendsingle vextendsingle t = π/ 2 = cos π = 1 . Thus f parenleftBig π 2 parenrightBig = 5 2 + C = 3 , and so f ( t ) = 5 2 cos 2 t + 1 2 . Consequently, f (0) = 2 . 004 10.0 points Consider the following functions: ( A ) F 1 ( x ) = cos 2 x 2 , ( B ) F 2 ( x ) = cos 2 x 4 , ( C ) F 3 ( x ) = sin 2 x 2 . Which are anti-derivatives of f ( x ) = sin x cos x ? 1. F 2 only 2. F 1 only 3. all of them 4. F 2 and F 3 only 5. none of them 6. F 3 only correct 7. F 1 and F 2 only 8. F 1 and F 3 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 ) Anti-derivative. 005 10.0 points Find the value of f ( π ) when f ( t ) = 2 3 cos 1 3 t 8 sin 2 3 t
mandel (tgm245) – HW12 – Radin – (56470) 3 and f ( π 2 ) = 3.

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