HW13.pdf - young(toy68 HW13 villafuerte altu(53785 This...

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young (toy68) – HW13 – villafuerte altu – (53785) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0points Find all functions g such that g ( x ) = x 2 + 4 x + 2 x . 1. g ( x ) = x parenleftbigg 1 5 x 2 + 4 3 x + 2 parenrightbigg + C 2. g ( x ) = 2 x ( x 2 + 4 x + 2 ) + C 3. g ( x ) = x ( x 2 + 4 x + 2 ) + C 4. g ( x ) = 2 x parenleftbigg 1 5 x 2 + 4 3 x + 2 parenrightbigg + C cor- rect 5. g ( x ) = 2 x ( x 2 + 4 x 2 ) + C 6. g ( x ) = 2 x parenleftbigg 1 5 x 2 + 4 3 x 2 parenrightbigg + C Explanation: After division g ( x ) = x 3 / 2 + 4 x 1 / 2 + 2 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 + 8 3 x 3 / 2 + 4 x 1 / 2 = 2 x parenleftbigg 1 5 x 2 + 4 3 x + 2 parenrightbigg is an antiderivative of g . Consequently, g ( x ) = 2 x parenleftbigg 1 5 x 2 + 4 3 x + 2 parenrightbigg + C with C an arbitrary constant. 002 10.0points Find the most general antiderivative, F , of the function f ( x ) = 6 x 2 16 x + 2 . 1. F ( x ) = 2 x 3 8 x 2 + 2 x 2. F ( x ) = 6 x 3 16 x 2 + 2 x + C 3. F ( x ) = 2 x 3 8 x 2 + 2 x + C correct 4. F ( x ) = 2 x 3 + 8 x 2 + 2 x + C 5. F ( x ) = 2 x 3 + 8 x 2 + 2 x Explanation: Since d dx x r = rx r 1 , the most general anti-derivative of f is the function F ( x ) = 6 parenleftbigg x 3 3 parenrightbigg 16 parenleftbigg x 2 2 parenrightbigg + 2 x + C with C an arbitrary constant. Consequently, F ( x ) = 2 x 3 8 x 2 + 2 x + C . 003 10.0points 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 ) 2 . Which are anti-derivatives of f ( x ) = sin( x ) cos( x ) ?
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young (toy68) – HW13 – villafuerte altu – (53785) 2 1. F 1 only 2. none of them 3. F 2 and F 3 only 4. F 1 and F 3 only 5. F 2 only 6. all of them 7. F 1 and F 2 only 8. F 3 only correct 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. 004 10.0points Find the most general function f such that f ′′ ( x ) = 48 cos(4 x ) . 1. f ( x ) = 3 cos(4 x ) + Cx + D correct 2. f ( x ) = 4 cos(4 x ) + Cx 2 + D 3. f ( x ) = 4 sin(4 x ) + Cx + D 4. f ( x ) = 3 sin( x ) + Cx + D 5. f ( x ) = 4 sin( x ) + Cx 2 + D 6. f ( x ) = 3 cos( x ) + Cx + D Explanation: When f ′′ ( x ) = 48 cos(4 x ) then f ( x ) = 12 sin(4 x ) + C with C an arbitrary contant. Consequently, the most general function f is f ( x ) = 3 cos(4 x ) + Cx + D with D also an arbitrary constant. 005 10.0points Find f ( x ) on parenleftBig π 2 , π 2 parenrightBig when f ( x ) = 7 + 6 tan 2 x and f (0) = 2. 1. f ( x ) = 4 + 7 x + 6 sec x 2. f ( x ) = 2 x 6 tan x 3. f ( x ) = 2 + x + 6 tan 2 x 4. f ( x ) = 2 + x + 6 tan x correct 5. f ( x ) = 8 x 6 sec x 6. f ( x ) = 4 + 7 x + 6 sec 2 x Explanation: The properties d dx (tan x ) = sec 2 x, tan 2 x = sec 2 x 1 , suggest that we rewrite f ( x ) as f ( x ) = 1 + 6 sec 2 x,
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young (toy68) – HW13 – villafuerte altu – (53785) 3 for then the most general anti-derivative of f is f ( x ) = x + 6 tan x + C, with C an arbitrary constant. But if f (0) = 2, then f (0) = C = 2 .
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