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# calc15 - pokharel(yp624 HW15 Radin(56520 This print-out...

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pokharel (yp624) – HW15 – Radin – (56520) 1 This print-out should have 17 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Find the value of f ( - 1) when f ( x ) = tan 1 x - 6 sin 1 x . 1. f ( - 1) = 15 4 π 2. f ( - 1) = 9 4 π 3. f ( - 1) = 17 4 π 4. f ( - 1) = 13 4 π 5. f ( - 1) = 11 4 π correct Explanation: Since tan 1 ( - 1) = - π 4 , sin 1 ( - 1) = - π 2 , we see that f ( - 1) = parenleftBig 3 - 1 4 parenrightBig π = 11 4 π . 002 10.0 points Find the exact value of sin 1 parenleftBig 3 2 parenrightBig in the interval parenleftBig 0 , π 2 parenrightBig . 1. π 4 2. π 7 3. π 6 4. π 3 correct 5. π 5 Explanation: We have to find x so that sin x = 3 2 , 0 < x < π 2 . Known trig values thus ensure that x = π 3 . 003 10.0 points Simplify the expression y = sin parenleftbigg tan 1 x 7 parenrightbigg by writing it in algebraic form. 1. y = x x 2 + 7 2. y = 7 x 2 + 7 3. y = x x 2 + 7 correct 4. y = x 2 + 7 7 5. y = x x 2 - 7 Explanation: The given expression has the form y = sin θ where tan θ = x 7 , - π 2 < θ < π 2 . To determine the value of sin θ given the value of tan θ , we can apply Pythagoras’ theorem to the right triangle 7 x θ radicalbig x 2 + 7

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pokharel (yp624) – HW15 – Radin – (56520) 2 From this it follows that y = sin θ = x x 2 + 7 . Alternatively, we can use the trig identity csc 2 θ = 1 + cot 2 θ to determine sin θ . 004 10.0 points Determine if lim x → ∞ tan 1 parenleftBigg 3 + 3 x 5 + x parenrightBigg exists, and if it does, find its value. 1. limit = π 6 2. limit = π 3 correct 3. limit = π 4 4. limit = π 2 5. limit does not exist 6. limit = 0 Explanation: Since lim x → ∞ 3 + 3 x 5 + x = 3 , we see that lim x → ∞ tan 1 parenleftBigg 3 + 3 x 5 + x parenrightBigg exists, and that the limit = tan 1 3 = π 3 . 005 10.0 points Determine the derivative of f ( x ) = 5 sin 1 ( x/ 4) . 1. f ( x ) = 20 16 - x 2 2. f ( x ) = 5 1 - x 2 3. f ( x ) = 5 16 - x 2 correct 4. f ( x ) = 20 1 - x 2 5. f ( x ) = 4 16 - x 2 6. f ( x ) = 4 1 - x 2 Explanation: Use of d dx sin 1 ( x ) = 1 1 - x 2 , together with the Chain Rule shows that f ( x ) = 5 radicalbig 1 - ( x/ 4) 2 parenleftBig 1 4 parenrightBig . Consequently, f ( x ) = 5 16 - x 2 . 006 10.0 points Find the derivative of f when f ( x ) = parenleftBig tan 1 (2 x ) parenrightBig 2 . 1. f ( x ) = 2 4 + x 2 tan 1 (2 x ) 2. f ( x ) = sec 2 (2 x ) tan(2 x ) 3. f ( x ) = 2 1 + 4 x 2 tan 1 (2 x )
pokharel (yp624) – HW15 – Radin – (56520) 3 4. f ( x ) = 4 4 + x 2 tan 1 (2 x ) 5. f ( x ) = 4 1 + 4 x 2 tan 1 (2 x ) correct 6. f ( x ) = 4 sec 2 (2 x ) tan(2 x ) Explanation: Since d dx tan 1 x = 1 1 + x 2 , the Chain Rule gives d dx tan 1 (2 x ) = 2 1 + 4 x 2 .

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calc15 - pokharel(yp624 HW15 Radin(56520 This print-out...

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