HW 15 Calc - garcia(jjg2564 – HW 15 – gualdani...

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Unformatted text preview: garcia (jjg2564) – HW 15 – gualdani – (56410) 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 the value of f (- 1) when f ( x ) = tan − 1 x- 2 sin − 1 x . 1. f (- 1) = 7 4 π 2. f (- 1) = 3 4 π correct 3. f (- 1) = 9 4 π 4. f (- 1) = 11 4 π 5. f (- 1) = 5 4 π Explanation: Since tan − 1 (- 1) =- π 4 , sin − 1 (- 1) =- π 2 , we see that f (- 1) = parenleftBig 1- 1 4 parenrightBig π = 3 4 π . 002 10.0 points Simplify the expression y = sin parenleftbigg tan − 1 x √ 6 parenrightbigg by writing it in algebraic form. 1. y = x x 2 + 6 2. y = x √ x 2 + 6 correct 3. y = √ x 2 + 6 √ 6 4. y = x √ x 2- 6 5. y = √ 6 √ x 2 + 6 Explanation: The given expression has the form y = sin θ where tan θ = x √ 6 ,- 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 √ 6 x θ radicalbig x 2 + 6 From this it follows that y = sin θ = x √ x 2 + 6 . Alternatively, we can use the trig identity csc 2 θ = 1 + cot 2 θ to determine sin θ . keywords: TrigFunc, TrigFuncExam, 003 10.0 points Determine if lim x →∞ tan − 1 parenleftbigg 5 + 4 x 4 + 2 x 2 parenrightbigg exists, and if it does, find its value. 1. limit = π 4 2. limit = 0 correct garcia (jjg2564) – HW 15 – gualdani – (56410) 2 3. limit = π 3 4. limit = π 2 5. limit = π 6 6. limit does not exist Explanation: Since lim x →∞ 5 + 4 x 4 + 2 x 2 = 0 , we see that lim x →∞ tan − 1 parenleftbigg 5 + 4 x 4 + 2 x 2 parenrightbigg exists, and that the limit = tan − 1 0 = 0 . 004 10.0 points Determine the derivative of f ( x ) = 5 sin − 1 ( x/ 2) . 1. f ′ ( x ) = 5 √ 4- x 2 correct 2. f ′ ( x ) = 10 √ 4- x 2 3. f ′ ( x ) = 5 √ 1- x 2 4. f ′ ( x ) = 10 √ 1- x 2 5. f ′ ( x ) = 2 √ 1- x 2 6. f ′ ( x ) = 2 √ 4- 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/ 2) 2 parenleftBig 1 2 parenrightBig . Consequently, f ′ ( x ) = 5 √ 4- x 2 . 005 10.0 points Find the derivative of f when f ( x ) = parenleftBig tan − 1 parenleftBig x 2 parenrightBigparenrightBig 2 ....
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HW 15 Calc - garcia(jjg2564 – HW 15 – gualdani...

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