FINAL Calc

FINAL Calc - Version 100 K Final Exam gualdani(56410 This...

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Version 100 – K Final Exam – gualdani – (56410) 1 This print-out should have 25 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Find the derivative of f when f ( x ) = 6 x 2 + 2 x 3 + 3 π . 1. f ( x ) = 6 parenleftBig 1 2 x 5 x 4 parenrightBig 2. f ( x ) = 3 parenleftBig 1 2 x 5 x 4 parenrightBig 3. f ( x ) = 6 parenleftBig 2 x 5 + 1 x 4 parenrightBig 4. f ( x ) = 3 parenleftBig 2 x 4 1 x 3 parenrightBig 5. none of the other answers 6. f ( x ) = 3 parenleftBig 2 x 4 + 1 x 3 parenrightBig 7. f ( x ) = 6 parenleftBig 2 x 5 1 x 4 parenrightBig correct Explanation: Since d dx ( x r ) = rx r 1 holds for all r , we see that f ( x ) = 12 x 6 x 4 . Consequently, f ( x ) = 6 parenleftBig 2 x 5 1 x 4 parenrightBig . keywords: derivatives, negative powers 002 10.0 points Find the derivative of f when f ( x ) = (2 3 x ) e 2 x +9 . 1. f ( x ) = (7 + 6 x ) e 2 x +9 2. f ( x ) = (7 6 x ) e 2 x +9 3. f ( x ) = (4 6 x ) e 2 x +9 4. f ( x ) = (1 6 x ) e 2 x +9 correct 5. f ( x ) = (1 + 6 x ) e 2 x +9 Explanation: By the Product and Chain Rules, f ( x ) = 3 e 2 x +9 + 2(2 3 x ) e 2 x +9 . Consequently, f ( x ) = (1 6 x ) e 2 x +9 . 003 10.0 points Find the derivative of f when f ( θ ) = ln (sin 5 θ ) . 1. f ( θ ) = 5 tan 5 θ 2. f ( θ ) = 5 cot 5 θ correct 3. f ( θ ) = tan 5 θ 4. f ( θ ) = 5 sin 5 θ 5. f ( θ ) = cot 5 θ 6. f ( θ ) = 1 cos 5 θ Explanation:

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Version 100 – K Final Exam – gualdani – (56410) 2 By the Chain Rule, f ( θ ) = 1 sin(5 θ ) d (sin 5 θ ) = 5 cos 5 θ sin 5 θ . Consequently, f ( θ ) = 5 cot 5 θ . 004 10.0 points Find f ( x ) when f ( x ) = x 2 ln ( 5 + x 2 ) . 1. f ( x ) = 2 x 8 x 3 (5 + x 2 ) 2 2. f ( x ) = 2 x + 4 x 3 5 + x 2 3. f ( x ) = 8 x x 3 5 + x 2 4. f ( x ) = 8 x + 2 x 3 (8 + x 2 ) 2 5. f ( x ) = 8 x + 2 x 3 5 + x 2 correct Explanation: By the Chain Rule, f ( x ) = 2 x 2 x 5 + x 2 . Consequently, after simplification, f ( x ) = 8 x + 2 x 3 5 + x 2 . 005 10.0 points Determine the derivative of f ( x ) = 5 sin 1 ( x/ 3) . 1. f ( x ) = 3 9 x 2 2. f ( x ) = 5 1 x 2 3. f ( x ) = 3 1 x 2 4. f ( x ) = 15 1 x 2 5. f ( x ) = 5 9 x 2 correct 6. f ( x ) = 15 9 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/ 3) 2 parenleftBig 1 3 parenrightBig . Consequently, f ( x ) = 5 9 x 2 . 006 10.0 points Find the derivative of f when f ( x ) = tan 1 ( e 6 x ) + e 6 x . 1. f ( x ) = e 6 x 1 + e 12 x 2. f ( x ) = 6 e 6 x 1 + e 12 x 3. f ( x ) = 6 e 6 x 1 e 12 x 4. f ( x ) = 6 e 6 x 1 + e 12 x correct 5. f ( x ) = 6 e 6 x 1 e 12 x Explanation:
Version 100 – K Final Exam – gualdani – (56410) 3 By the Chain Rule, f ( x ) = 6 e 6 x 1 + e 12 x 6 e 6 x since d dx tan 1 x = 1 1 + x 2 and ( e 6 x ) 2 = e 12 x . The expression for f can now be simplified by bringing the right hand side to a common denominator, for then f ( x ) = 6 e 6 x 6 e 6 x (1 + e 12 x ) 1 + e 12 x = 6 e 6 x 6 e 6 x 6 e 6 x 1 + e 12 x .

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