Final Exam - Version 013 – K Final Exam – gualdani...

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Unformatted text preview: Version 013 – K Final Exam – gualdani – (56455) 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 4 + 4 x 3 + 3 π . 1. f ′ ( x ) = 3 parenleftBig 1 − 2 x 7 x 4 parenrightBig 2. f ′ ( x ) = 12 parenleftBig 2 x 7 + 1 x 4 parenrightBig 3. f ′ ( x ) = 12 parenleftBig 2 x 7 − 1 x 4 parenrightBig correct 4. none of the other answers 5. f ′ ( x ) = 3 parenleftBig 2 x 6 + 1 x 3 parenrightBig 6. f ′ ( x ) = 3 parenleftBig 2 x 6 − 1 x 3 parenrightBig 7. f ′ ( x ) = 12 parenleftBig 1 − 2 x 7 x 4 parenrightBig Explanation: Since d dx ( x r ) = rx r − 1 holds for all r , we see that f ′ ( x ) = 24 x 3 − 12 x 4 . Consequently, f ′ ( x ) = 12 parenleftBig 2 x 7 − 1 x 4 parenrightBig . keywords: derivatives, negative powers 002 10.0 points Find the derivative of f when f ( x ) = (2 − 5 x ) e 3 x +3 . 1. f ′ ( x ) = (11 − 15 x ) e 3 x +3 2. f ′ ( x ) = (11 + 15 x ) e 3 x +3 3. f ′ ( x ) = (6 − 15 x ) e 3 x +3 4. f ′ ( x ) = (1 − 15 x ) e 3 x +3 correct 5. f ′ ( x ) = (1 + 15 x ) e 3 x +3 Explanation: By the Product and Chain Rules, f ′ ( x ) = − 5 e 3 x +3 + 3(2 − 5 x ) e 3 x +3 . Consequently, f ′ ( x ) = (1 − 15 x ) e 3 x +3 . 003 10.0 points Find the derivative of f when f ( θ ) = ln(sin 6 θ ) . 1. f ′ ( θ ) = 6 sin 6 θ 2. f ′ ( θ ) = cot6 θ 3. f ′ ( θ ) = 6 cot6 θ correct 4. f ′ ( θ ) = 6 tan6 θ 5. f ′ ( θ ) = − tan 6 θ 6. f ′ ( θ ) = 1 cos 6 θ Explanation: Version 013 – K Final Exam – gualdani – (56455) 2 By the Chain Rule, f ′ ( θ ) = 1 sin(6 θ ) d dθ (sin 6 θ ) = 6 cos6 θ sin 6 θ . Consequently, f ′ ( θ ) = 6 cot6 θ . 004 10.0 points Find f ′ ( x ) when f ( x ) = x 2 − ln ( 4 + x 2 ) . 1. f ′ ( x ) = 6 x + 2 x 3 (6 + x 2 ) 2 2. f ′ ( x ) = 6 x − x 3 4 + x 2 3. f ′ ( x ) = 2 x − 6 x 3 (4 + x 2 ) 2 4. f ′ ( x ) = 6 x + 2 x 3 4 + x 2 correct 5. f ′ ( x ) = 2 x + 3 x 3 4 + x 2 Explanation: By the Chain Rule, f ′ ( x ) = 2 x − 2 x 4 + x 2 . Consequently, after simplification, f ′ ( x ) = 6 x + 2 x 3 4 + x 2 . 005 10.0 points Determine the derivative of f ( x ) = 2 sin − 1 ( x/ 3) . 1. f ′ ( x ) = 6 √ 1 − x 2 2. f ′ ( x ) = 2 √ 9 − x 2 correct 3. f ′ ( x ) = 3 √ 9 − x 2 4. f ′ ( x ) = 2 √ 1 − x 2 5. f ′ ( x ) = 3 √ 1 − x 2 6. f ′ ( x ) = 6 √ 9 − x 2 Explanation: Use of d dx sin − 1 ( x ) = 1 √ 1 − x 2 , together with the Chain Rule shows that f ′ ( x ) = 2 radicalbig 1 − ( x/ 3) 2 parenleftBig 1 3 parenrightBig . Consequently, f ′ ( x ) = 2 √ 9 − x 2 ....
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Final Exam - Version 013 – K Final Exam – gualdani...

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