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M 408L HW01 - kodeih(njk359 HW01 Henry(54974 This print-out...

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kodeih (njk359) – HW01 – Henry – (54974) 1 This print-out should have 22 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 lim x → − 3 6 x + 3 parenleftbigg 3 x 2 + 6 1 5 parenrightbigg . 1. limit = 4 5 2. limit = 12 25 correct 3. limit = 2 5 4. limit does not exist 5. limit = 6 25 Explanation: After the second term in the product is brought to a common denominator it becomes 15 x 2 6 5( x 2 + 6) = 9 x 2 5( x 2 + 6) . Thus the given expression can be written as 6(9 x 2 ) 5( x + 3)( x 2 + 6) = 6(3 x ) 5( x 2 + 6) so long as x negationslash = 3. Consequently, lim x → − 3 6 x + 3 parenleftbigg 3 x 2 + 6 1 5 parenrightbigg = lim x → − 3 6(3 x ) 5( x 2 + 6) . By properties of limits, therefore, limit = 12 25 . 002 10.0 points Find the derivative of f when f ( x ) = 1 + 2 sin x cos x . 1. f ( x ) = 1 2 cos x sin 2 x 2. f ( x ) = 2 sin x + 1 cos 2 x 3. f ( x ) = 2 cos x sin 2 x 4. f ( x ) = sin x 2 cos 2 x 5. f ( x ) = 2 + cos x sin 2 x 6. f ( x ) = 1 + 2 cos x sin 2 x 7. f ( x ) = 2 + sin x cos 2 x correct 8. f ( x ) = 2 sin x 1 cos 2 x Explanation: By the quotient rule, f ( x ) = 2 cos 2 x + sin x (1 + 2 sin x ) cos 2 x = 2(sin 2 x + cos 2 x ) + sin x cos 2 x . But cos 2 x + sin 2 x = 1. Consequently, f ( x ) = 2 + sin x cos 2 x . 003 10.0 points Find the derivative of f when f ( x ) = 7 x cos 2 x 4 sin 2 x . 1. f ( x ) = 14 x sin 2 x cos 2 x
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kodeih (njk359) – HW01 – Henry – (54974) 2 2. f ( x ) = 8 cos 2 x + 14 x sin 2 x 3. f ( x ) = 14 x sin 2 x cos 2 x correct 4. f ( x ) = 8 cos 2 x x sin 2 x 5. f ( x ) = 14 x sin 2 x 8 cos 2 x Explanation: Using formulas for the derivatives of sine and cosine together with the Product and Chain Rules, we see that f ( x ) = 7 cos 2 x 14 x sin 2 x 8 cos 2 x = 14 x sin 2 x cos 2 x . 004 10.0 points Find f ( x ) when f ( x ) = 1 x 2 6 x . 1. f ( x ) = x 3 (6 x x 2 ) 1 / 2 2. f ( x ) = 3 x ( x 2 6 x ) 1 / 2 3. f ( x ) = x 3 ( x 2 6 x ) 3 / 2 4. f ( x ) = 3 x (6 x x 2 ) 3 / 2 5. f ( x ) = x 3 (6 x x 2 ) 3 / 2 6. f ( x ) = 3 x ( x 2 6 x ) 3 / 2 correct Explanation: By the Chain Rule, f ( x ) = 1 2( x 2 6 x ) 3 / 2 (2 x 6) . Consequently, f ( x ) = 3 x ( x 2 6 x ) 3 / 2 . 005 10.0 points Determine f ( x ) when f ( x ) = 3 sec 2 x 2 tan 2 x . 1. f ( x ) = 10 sec 2 x tan x 2. f ( x ) = 2 tan 2 sec x 3. f ( x ) = 2 tan 2 sec x 4. f ( x ) = 10 tan 2 sec x 5. f ( x ) = 2 sec 2 x tan x 6. f ( x ) = 2 sec 2 x tan x correct Explanation: Since d dx sec x = sec x tan x, d dx tan x = sec 2 x, the Chain Rule ensures that f ( x ) = 6 sec 2 x tan x 4 tan x sec 2 x . Consequently, f ( x ) = 2 sec 2 x tan x .
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