HW01 solution

# HW01 solution - mao(tm23477 HW01 Radin(56570 This print-out...

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mao (tm23477) – HW01 – Radin – (56570) 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 Determine if lim x 4 x + 5 3 x 4 exists, and if it does, find its value. 1. limit = 1 6 correct 2. limit = 1 3 3. limit = 5 4. limit = 6 5. limit = 1 7 6. limit does not exist Explanation: Since ( x + 5 3)( x + 5 + 3) = ( x + 5) 9 = x 4 , we see by rationalizing the numerator that x + 5 3 x 4 = x 4 ( x 4)( x + 5 + 3) = 1 x + 5 + 3 provided x negationslash = 4. On the other hand, lim x 4 x + 5 + 3 = 6 . Consequently, by properties of limits, lim x 4 x + 5 3 x 4 exists and has limit = 1 6 . 002 10.0 points Find the value of lim x → − 4 7 x + 4 parenleftbigg 4 x 2 + 8 1 6 parenrightbigg . 1. limit does not exist 2. limit = 7 12 3. limit = 7 36 4. limit = 7 24 5. limit = 7 18 correct Explanation: After the second term in the product is brought to a common denominator it becomes 24 x 2 8 6( x 2 + 8) = 16 x 2 6( x 2 + 8) . Thus the given expression can be written as 7(16 x 2 ) 6( x + 4)( x 2 + 8) = 7(4 x ) 6( x 2 + 8) so long as x negationslash = 4. Consequently, lim x → − 4 7 x + 4 parenleftbigg 4 x 2 + 8 1 6 parenrightbigg = lim x → − 4 7(4 x ) 6( x 2 + 8) . By properties of limits, therefore, limit = 7 18 . 003 10.0 points

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mao (tm23477) – HW01 – Radin – (56570) 2 Find the derivative of f when f ( x ) = 1 2 cos x sin 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 + sin x cos 2 x 4. f ( x ) = 2 cos x sin 2 x correct 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 1 cos 2 x 8. f ( x ) = sin x 2 cos 2 x Explanation: By the quotient rule, f ( x ) = 2 sin 2 x cos x (1 2 cos x ) sin 2 x = 2(sin 2 x + cos 2 x ) cos x sin 2 x . But cos 2 x + sin 2 x = 1. Consequently, f ( x ) = 2 cos x sin 2 x . 004 10.0 points Find the derivative of f when f ( x ) = 4 x cos 3 x 2 sin 3 x . 1. f ( x ) = 6 cos 3 x 2 x sin 3 x 2. f ( x ) = 12 x sin 3 x 2 cos 3 x correct 3. f ( x ) = 12 x sin 3 x 2 cos 3 x 4. f ( x ) = 12 x sin 3 x 6 cos 3 x 5. f ( x ) = 6 cos 3 x + 12 x sin 3 x Explanation: Using formulas for the derivatives of sine and cosine together with the Product and Chain Rules, we see that f ( x ) = 4 cos 3 x 12 x sin 3 x 6 cos 3 x = 12 x sin 3 x 2 cos 3 x . 005 10.0 points Find f ( x ) when f ( x ) = 1 4 x x 2 . 1. f ( x ) = 2 x (4 x x 2 ) 3 / 2 2. f ( x ) = x 2 ( x 2 4 x ) 3 / 2 3. f ( x ) = 2 x ( x 2 4 x ) 1 / 2 4. f ( x ) = x 2 (4 x x 2 ) 3 / 2 correct 5. f ( x ) = x 2 (4 x x 2 ) 1 / 2 6. f ( x ) = 2 x ( x 2 4 x ) 3 / 2 Explanation: By the Chain Rule, f ( x ) = 1 2(4 x x 2 ) 3 / 2 (4 2 x ) .
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