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Unformatted text preview: Perez (nap563) HW01 Zheng (56555) 1 This printout should have 22 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. 001 10.0 points Determine if lim x 3 x + 6 3 x 3 exists, and if it does, find its value. 1. limit = 6 2. limit = 5 3. limit does not exist 4. limit = 1 6 correct 5. limit = 1 7 6. limit = 1 3 Explanation: Since ( x + 6 3)( x + 6 + 3) = ( x + 6) 9 = x 3 , we see by rationalizing the numerator that x + 6 3 x 3 = x 3 ( x 3)( x + 6 + 3) = 1 x + 6 + 3 provided x negationslash = 3. On the other hand, lim x 3 x + 6 + 3 = 6 . Consequently, by properties of limits, lim x 3 x + 6 3 x 3 exists and has limit = 1 6 . 002 10.0 points Find the value of lim x 3 8 x + 3 parenleftbigg 3 x 2 + 6 1 5 parenrightbigg . 1. limit = 16 25 correct 2. limit does not exist 3. limit = 8 15 4. limit = 16 15 5. limit = 8 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 8(9 x 2 ) 5( x + 3)( x 2 + 6) = 8(3 x ) 5( x 2 + 6) so long as x negationslash = 3. Consequently, lim x 3 8 x + 3 parenleftbigg 3 x 2 + 6 1 5 parenrightbigg = lim x 3 8(3 x ) 5( x 2 + 6) . By properties of limits, therefore, limit = 16 25 . 003 10.0 points Perez (nap563) HW01 Zheng (56555) 2 Find the derivative of f when f ( x ) = 1 2 cos x sin x . 1. f ( x ) = 2 sin x + 1 cos 2 x 2. f ( x ) = 2 cos x sin 2 x correct 3. f ( x ) = 2 + sin x cos 2 x 4. f ( x ) = 2 sin x 1 cos 2 x 5. f ( x ) = 1 2 cos x sin 2 x 6. f ( x ) = sin x 2 cos 2 x 7. f ( x ) = 2 + cos x sin 2 x 8. f ( x ) = 1 + 2 cos x sin 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 ) = 5 x cos 3 x 6 sin 3 x . 1. f ( x ) = 15 x sin3 x 18cos 3 x 2. f ( x ) = 15 x sin3 x 13 cos3 x correct 3. f ( x ) = 18cos 3 x + 15 x sin3 x 4. f ( x ) = 15 x sin 3 x 13 cos 3 x 5. f ( x ) = 18cos 3 x 13 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 ) = 5 cos3 x 15 x sin 3 x 18cos 3 x = 15 x sin3 x 13 cos3 x ....
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This note was uploaded on 03/17/2010 for the course MATH 408 L taught by Professor Zheng during the Spring '10 term at University of Texas at Austin.
 Spring '10
 ZHENG
 Math, Calculus

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