HW01-solutions - nunez (djn358) HW01 Radin (54915) 1 This...

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Unformatted text preview: nunez (djn358) HW01 Radin (54915) 1 This print-out should have 22 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. HW01 is a M408K review. If you have trouble with these problems, check out the re- fresher courses at SLCC listed on the first day handout. You can find the first day handout at www.ma.utexas.edu, Academics, Courses, M408L. HW01 overlaps HW02 and HW03 which cover new material. 001 10.0 points Find the value of lim x 3 7 x + 3 parenleftbigg 3 x 2 + 6 1 5 parenrightbigg . 1. limit does not exist 2. limit = 14 15 3. limit = 7 15 4. limit = 7 25 5. limit = 14 25 correct 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 7(9 x 2 ) 5( x + 3)( x 2 + 6) = 7(3 x ) 5( x 2 + 6) so long as x negationslash = 3. Consequently, lim x 3 7 x + 3 parenleftbigg 3 x 2 + 6 1 5 parenrightbigg = lim x 3 7(3 x ) 5( x 2 + 6) . By properties of limits, therefore, limit = 14 25 . 002 10.0 points Find the derivative of f when f ( x ) = 1 + 2 cos x sin x . 1. f ( x ) = 2 cos x sin 2 x 2. f ( x ) = 2 sin x 1 cos 2 x 3. f ( x ) = 1 2 cos x sin 2 x 4. f ( x ) = 2 + sin x cos 2 x 5. f ( x ) = sin x 2 cos 2 x 6. f ( x ) = 2 sin x + 1 cos 2 x 7. f ( x ) = 2 + cos x sin 2 x correct 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 . 003 10.0 points nunez (djn358) HW01 Radin (54915) 2 Find the derivative of f when f ( x ) = 4 x cos 3 x 8 sin 3 x . 1. f ( x ) = 12 x sin3 x 20 cos3 x correct 2. f ( x ) = 24 cos 3 x 20 x sin3 x 3. f ( x ) = 12 x sin 3 x 20 cos3 x 4. f ( x ) = 12 x sin3 x 24cos 3 x 5. f ( x ) = 24 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 sin3 x 24cos 3 x = 12 x sin 3 x 20 cos 3 x . 004 10.0 points Find f ( x ) when f ( x ) = 1 8 x x 2 . 1. f ( x ) = x 4 ( x 2 8 x ) 3 / 2 2. f ( x ) = x 4 (8 x x 2 ) 1 / 2 3. f ( x ) = x 4 (8 x x 2 ) 3 / 2 correct 4. f ( x ) = 4 x (8 x x 2 ) 3 / 2 5. f ( x ) = 4 x ( x 2 8 x ) 1 / 2 6. f ( x ) = 4 x ( x 2 8 x ) 3 / 2 Explanation: By the Chain Rule, f ( x ) = 1 2(8 x x 2 ) 3 / 2 (8 2 x ) ....
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This note was uploaded on 09/09/2011 for the course MATH 408L taught by Professor Gogolev during the Spring '09 term at University of Texas at Austin.

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HW01-solutions - nunez (djn358) HW01 Radin (54915) 1 This...

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