HW13-solutions - white (taw933) HW13 ben-zvi (55600) 1 This...

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Unformatted text preview: white (taw933) HW13 ben-zvi (55600) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Determine which, if any, of f ( x ) = 256(16 3 x ) , g ( x ) = parenleftbigg 1 16 parenrightbigg 3 x 2 , h ( x ) = 256(4 6 x ) , define the same function. 1. none of f, g, or h 2. only g, f 3. f, g, and h correct 4. only f, h 5. only g, h Explanation: By the laws of Exponents f ( x ) = 256(16 3 x ) = 16 2 (16 3 x ) = 16 2 3 x , while g ( x ) = parenleftbigg 1 16 parenrightbigg 3 x 2 = 16 (3 x 2) = 16 2 3 x and h ( x ) = 256(4 6 x ) = 16 2 ( 4 2 ) 3 x = 16 2 (16 3 x ) = 16 2 3 x Consequently, all of f, g, and h define the same function. 002 10.0 points Determine if lim x parenleftbigg e 2 x + 2 e 2 x 3 e 2 x + 5 e 2 x parenrightbigg exists, and if it does, find its value. 1. limit = 2 5 correct 2. limit = 0 3. limit = 2 3 4. limit = 1 3 5. limit does not exist 6. limit = 1 5 Explanation: Since lim x e ax = , lim x e ax = 0 when a > 0, evaluating the limit directly gives lim x parenleftbigg e 2 x + 2 e 2 x 3 e 2 x + 5 e 2 x parenrightbigg = , which doesnt make any sense. (And we cant just cancel the s because infinities dont work like that.) So we try to get rid of terms that go to and leave terms that go to zero. To achieve this, multiply top and bottom by e 2 x . Then e 2 x e 2 x parenleftbigg e 2 x + 2 e 2 x 3 e 2 x + 5 e 2 x parenrightbigg = e 4 x + 2 3 e 4 x + 5 , and so lim x e 2 x + 2 e 2 x 3 e 2 x + 5 e 2 x = lim x e 4 x + 2 3 e 4 x + 5 . white (taw933) HW13 ben-zvi (55600) 2 But, as we noted, lim x e 4 x = 0 , so by properties of limits, lim x ( e 4 x + 2) = 2 , lim x (3 e 4 x + 5) = 5 . By properties of limits again, therefore, lim x parenleftbigg e 2 x + 2 e 2 x 3 e 2 x + 5 e 2 x parenrightbigg exists and the limit = 2 5 . keywords: exponential function, limit at in- finity, properties of limits, 003 10.0 points Find the derivative of f ( x ) = ( x 2 + x- 5) e x . 1. f ( x ) = (6- x + x 2 ) e x 2. f ( x ) = (6 + x- x 2 ) e x correct 3. f ( x ) = (6- x- x 2 ) e x 4. f ( x ) = (5 + x- x 2 ) e x 5. f ( x ) = (5- x + x 2 ) e x 6. f ( x ) = (5 + x + x 2 ) e x Explanation: By the Product Rule, f ( x ) = (2 x + 1) e x- ( x 2 + x- 5) e x ....
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HW13-solutions - white (taw933) HW13 ben-zvi (55600) 1 This...

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