HW05.pdf - young(toy68 HW05 villafuerte altu(53780 This...

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young (toy68) – HW05 – villafuerte altu – (53780) 1 This print-out should have 26 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0points Let f be a continuous function on [ 3 , 1] such that f ( 3) = 1 , f (1) = 7 . Which of the following is a consequence of the Intermediate Value Theorem without further restrictions on f ? 1. f ( c ) = 1 for some c in ( 3 , 1) 2. f ( c ) = 0 for some c in ( 3 , 0) 3. 1 f ( x ) 7 for all x in ( 3 , 1) 4. f ( c ) = 1 for some c in ( 3 , 1) correct 5. f (0) = 0 Explanation: Because f is continuous on [ 3 , 1] the In- termediate Value Theorem ensures that for each M, 1 < M < 7 , there exists at least one choice of c in ( 3 , 1) for which f ( c ) = M . In particular, f ( c ) = 1 for some c in ( 3 , 1) . But without further restrictions on f , none of the other properties need hold. 002 10.0points Let f be a function such that lim h 0 f (1 + h ) = 2 , and lim h 0 f (1 + h ) f (1) h = 3 . Which of the following statements are true? A. f is differentiable at x = 1 , B. f (1) = 3 , f (1) = 2 , C. f has a removable discontinuity at x = 1 . 1. B only 2. A and B only 3. none are true 4. C only 5. A and C only 6. A only correct 7. all are true 8. B and C only Explanation: A. True: by definition. B. False: by definition, f is differentiable at x = 1 and f (1) = 3. C. False: f is continuous at x = 1. 003 10.0points If f ( x ) = 5 x 2 + 2 x , which of the following determines the deriva- tive of f ( x )? 1. lim h 0 parenleftbigg 5 x 2 + 2 x + h + x 2 x h parenrightbigg 2. lim h 0 5( x + h ) 2 + 2( x + h ) + 5 x 2 2 x h correct 3. 5( x + h ) 2 + 2( x + h ) 5 x 2 + 2 x h 4. lim x 0 parenleftbigg 5 x 2 + 2 x + hx 2 x h parenrightbigg 5. lim x 0 5( x + h ) 2 + 2( x + h ) + 5 x 2 2 x h
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young (toy68) – HW05 – villafuerte altu – (53780) 2 6. 5( x + h ) 2 + 2( x + h ) + 5 x 2 2 x h Explanation: When f ( x ) = 5 x 2 + 2 x then f ( x + h ) = 5( x + h ) 2 + 2( x + h ) so the derivative of f ( x ) is given by f ( x ) = lim h 0 f ( x + h ) f ( x ) h = lim h 0 5( x + h ) 2 + 2( x + h ) + 5 x 2 2 x h . 004 10.0points Find the value of f ( a ) when f ( x ) = 5 x + 6 . 1. f ( a ) = 5 5 a + 6 2. f ( a ) = 5 5 a + 6 2 3. f ( a ) = 5 5 a + 6 4. f ( a ) = 5 2 5 a + 6 correct 5. f ( a ) = 5 5 a + 6 6. f ( a ) = 5 2 5 a + 6 Explanation: By definition, f ( a ) = lim x a f ( x ) f ( a ) x a . Now, for the given f , f ( x ) = 5 x + 6 , while f ( a ) = 5 a + 6 . Thus f ( x ) f ( a ) = 5 x + 6 5 a + 6 , which after rationalization becomes (5 x + 6) (5 a + 6) 5 x + 6 + 5 a + 6 = 5( x a ) 5 x + 6 + 5 a + 6 . Consequently, f ( x ) f ( a ) x a = 5 5 x + 6 + 5 a + 6 , and so f ( a ) = 5 2 5 a + 6 . 005 10.0points If f is a function having 2 4 2 4 2 4 2 4 as its graph, which of the following is the graph of the derivative f of f ? 1. 2 4 2 4 2 4 2 4 correct
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