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Quest HW 4 - roberts(eer474 Quest HW 4 seckin(56425 This...

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roberts (eer474) – Quest HW 4 – seckin – (56425) 1 This print-out should have 17 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points What is the significance of the expression f (1 + h ) f (1) h in the following graph of f when h = 7 2 ? 1 2 3 4 5 1 2 3 4 5 P Q R S T U 1. slope of line through P and U 2. slope of line through P and R 3. equation of line through P and T 4. length of line segment PR 5. equation of line through P and R 6. slope of tangent line at P 7. length of line segment PT 8. slope of line through P and T correct 9. equation of line through P and U 10. length of line segment PU Explanation: When h = 7 2 the expression f (1 + h ) f (1) h is the ratio of the rise and the run between the points P and T . Thus the expression is the slope of line through P and T . 002 10.0 points If f is a differentiable function, then f ( a ) is given by which of the following? I. lim h 0 f ( a + h ) f ( a ) h II. lim x a f ( x ) f ( a ) x a III. lim x a f ( x + h ) f ( x ) h 1. I, II, and III 2. I only 3. II only 4. I and III only 5. I and II only correct Explanation: Both of f ( a ) = lim h 0 f ( a + h ) f ( a ) h and f ( a ) = lim x a f ( x ) f ( a ) x a are valid definitions of f ( a ). By contrast, lim x a f ( x + h ) f ( x ) h = f ( a + h ) f ( a ) h because f is continuous. Consequently, f ( a ) is given only by I and II . 003 10.0 points
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roberts (eer474) – Quest HW 4 – seckin – (56425) 2 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 (1) = 2 , f (1) = 3 , B. f is continuous at x = 1 , C. f is differentiable at x = 1 . 1. all are true correct 2. none are true 3. A and B only 4. A and C only 5. C only 6. A only 7. B and C only 8. B only Explanation: A. True: by definition, f is differentiable at x = 1, hence also continuous at x = 1. B. True: f is differentiable at x = 1, so also continuous at x = 1. C. True: by definition. 004 10.0 points Let f be the function defined by f ( x ) = 7 x ( x 3 + | x 3 | ) 2 . Determine if lim h 0 f (1 + h ) f (1) h exists, and if it does, find its value. 1. limit doesn’t exist 2. limit = 9 3. limit = 8 4. limit = 7 correct 5. limit = 10 6. limit = 11 Explanation: Since f ( x ) = braceleftbigg 7 x, x < 3, 7 x 4( x 3) 2 , x 3, we see that lim h 0 f (1 + h ) f (1) h = f (1) because 1 , 1 + h < 3 for all small h . Conse- quently, limit = 7 . 005 10.0 points If f is a function having 2 4 6 2 4 2 4 6 8 10 2 as its graph, which of the following could be the graph of f ?
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