Quest HW 4 - roberts (eer474) – Quest HW 4 – seckin –...

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Unformatted text preview: 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 → 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 → 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 roberts (eer474) – Quest HW 4 – seckin – (56425) 2 Let f be a function such that lim h → f (1 + h ) = 2 , and lim h → 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 → 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 → f (1 + h ) − f (1) h = f ′ (1) because 1 , 1 + h < 3 for all small h . Conse- quently, limit = 7 ....
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This note was uploaded on 11/08/2010 for the course MATH 408K taught by Professor Gualdani during the Spring '09 term at University of Texas at Austin.

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Quest HW 4 - roberts (eer474) – Quest HW 4 – seckin –...

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