Math1011_Prelim1B - Math1011 Fall 2009 Prelim I-B1 Please...

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Math1011 Fall 2009 Prelim I-B1 Please answer the following questions about the function f(x) whose graph is shown in the figure below. Is f(x) a one-to-one function? Why or why not? On what intervals between -5 and 5 is f(x) increasing? For which values of c is lim ( ) 1? x c f x = − For which values of c is lim ( ) lim ( )? x c x c f x f x + At which values c between -5 and 5 is f(x) continuous?
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Math1011 Fall 2009 Prelim B1 Please answer the following questions about the function f(x) whose graph is shown in the figure below. Is f(x) a one-to-one function? Why or why not? A function is one-to-one if f(a) f(b) for a b. f(x) is NOT one-to-one by counter example, as f(0) = f(4) = -1. We could also say f(x) fails the horizontal line test for y = 2. On what intervals between -5 and 5 is f(x) increasing? f(x) is increasing on [-5,-1] and [2, 4). For which values of c is lim ( ) 1? x c f x = − 4 c=0. Note: lim ( ) so 4 is NOT a correct answer. x f x = ∞ For which values of c is lim ( ) lim ( )? x c x c f x f x + c=-1, c = 2 and one could argue c = 4 as well. At which values c between -5 and 5 is f(x) continuous? f(x) is continuous on [-5,-1), (-1,2), (2,4), and (4,5]. (Doc #1011p.25.03)
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Math1011 Spring 2005 Prelim B2 Let f be the function graphed below: Is f continuous at 3? Justify your answer using the definition of continuity. At what numbers on is f discontinuous? x 4+ What is lim ( )? f x x 4- What is lim ( )? f x x 4 What is lim ( )? f x Note: stands for the set of Real Numbers. How would the answers differ if asked on the domain of f?
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Math1011 Spring 2005 Prelim B2 Let f be the function graphed below: Is f continuous at 3? Justify your answer using the definition of continuity. 3 3 Definition of Continuity: lim ( ) ( ) lim ( ) 0 but (3) 1 As lim ( ) (3), f(x) is not continuous at x = 3. x a x x f x f a f x f f x f = = = − At what numbers on is f discontinuous? x = 1 (jump discontinuity) x = 3 (removable discontinuity) x = 4 (infinite discontinuity) x 4+ What is lim ( )? f x - x 4- What is lim ( )? f x x 4 What is lim ( )? f x DNE (left sided limit does not equal right sided limit) How would the answers differ if asked on the domain of f? 4 is not on the domain of f and thus would not be included. (Doc #011p.14.01A)
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Math1011 Spring 2002 Prelim B3 Draw the graph of a function y = f(x) such that the following conditions are satisfied: At x = 1: f (1) = 0 x 1 lim ( ) 2 f x → + = x 1 lim ( ) does not exist f x At x = 5: x 5 lim ( ) 3 f x = f is not continuous at x = 5 At x = 9: slope of tangent line is 1 at x = 9 -4 -2 0 2 4 6 8 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12
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Math1011 Spring 2002 Prelim B3 Draw the graph of a function y = f(x) such that the following conditions are satisfied: At x = 1: f (1) = 0 x 1 lim ( ) 2 f x → + = x 1 lim ( ) does not exist f x At x = 5: x 5 lim ( ) 3 f x = f is not continuous at x = 5 At x = 9: slope of tangent line is 1 at x = 9 (Doc #011p.26.03t)
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Math1011 Fall 2002/Fall 2006 Prelim B4 5 3 Prove that there is a positive number c such that c 2 10 c c + + = 2 If ( ) 10sin , show that there is a number c such that ( ) 1000. Give complete reasons for your answer.
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