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final-00

# final-00 - Faculty of Arts and Science University of...

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Faculty of Arts and Science University of Toronto MAT 137Y1Y Calculus! April/May Examinations; April 17, 2000 Time Alloted: 3 hours Instructors: G. Baumgartner, O. Calin, T. Haines, V. Jurdjevic, S. Lillywhite, R. Martinez No aids allowed. (9%) 1. Given the sketch of the function f below, indicate on a chart whether f , f , and f are positive, negative, or zero at the points x a , x b , x c , and x d . d a b c (8%) 2. Applying the ε , δ definition of limit, prove lim x 3 x 2 4 5 (7%) 3. Let f x 2sin x x π 2 A sin x B π 2 x π 2 cos x x π 2 Find the values of A and B such that f is continuous, or show that the values do not exist. 1

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4. Let f x sin x x x 0 1 x 0 (7%) (a) Use the definition of derivative to compute f 0 . (6%) (b) Show there exists c 0 π such that f c 1 π . (7%) 5. Let f be defined by the following graph. a 2a f If F x x 0 f t dt , draw the graph for F . ( f x 0 for all x a 2 a .)
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final-00 - Faculty of Arts and Science University of...

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