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Unformatted text preview: 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. a b c d (8%) 2. Applying the , definition of limit, prove x 3 (7%) 3. Let 2 Find the values of A and B such that f is continuous, or show that the values do not exist. 1 x 2 f x 2 sin x A sin x B cos x x lim x2 4 5 2 x 2 (7%) (a) Use the definition of derivative to compute f 0 . " ! (6%) (b) Show there exists c 0 such that f c (7%) 5. Let f be defined by the following graph. f a 0 % & 2a ! 6. Determine the following integrals. x2 ' % (6%) (ii) n 2 10. Evaluate the following sums. (5%) (i) n 1 0 nn 2 3 . 2 (5%) 9. Determine whether the series n2 0 2 n converges or diverges. n ' ) x log x, compute the Taylor polynomial P5 1 (6%) 8. For the function f x polynomial of degree 5 for f at a 1). ) ( f % &" (8%) 7. Find the derivative of the function F x % (6%) (i) dx a a2 ex dx. 1 e2x 0. x4 0 cos137 t dt. x (that is, the Taylor " If F x f t dt, draw the graph for F. ( f x 0 for all x x $ # $ # 4. Let f x x 0 sin x x 1 x 0 1 . a 2a .) n 1 0 3 ' (8%) 12. For what values of x does the series 1 n xn converge? n (8%) 11. Find the power series expansion at a 0 for the function f x 4 n 1 0 2 31 (4%) (ii) 5 5 4 5 n 1 . x2 e2x . ...
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This note was uploaded on 10/14/2010 for the course MAT MAT 137 taught by Professor Unknown during the Spring '10 term at Touro CA.

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