final-98 - (5 b Suppose a n 1 2 for all n Find the radius...

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Faculty of Arts and Science University of Toronto April/May Examinations MAT 137Y Calculus! Monday April 27, 1998 Time Alloted: 180 minutes Examiners: M.D. Choi, F. Recio, P. Rosenthal, F. Sottile (15%) 1. Sketch the graph of the function f x 3 x 5 5 x 3 ; clearly stating the following properties of the graph whenever appropriate: asymptotes, crit- ical points, local and global extrema, inflection points, intervals where the function is de- creasing or increasing, and intervals where the function is concave up or concave down. 2. Evaluate the following integrals. (8%) a. x x 1 x 2 dx (8%) b. e 1 x log x dx (8%) 3. Find the arclength of the curve f x 1 3 x 2 2 3 2 for x 1 3 . 4. Classify each of the following series as convergent or divergent and justify your answer. (7%) a. n 1 1 n 1 n 2 (7%) b. n 1 n ! 2 2 2 n ! (7%) c. n 2 1 n log n log n 2 5. Suppose f x is given as the power series n 0 a n x n with non-zero radius of convergence. (4%) a. Express f x as a power series.
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Unformatted text preview: (5%) b. Suppose a n 1 2 for all n . Find the radius of convergence for the power series of f x . 1 (6%) c. Using parts (a) and (b), compute the exact value of the sum ∞ ∑ n n 2 n (Recall that n 2 n n 1 2 n .) 6. Suppose f x M for all x a b , where M is some constant. (4%) a. Apply a theorem to show that f b f a M b a (5%) b. Use part (a) to show that log 66 log8 1 64 (8%) 7. Suppose ∞ ∑ n 1 a n and ∞ ∑ n 1 b n are two series that both converge, and a n b n 0 for all n . Must the series ∞ ∑ n 1 a n b n also converge? Justify your answer with an appropriate proof or coun-terexample. (8%) 8. Give the ε-N definition of the statement a n converges to L Prove, using the definition, that the sequence a n defined by a n n n 3 1 converges to zero. 2...
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final-98 - (5 b Suppose a n 1 2 for all n Find the radius...

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