final-96

# final-96 - 6(12 a Determine with justiﬁcation whether...

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Faculty of Arts and Science University of Toronto April/May Examinations MAT 137Y Calculus! Monday May 6, 1996 Time Alloted: 180 minutes (12) 1. What is the largest area of a rectangle having two vertices on a semicircle of radius 10 and one side along the diameter? (15) 2. Make a careful sketch of the function y log 8 x x 2 , showing whenever possible all in- tercepts, asymptotes, relative and absolute maxima and minima, inﬂection points, regions where the function is increasing and decreasing, and regions of concavity. (12) 3. Evaluate the following integrals. a. x 2 9 x dx b. 1 x 1 2 x 1 4 dx (10) 4. Give an ε δ proof using limits to show that f x x 3 x 5 is continuous at x 3. (10) 5. Find the volume of the solid obtained by revolving the area under y sin x for x 2 π 3 π about the y -axis.

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Unformatted text preview: 6. (12) a. Determine, with justiﬁcation, whether each of the following series converges. If any of the convergent series contains negative terms, distinguish between absolute and condi-tional convergence. i) ∞ ∑ n 2 1 n log n ii) ∞ ∑ n 1 1 n 1 n 1 n 2 iii) ∞ ∑ n 1 sin 1 n 2 1 (6) b. Suppose ∞ ∑ n 1 a n converges, where a n 0 for all n . Does ∞ ∑ n 1 a 2 n converge? Justify your answer or give a suitable counterexample. (15) 7. a. Express the function f x x t 2 sin t dt as a power series about a 0. b. What is the radius of convergence of the series ∞ ∑ n 1 nx n n ! 2 n ? (8) 8. Suppose x 3 f t dt x cos π x 6 . Compute f 8 . 2...
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## This note was uploaded on 10/14/2010 for the course MAT MAT taught by Professor Unknown during the Spring '10 term at Touro CA.

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final-96 - 6(12 a Determine with justiﬁcation whether...

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