final-97

# final-97 - ∑ n 1 x n n 1 6(8 a Find the continuous...

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Faculty of Arts and Science University of Toronto April/May Examinations MAT 137Y Calculus! Thursday May 1, 1997 Time Alloted: 180 minutes Examiners: E. Hironaka, A. Hwang, B. Khesin, P. Rosenthal (12) 1. An aluminum can – in the form of a cylinder – is to contain 216 cm 3 of soup. Determine the radius of the base in order to minimize the amount of aluminum. (In other words, the total surface area of the side, top, and bottom is to be minimized.) Verify your answer. 2. Evaluate the following integrals. (9) a. x 3 x 2 π x dx (10) b. x 3 3 x 2 5 x 6 x 2 2 x 5 dx (10) 3. Find the volume of the solid obtained by revolving the region under the curve y sin x x for x 0 1 for x 0 for x 0 π 2 about the y -axis. 4. Classify each of the following series as convergent or divergent and justify your answer. (5) (a) n 2 1 n 2 3 (5) (b) n 1 1 n n n 1 (5) (c) n 1 1 n 1 1 n n 5. Find the radius of convergence of each of the following power series: (8) a. n 1 n n x n (8) b.

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Unformatted text preview: ∑ n 1 x n ! n 1 6. (8) a. Find the continuous function f which satisﬁes 2 x 3 f t dt 2 x 2 4 x 3 2 (2) b. Suppose g x is a continuous function such that 2 x 3 g t dt 2 x 2 4 x C for some constant C . Show that C 3 2 . (10) 7. Suppose ∞ ∑ n 1 a n and ∞ ∑ n 1 b n are two series that both converge absolutely . Must the series ∞ ∑ n 1 a n b n also converge absolutely ? Show that your answer is correct. (8) 8. Answer EITHER part (a) or (b), but NOT BOTH. (a) Suppose that f is continuous and non-negative on a b and that f x 0 for some x a b . Show that b a f 0. (b) Give the formal ( ε N ) deﬁnition of the statement: the sequence a n converges to L By applying the deﬁnition, give a rigourous proof that the sequence 3 n 2 converges to zero. 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-97 - ∑ n 1 x n n 1 6(8 a Find the continuous...

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