final-04

# final-04 - (3(b Using part(a ﬁnd a power series expansion...

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Faculty of Arts and Science University of Toronto MAT 137Y1Y Calculus! April/May Examinations; April 29, 2004 Time Alloted: 3 hours Instructors: V. Blomer, K. Consani, M. Harada, G. Leuschke, D. Miller, M. Pinsonnault, P. Rosenthal, S. Uppal, R. Wendt (10%) 1. Find the volume generated by rotating the region bounded by the curves y = x 2 , y = 0 , x = 2 , x = 3 about the line x = 2. 2. Evaluate the following expressions. (7%) (i) Z x 3 1 - x 2 dx . (7%) (ii) Z x arctan xdx . (6%) (iii) lim x π 4 - ( tan x ) tan2 x . (10%) 3. Find the points on the ellipse 4 x 2 + y 2 = 4 that are farthest away from the point ( 1 , 0 ) . Make sure to verify that the distance is maximized. 4. Evaluate the following sums. (5%) (i) n = 1 1 n ( n + 3 ) . (5%) (ii) 1 - 2 + ( 2 ) 2 2! - ( 2 ) 3 3! + ( 2 ) 4 4! -··· . 5. Determine the convergence or divergence of each of the following series. (5%) (i) k = 2 1 k ( ln k ) 3 / 2 . (5%) (ii) k = 0 k ! 3 · 7 · 11 · 15 ··· ( 4 k + 3 ) . 6. Let f ( x ) = 1 1 + x . (5%) (a) By integrating the power series of f ( x ) term-by-term, ﬁnd a power series expansion in powers of x for the function g ( x ) = ln ( 1 + x ) ; be sure to include the general term.

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Unformatted text preview: (3%) (b) Using part (a), ﬁnd a power series expansion for the function h ( x ) = x ln ( 1 + x 2 ) . 1 (2%) (c) Find the radius of convergence for the power series obtained in part (b). (4%) (d) Find h ( 999 ) ( ) . (8%) 7. Suppose f is an odd function and is differentiable everywhere. Prove that for every positive number b , there exists a number c ∈ (-b , b ) such that f ( c ) = f ( b ) / b . (8%) 8. Show that f deﬁned by f ( x ) = ± e-1 | x | , x 6 = , , x = is differentiable at 0. 9. (5%) (a) Find the Taylor polynomial P 8 ( x ) for the function f ( x ) = cos x . (5%) (b) It follows that P 8 ( 1 ) = 4357 8064 estimates cos1 to within some error. Give the best estimate of the difference between cos1 and P 8 ( 1 ) that can be obtained using the formula for R 8 ( x ) given in class. 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-04 - (3(b Using part(a ﬁnd a power series expansion...

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