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Unformatted text preview: Faculty of Arts and Science University of Toronto MAT 137Y1Y Calculus! April/May Examinations; May 2, 2001 Time Alloted: 3 hours Instructors: A. del Junco, A. Igelfeld, J. Lansky, G. Maschler, A. Tourin, S. Uppal No aids allowed. 1. Evaluate the following integrals. (8%) (a) (8%) (b) (10%) 2. Find the volume of the solid generated by revolving the region under the curve y about the x-axis. (8%) 3. Prove for all natural numbers n: x 1 xx 2 i 1 (10%) 4. Sand in an hourglass falls at a rate of 2 cm3 min. The sand gathers in the shape of a cone whose base radius is twice as large as its height. (Recall that the volume of a cone of base 1 2 radius r and height h is V 3 r h.) How fast is the area of the base changing when the radius of the base is 2 cm? (4%) (b) Show that the equation p x x6 5x4 15x2 1 0 has exactly two solutions. (If p x had three or more solutions, what can you say about p x ?) 1 % (3%) (a) Show that f x is differentiable at x evaluate the limit.) ! " 6. Let f x $ e 0 1 x2 x x 0 0 0 and find f 0 . (Hint: Use a substitution to (4%) (a) Show that there are at least two solutions to the equation p x 5. Consider the polynomial p x x6 5x4 15x2 1. 0. i n i! n 1! 1 # x 2x 1 1 5 x4 ln x dx. 1 3 dx. 1 x 4 (3%) (b) Find all asymptotes of f x . (4%) (c) Using the result from part (b) and the Taylor polynomials for e x and cos x, evaluate n 0 (3%) (c) You should recognize the power series for the function g x x as the derivative of a power series which you are familiar with. Use this to express g x as a rational function. (3%) (d) Using the result of part (c), express f x as a rational function. 1 2 22 22 32 23 42 24 9. . x a (3%) (b) Suppose S is a bounded, non-empty set of real numbers. Define what it means for a number M to be a least upper bound of S. (2%) (c) Suppose f x is defined on 0 1 and f x is increasing and bounded on 0 1 . Show that lim f x lub f x : x 0 1 . x 1 2 (3%) (a) Define the statement lim f x L. (3%) (e) Find the sum of the series . n 0 0 n 0 (3%) (b) Letting f x n 1 2 xn , show that g x f t dt n 1 xn n 0 x (3%) (a) Find the radius of convergence of the power series 8. Consider the power series n 1 2 xn 1 22 x 3 2 x2 x 0 ex 2 1 x2 x4 2 . 1 2 xn . n lim 1 x2 1 x2 2 1 cos x (2%) (b) Find the Taylor polynomial at 0 of degree 4 for g x (4%) (a) Find the Taylor polynomial at 0 of degree 2 for f x . 7. Consider the function f x 1 x. ! (4%) (d) Sketch the graph of f x $ e 0 1 x2 x x 0 0 1 x2 . 1 ! (5%) (c) Find all relative extrema and inflection points of f x $ e 0 # # 1 x2 x x 0 0 . ...
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