test2-05 - Department of Mathematics, University of Toronto...

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Department of Mathematics, University of Toronto Term Test 2 – January 18, 2006 MAT 137Y, Calculus! Time Alloted: 1 hour 50 minutes Examiners: I. Alexandrova, M. Harada, V. Ivrii, G. Lynch, M. Saprykina, A. Savage 1. (10%) (a) Find a constant k for which lim x 0 sin x - x - kx 3 x 5 exists and the limit is also non-zero, and determine the value of the limit. (10%) (b) Find the equation of all tangent lines to the curve y = x 2 that intersect the point ( 1 4 , - 3 2 ) . (12%) 2. A poster is to have an area of 180 square inches with 1 inch margins at the bottom and sides and a 2 inch margin at the top. What dimensions will give the largest printed area? Make sure to verify that your answer yields a maximum. 3. Consider the function f ( x ) = x x 2 + 9 . (4%) (a) Show that f 00 ( x ) = 2 x ( x 2 - 27 ) ( x 2 + 9 ) 3 . (4%) (b) Find the domain, all intercepts, and asymptotes. (6%) (c) Locate all critical points of f , determine and clearly indicate the intervals for which f
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This note was uploaded on 04/09/2010 for the course MAT 137 taught by Professor Uppal during the Spring '08 term at University of Toronto- Toronto.

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test2-05 - Department of Mathematics, University of Toronto...

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