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test2-0708 - Department of Mathematics University of...

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Department of Mathematics, University of Toronto Term Test 2 – January 16, 2008 MAT 137Y, Calculus! Time Alloted: 1 hour 50 minutes Examiners: G. Amir, B. Khesin, R. Stanczak, B. Stephens, J. Sylvestre, S. Uppal (15%) 1. A farmer with 1500 feet of fencing wants to enclose a rectangular area and then divide it into four pens (fenced enclosures) with fencing parallel to one side of the rectangle. Find the largest possible total area of the four pens; be sure to verify that the area is indeed maximized. 2. (10%) (i) Find the values of a and b such that lim x 0 sin2 x + ax 3 + bx x 3 = 0. (10%) (ii) The length of a rectangle is increasing at a rate of 7 cm/sec and the width is decreasing at 2 cm/sec. When the length is 20 cm and the width is 6 cm, determine whether the area of the rectangle is increasing or decreasing, and find the rate at which the area is changing. 3. Let f ( x ) = 2 x - x . (5%) (a) Find the domain of f ( x ) and, if any, the intercepts and asymptotes. (5%) (b) Find the intervals for which f is increasing and decreasing, and identify all local ex-
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