final-06

final-06 - Faculty of Arts and Science University of...

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Faculty of Arts and Science University of Toronto MAT 137Y1Y Calculus! April/May Examinations; May 4, 2006 Time Alloted: 3 hours Examiners: I. Alexandrova, M. Harada, V. Ivrii, G. Lynch, M. Sparykina, A. Savage 1. Evaluate the following expressions; simplify your answer whenever possible. (5%) (i) lim x p x 2 + 6 x - x . (5%) (ii) lim x 0 + ( cos x ) 1 / x 2 . (5%) (iii) Z sin x + sec x tan x dx . (5%) (iv) Z xe x dx . (5%) (v) Z x 2 1 - x 2 dx . (5%) (vi) d 2 dx 2 Z x 0 ± Z cos ( u 2 ) 1 p 3 + t 4 dt ! du . (8%) 2. Consider all rectangles ABCD such that the distance between A and the midpoint of BC is equal to 1. What is the maximum possible area of this rectangle? Be sure to verify that the area of this rectangle is indeed maximized. (7%) 3. Show that the equation 1 + 2 x + x 3 + 4 x 5 = 0 has exactly one real root. 4. Consider the function f ( x ) = 1 / ( 1 + e - x ) , which is deﬁned for all x . (4%)

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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-06 - Faculty of Arts and Science University of...

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