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test3-04 - f x = π cos 2 1-e x f = 1(5(a Show that f has...

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Department of Mathematics, University of Toronto Term Test 3 – March 9, 2005 MAT 137Y, Calculus! Time Alloted: 1 hour 50 minutes Examiners: P. Blue, M. Branker, D. Cheliotis, N. Derzko, G. Karali, A. Igelfeld, F. Latremoliere, S. Uppal 1. Evaluate the following integrals. (10%) (i) sec 2 x tan x + 1 dx . (10%) (ii) 9 - x 2 dx . (10%) (iii) 2 x 2 - x + 4 x ( x 2 + 4 ) dx . 2. (10%) (i) Find the volume generated by rotating the region bounded by the curves y = e x , y = e - x , x = 1 about the y -axis. (10%) (ii) Sketch the region bounded by the curves y = x 2 , y = 2 - x , y = - x , and find the area of the region. 3. Evaluate the following limits. (8%) (i) lim x 0 3 x - 1 2 x - 1 . (8%) (ii) lim x e + ( ln x ) 1 / ( x - e ) . 4. Suppose
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Unformatted text preview: f ( x ) = π + cos 2 ( 1-e x ) , f ( ) = 1 . (5%) (a) Show that f has an inverse. (7%) (b) Find ( f-1 ) ( 1 ) . 1 5. (8%) (a) Find all continuous functions f ( t ) which satisfy Z x 2 f ( t ) dt = e 2 x 2-1; and verify your answer; otherwise show that such a function does not exist. (4%) (b) Does there exist a continuous function f which satisfies Z x f ( t ) dt = e x ? (10%) 6. Show for all real numbers x and y that if x 6 = y , then | tan-1 x-tan-1 y | < | x-y | . 2...
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