test3-04 - (8%) 4. Find d 2 dx 2 Z x Z sin t 1 e u 2 du dt...

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Department of Mathematics, University of Toronto Term Test 3 – March 10, 2004 MAT 137Y, Calculus! Time Alloted: 1 hour 50 minutes Examiners: V. Blomer, K. Consani, M. Harada, G. Leuschke, D. Miller, M. Pinsonnault, P. Rosenthal, S. Uppal, R. Wendt 1. Compute the following integrals. (10%) (i) Z 5 2 + x dx . (10%) (ii) Z x 3 ( ln x ) 2 dx . (10%) (iii) Z dx x 2 x 2 + 16 . (10%) (iv) Z 4 x ( x - 1 ) 2 ( x + 1 ) dx . 2. Compute the following limits. (8%) (i) lim x 1 + ( ln x ) tan ± π x 2 ² . (8%) (ii) lim x ( e x + x ) 1 / x . (12%) 3. A hole of radius r is drilled through the center of a sphere of radius R (where R > r ). Find the volume of the remaining portion of the sphere.
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Unformatted text preview: (8%) 4. Find d 2 dx 2 Z x Z sin t 1 e u 2 du dt . 5. Suppose f ( x ) = 2 x + cos x . (6%) (a) Show that f ( x ) is one-to-one. (6%) (b) Find ( f-1 ) ( 1 ) . 6. Suppose f is continuous on [ a , b ] . (8%) (a) Prove there exists c [ a , b ] such that Z c a f ( t ) dt = Z b c f ( t ) dt . (4%) (b) Must there exist c ( a , b ) such that Z c a f ( t ) dt = Z b c f ( t ) dt ? Justify your answer with an appropriate proof or counterexample. 1...
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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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