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# 1011T3 - (3(c(4 Find g-1(3 where g x = h x 1 5(11 Determine...

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Department of Mathematics, University of Toronto Term Test 3 - March 16, 2011 MAT137Y, Calculus! Time Alloted: 1 hour 50 minutes 1. Evaluate the following integrals. (a) (7%) Z e 1 ln( x ) x 3 dx (b) (9%) Z 1 (1 - x 2 ) 5 / 2 dx (c) (8%) Z x 2 ( x - 1) 2 ( x + 1) dx 2. (a) (8%) Evaluate lim x →∞ cos 1 x x . (b) (8%) Find the equation for the tangent line for the curve y = (sec( x )) 1 /x at x = π 3 . 3. Determine if the improper integral converges or diverges. (a) (8%) Z 1 0 e x x dx . (b) (7%) Z e dx x ln( x ) 4. Suppose that h is a function such that h 0 ( x ) = sin 2 (sin( x + 1)) + 1 and h (0) = 3. (a) (4%) Show that h has an inverse. (b) (4%) Find ( h -
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Unformatted text preview: (3). (c) (4%) Find ( g-1 ) (3) where g ( x ) = h ( x + 1). 5. (11%) Determine the volume of the solid of revolution obtained by revolving the region bounded by y = 3 x-x 2 and y = x 2-3 x about the y-axis. 6. (10%) Find all diﬀerentiable functions f satisfying Z x f ( t ) dt = ( f ( x )) 2 + C for a ﬁxed constant C . 7. (12%) Cobalt-60 has a half-life of 5.3 years. If you have 100 grams of cobalt-60 now, how much was there three years ago?...
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