0910&0809T3 - Department of Mathematics, University of...

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Unformatted text preview: Department of Mathematics, University of Toronto Term Test 3 - March 2010 MAT137Y, Calculus! Time Alloted: 1 hour 50 minutes 1. Evaluate the following integrals. (i) (8%) Z dx x (ln x ) 5 / 2 (ii) (8%) Find Z 1 arctan( x ) dx (iii) (10%) Z dx (2 + x 2 ) 2 (iv) (10%) Z 1- x + 2 x 2- x 3 x ( x 2 + 1) 2 dx 2. (i) (10%) Find the equation of the tangent line for the curve f ( x ) = (tan x ) 1 /x at x = 4 . (ii) (10%) Evaluate F ( x ) if F ( x ) = Z R x 1 e t 2 dt 3 1 1 + sin 6 t + t 2 dt 3. Suppose h is a function such that h ( x ) = e sin ( x +1) and h (0) = 3. (a) (5%) Show that h has an inverse. (b) (5%) Find ( h- 1 ) (3). (c) (5%) Find ( g- 1 ) (3), where g ( x ) = h ( x + 1). 4. (14%) An ellipsoid is obtained when the ellipse consisting of all points ( x,y ) with x 2 a 2 + y 2 b 2 = 1 is rotated around the x-axis. Find the volume of the ellipsoid and simplify your answer. 5. (i) (8%) Prove that arctan x + arctan ( 1 x ) = 2 for any value of x > 0....
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0910&0809T3 - Department of Mathematics, University of...

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