sol2-04 - MAT 137Y, 2004-2005, Solutions to Term Test 2 1....

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Unformatted text preview: MAT 137Y, 2004-2005, Solutions to Term Test 2 1. For the following, simplify your answers unless otherwise instructed. (8%) (i) Evaluate lim x cos5 x- cos3 x x 2 . Let L be the limit above. Then by LHopitals Rule, L H = lim x - 5sin5 x + 3sin3 x 2 x H = lim x - 25cos5 x + 9cos3 x 2 =- 8 . (8%) (ii) Find the derivative of f ( x ) = - x , where x < 0, using the definition of derivative . f ( x ) = lim h f ( x + h )- f ( x ) h = lim h p- ( x + h )- - x h p- ( x + h ) + - x p- ( x + h ) + - x = lim h - ( x + h )- (- x ) h ( p- ( x + h ) + - x ) = lim h - h h ( p- ( x + h ) + - x ) = lim h - 1 p- ( x + h ) + - x =- 1 2 - x . (8%) (iii) Find the equation of the tangent line to the curve 2 ( x 2 + y 2 ) 2 = 25 ( x 2- y 2 ) at the point ( 3 , 1 ) . Implicitly differentiating gives us 4 ( x 2 + y 2 ) ( 2 x + 2 yy ) = 50 x- 50 yy . Sticking in x = 3, y = 1 gives 40 ( 6 + 2 y ) = 150- 50 y or 130 y =- 90, so y =- 9 13 . Hence the equation of the tangent line is y- 1 =- 9 13 ( x- 3 ) . (8%) (iv) Use Newtons Method with x 1 = 2 to find the next approximation x 2 to the root of x 4- 20 = 0. Applying Newtons Method to f ( x ) = x 4- 20, we have x 2 = x 1- f ( x 1 ) f ( x 1 ) = 2- 2 4- 20 4 2 3 = 2 + 4 32 = 17 8 ....
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This note was uploaded on 04/09/2010 for the course MAT 137 taught by Professor Uppal during the Spring '08 term at University of Toronto- Toronto.

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sol2-04 - MAT 137Y, 2004-2005, Solutions to Term Test 2 1....

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