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

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MAT 137Y, 2004-2005, Solutions to Term Test 2 1. For the following, simplify your answers unless otherwise instructed. (8%) (i) Evaluate lim x 0 cos5 x - cos3 x x 2 . Let L be the limit above. Then by L’Hˆopital’s Rule, L H = lim x 0 - 5sin5 x + 3sin3 x 2 x H = lim x 0 - 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 0 f ( x + h ) - f ( x ) h = lim h 0 - ( x + h ) - - x h · - ( x + h )+ - x - ( x + h )+ - x = lim h 0 - ( x + h ) - ( - x ) h ( - ( x + h )+ - x ) = lim h 0 - h h ( - ( x + h )+ - x ) = lim h 0 - 1 - ( 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 Newton’s Method with x 1 = 2 to find the next approximation x 2 to the root of x 4 - 20 = 0. Applying Newton’s 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 . (12%) 2. A car leaves an intersection at noon and travels due south at a speed of 80 km/h. Another car has been heading due east at 60 km/h and reaches the same intersection at 1:00 p.m. At what time were the cars closest together? Verify that your answer is correct by showing that at that given time the distance is

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