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final05fall - Final Examination MATH 262 1 Find the...

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Final Examination MATH 262 December 12, 2005. 1. Find the interval of convergence of the series X n =0 ( x - 2) n 2 2 n + 1 and determine whether the series converges absolutely or conditionally at the endpoints. 2. (a) Find the Maclaurin expansion for the function F ( x ) = Z x 0 sin( t 2 ) dt and compute F (1) to 2 decimal places. (b) Find the Taylor series expansion of f ( x ) = x - 1 1 + ( x - 1) 2 about x = 1 and use this to compute f (9) (1) and f (10) (1). 3. Find the solution of the differential equation (1 - x 2 ) y 00 - xy 0 + 9 y = 0 satisfying y (0) = 0, y 0 (0) = 1. 4. Given the curve r = t 2 i + t 2 j + t 3 k , (a) find the arc length of the curve from t = 0 to t = 1; (b) find the unit tangent vector ˆ T , principal normal ˆ N , and binormal ˆ B of the curve at the point t = 1; (c) find the curvature κ and torsion τ of the curve at point t = 1. 1
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Final Examination MATH 262 December 12, 2005. 5. Show that the function f ( x, y ) = ( ax + by 2 x 2 + y 2 if ( x, y ) 6 = (0 , 0) , 0 if ( x, y ) = (0 , 0) is continuous at (0 , 0) if a = 0 but is not continuous at (0 , 0) if a 6 = 0.
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