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Fall 2008

# Fall 2008 - M ath 2 62 F all 2 008 Dec 1 7,9:00am 1 2:00pm...

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Math 262 / Fall 2008 Dec. 17, 9:00am - 12:00pm Problem 1 (10 marks). Find the interval of convergence, including the end points, of the foliowing power series: ,.,,, Efr{,-,)" * ^3n rn \-' \ " . / , / t - , 4^ +tL 1 Problem 2 (10 marks). Consider the function /(r) Use a series to approximate the value of /(0.t) with error at most 10-5. S,/Problem 3 (10 marks). Find the recunence relation for the solutions to the dif- ferential equation (7 - ,)'u" *'y = 0 abor.rt the origin. Give the first three l:tolr-zero terrns of two linearly independent solutioris to tlie equa|iotr' Lr/ probnem 4 (10 marks). Cornplete both of the foliowing problenrs: (a) Find the length of the curve r(t) :6t i +3* j+t3 k from f :0 to t:I. (b) Compute the curvature function n.(t) for the curve r(t) : (t + cost)i + (t - cost)j + r4sint k, ' Problem 5 (10 marks). Consider the following pair of equations: t u * Y u * Y 2 n u : 3 'u2 +'u' - r'y *'u'ux : 2 (a) Show that the aboveecluations inplicitly define functions 'A : Y(x,u) ancl ,u : V(r,u) near the point (1,1,1,1). (Write a sentence
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