# 01spex3 - Math 222 Exam III Answers I(30 points formulas...

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Math 222, Exam III, March 30, 2001 Answers I. (30 points.) Complete the following paragraph by writing in the correct formulas. The Taylor series of a function f ( x ) about the point a is the infinite series X k =0 f ( k ) ( a )( x - a ) k k ! It may be used to compute f ( x ) when x is sufficiently close to a . The n th degree Taylor polynomial of f ( x ) about the point a is the polynomial f n ( x ) of degree n which best approximates f ( x ) for x near a . It is f n ( x ) = n X k =0 f ( k ) ( a )( x - a ) k k ! The n th remainder (error) is defined by f ( x ) = f n ( x ) + R n ( x, a ) . Three formulas for the remainder are R n ( x, a ) = X k = n +1 f ( k ) ( a )( x - a ) k k ! R n ( x, a ) = Z x a ( x - t ) n n ! f ( n +1) ( t ) dt and R n ( x, a ) = f ( n +1) ( c )( x - a ) n +1 ( n + 1)! for some number c with a < c < x or x < c < a . The last is called Lagrange’s formula for the remainder. 1

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II. (30 points.) (1) Find all solutions of dy dx + 2 y = 0 . Answer: By separation of variables dy y = - 2 dx so ln y = - 2 x + C so y = y 0 e - 2 x where y 0 = e C . (2) Find all solutions of dy dx + 2 y = e - x .
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