written assignment 4

written assignment 4 - a b and x we can finally plug these...

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JP Rodriguez Prof. Muralee Calc 21-410 6 November 2007 Written Assignment #4 A) For values of x near a = 0, finda quadratic polynomial, p ( x ) = Ax 2 + Bx + C , that is a quadratic approximation for the function f ( x ) = e - x^2 Since we know that f ( a ) needs to equal p ( a ), as well as the first and second derivatives of each needing to be equal to each other, we need to find the derivatives of the functions f ( x ) = e - x^2 p ( x ) = Ax 2 + Bx + C f’ ( x ) = -2 xe -x^2 p’ ( x ) = 2 Ax + B f’’ ( x ) = e - x^2 (2 x 2 -1) p’’ ( x ) = 2 A We start by working backwards from the second derivative of each function. We will set them equal to one another and plug in 0 for x . 2 a = e - x^2 (2 x 2 -1) 2 a = 2 e -(0)^2 (2(0) 2 -1) 2 a = 2(1)(-1) 2 a = -2 a = -1 Now that we have a value for a , we can now set the first derivative of both functions equal to one another, with a = -1 and x = 0. 2 ax + b = -2 xe -x^2 2(-1)(0) + b = -2(0) e -(0)^2 0 + b = 0 b = 0 Now that we have values for
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Unformatted text preview: a, b, and x, we can finally plug these three values into f ( a ) and p ( a ) and set both functions equal to one another. ax 2 + bx + c = e-x^2 (-1)(0) 2 + (0)(0) + c = e-(0)^2 0 + 0 + c = 1 c = 1 Now that we have values for a, b, and c, we can plug it into p ( x ) to get: p ( x ) = ax 2 + bx + c p ( x ) = (-1) x 2 + (0) x + 1 p ( x ) = -x 2 + 1 And that is our quadratic approximation for f ( x ). B) On separate sheet C) Use the function p to approximate the value of f at x = ½ and x = 2 p ( 1/2 ) = -(1/2) 2 + 1 = (1/4) + 1 = 1.25 p ( 2 ) = -(2) 2 + 1 = -4 + 1 = -3 D) For values of a near x = 0, p ( x ) is a good approximation of f ( x ), but as values for a move farther from 0 (in either direction), p ( x ) becomes a progressively worse approximation...
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This note was uploaded on 04/13/2008 for the course MATH 021 taught by Professor Muralee during the Spring '08 term at Lehigh University .

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written assignment 4 - a b and x we can finally plug these...

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