RedAssign12[1].3 - function is the tangent line...

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Unit: Sequences and Series Module: Polynomial Approximations of Elementary Functions Polynomial Approximation of Elementary Functions www.thinkwell.com info@thinkwell.com Copyright 2001, Thinkwell Corp. All Rights Reserved. Polynomials can approximate complicated functions. The tangent line approximation of f ( x ) at x = c is y = f ( c ) + f ΄ ( c )( x c ). Near x = c , the tangent line is a good approximation to the curve of f ( x ). However, for values further away from c the approximation is not so good. Although it seems very simple, sine is a difficult function to evaluate. Sure, you can evaluate the sine of special angles, but what is the sine of a number like four? Sine may be hard to evaluate, but polynomials are not. Polynomials can be evaluated with just addition and multiplication. And one of the simplest polynomials is the expression of a line. A simple, yet accurate method for approximating a
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Unformatted text preview: function is the tangent line approximation . You can use the point-slope form of a line to determine the formula for the tangent line. To find the tangent line approximation of a function f at a point c , you will need to know the slope of the line tangent to the curve at c and the value of f at c . Since you know the value of sine of zero, you can find the tangent line approximation of sine at zero. Just plug in the values of sine, its derivative, and the point zero. Now you can approximate other values of sine near zero. Inside this circle you can see that the line matches up well with the graph of sine. The further away you go from the origin, the worse the approximation becomes. But there are other polynomials besides linear ones. Maybe a quadratic polynomial, which describes a parabola, will fit the curve of sine better....
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This note was uploaded on 04/27/2010 for the course MATH 172 taught by Professor Hyon during the Spring '10 term at Community College of Denver.

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