RedAssign12[1].4 - The third step is to plug the results...

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Unit: Sequences and Series Module: Taylor and Maclaurin Polynomials Taylor Polynomials www.thinkwell.com info@thinkwell.com Copyright 2001, Thinkwell Corp. All Rights Reserved. 1601 –rev 06/15/2001 Higher-degree polynomial approximations result in more accurate representations. To find a Taylor polynomial , find all the necessary derivatives and evaluate them at x = c , then insert the values into the Taylor polynomial function. Higher-degree polynomial approximations can be represented using sigma notation ( Σ ). There is a mathematical convention that zero factorial equals one. These polynomial approximations are called Taylor polynomials . When deriving the Taylor polynomial for a given function, the first step is to find the derivatives of the function. The second step is to evaluate the derivatives at the point about which the approximation is centered.
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Unformatted text preview: The third step is to plug the results into the formula for the Taylor polynomial. When you derive the fourth Taylor polynomial, you get the lower polynomials for free. When you look at the graphs of the approximations you can tell that each one does a better job than the previous. The linear approximation was only good for values very close to the center x = 1. The quartic approximation improves on all the previous approximations, as it hugs the curve of the natural log function much better. Although you know the natural log of one is zero, you might not know the natural log of 1.2. By plugging in 1.2, you can use your Taylor polynomial to approximate the value of ln (1.2). The result agrees with the actual value of 0.18232155 to three significant figures....
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