Lab 1: Tayloring e - How does your calculator compute TDEC 180 (Computation Lab III) Spring 2005-06 Jeremy Johnson Name: Partner: Section: The purpose of this lab is to illustrate how to use a polynomial approximation to estimate the value of a function. A crucial part of the approximation is providing a bound on the error of the approximation. The tools presented in this lab are commonly used to provide approximations to functions such as sin(x), cos(x), log(x), and exp(x). If we know the value of a function at a point and the values of all of its derivatives at that point, then it is possible to construct a polynomial of degree n that has the same value at the point and the same values for its first n derivatives at that point. Moreover, this polynomial can be used to approximate the function near the point. Such a polynomial is called a Taylor polynomial and Taylor's theorem (we will use a version called Lagrange's bound) tells us the error when using this polynomial to approximate values of the function. In this lab, you will use Taylor polynomials, and the corresponding error bound, to approximate e=exp(1) to n digits.

The Taylor series, about x= 0, (Maclaurin Series) for 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this series is truncated, the 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-th Maclaurin polynomial is obtained: 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theorem says that the truncation error, 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some number 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 all 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then 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. See sections 10.7 and 10.8 of the text for further discussion on Taylor polynomials and Taylor's theorem. The following example shows how to use Maple to compute the Taylor series for sin(x) diff(sin(x),x);

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

diff(sin(x),x,x);

NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiQtSSNtb0dGJTYzUSomdW1pbnVzMDtGKC8lJWZvcm1HUSdwcmVmaXhGKC8lJmZlbmNlR1EmZmFsc2VGKC8lKnNlcGFyYXRvckdGNS8lJ2xzcGFjZUdRJDBlbUYoLyUncnNwYWNlR0Y6LyUpc3RyZXRjaHlHRjUvJSpzeW1tZXRyaWNHRjUvJShtYXhzaXplR1EpaW5maW5pdHlGKC8lKG1pbnNpemVHUSIxRigvJShsYXJnZW9wR0Y1LyUubW92YWJsZWxpbWl0c0dGNS8lJ2FjY2VudEdGNS8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYoLyUlc2l6ZUdRIzEyRigvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRigvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYoLUYkNiUtSSNtaUdGJTY5USRzaW5GKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdGUi8lJWJvbGRHRjUvJSdpdGFsaWNHRjUvJSp1bmRlcmxpbmVHRjUvJSpzdWJzY3JpcHRHRjUvJSxzdXBlcnNjcmlwdEdGNS8lK2ZvcmVncm91bmRHRlUvJStiYWNrZ3JvdW5kR0ZYLyUnb3BhcXVlR0Y1LyUrZXhlY3V0YWJsZUdGNS8lKXJlYWRvbmx5R1EldHJ1ZUYoLyUpY29tcG9zZWRHRjUvJSpjb252ZXJ0ZWRHRjUvJStpbXNlbGVjdGVkR0Y1LyUscGxhY2Vob2xkZXJHRjUvJTBmb250X3N0eWxlX25hbWVHRk8vJSptYXRoY29sb3JHRlUvJS9tYXRoYmFja2dyb3VuZEdGWC8lK2ZvbnRmYW1pbHlHRltvLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGKC8lKW1hdGhzaXplR0ZSLUYtNjNRMCZBcHBseUZ1bmN0aW9uO0YoL0YxUSZpbmZpeEYoRjNGNkY4RjtGPUY/RkFGREZHRklGS0ZNRlBGU0ZWLUYkNiUtRi02M1EiKEYoRjAvRjRGYnBGNi9GOVEudGhpbm1hdGhzcGFjZUYoL0Y8RmRyL0Y+RmJwRj9GQUZERkdGSUZLRk1GUEZTRlYtRiQ2Iy1GZm42OVEieEYoRmluRlxvRl5vL0Zhb0ZicEZib0Zkb0Zmb0Zob0Zqb0ZccEZecEZgcEZjcEZlcEZncEZpcEZbcUZdcUZfcUZhcS9GZHFRJ2l0YWxpY0YoRmZxLUYtNjNRIilGKC9GMVEocG9zdGZpeEYoRmJyRjZGY3IvRjxRMnZlcnl0aGlubWF0aHNwYWNlRihGZnJGP0ZBRkRGR0ZJRktGTUZQRlNGVjcjLCQtSSRzaW5HNiQlKnByb3RlY3RlZEdGKjYjSSJ4R0YoISIi

diff(sin(x),x,x,x);

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

x$3;

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

diff(sin(x),x$3);

NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiQtSSNtb0dGJTYzUSomdW1pbnVzMDtGKC8lJWZvcm1HUSdwcmVmaXhGKC8lJmZlbmNlR1EmZmFsc2VGKC8lKnNlcGFyYXRvckdGNS8lJ2xzcGFjZUdRJDBlbUYoLyUncnNwYWNlR0Y6LyUpc3RyZXRjaHlHRjUvJSpzeW1tZXRyaWNHRjUvJShtYXhzaXplR1EpaW5maW5pdHlGKC8lKG1pbnNpemVHUSIxRigvJShsYXJnZW9wR0Y1LyUubW92YWJsZWxpbWl0c0dGNS8lJ2FjY2VudEdGNS8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYoLyUlc2l6ZUdRIzEyRigvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRigvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYoLUYkNiUtSSNtaUdGJTY5USRjb3NGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdGUi8lJWJvbGRHRjUvJSdpdGFsaWNHRjUvJSp1bmRlcmxpbmVHRjUvJSpzdWJzY3JpcHRHRjUvJSxzdXBlcnNjcmlwdEdGNS8lK2ZvcmVncm91bmRHRlUvJStiYWNrZ3JvdW5kR0ZYLyUnb3BhcXVlR0Y1LyUrZXhlY3V0YWJsZUdGNS8lKXJlYWRvbmx5R1EldHJ1ZUYoLyUpY29tcG9zZWRHRjUvJSpjb252ZXJ0ZWRHRjUvJStpbXNlbGVjdGVkR0Y1LyUscGxhY2Vob2xkZXJHRjUvJTBmb250X3N0eWxlX25hbWVHRk8vJSptYXRoY29sb3JHRlUvJS9tYXRoYmFja2dyb3VuZEdGWC8lK2ZvbnRmYW1pbHlHRltvLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGKC8lKW1hdGhzaXplR0ZSLUYtNjNRMCZBcHBseUZ1bmN0aW9uO0YoL0YxUSZpbmZpeEYoRjNGNkY4RjtGPUY/RkFGREZHRklGS0ZNRlBGU0ZWLUYkNiUtRi02M1EiKEYoRjAvRjRGYnBGNi9GOVEudGhpbm1hdGhzcGFjZUYoL0Y8RmRyL0Y+RmJwRj9GQUZERkdGSUZLRk1GUEZTRlYtRiQ2Iy1GZm42OVEieEYoRmluRlxvRl5vL0Zhb0ZicEZib0Zkb0Zmb0Zob0Zqb0ZccEZecEZgcEZjcEZlcEZncEZpcEZbcUZdcUZfcUZhcS9GZHFRJ2l0YWxpY0YoRmZxLUYtNjNRIilGKC9GMVEocG9zdGZpeEYoRmJyRjZGY3IvRjxRMnZlcnl0aGlubWF0aHNwYWNlRihGZnJGP0ZBRkRGR0ZJRktGTUZQRlNGVjcjLCQtSSRjb3NHNiQlKnByb3RlY3RlZEdGKjYjSSJ4R0YoISIi

sin(0); cos(0);

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

seq(diff(sin(x),x$n),n=1..10);

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

seq(eval(subs(x=0,diff(sin(x),x$n))),n=1..10);

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

seq(eval(subs(x=0,diff(sin(x),x$n)))/n!,n=1..10);

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

series(sin(x),x);

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

series(sin(x),x,10);

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

S3 := series(sin(x),x,5);

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

P3 := convert(S3,polynom);

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

S5 := series(sin(x),x,7);

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

P5 := convert(S5,polynom);

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

The following plots show that even small degree Taylor polynomials provide a good approximation, though as the degree increases the approximation gets better. Moreover, as the point x moves further from 0, the point we expanded about, the error increases. Notice also, that since the series is alternating in sign, successive approximations alternately undershoot and overshoot the function they are approximating. with(plots):

Warning, the name changecoords has been redefined

plot3 := plot(P3,x=0..Pi): plot5 := plot(P5,x=0..Pi): plot0 := plot(sin(x),x=0..Pi): display({plot3,plot5,plot0});

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

In order to use these Taylor polynomials to approximate sin(x), we need to get a bound on the error. 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 all 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, Taylor's theorem implies that 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 fact, we can use 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 the bound, when 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 odd, since the next term in the series is zero. evalf(sin(Pi/4));