8.9 Applications of Taylor Polynomials

Mathematica script by Chris Parrish,
cparrish@sewanee.edu

Sources and references for some of these problems include
James Stewart, "Calculus: Concepts and Contexts," Second Edition, Brooks/Cole, 2001
Deborah Hughes-Hallett, Andrew M. Gleason, et. al., "Calculus," Second Edition, John Wiley & Sons, 1998
Robert Fraga, ed., "Calculus Problems for a New Century," The Mathematical Association of America, 1993
Selwyn Hollis, "CalcLabs with
Mathematica" for Stewart's "Single Variable Calculus, Concepts and Contexts, Second Edition," Brooks/Cole, 2001

3rd-degree Taylor polynomial for Log[1 + 2x]
Centered on x = 1

Stewart, 8.9.16, page 633

Let's find the 3rd degree Taylor Polynomial for ln(1+2x) centered on x = 1.

In[459]:=

In[460]:=

Clear[f,taylor3,x,x0,a,b,n,max,error,errorBound]

x0 = 1;
n =3;

f[x_] := Log[1 + 2x]

taylor3[x_] = Normal[Series[f[x],{x,x0,n}]]

Out[464]=

In[465]:=

a = -0.5;
b = 5;

Plot[{f[x],taylor3[x]},{x,a,b},
PlotLabel -> "Taylor Polynomial Approximation",
PlotStyle ->{Orchid,ForestGreen}];

If we confine x to the interval [0.5,1.5] then |x - 1| <= 0.5, so by Taylor's Inequality, we have ...

In[468]:=

Out[468]=

... where "max" is an upper bound for the fourth derivative of f over the interval [0.5,1.5].

In[469]:=

It seems that the largest (negative) value for f'''' over the interval [0.5,1.5] is taken on at the endpoint x = 0.5.

In[470]:=

Out[470]=

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Let's check that result by graphing the absolute value of the difference of f(x) and that particular Taylor Polynomial over the interval [0.5,1.5].

In[473]:=

The largest actual error over the interval [0.5,1.5] occurs at the endpoint x = 0.5.

In[475]:=

Out[476]=

That largest actual error is considerably less than the upper bound for the error that we obtained from Taylor's Inequality.

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Created by Mathematica  (May 5, 2004)