8.7.3 Finding and Using Taylor Series

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

Problems from Hughes-Hallett, Gleason, et al.

Taylor polynomials for -Log[1 - 2x] near x = 0

Hughes-Hallett, Gleason, et al, Second Edition, HHG IM 9.3.4

In[367]:=

Clear[f,taylor,x,x0,a,b,n]

x0 = 0;

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

taylor5[x_] = Normal[Series[f[x],{x,x0,5}]];
taylor10[x_] = Normal[Series[f[x],{x,x0,10}]];
taylor15[x_] = Normal[Series[f[x],{x,x0,15}]];
taylor20[x_] = Normal[Series[f[x],{x,x0,20}]]

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In[374]:=

a = -0.8;
b = 0.499;

Plot[{f[x],taylor5[x],taylor10[x],taylor15[x],taylor20[x]},{x,a,b},
PlotLabel -> "Taylor Polynomial Approximations",
PlotStyle -> {Red,Green,ForestGreen,Blue,Indigo}];

Let's just plot the polynomials, in order to get beyond the point where f[x] blows up.

In[377]:=

a = -0.8;
b = +0.8;

Plot[{taylor5[x],taylor10[x],taylor15[x],taylor20[x]},{x,a,b},
PlotLabel -> "Taylor Polynomial Approximations",
PlotStyle -> {Green,ForestGreen,Blue,Indigo}];

Some Taylor polynomials near x = 0

Hughes-Hallett, Gleason, et al, Second Edition, 9.3.15, page 449

In[380]:=

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In[390]:=

a = -2;
b = 2;

Plot[{taylorF[x],taylorG[x],taylorH[x],taylorK[x]},{x,a,b},
PlotLabel -> "Taylor Polynomials",
PlotStyle -> {Green,ForestGreen,Blue,Indigo}];

Who's who?

Taylor polynomial for Exp[-] and 1/(1 + ) near x = 0

Hughes-Hallett, Gleason, et al, Second Edition, 9.3.16, page 449

In[393]:=

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In[400]:=

a = -1;
b = 1;

Plot[{taylorF[x],taylorG[x]},{x,a,b},
PlotLabel -> "Taylor Polynomial Approximation",
PlotStyle -> {Green,ForestGreen}];

Taylor polynomial for 1/(2+x) in terms of

Hughes-Hallett, Gleason, et al, Second Edition, 9.3.17, page 449

In[403]:=

Clear[f,taylor,x,x0,a,b,n]

x0 = 0;
n = 4;

f[x_] := 1/(2 + x)

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

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Created by Mathematica  (May 5, 2004)