Lauren Hensley

p. 181 # 6

Plot[{Sin[x], x, x - x^3/6, x - x^3/6 + x^5/Factorial[5], x - x^3 /6 + x^5/Factorial[5] - x^7/Factorial[7]}, {x, 0, 2*Pi}, PlotLegends->"Expressions"]

p. 178 # 3

Let’s look at the 4th degree approximation...

approximation[x_] := (x + x^2/2 + x^3/6  + x^4/Factorial[4]);

actual[x_] := Sum[x^n/Factorial[n], n]

Plot[{Exp[x] - 1, approximation[x]}, {x, 0, 0.1}]

relativeError[x_] := approximation[x]/(Exp[x]-1)- 1;

Plot[relativeError[x], {x, 0, 0.1}]

Therefore, the maximum of the absolute relative error is at the endpoint x = 0.1.

maximum =Abs[relativeError[0.1]]

This maximum < 10^(-5).