Sections 11.11: p. 798, #2, 3, 27

#2:

f[x_]:=1/x
f0[x_]:=f[1]
f1[x_]:=f[1]+f'[1]/1*(x-1)
f2[x_]:=f[1]+f'[1]/1*(x-1)+f''[1]/2*(x-1)^2
f3[x_]:=f[1]+f'[1]/1*(x-1)+f''[1]/2*(x-1)^2+f'''[1]/Factorial[3]*(x-1)^3
f3[x]
Plot[{f[x],f0[x],f1[x],f2[x],f3[x]},{x,0,2}]

{f[0.9],f0[0.9],f1[0.9],f2[0.9],f3[0.9]}
{f[1.3],f0[1.3],f1[1.3],f2[1.3],f3[1.3]}

Not too surprisingly, the taylor polynomials do a better and better job of catching the behavior of f.

#3:

f[x_]:=1/x
f3[x_]:=f[2]+f'[2]/1*(x-2)+f''[2]/2*(x-2)^2+f'''[2]/Factorial[3]*(x-2)^3
f3[x]
Plot[{f[x],f3[x]},{x,0,4}]

#27:

f[x_]:=Sin[x]
fPoly[x_]:=x-x^3/6
errorBound[x_]:=1/Factorial[5]*Abs[x]^5
Solve[errorBound[x]==.01,x]
(.01*Factorial[5])^(1/5)
Plot[{Abs[f[x]-fPoly[x]],.01,errorBound[x]},{x,0,1.1}]