Manipulate[Plot[{x/2+x^2,x/2-x^2,x/2,x/2+x^2Sin[1/x]},{x,-r,r},PlotStyle->{LightGray,LightGray,LightGray,Blue}],{r,.001,1}]

Manipulate[Plot[{1/2-Cos[1/x]+2x Sin[1/x]},{x,-r,r}],{r,.0001,1}]

h[x_]:=TriangleWave[{0,1},x/2-1/4];
g[n_,x_]:=Sum[1/2^k h[2^k x],{k,0,n}];
Plot[h[x],{x,-4,4}]
Plot[g[6,x],{x,-4,4},PlotRange->{0,2}]

n=2;
s = 1+Sum[x^k/(k!), {k,1,n}]
Plot[{Exp[x],s},{x,-2,2}]

Plot[Exp[-1/x^2],{x,-1,1}]

p = Table[Normal[Series[Sqrt[1-y],{y,0,n}]],{n,1,10}]

Plot[{Sqrt[1-y],p[[1]],p[[2]]},{y,-1,1}]

NMaximize[{Abs[Sqrt[1-y] - p[[3]]], -1≤ y ≤1},y]

Plot[{Abs[x], p[[4]]/.{y->(1-x^2)},p[[10]]/.{y->(1-x^2)}},{x,-1,1}]