f[x_]=Sin[Exp[x]-2]; 
dfdx[x_]=D[f[x],{x,1}]
d4fdx4[x_]=D[f[x],{x,4}]

x0=0.8
x1=1.0
f0=f[x0]
f1=f[x1]
f00=dfdx[x0]
f11=dfdx[x1]
f01=(f1-f0)/(x1-x0)
f001=(f01-f00)/(x1-x0)
f011=(f11-f01)/(x1-x0)
f0011=(f011-f001)/(x1-x0)

p[x_]=f0+f00*(x-x0)+f001*(x-x0)^2+f0011*(x-x0)^2*(x-x1)
Plot[{p[x]},{x,x0,x1},PlotStyle->{Red}]

p[x0]
D[p[x],{x,1}]/.x->x0
p[x1]
D[p[x],{x,1}]/.x->x1

a=.9
approx=p[a]
exact=f[a]
error=Abs[approx-exact]

Plot[{Abs[d4fdx4[x]]},{x,x0,x1},PlotStyle->{Red}]

bound=d4fdx4[x1]/Factorial[4]*(a-x0)^2*(a-x1)^2
Abs[f[a]-p[a]]

b[x_]=Abs[d4fdx4[x1]/Factorial[4]*(x-x0)^2*(x-x1)^2]

Plot[{Abs[f[x]-p[x]], b[x]},{x,x0,x1},PlotStyle->{Red, Blue}]