Problem :
Compute the Taylor polynomial at 0 of degree 3 for f (x) = sin(e^{x}).
The first few derivatives are
f'(x)  =  cos(e^{x})e^{x} 

f''(x)  =  cos(e^{x})e^{x}  sin(e^{x})e^{2x} 

f'''(x)  =  f''(x)  [sin(e^{x})2e^{2x} + cos(e^{x})e^{3x}] 

Substituting
0 for
x yields
f (0) = sin(1),
f'(0) = cos(1),
f''(0) = cos(1)  sin(1),
f'''(0) =  3 sin(1), so the desired Taylor polynomial is:
sin(1) + cos(1)x + + x^{3} 

Problem :
Approximate log(0.8) using the Taylor polynomial of degree 3 at x = 1 for
log(x).
The desired Taylor polynomial is
(x  1)  + 

Substituting
0.8 for
x, we have
log(0.8) 0.222.
Problem :
Find the Taylor series for sin(x) + cos(x).
Adding termbyterm, we have