Determine a fourth-degree Taylor polynomial matching the function e^x at x_0=1 .

The formula for the Taylor polynomial of degree n centered at x_0 , approximating a function f(x) possessing n derivatives at x_0 , is given by
p_n(x)=f(x_0)+f'(x_0)(x-x_0)+f''(x_0)/(2!)(x-x_0)^2+f'''(x_0)/(3!)(x-x_0)^3+...
p_n(x)=sum_(j=0)^n (f^((j))(x_0))/(j!) (x-x_0)^j
For f(x)=e^x, f^((j))(2)=e^2 for all j .
Therefore to fourth order in x about the point x_0=2
e^x~~p_4(x)=e^2+e^2(x-2)+e^2/2(x-2)^2+e^2/(3!)(x-2)^3+e^2/(4!)(x-2)^4
Notice the graph below. The function p_4(x)~~e^x (red) the most at x=2 . It will become more and more approximate to e^x the higher order the approximation.
http://mathworld.wolfram.com/TaylorSeries.html