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.