2. Taylor Series and Numerical Differentiation (25\%) 2.1. Write the Taylor expansion approximation up to the second order term for a general function f around a given point a for a distance h. Write this approximation for the exponential function f(x)=ex around point a as a function of h. 2.2. Using the Taylor approximation in Part 2.1, write the expressions for the following numerical differentiation schemes: (i) forward, (ii) backward, and (iii) central difference. 2.3. Now, compute the approximations (i)-(iii) in Part 2.2 for the exponential function in Part 2.1 at the point a=1 as a function of h. For each case, create a table that shows the approximation error as a function h (use a small range for h ) and plot the results with approximation error on the y-axis and h on the x-axis. Compare the results for each case and comment on your results. 2.4. Using the Taylor expansion in Part 2.1, write an approximation for the second order derivative of a general function f around a given point a (within a distance h ). Use this expansion to show that the second order differentiation can be expressed as. f(a)h2f(a+h)2f(a)+f(ah) Write the second order differentiation for the exponential function at the point a=1 as a function of h and plot the approximation error as a function h