Chapter 23, Problem 19P

The objective of this problem is to compare second-order accurate forward, backward, and centered finite-difference approximations of the first derivative of a function to the actual value of the derivative. This will be done for

$f\left(x\right)=e^{-2x}-x$

(a) Use calculus to determine the correct value of the derivative at $x=2$.

(b) To evaluate the centered finite-difference approximations, start with. Thus, for the first evaluation, the x values for the $x=0.5$ centered difference approximation will be $x=2\pm 0.5$ or $x=1.5$ and 2.5 Then, decrease in increments of 0.01 down to a minimum value of $\Delta x=0.01$.

(c) Repeat part (b) for the second-order forward and backward differences. (Note that these can be done at the same time that the centered difference is computed in the loop.)

(d) Plot the results of (b) and (c) versus x. Include the exact result on the plot for comparison.