21.19 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(x) = e^{-2x} - x
(a) Use calculus to determine the correct value of the derivative at x = 2.
(b) Develop an M-file function to evaluate the centered finite-difference approximations, starting with ?x = 0.5. Thus, for the first evaluation, the x values for the centered difference approximation will be x = 2 ± 0.5 or x = 1.5 and 2.5. Then, decrease in increments of 0.1 down to a minimum value of ?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.
21.34 An nth-order rate law is often used to model chemical reactions that solely depend on the concentration of a single reactant:
dc/dt = -kc^n
where c = concentration (mole), t = time (min), n = reaction order (dimensionless), and k = reaction rate (min^-1 mole^{1-n}). The differential method can be used to evaluate the parameters k and n. This involves applying a logarithmic transform to the rate law to yield,
log(-dc/dt) = log k + n log c
Therefore, if the nth-order rate law holds, a plot of the log(-dc/dt) versus log c should yield a straight line with a slope of n and an intercept of log k. Use the differential method and linear regression to determine k and n for the following data for the conversion of ammonium cyanate to urea:
