4.1 Cramer’s rule and matrix inversion-multiplication offer alternative techniques to solve a system of linear algebraic equations. Conduct a literature search to collect information about these two techniques and the elimination and iteration techniques discussed in this chapter. Compare the various techniques regarding the complexity of algorithms, ease of implementation, and potential errors.
4.2 The Newton-Raphson technique may not converge to a solution. Inspecting equation 4.16, in what other possible way can the technique fail?
4.3 Roots of any equation can be found using what is known as the bracketing technique. Conduct a literature search and explain the principle behind such solution techniques.
4.4 The following data were obtained in an experiment where the concentration of a substance was monitored as a function of time. Calculate the first derivative of the concentration with respect to time for all possible times using the forward difference formula. Can the second derivative also be calculated numerically?
4.5 What is the area under the concentration-time curve obtained from the data shown for problem 4.4? Use the trapezoid method. An alternative technique is to use the rectangle method. What is the difference in t he areas if the area is calculated using the rectangle method?