Differentiation questions

Exploring errors

  1. Write a program that defines a function \(f(x)\) returning the value \(x(x-1)\), then uses the backwards difference method to calculate the numerical derivative of \(f(x)\) at the point \(x=1\), with \(h=10^{-2}\).

  2. Calculate the exact value of the derivative and compare this to the answer your program gives. The two will not agree perfectly - why not?

  3. Repeat the calculation for \(h=10^{-2},10^{-4},10^{-6},10^{-8},10^{-10},10^{-12},10^{-14}\). What do you observe about the accuracy of the calculation? Why does it behave in this way?

The method of finite differences for second order derivatives

In the tutorial we give an expression for calculating second order derivatives using the finite difference method:

\[\begin{equation} \frac{\mathrm{d} ^2f}{\mathrm{d} x^2} \simeq \frac{f(x+h)-2f(x)+f(x-h)}{h^2}. \end{equation}\]

Using the fact that a second derivative is, by definition, a derivative of a derivative, and by applying the central difference method multiple times, derive this expression.