Question

Approximating derivatives

$$ \begin{aligned} f^{\prime}(x) & \approx\left(\delta_{+} f\right)(x)=\frac{f(x+h)-f(x)}{h} & \text { Forward difference } \\ f^{\prime}(x) & \approx\left(\delta_{-} f\right)(x)=\frac{f(x)-f(x-h)}{h} & \text { Backward difference } \\ f^{\prime}(x) & \approx(\delta f)(x)=\frac{f(x+h)-f(x-h)}{2 h} & \text { Centered difference for 1st derivative } \\ f^{\prime \prime}(x) & \approx\left(\delta^{2} f\right)(x)=\frac{f(x+h)-2 f(x)+f(x-h)}{h^{2}} & \text { Centered difference for 2nd derivative } \end{aligned} $$

1. a. Use Taylor's polynomials to derive the centered difference approximation for the first derivative:

$$ f^{\prime}(x) \approx \delta f(x)=\frac{f(x+h)-f(x-h)}{2 h}, $$

include the error in the form \(k h^{m}+O\left(h^{n}\right)\), for some \(m\) and \(n\), with \(m<n\)(find \(m, n\), and \(k\) explicitly).

b. Let \(f(1)=0\) and \(f(1.4)=0.659\). Use the centered difference approximation (with \(h=0.2\) ) to approximate \(f^{\prime}(x)\) at \(x=1.2\) (i.e. \(x-h=1, x=1.2\), and \(x+h=1.4\) ).

c. The above data were generated using the function \(f(x)=x^{2} \ln x\). Derive a bound for the error in the approximation of the derivative computed in part b. What condition on the function \(f\) is needed to guarantee this accuracy?

2. Write a short program that uses formulas (1), (3) and (4) to approximate \(f^{\prime}(1)\) and \(f^{\prime \prime}(1)\) for \(f(x)=e^{x}\) with \(h=1,2^{-1}, 2^{-2}, \ldots, 2^{-60} .\) Format your output in columns as follows:

\(h \quad\left(\delta_{+} f\right)(1)\) error \((\delta f)(1)\) error \(\left(\delta^{2} f\right)(1)\) error

Indicate the values of \(h\) (in each column) that give the least error (where, as usual, the error is the absolute value of the difference between the exact value, of the derivative, and the approximate value).

Answer 1