Chapter 2, Problem 5RWDT

To determine

To prove : If $\underset{h\to 0}{\lim }\frac{f\left(a+h\right)-f\left(a-h\right)}{2h}=f\text{'}(a)$ exists, prove it graphically.

Explanation of Solution

Given information :

The numerical derivative uses the symmetric difference quotient $\frac{f\left(a+h\right)-f\left(a-h\right)}{2h}$ with a small value of $h$ , to approximate $f\text{'}\left(a\right)$ .

Also, $\underset{h\to 0}{\lim }\frac{f\left(a+h\right)-f\left(a-h\right)}{2h}$ exists.

Formula used :

A function $y=f\left(x\right)$ is differentiable on a closed interval $\left[a,b\right]$ if it has a derivative at every interior point of the interval, and if the limits

  $\begin{array}{l}\underset{h\to 0^{+}}{\lim }\frac{f\left(a+h\right)-f\left(a\right)}{h}\\ \underset{h\to 0^{-}}{\lim }\frac{f\left(a+h\right)-f\left(b\right)}{h}\\ \end{array}$

Exist at the end point.

Proof:

  

The symmetric difference quotient is the slope between two points on the graph, which is the slope of a secant line to the curve, where the two points are equally to the right and the left of the point of interest.