Assuming the limit exists, the definition of the derivative $f^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ implies that if h is small, then an approximation to $f^{\prime}(a)$ is given by
$$f^{\prime}(a) \approx \frac{f(a+h)-f(a)}{h}$$
If $h>0,$ then this approximation is called a forward difference quotient; if $h<0,$ it is a backward difference quotient. As shown in the following exercises, these formulas are used to approximate $f^{\prime}$ at $a$ point when $f$ is a complicated function or when $f$ is represented by a set of data points.
Another way to approximate derivatives is to use the centered difference quotient:
$$
f^{\prime}(a) \approx \frac{f(a+h)-f(a-h)}{2 h}
$$
Again, consider $f(x)=\sqrt{x}$
a. Graph $f$ near the point (4,2) and let $h=1 / 2$ in the centered difference quotient. Draw the line whose slope is computed by the centered difference quotient and explain why the centered difference quotient approximates $f^{\prime}(4)$
b. Use the centered difference quotient to approximate $f^{\prime}(4)$ by completing the table.
(TABLE CANNOT COPY)
c. Explain why it is not necessary to use negative values of $h$ in the table of part (b).
d. Compare the accuracy of the derivative estimates in part (b) with those found in Exercise 62