Chapter 3.1, Problem 63E

Another way to approximate derivatives is to use the centered difference quotient:

     $f^{\prime }\left(a\right)\approx \frac{f\left(a+h\right)-f\left(a-h\right)}{2h}$

Again consider $f\left(x\right)=\sqrt{x}$.

  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′(4).

  b.    Use the centered difference quotient to approximate f′(4) by completing the table.

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.