Chapter 3.1, Problem 62E

To determine

(a) We have to explain graphically the reason of

$\frac{f\left(-h\right)-f\left(0\right)}{-h}\le f^{\text{'}}\left(0\right)\le \frac{f\left(h\right)-f\left(0\right)}{h}$

(b) Using (a) we need to show that $0.69314\le f^{\text{'}}\left(0\right)\le 0.69315$

(c) We need to determine $f^{\text{'}}\left(x\right)$ to 4 decimal places for $x=1,2,3,4$

(d) We need to determine the ratio $\frac{f^{\text{'}}\left(x\right)}{f^{\text{'}}\left(0\right)}$ for $x=1,2,3,4$. We need to determine an approximate formula for $f^{\text{'}}\left(x\right)$