Chapter 8.1, Problem 9E

a. Construct the Taylor polynomial $p_3\left(x\right)$ of degree $3$ for the function $f\left(x\right)=\ln x$ around $x=1$.

b. Using the error formula $\left(6\right)$, show that

$\left|\ln \left(1.5\right)-p_3\left(1.5\right)\right|\le \frac{{\left(0.5\right)}^4}{4}=0.015625$.

c. Compare the estimate in part (b) with the actual error by calculating $\left|\ln \left(1.5\right)-p_3\left(1.5\right)\right|$.

d. Sketch the graphs of $\ln \text{x}$ and $p_3\left(x\right)$ (on the sane axes) for $0<x<2$.