Chapter 9.7, Problem 39ES

Let $p_1(x)$ and $p_2(x)$ be the local linear and local quadratic approximates of $f(x)=e^{\sin x}\text{at}x=0.$

(a) Use a graphing utility to generate the graphs of $f(x),p_1(x),$ and $p_2(x),$ on the same screen for $-1\le x\le 1.$

(b) Construct a table of values of $f(x),p_1(x),$ and $p_2(x)$ for $x=-1.00,-0.75,-0.50-0.25,0,0.25,0.50,0.75,\mathrm{1.00.}$ Round the values to three decimal places.

(c) Generate the graph of $\left|f(x)-p_1(x)\right|,$ and use the graph to determine an interval on which $p_1(x)$ approximates $f(x)$ with an error of at most $\pm \mathrm{0.01.}$ [Suggestion Review the discussion to Figure 3.5.4.]

(d) Generate the graph of $\left|f(x)-p_2(x)\right|,$ and use the graph to determine an interval on which $p_2(x)$ approximates $f(x)$ with an error of at most $\pm \mathrm{0.01.}$