Chapter 2, Problem 6PS

Finding Polynomials

(a) Find the polynomial $P_1(x)=a_0+a_{1}x$ whose value and slope agree with the value and slope of $f(x)=\cos x$ at the point $x=0$.

(b) Find the polynomial $P_2(x)=a_0+a_{1}x+a_2{x}^2$ whose value and first two derivatives agree with the value and first two derivatives of $f(x)=\cos x$ at the point $x=0$. This polynomial is called the second-degree Taylor polynomial of $f(x)=\cos x\text{at}x=0$.

(c) Complete the table comparing the values of $f(x)=\cos x\text{and}{P}_2(x)$. What do you observe?

x	-1.0	-0.1	-0.001	0	0.001	0.1	1.0
$\cos x$
$P_2(x)$

(d) Find the third-degree Taylor polynomial of $f(x)=\sin x\text{at}x=0$.