A Taylor polynomial from a table, with…

Name: ______________________ Date: ____________ A Taylor polynomial from a table, with two error bounds (9 points)

Table

The function

h

has derivatives of all orders for

x > -1

. The table gives values of

h

and its first three derivatives at

x = 0

and

x = 1

. The fourth derivative of

h

h^{(4)}(x)

= \frac{960}{(x + 1)^6}

. The Taylor series for

h

about

x = 1

\sum_{n=0}^{\infty} \frac{(-1)^n (n + 1)(x - 1)^n}{2^{n-1}}

, and it converges to

h(x)

for

-1 < x < 3

Values of $h$ , $h'$ , $h''$ and $h'''$
$x$	$h(x)$	$h'(x)$	$h''(x)$	$h'''(x)$
$0$	$8$	$-16$	$48$	$-192$
$1$	$2$	$-2$	$3$	$-6$

(a) Write the third-degree Taylor polynomial $P_3(x)$ for $h$ about $x = 1$ . [2 points]
(b) Use $P_3(x)$ to estimate $h(1.2)$ . [1 point]
(c) Use the Lagrange error bound to show that $\left|h(1.2) - P_3(1.2)\right| < \frac{1}{500}$ . [3 points]
(d) Explain why the alternating series error bound can be used for the estimate in part (b), and use it to find a bound on $\left|h(1.2) - P_3(1.2)\right|$ . Can the alternating series error bound be used in the same way to bound $\left|h(0.8) - P_3(0.8)\right|$ ? Explain. [3 points]

A Taylor polynomial from a table, with two error bounds