Chapter 4.4, Problem 60SB

Newton’s Method In calculus you will learn that if

$P\left(x\right)\text{ = }{a}_n{x}^n\text{ + }{a}_{n-\text{ 1 }}{x}^{n-\text{ 1}}\text{ + }\mathrm{...}\text{ + }{a}_{1}x\text{ + }{a}_0$

is a polynomial function, then the derivative of is

$P^{\text{'}}\left(x\right)\text{ = }n{a}_n{x}^{n-\text{ 1}}\text{ + }\left(n-\text{ 1}\right)a_{n-\text{ 1 }}{x}^{n-\text{ 2}}\text{ + }\mathrm{...}\text{ + 2}{a}_{2}x\text{ + }{a}_1$

Newton’s Method is an efficient method for approximating the $x\text{-intercepts}$ (or real zeros) of a function, such as $p\left(x\right)$. The following steps outline Newton’s Method.

STEP 1: Select an initial value $x_0$ that is somewhat close to the $x\text{-intercept}$ being sought.

STEP 2: Find values for $x$ using the relation

$x_{n\text{ + 1}}\text{ = }{x}_n-\frac{P\left(x_n\right)}{P\text{'}\left(x_n\right)}n\text{ = 1, 2,}\mathrm{...}$

until you get two consecutive values $x_n$ and $x_{n\text{ + 1}}$ that agree to whatever decimal place accuracy you desire.

STEP 3: The approximate zero will be $x_{n\text{ + 1}}$.

Consider the polynomial $P\left(x\right)\text{ = }{x}^3-\text{ 7}x-\text{ 40}$.

a. Evaluate $p\left(5\right)$ and $p\left(-3\right)$.

b. What might we conclude about a zero of $p$? Explain.

c. Use Newton’s Method to approximate an $x\text{-intercept}$, $r$, $-3\text{ < }r\text{ < 5}$, of $p\left(x\right)$ to four decimal places.

d. Use a graphing utility to graph $p\left(x\right)$ and verify your answer in part .

e. Using a graphing utility, evaluate $p\left(r\right)$ to verify your result.