4.5.1Do you know your roots?

Newton's Method

4-168.

Based on the fragment of a function shown at right, make a reasonable estimate for the root ( $x$ -intercept) of the function. Justify your estimate.

First quadrant, increasing concave up curve from approximate point (3.2, comma 2.2) to approximate point (4, comma 3.8).

4-169.

Locate the graph of $y = f\left(x\right)$ on the Lesson 4.5.1. Resource Page .

Determine the integer closest to the root ( $x$ -intercept) of $f\left(x\right)$ and label it $x_{1}$ .
Use a tangent line at the point $\left(x_{1}, f\left(x_{1}\right)\right)$ to make another approximation of the root. Label this point $x_{2}$ . Which point $\left(x_{1} \text{ or } x_{2}\right)$ is a better approximation?
Repeat the process again: use a tangent line at the point $\left(x_{2}, f\left(x_{2}\right)\right)$ to make another approximation of the root. Label this point $x_{3}$ .
Describe your observations from this process.

Your teacher will provide you with a model.

4-170.

The process for determining the root of a function that you used in problem 4-169 is called Newton’s Method. Newton’s Method starts with an $x$ -value, called $x_{1}$ , reasonably close to the root, $r$ . It then uses this $x$ -value to find another $x$ -value, $x_{2}$ , closer to the root, which itself is used to find a better approximation, and so on. To find the next $x$ -value, we determine the root of the line tangent to the curve at the previous root approximation. For example, $x_{2}$ is the root of the line tangent to $f$ at $\left(x_{1}, f\left(x_{1}\right)\right)$ .

If $f\left(x\right) = x^{3} + x^{2} – 1$ and $x_{1} = 1$ , calculate $x_{2}, x_{3}$ , and $x_{4}$ .
Explain what happens to $x_{n}$ as $n$ increases.

Math Notes

Newton's Method for Approximating Roots

Newton’s Method uses an iterative process to approximate a root ( $x$ -intercept) of a function.

$⇒$ Begin with a point $x_{1}$ near the root. Write the equation of its tangent line at $x_{1}: y – f\left(x_{1}\right) = f ′\left(x_{1}\right)\left(x – x_{1}\right)$ .

The $x$ -intercept of this tangent line is $x_{2}$ .
Solve for $x_{2}$ by setting $y = 0$ .

$0 − f\left(x_{1}\right) = f ′\left(x_{1}\right)\left(x_{2} − x_{1}\right),$ so $x _ { 2 } = x _ { 1 } - \frac { f ( x _ { 1 } ) } { f ^ { \prime } ( x _ { 1 } ) }$ .

$⇒$ To calculate $x_{3}$ , solve for the root of the line tangent to $f$ at $x_{2}$ . This process continues until the desired accuracy is met.

Therefore, each new approximation, $x_{n+1}$ , can be found using $x _ { n + 1 } = x _ { n } - \frac { f ( x _ { n } ) } { f ^ { \prime } ( x _ { n } ) }$ . Newton’s Method was very important before calculators were invented and represents an excellent example of an iterative process. For most functions it works well, but it is always possible to find examples where it does not.

4-171.

Suppose you want to approximate $\sqrt { N }$ by using Newton’s Method to determine the positive root of $f\left(x\right) = x^{2} – N$ . If $x_{n}$ is the $n$ ^th approximation, write an equation for $x_{n+1}$ .

4-172.

Sketch $f\left(x\right)=x+\cos\left(x\right)$ . Homework Help ✎

Use the Intermediate Value Theorem to show there is a root between $–1$ and $0$ .
Let $x_{1} = –1$ . Use Newton’s Method to calculate $x_{3}$ .
How close is $x_{3}$ to the actual root of $f$ ? Calculate the error.

4-173.

Use the equation for Newton’s Method in the Math Notes box in this lesson to approximate the root of $f\left(x\right)=$ $16x^3-24x^2+12x-1$ in the interval $[0, 1]$ , accurate to three decimal places. Homework Help ✎

4-174.

Examine the equation $\int _ { 5 } ^ { 16 } H ( t ) d t = 70$ where $H$ represents the rate that human hair grows in inches as a function of time and $t$ is measured in years.

Write a complete description about what the integral will compute. In other words, in the context of the problem, what is accumulated? Use correct units and be sure to mention the meaning of the bounds in your description. Homework Help ✎

4-175.

Rewrite each of the following expressions using a single trigonometric function. You may wish to review your trigonometric identities from Chapter 1. Homework Help ✎

$10\sin\left(3x\right)\cos\left(3x\right)$
$\sin(x)\cos(3x)-\sin(3x)\cos(x)$
$\cos^4\left(x\right)-\sin^4\left(x\right)$
$\tan(x)+\cot(x)$

4-176.

Solve for $x$ in each equation. Assume $x$ is measured in radians. Hint: First use a substitution to make the problem easier. Homework Help ✎

$2\sin^2\left(x+1\right)-\sin\left(x+1\right)-1=0$
$( x ^ { 2 } - 2 x + 1 ) - 3 \sqrt { x ^ { 2 } - 2 x + 1 } = - 2$

4-177.

Use the definition of a derivative as a limit to determine $\frac { d } { d x } ( \frac { 2 } { x ^ { 2 } } )$ . Homework Help ✎

4-178.

Using your graphing calculator, determine all relative maxima and minima points of inflection, and discontinuities of the curve $y = \frac { \operatorname { sin } ( x ) } { x }$ between $–2π ≤ x ≤ 2π$ . 4-178 HW eTool Homework Help ✎.

Calculus Third Edition

Newton's Method for Approximating Roots