1. 1 4.8 – Newton’s Method

2. 2 The Situation Let’s find the x-intercept of function graphed using derivatives and tangent lines. |x1|x1 |x2|x2 |x3|x3 Continuing, the x- intercept of the tangent lines coincides with the x-intercept of the function. http://math.fullerton.edu/mathews/a2001/animations/rootfind ing/NewtonMethod/NewtonMethod.html

3. 3 The Algorithm An algorithm can be developed for this process. Simply find the equation of the tangent line. |x1|x1 |x2|x2 ● (x 2, 0) ● (x 1, f (x 1 ))

4. 4 Newton’s Method for Finding Roots If x n is a term in the sequence, x n - 1 is the term before it and x n + 1 is the term after it. In general, Where x 1 is your initial guess at the root.

5. 5 Example 1. Determine x 2 and x 3 for x 5 + 2 = 0 using Newton’s method and an initial guess of x 1 = -1. Develop an algorithm on your calculator that will automatically generate the values of the terms. You will use the equation editor, the numerical derivative, and storage features of your calculator.

6. 6 Examples 2. Determine the root of x 4 + x – 4 = 0 on the interval [1, 2] correct to six decimal places and using Newton’s method. 3. Find all roots to the equation.