1. 4.8 Newton’s Method Mon Nov 9 Do Now Find the equation of a tangent line to f(x) = x^5 – x – 1 at x = 1

2. Newton’s Method Newton’s Method is a procedure for finding numerical approximations to zeros of functions by using the tangent line to get closer to the zero

3. Newton’s Method To approximate a root of f(x) = 0: 1) Choose initial guess x0 (close to the zero if possible) 2) Generate successive approximations x1, x2,…, where

4. Ex Calculate the first three approximations to a root of f(x) = x^2 – 5 using initial guess x =2

5. How many iterations to use? If 2 successive iterations agree to N decimal places, then that approximation is accurate to N decimal places

6. Ex Approximate the cube root of 5, accurate to 3 decimal places using Newton’s Method

7. Note Your initial guess is important! If your initial guess is not close enough, Newton’s Method could take you the wrong way! (or a different root)

8. Ex Using Newton’s Method, find a root of f(x) = x^4 – 6x^2 + x + 5 using initial guess x = 0

9. Ex Using Newton’s Method, find a root of f(x) = x^4 – 6x^2 + x + 5 using initial guess x = -1

10. Closure Apply Newton’s method 2 times to y = x^2 – 6 with initial guess 2. HW: P.272 #3 7 9 11 17 25