4.8 Newton’s MethodMon Nov 9

Do NowFind the equation of a tangent line

to f(x) = x^5 – x – 1 at x = 1

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

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

Ex

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

How many iterations to use?

• If 2 successive iterations agree to N decimal places, then that approximation is accurate to N decimal places

Ex

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

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)

Ex

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

Ex

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

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