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