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Section 3.8 Newton’s Method:

Section 3.8 Newton’s Method: . A technique for approximating the real zeros of a Function. What is the derivative of a function? . What is the derivative of a function? f’(x 1 ) = m of the tangent line at some x 1 . Using the point slope formula: y-y 1 =m(x-x 1 ).

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Section 3.8 Newton’s Method:

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  1. Section 3.8 Newton’s Method: A technique for approximating the real zeros of a Function.

  2. What is the derivative of a function?

  3. What is the derivative of a function? f’(x1) = m of the tangent line at some x1.

  4. Using the point slope formula: y-y1=m(x-x1) y - y1= f’(x1)(x - x1)

  5. Using the point slope formula: y-y1=m(x-x1) y - y1= f’(x1)(x - x1) y - f(x1) = f’(x1)(x - x1)

  6. Using the point slope formula: y-y1=m(x-x1) y - y1= f’(x1)(x - x1) y - f(x1) = f’(x1)(x - x1) y = f(x1)+ f’(x1)(x - x1)

  7. Let y = 0, since we are looking for the zeros of the function. 0 = f(x1)+ f’(x1)(x - x1)

  8. Let y = 0, since we are looking for the zeros of the function. 0 = f(x1)+ f’(x1)(x - x1) Distribute f’(x1) 0 = f(x1)+ f’(x1)x - f’(x1)x1

  9. Let y = 0, since we are looking for the zeros of the function. 0 = f(x1)+ f’(x1)(x - x1) 0 = f(x1)+ f’(x1)x - f’(x1) x1 Isolate f’(x1)x f’(x1)x = f’(x1) x1- f(x1)

  10. Let y = 0, since we are looking for the zeros of the function. 0 = f(x1)+ f’(x1)(x - x1) 0 = f(x1)+ f’(x1)x - f’(x1) x1 f’(x1)x = f’(x1) x1- f(x1) Solve for x x = [f’(x1)/ f’(x1)] x1- [ f(x1)/ f’(x1)]

  11. Let y = 0, since we are looking for the zeros of the function. 0 = f(x1)+ f’(x1)(x - x1) 0 = f(x1)+ f’(x1)x - f’(x1) x1 f’(x1)x = f’(x1) x1- f(x1) x = [f’(x1)/ f’(x1)] x1- [ f(x1)/ f’(x1)] x = x1- f(x1)/ f’(x1)

  12. If we do this again and again we have the process which is called Newton’s Method.

  13. Newton’s Method for Approximating the Zeros of a Function: Let f(c) = 0, where f is differentiable on an open interval containing c. Then, to approximate c, use the following steps. • Make an initial estimate that is close to c. (A graph is helpful) • Determine a new approximation xn+1 = xn- f(xn)/ f’(xn) • If |xn – xn+1| is within the desired accuracy, let xn+1 serve as the final approximation. Otherwise, return to Step 2 and calculate a new approximation. Each successive application of this procedure is called an iteration.

  14. Graph of f(x) = 3(x-1)1/2-x

  15. Newton’s Method: f(x) = 3(x-1)1/2-x

  16. Graph of f(x) = x3+3

  17. Newton’s Method: f(x) = x3+3

  18. Assignment: Worth One Quiz Grade • Create an Excel spreadsheet for Section 3.8 problems 6, 10, and 14. All cells must be linked so that if I change the initial x-coordinate your spreadsheet reflects the change. • Find a graphing utility in Excel, graph these functions and include a copy on your file. • This assignment is Due Thursday , October 20 before 9:30AM. No extensions. • Email your final Excel file to me by the due date and time.

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