1. A technique for approximating the real zeros of a Function.

2. What is the derivative of a function?

3. f’(x 1 ) = m of the tangent line at some x 1.

4. y - y 1 = f’(x 1 )(x - x 1 )

5. y - f(x 1 ) = f’(x 1 )(x - x 1 )

6. y - y 1 = f’(x 1 )(x - x 1 ) y - f(x 1 ) = f’(x 1 )(x - x 1 ) y = f(x 1 ) + f’(x 1 )(x - x 1 )

7. 0 = f(x 1 ) + f’(x 1 )(x - x 1 )

8. Distribute f’(x 1 ) 0 = f(x 1 ) + f’(x 1 )x - f’(x 1 )x 1

9. 0 = f(x 1 ) + f’(x 1 )(x - x 1 ) 0 = f(x 1 ) + f’(x 1 )x - f’(x 1 ) x 1 Isolate f’(x 1 )x f’(x 1 )x = f’(x 1 ) x 1 - f(x 1 )

10. 0 = f(x 1 ) + f’(x 1 )(x - x 1 ) 0 = f(x 1 ) + f’(x 1 )x - f’(x 1 ) x 1 f’(x 1 )x = f’(x 1 ) x 1 - f(x 1 ) Solve for x x = [f’(x 1 )/ f’(x 1 )] x 1 - [ f(x 1 )/ f’(x 1 )]

11. 0 = f(x 1 ) + f’(x 1 )(x - x 1 ) 0 = f(x 1 ) + f’(x 1 )x - f’(x 1 ) x 1 f’(x 1 )x = f’(x 1 ) x 1 - f(x 1 ) x = [f’(x 1 )/ f’(x 1 )] x 1 - [ f(x 1 )/ f’(x 1 )] x = x 1 - f(x 1 )/ f’(x 1 )

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

13. Let f(c) = 0, where f is differentiable on an open interval containing c. Then, to approximate c, use the following steps. 1. Make an initial estimate that is close to c. (A graph is helpful) 2. Determine a new approximation x n+1 = x n - f(x n )/ f’(x n ) 3. If |x n – x n+1 | is within the desired accuracy, let x n+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.

15. Problem 7xnxn f(x n )f ’(x n )f(x n )/f ’(x n )x n -f(x n )/f ’(x n ) f(x)=3*(x-1) 1/2 -x 1.1000-0.15133.7434-0.04041.1404 -0.01623.0029-0.00541.1458 -0.00022.9280-0.00011.1459 0.00002.92710.0000 x =1.1459 7.10000.3095-0.3927-0.78817.8881 -0.0145-0.42850.03397.8542 0.0000-0.42710.00017.8541 0.0000-0.42710.0000 x =7.8541

17. Problem 9xnxn f(x)f ’(x n )f(x n )/f ’(x n )x n -f(x n )/f '(x n ) f(x)=x 3 +3-2.0000-5.000012.0000-0.4167-1.5833 -0.96937.5208-0.1289-1.4544 -0.07686.3463-0.0121-1.4424 -0.00066.2411-0.0001 x =-1.4424

18. 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.