## Practice Problems

### Problems

1. The graph of is given in Fig. 15. Estimate the locations of and when Newton's method is applied to with the given starting value

3. The function in Fig. 17 has several roots. Which root do the iterates of Newton's method converge to if we start with ?

5. What happens to the iterates if we apply Newton's method to the function in Fig. 19 and start with ?

7. What happens if we apply Newton's method to a function and start with a maximum of ? In problems 8 and 9, a function and a value for are given. Apply Newton's method to find and .

In problem 11, use Newton's method to find a root or solution, accurate to 2 decimal places, of the given functions using the given starting points.

In problems 13-15, use Newton's method to find all roots or solutions, accurate to 2 decimal places. It is helpful to examine a graph of the function to determine a "good" starting value .

17. Use Newton's method to devise an algorithm for approximating the cube root of a number .

19. The iterates of numbers using the Simple Chaotic Algorithm have a number of properties.

(a) Verify that the iterates of are all equal to .

(b) Verify that if , and, in general, , then the nth iterate of is 0 (and so are all of the iterates beyond the nth iterate).

21. is called a "stretch and fold" function.

(a) Describe what does to the points in the interval .

(b) Examine and describe the behavior of the iterates of , and .

(c) Examine and describe the behavior of the iterates of , and .

(d) Do the iterates of lead to chaotic behavior?

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-3.8-Newtons-Method.pdf

This work is licensed under a Creative Commons Attribution 3.0 License.