Practice Problems

Work through the odd-numbered problems 1-21. Once you have completed the problem set, check your answers.


1. The graph of y=f(x) is given in Fig. 15. Estimate the locations of x_{1} and \mathrm{x}_{2} when Newton's method is applied to \mathrm{f} with the given starting value \mathrm{x}_{0}

3. The function in Fig. 17 has several roots. Which root do the iterates of Newton's method converge to if we start with \mathrm{x}_{0}=1? \mathrm{x}_{0}=5 ?

5. What happens to the iterates if we apply Newton's method to the function in Fig. 19 and start with \mathrm{x}_{0}=1 ? \mathrm{x}_{0}=5?

7. What happens if we apply Newton's method to a function \mathrm{f} and start with \mathrm{x}_{0}= a maximum of \mathrm{f} ? In problems 8 and 9, a function and a value for x_{0} are given. Apply Newton's method to find x_{1} and x_{2}.

9. f(x)=x^{4}-x^{3}-5 and x_{0}=2

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.

11. f(x)=x-\cos (x) and x_{0}=0.7

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 \mathrm{x}_{0}.

13. \frac{x}{x+3}=x^{2}-2

15. x=\sqrt[5]{3}

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

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

(a) Verify that the iterates of \mathrm{x}=0 are all equal to \mathrm{0}.

(b) Verify that if \mathrm{x}=1 / 2,1 / 4,1 / 8, and, in general, 1 / 2^{\mathrm{n}}, then the nth iterate of \mathrm{x} is 0 (and so are all of the iterates beyond the nth iterate).

21. \mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}2 \mathrm{x} & \text { if } 0 \leq \mathrm{x} < 1 / 2 \\ 2-2 \mathrm{x} & \text { if } 1 / 2 \leq \mathrm{x} \leq 1\end{array} \quad\right. is called a "stretch and fold" function.

(a) Describe what \mathrm{f} does to the points in the interval [0,1].

(b) Examine and describe the behavior of the iterates of 2 / 3,2 / 5,2 / 7, and 2 / 9.

(c) Examine and describe the behavior of the iterates of \mathrm{.10, \,.105}, and \mathrm{.11}.

(d) Do the iterates of \mathrm{f} lead to chaotic behavior?

Source: Dale Hoffman,
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