Practice Problems

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

Problems

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).