Practice 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).

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