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

1. See Fig. 15

3. $x_{0}=1:$ a.      $x_{0}=5:$ b.

5. $\begin{aligned} &x_{0}=1: 1,2,1,2,1, \ldots \\ &x_{0}=5: x_{1} \text { is undefined since } f^{\prime}(5)=0 \end{aligned}$

7. If $\mathrm{f}$ is differentiable, then $\mathrm{f}^{\prime}\left(\mathrm{x}_{0}\right)=0$ and $\mathrm{x}_{1}=\mathrm{x}_{\mathrm{o}}-\frac{\mathrm{f}\left(\mathrm{x}_{\mathrm{o}}\right)}{\mathrm{f}^{\prime}\left(\mathrm{x}_{\mathrm{O}}\right)}$ is undefined.

9. $f(x)=x^{4}-x^{3}-5, x_{0}=2. f^{\prime}(x)=4 x^{3}-3 x^{2}$.

Then $x_{1}=2-\frac{3}{20}=\frac{37}{20}=1.85$ and $\quad x_{2}=1.85-\frac{1.85^{4}-1.85^{3}-5}{4(1.85)^{3}-3(1.85)^{2}} \approx 1.824641$.

11. $f(x)=x-\cos (x), f^{\prime}(x)=1+\sin (x), x_{0}=0.7.$ Then $x_{1}=0.7394364978, x_{2}=0.7390851605$, and root $\approx 0.74$

13. $\frac{x}{x+3}=x^{2}-2$ so we can use $f(x)=x^{2}-2-\frac{x}{x+3}$. If $x_{0}=-4$, then the iterates $x_{n} \rightarrow-3.3615$. If $x_{0}$ $=-2$, then $x_{n} \rightarrow-1.1674$. If $x_{0}=2$, then the iterates $x_{n} \rightarrow 1.5289$.

15. $\mathrm{x}^{5}-3=0$ and $\mathrm{x}_{\mathrm{O}}=1.$ Then $\mathrm{x}_{\mathrm{n}} \rightarrow 1.2457$.

17. $f(x)=x^{3}-A$ so $f^{\prime}(x)=3 x^{2}$.

Then $x_{n+1}=x_{n}-\frac{\left(x_{n}\right)^{3}-A}{3\left(x_{n}\right)^{2}}=x_{n}-\frac{x_{n}}{3}+\frac{A}{3\left(x_{n}\right)^{2}}=\frac{1}{3}\left\{2 x_{n}+\frac{A}{\left(x_{n}\right)^{2}}\right\}$.

19. (a) $2(0)-\operatorname{INT}(2(0))=0-0=0$.

 (b) $2(1 / 2)-\operatorname{INT}(2(1 / 2))=1-1=0 .$

      $2(1 / 4)-\operatorname{INT}(2(1 / 4))=1 / 2-0=1 / 2 \rightarrow 0$

      $2(1 / 8)-\operatorname{INT}(2(1 / 8))=1 / 4-0=1 / 4 \rightarrow 1 / 2 \rightarrow 0$

       $2\left(1 / 2^{\mathrm{n}}\right)-\operatorname{INT}\left(2\left(1 / 2^{\mathrm{n}}\right)\right)=1 / 2^{\mathrm{n}-1}-0=1 / 2^{\mathrm{n}-1} \rightarrow 1 / 2^{\mathrm{n}-2} \rightarrow \ldots \rightarrow 1 / 4 \rightarrow 1 / 2 \rightarrow 0 .$

21. (a) If $0 \leq x \leq 1 / 2$, then $f$ stretches $x$ to twice its value, $2 x$.