Chapter 12.8, Problem 35E

(a)

To determine

To find:the value of the roots through calculator.

Answer to Problem 35E

From calculator the value of the under root can be measured.

Explanation of Solution

Given:

  $\sqrt{2},\sqrt{\sqrt{2}},\sqrt{\sqrt{\sqrt{2}}}\text{ and }\sqrt{\sqrt{\sqrt{\sqrt{2}}}}.$

Concept used:

Iteration method:

Let the given equation be $f\left(x\right)=0$ and the value of $x$ to be determined. By using the iteration method and it can find the roots of the equation.

To find the root of the equation first it had to write equation like below:

  $x=\pi \left(x\right)$ .

Let $x=x_0$ be an initial approximation of the required root $\alpha$ then the first approximation $x_1$ is given by:

  $x_1=\pi \left(x_0\right)$ .

Similarly,

  $\begin{array}{l}{x}_2=\pi \left(x_1\right).\\ x_3=\pi \left(x_2\right).\\ x_4=\pi \left(x_3\right).\\ x_n=\pi \left(x_{n-1}\right).\end{array}$

Calculation:

The value of the under root can be measured as:

  $\begin{array}{l}\sqrt{2}=\mathrm{1.41421356.}\\ \sqrt{\sqrt{2}}=\mathrm{1.189207115.}\\ \sqrt{\sqrt{\sqrt{2}}}=\mathrm{1.090507733.}\\ \sqrt{\sqrt{\sqrt{\sqrt{2}}}}=\mathrm{1.044273782.}\end{array}$

Hence, from calculator the value of the under root can be measured.

(b)

To determine

To find:the definition of an iteration function $f\left(z\right)$ that models the situation describes in part a.

Answer to Problem 35E

  $f\left(z\right)=\sqrt{z},z_0=2.$

Explanation of Solution

Given:

  $\sqrt{2},\sqrt{\sqrt{2}},\sqrt{\sqrt{\sqrt{2}}}\text{ and }\sqrt{\sqrt{\sqrt{\sqrt{2}}}}.$

Concept used:

Iteration method:

Let the given equation be $f\left(x\right)=0$ and the value of $x$ to be determined. By using the iteration method and it can find the roots of the equation.

To find the root of the equation first it had to write equation like below:

  $x=\pi \left(x\right)$ .

Let $x=x_0$ be an initial approximation of the required root $\alpha$ then the first approximation $x_1$ is given by:

  $x_1=\pi \left(x_0\right)$ .

Similarly,

  $\begin{array}{l}{x}_2=\pi \left(x_1\right).\\ x_3=\pi \left(x_2\right).\\ x_4=\pi \left(x_3\right).\\ x_n=\pi \left(x_{n-1}\right).\end{array}$

Calculation:

The function is dependent on $z$ .

  $f\left(z\right)=\sqrt{z},z_0=2.$

Hence, the function is $f\left(z\right)=\sqrt{z},z_0=2.$

(c)

To determine

To find:the determination of the limit of $f\left(z\right)$ as the number of iterations approaches infinity values of $z_0>0$ .

Answer to Problem 35E

The limit of $f\left(z\right)$ as the number of iterations approaches infinity as $1$ .

Explanation of Solution

Given:

  $\sqrt{2},\sqrt{\sqrt{2}},\sqrt{\sqrt{\sqrt{2}}}\text{ and }\sqrt{\sqrt{\sqrt{\sqrt{2}}}}.$

Concept used:

Iteration method:

Let the given equation be $f\left(x\right)=0$ and the value of $x$ to be determined. By using the iteration method and it can find the roots of the equation.

To find the root of the equation first it had to write equation like below:

  $x=\pi \left(x\right)$ .

Let $x=x_0$ be an initial approximation of the required root $\alpha$ then the first approximation $x_1$ is given by:

  $x_1=\pi \left(x_0\right)$ .

Similarly,

  $\begin{array}{l}{x}_2=\pi \left(x_1\right).\\ x_3=\pi \left(x_2\right).\\ x_4=\pi \left(x_3\right).\\ x_n=\pi \left(x_{n-1}\right).\end{array}$

Calculation:

The limit of $f\left(z\right)$ as the number of iterations approaches infinity as $1$ .

Hence, the limit of $f\left(z\right)$ as the number of iterations approaches infinity as $1$ .