Chapter 4, Problem 9RCC

a.

To determine

The Newton’s method of obtaining the second approximation by using a graph.

Explanation of Solution

Given information:

The initial approximation is $x_1$ to a root of the equation $f(x)=0$ .

Calculations:

Newton’s method of obtaining the second approximation is done by using the formula,

  $\begin{array}{l}{x}_{n+1}=x_n-\frac{f(x_n)}{f\text{'}(x_n)}\\ x_2=x_1-\frac{f(x_1)}{f\text{'}(x_1)}\end{array}$

For getting the third root the same formula is followed. The process is kept on repeating and it finally converges to a desired root.

  

b.

To determine

The expression for $x_2$ in terms of $x_1,f(x_1)\text{ and }f\text{'}(x_1)$

Explanation of Solution

Given information:

The initial approximation is $x_1$ to a root of the equation $f(x)=0$ .

Calculations:

Newton’s method of obtaining the second approximation is done by using the formula,

  $\begin{array}{l}{x}_{n+1}=x_n-\frac{f(x_n)}{f\text{'}(x_n)}\\ x_2=x_1-\frac{f(x_1)}{f\text{'}(x_1)}\end{array}$

For getting the third root the same formula is followed. The process is kept on repeating and it finally converges to a desired root.

c.

To determine

The expression for $x_{n+1}$ in terms of $x_n,f(x_n)\text{ and }f\text{'}(x_n)$

Explanation of Solution

Given information:

The initial approximation is $x_1$ to a root of the equation $f(x)=0$ .

Calculations:

Newton’s method of obtaining the second approximation is done by using the formula,

  $\begin{array}{l}{x}_{n+1}=x_n-\frac{f(x_n)}{f\text{'}(x_n)}\\ x_2=x_1-\frac{f(x_1)}{f\text{'}(x_1)}\end{array}$

For getting the third root the same formula is followed. The process is kept on repeating and it finally converges to a desired root.

d.

To determine

The circumstances in which Newton’s method is likely to fail or work very ssslow.

Explanation of Solution

Given information:

The initial approximation is $x_1$ to a root of the equation $f(x)=0$ .

Calculations:

Newton’s method of obtaining the approximation is done by using the formula,

  $x_{n+1}=x_n-\frac{f(x_n)}{f\text{'}(x_n)}$

Several approximations are taken and the process is kept on repeating until it finally converges to a desired root. Under certain circumstances the method fails or works very slowly. This happens when $f\text{'}(x_1)$ is close to zero. Also there can be certain situation when the approximation is outside the domain of $f$ . In such cases the method will fail.