BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 4, Problem 9RCC

a.

To determine

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

Expert Solution

Explanation of Solution

Given information:

The initial approximation is x1 to a root of the equation f(x)=0 .

Calculations:

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

  xn+1=xnf(xn)f'(xn)x2=x1f(x1)f'(x1)

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

  Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 4, Problem 9RCC

b.

To determine

The expression for x2 in terms of x1,f(x1) and f'(x1) 

Expert Solution

Explanation of Solution

Given information:

The initial approximation is x1 to a root of the equation f(x)=0 .

Calculations:

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

  xn+1=xnf(xn)f'(xn)x2=x1f(x1)f'(x1)

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 xn+1 in terms of xn,f(xn) and f'(xn) 

Expert Solution

Explanation of Solution

Given information:

The initial approximation is x1 to a root of the equation f(x)=0 .

Calculations:

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

  xn+1=xnf(xn)f'(xn)x2=x1f(x1)f'(x1)

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.

Expert Solution

Explanation of Solution

Given information:

The initial approximation is x1 to a root of the equation f(x)=0 .

Calculations:

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

  xn+1=xnf(xn)f'(xn)

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'(x1) 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.

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