Calculating Newton's Method: Step-by-Step Guide

• jisbon
Then use the result to approximate the value of ##c## using Newton's method.In summary, the conversation discusses using Newton's method to approximate a stationary value of a given function. The method involves using the formula x_{n+1}=x_{n}-\frac{f(x_{n})}{f'(x_{n})} and starting with an initial value of x_{0}. The output of the first iteration, x_{1}, is then used in subsequent iterations. The conversation confirms that the first iteration is correct and advises to proceed by evaluating cos(0) and sin(0) and substituting in the value for x_{0}. Then, the result can be used to approximate the value of c using Newton's
jisbon
Homework Statement
Let $$f(x) = \begin{cases} \dfrac{x^3-1}{\sqrt{x}-1}, & x > 1\\ cos(x-1)-x^2, & x \leq 1\end{cases}$$

Use Newton's method with ##x_{0} =1##, compute the second iterate to approximate value ##c## where ##c## is a stationary value that lies in the x-axis for some ##0<c<1##
Relevant Equations
-
Since the Newton's method is as follows:

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

$$x_{1}=x_{0}-\frac{cos(0)-1}{-sin(0)-2}$$

Is this correct? What should I proceed on from here?

Last edited by a moderator:
jisbon said:
Homework Statement: Let $$f(x) = \begin{cases} \dfrac{x^3-1}{\sqrt{x}-1}, & x > 1\\ cos(x-1)-x^2, & x \leq 1\end{cases}$$

Use Newton's method with ##x_{0} =1##, compute the second iterate to approximate value ##c## where ##c## is a stationary value that lies in the x-axis for some ##0<c<1##
Homework Equations: -

Since the Newton's method is as follows:

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

$$x_{1}=x_{0}-\frac{cos(0)-1}{-sin(0)-2}$$

Is this correct?
Yes, as far as you went
jisbon said:
What should I proceed on from here?
Evaluate cos(0) and sin(0) and substitute in the value for ##x_0##.

1. What is Newton's Method?

Newton's Method is an algorithm used to find the roots of a function. It is also known as the Newton-Raphson Method, named after mathematicians Isaac Newton and Joseph Raphson.

2. How does Newton's Method work?

The method involves taking an initial guess for the root of a function and then repeatedly applying a simple formula to refine the guess. This process continues until the desired level of accuracy is reached. The formula is: xn+1 = xn - f(xn) / f'(xn), where xn is the current guess and f(xn) and f'(xn) are the function and its derivative evaluated at xn.

3. Why is Newton's Method useful?

Newton's Method is a powerful tool for finding the roots of complex functions. It is often used in numerical analysis and optimization problems. It is also useful in finding the maximum or minimum of a function.

4. What are the limitations of Newton's Method?

Newton's Method may not always converge to the desired root, especially if the initial guess is far from the actual root. It also requires knowledge of the function and its derivative, which may not always be readily available. In addition, the method may produce multiple solutions or fail to find a solution if the function has multiple roots or is not continuous.

5. Can Newton's Method be used for all types of functions?

No, Newton's Method is best suited for finding the roots of continuous and differentiable functions. It may not work well for functions with sharp turns or discontinuities. Additionally, the method may fail to converge if the function is very flat near the root.

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