Solving e-x=x Using Newton's Method

You should be able to find the derivative, f'(x), yourself. In summary, to solve the equation e^(-x)= x to three decimal places using Newton's method, choose an initial guess of x_0= 0 or x_0= 1 and use the iteration formula x_{n+1}= x_n- (e^{-x_n}- x_n)/(1+ e^{-x_n}). Continue until you have reached desired accuracy.
  • #1
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Homework Statement



Find, to three decimal places, the value of x such that e-x = x. Use Newton's method.

Homework Equations





The Attempt at a Solution


I looked up what Newton's method was and I found that it was

f(x)= [tex]\int x[/tex] = f(xo) + f](xo)(x-xo)

But I don't understand how I can apply it to this problem. Could someone just help me set up the problem, and I should be able to figure the problem from there. Thanks!
 
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  • #2
You are trying to solve f(x)=e^x-x=0. Newton's method tells you how to turn an initial guess for a solution x0 into a better approximation of a solution, x1. x1=x0-f'(x0)/f(x0). Take an initial guess at the solution and try it out.
 
  • #3
toasted said:

Homework Statement



Find, to three decimal places, the value of x such that e-x = x. Use Newton's method.

Homework Equations





The Attempt at a Solution


I looked up what Newton's method was and I found that it was

f(x)= [tex]\int x[/tex] = f(xo) + f](xo)(x-xo)

But I don't understand how I can apply it to this problem. Could someone just help me set up the problem, and I should be able to figure the problem from there. Thanks!
That is certainly NOT "Newton's method" for solving equations. The left side looks like the formula for the tangent line at [itex]x_0[/itex] and it is certainly not equal to f(x) (I don't know how the integral got in there).

As Dick said, to solve f(x)= 0, choose some starting value [itex]x_0[/itex] and iterate:
[tex]x_{n+1}= x_n- \frac{f(x_n)}{f'(x_n)}[/itex]
until you have sufficient accuracy.

Here, [itex]f(x)= e^{-x}- x[/itex] and either [itex]x_0= 0[/itex] or [itex]x_0= 1[/itex] will do.
 

1. What is Newton's Method?

Newton's Method is an iterative algorithm used to find the roots of a function. It involves using the derivative of the function to approximate the root and then repeating the process with the new approximation until a desired level of accuracy is reached.

2. How is Newton's Method used to solve e-x=x?

To solve e-x=x using Newton's Method, we first need to rearrange the equation to the form f(x)=0, where f(x)=e-x-x. Then, we can use the formula xn+1 = xn - f(xn)/f'(xn) to iteratively approximate the root until a desired level of accuracy is reached.

3. What are the benefits of using Newton's Method to solve equations?

Newton's Method is a fast and efficient method for finding roots of a function. It often requires fewer iterations than other methods and can converge to the root quickly, making it a popular choice for solving equations.

4. Are there any limitations to using Newton's Method?

Yes, there are some limitations to using Newton's Method. It may not always converge to the root if the initial guess is not close enough, or if the function has multiple roots. It also requires knowledge of the derivative of the function, which may not always be readily available.

5. How can I check the accuracy of the solution obtained using Newton's Method?

To check the accuracy of the solution, you can use the error formula |xn+1 - xn| to calculate the difference between successive approximations. The smaller the error, the closer the approximation is to the root. Additionally, you can plug in the solution into the original equation and see if it satisfies the equation.

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