How do you use Newton's method to find the approximate solution to the equation #e^x=1/x#?

Question

Aurora Alvarado · Accepted Answer

#x=0.56714329# to 8dp.
We want to solve:

# e^x=1/x => e^x -1/x =0 #

Let #f(x) = e^x -1/x# Then our aim is to solve #f(x)=0#.

First let us look at the graphs:
graph{e^x -1/x [-5, 5, -10, 10]}

We can see there is one solution in the interval #0 le x le 1#. Let us start with an initial approximation #x=1#.

To find the solution numerically, using Newton-Rhapson method we will need the derivative #f'(x)#.

# \ \ \ \ \ \ \f(x) = e^x-1/x #
# :. f'(x) = e^x+1/x^2 #

The Newton-Rhapson method uses the following iterative sequence

# { (x_1,=1), ( x_(n+1), = x_n - f(x_n)/(f'(x_n)) ) :} #

Then using excel working to 8dp we can tabulate the iterations as follows:

We could equally use a modern scientific graphing calculator as most new calculators have an " Ans " button that allows the last calculated result to be used as the input of an iterated expression.

And we conclude that the solution is #x=0.56714329# to 8dp.

Axel Alvarez · Answer

undefined To use Newton's method to find the approximate solution to the equation $ e^x = \frac{1}{x} $, follow these steps:

1. Start with an initial guess $ x_0 $ for the solution.

2. Find the derivative of the function $ f(x) = e^x - \frac{1}{x} $. The derivative is $ f'(x) = e^x + \frac{1}{x^2} $.

3. Use the formula for Newton's method to update the guess:
$$ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} $$

4. Repeat step 3 until the value of $ x $ converges to the desired accuracy.

5. The final value of $ x $ obtained after convergence is the approximate solution to the equation $ e^x = \frac{1}{x} $.

By following these steps, you can use Newton's method to find the approximate solution to the given equation.