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

1 Answer
Steve M
Feb 22, 2017

#x=0.56714329# to 8dp.

Explanation:

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:
enter image source here

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.

Related questions
See all questions in Using Newton's Method to Approximate Solutions to Equations
Impact of this question
7598 views around the world
You can reuse this answer
Creative Commons License