Question

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

Answer 1

`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:

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.