Question

How do you use Newton's method to find the approximate solution to the equation `e^x+x=4`?

Answer 1

`x=1.0737` to 4dp

Explanation:

If `e^x+x=4 => e^x+x-4=0`

Let `f(x) = e^x+x-4` Then our aim is to solve `f(x)=0` in the interval `-oo lt x lt oo`

First let us look at the graphs:
graph{e^x+x-4 [-15, 15, -20, 15]}

We can see there is one solution in the interval `0 le x le 2`.

We can find the solution numerically, using Newton-Rhapson method

`\ \ \ \ \ \ \f(x) = e^x+x-4`
`:. f'(x) = e^x+1`

The Newton-Rhapson method uses the following iterative sequence

`{ (x_0,=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=1.0737` to 4dp