How do you use Newton's method to find the approximate solution to the equation #e^x=1/x#?
#x=0.56714329# to 8dp.
We want to solve:
# e^x=1/x => e^x 1/x =0 #
Let
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
To find the solution numerically, using NewtonRhapson method we will need the derivative
# \ \ \ \ \ \ \f(x) = e^x1/x #
# :. f'(x) = e^x+1/x^2 #
The NewtonRhapson 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
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To use Newton's method to find the approximate solution to the equation ( e^x = \frac{1}{x} ), follow these steps:

Start with an initial guess ( x_0 ) for the solution.

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

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

Repeat step 3 until the value of ( x ) converges to the desired accuracy.

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.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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