# How do you use Newton's method to find the approximate solution to the equation #tanx=e^x, 0<x<pi/2#?

Let

First let us look at the graphs:

graph{tanx-e^x [-1, 5, -15, 15]}

We can see there is one solution in the interval

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

Then using excel working to 6dp we can tabulate the iterations as follows:

And we conclude that the remaining solution is

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To use Newton's method to find the approximate solution to the equation ( \tan(x) = e^x ), within the interval ( 0 < x < \frac{\pi}{2} ), follow these steps:

- Start with an initial guess for the solution, ( x_0 ), within the specified interval.
- Calculate the derivative of the function ( f(x) = \tan(x) - e^x ), which is ( f'(x) = \sec^2(x) - e^x ).
- Use the formula ( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} ) iteratively, where ( x_{n+1} ) is the next approximation, and ( x_n ) is the current approximation.
- Repeat the iteration until the desired level of accuracy is achieved.

Note: Since the initial guess is crucial for the convergence of Newton's method, it's important to choose it wisely to ensure convergence towards the desired solution within the specified interval.

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When evaluating a one-sided 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 one-sided 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 one-sided 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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