# How do you use Newton's method to find the approximate solution to the equation #x^4=x+1,x>0#?

# x_1 = 1.2207440846 #

We have:

# x^4 = x+1 => x^4-x-1=0#

Let:

# f(x) = x^4-x-1 #

Our aim is to solve

graph{x^4-x-1 [-4, 4, -5, 10]}

We can see that there are two solutions; one solution in the interval

To find the solution numerically, using Newton-Rhapson method we use the following iterative sequence

# { (x_1,=x_0), ( x_(n+1), = x_n - f(x_n)/(f'(x_n)) ) :} #

Therefore we need the derivative:

# \ \ \ \ \ \ \f(x) = x^4-x-1 #

# :. f'(x) = 4x^3-1 #

Then using excel working to 10dp we can tabulate the iterations as follows:

Initial Value

Incidentally, we can also find the other solution

Initial Value

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 (to 10dp) are:

# x_1 = 1.2207440846 #

# x_2 = -0.7244919590 #

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To use Newton's method to find the approximate solution to the equation ( x^4 = x + 1 ) where ( x > 0 ), follow these steps:

- Start with an initial guess ( x_0 ) for the solution.
- Use the formula ( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} ) to iterate and improve the approximation, where ( f(x) = x^4 - x - 1 ) is the function and ( f'(x) ) is its derivative.
- Calculate ( f(x_n) ) and ( f'(x_n) ) at each iteration and update the value of ( x ) until you reach the desired level of accuracy.

Repeat these steps until the difference between consecutive approximations is sufficiently small or until the desired level of accuracy is achieved.

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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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