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

Question

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

Ezekiel Alexander · Accepted Answer

#x=-0.72449# to 5dp

We have:

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

Let #f(x) = x^4-x-1# Then our aim is to solve #f(x)=0#

First let us look at the graph:
graph{x^4-x-1 [-3, 3, -5, 8]}

We can see there is one solution in the interval # -1 < x < 0 # and a solution in # 1 < x < 2 #

In order to find the solution numerically, using Newton-Rhapson method, we need the derivative #f'(x)#

# f(x) = x^4-x-1 => f'(x) = 4x^3-1 #,
and the Newton-Rhapson method uses the following iterative sequence

# { (x_0,=0), ( x_(n+1), = x_n - f(x_n)/(f'(x_n)) ) :} #

Then using excel working to 5dp we can tabulate the iterations as follows:

And we conclude that a solution is #x=-0.72449# to 5dp

Charles Arocha · Answer

undefined To use Newton's method to find an approximate solution to the equation $x^4 = x + 1$, where $x < 0$, follow these steps:

1. Choose an initial guess $x_0$ for the solution.
2. Calculate the derivative $f'(x)$ of the function $f(x) = x^4 - x - 1$.
3. Use the formula for Newton's method iteration: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$.
4. Iterate using the formula until you reach a satisfactory approximation, i.e., until $|x_{n+1} - x_n|$ is sufficiently small.

Given that $x < 0$, choose a suitable initial guess for $x_0$, then apply the Newton's method iteration formula to find the approximate solution. Repeat the process until convergence is achieved.

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

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