How do you use Newton's method to find the approximate solution to the equation #x^3+5x-10=0#?

Question

Cameron Alexander · Accepted Answer

#x=1.423318 # to 6dp

Let #f(x) = x^3+5x-10# Then our aim is to solve #f(x)=0#

First let us look at the graphs:
graph{x^3+5x-10 [-5, 5, -20, 15]}

We can see there is one solution in the interval # 1 < x < 2 #.

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

# f(x) = x^3+5x-10 => f'(x) = 3x^2+5 #, and using the Newton-Rhapson method we use the following iterative sequence

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

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

And we conclude that the remaining solution is #x=1.423318 # to 6dp

Paisley Johnson · Answer

undefined To use Newton's method to find the approximate solution to the equation $x^3 + 5x - 10 = 0$:

1. Choose an initial guess, $x_0$.
2. Iterate using the formula: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$, where $f(x)$ is the function and $f'(x)$ is its derivative.
3. Repeat the iteration until the desired level of accuracy is reached or until the process converges.

For the given equation, $f(x) = x^3 + 5x - 10$. Taking the derivative, $f'(x) = 3x^2 + 5$.

Choose an initial guess, $x_0$, and substitute it into the iteration formula until convergence is achieved.