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Newton’s method is a means of finding a zero of a function $f(x)$, using the linearization of the function at a point.
Specifically, if $x_0$ is an initial guess for where a zero of $f$ may lie, we compute the linearization of $f$ at $x_0$, getting
$f(x_0)+f^\prime(x_0)(x-x_0)$, and then solve for where this linearization is zero.
This new value, $x_1=x_0-f(x_0)/f^\prime(x_0)$ is our new guess for a zero of $f$.
In particular, if $x_0$ is close enough to an actual zero that the linearization is a good approximation on the entire interval
$[x_0,x_1]$, then this new guess $x_1$ is a much better approximation to a zero of $f$.
Iterating this process gives a sequence of points $x_0,x_1,x_2,\ldots$, which can be used to numerically find
zeroes of even quite complicated functions!
You may enter a function in the $y=$ box, and optionally use the letters $a,b,c$ as parameters, which can be adjusted in the parameters menu. Adjust the slider for $x_0$ to see which zero of $f(x)$ Newton’s method converges to!