# Newton's method for Calc I

Newton's method is a technique to find numerical approximations to roots of functions. Given an initial guesss $$x_1$$, Newton's method improves this guess by applying the function $N(x) = x - \frac{f(x)}{f'(x)}.$ This produces $$x_2 = N(x_1)$$. We then plug that back in to get $$x_3$$ and continue. More generally, we produce a sequence $$(x_k)$$ via $$x_k=N(x_{k-1})$$.

## Example

Let $$f(x) = x^3-x-1$$. It's evident from a graph that there's one root.

Newton's method works by riding the tangent line from an initial guess. If we note that $$x_1=2$$ is pretty close to the root, we compute $$x_2 = N(x_2)$$, where $N(x) = x - \frac{f(x)}{f'(x)} = x - \frac{x^3-x-1}{3x^2-1}.$ Thus, $$x_2 = N(2) = 2-5/11 \approx 1.54545$$. Geometrically, this point is obtained by riding the tangent line to the $$x$$-axis:

If we do that again, we end up even closer to the root:

That's why we iterate!

## Resources for Newton's method and solving equations

### Performing Newton's method

To actually perform several Newton iterates, it makes sense to use a programming tool. You can do it with Sage like this.