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.

Exercises

1. Use a numerical tool to solve the following equations. Be sure to find all solutions
1. $$x^5-x-1 = 0$$
2. $$x^5-2x-1 = 0$$
3. $$\sin(3x) = x/2$$
2. For each of the following functons, take three Newton steps from the given initial point
1. $$f(x) = x^2 - 2$$ from $$x_1 = 2.0$$
2. $$f(x) = \sin(x)$$ from $$x_1 = 3.0$$
3. $$f(x) = x^5-x-1$$ from $$x_1 = 1.0$$
3. The equation $$\sin(x)=x/9$$ has 7 solutions. We want to find an approximation to the largest solution. Use a graph to find a good initial approximation for your $$x_1$$ and apply Newton's method from that point obtain the approximation.