Okay, in this video, I just want to do a quick discussion about Newton's method and how it may not work. So, this is the Newton's method formula. Hopefully, you have seen other examples on how to compute things. This is a basic geometric idea. Given the following equation and initial guess, we want to know why Newton's method would be bad. Suppose we want to find the zeros or the roots of the equation (X - 2)^2 - 1, and we start with the first guess of 2. We want to plug in a number that gives us something close to the solution. If we plug 2 into the equation, we'll get (2 - 2)^2 - 1, which equals -1. In one of the other videos, -1 was actually a good initial guess, so we started using it and everything worked out great. Now, what's the issue? Why would x = 2 be bad here? Let's graph the function real quick. X^2 is a parabola, and when we have (X - 2) on the inside, it shifts the parabola two units to the right. Then, when we subtract 1, it moves the parabola down one unit. The vertex of our parabola would be at (2, -1), and it will cross the x-axis somewhere. However, let's talk about why things can go wrong geometrically. Newton's method looks at a tangent line at the initial guess. The goal is to pick a point where the tangent line will hit the x-axis. If that happens, the x-intercept becomes the new point for the next iteration. So, let's say we choose a different initial guess, like 2.6. Then, the tangent line would look something like this. By using Newton's method, we can find the x-intercept of the tangent line, which becomes the new...

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