MAIN POINTS
The first section of the reading derives the Newton method by creating a Taylor polynomial where f(p) = 0. If the assumption is made that |p-p0| is very small, then only the initial taylor terms are needed— the polynomial is made to equal to 0, and then solved for p. (Because p=x, this is essentially solving for x). After explaining the process as one of tangents (which I didn't quite get), the text describes the algorithm for using the Newton method, including a discussion of Tolerance. It also mentions that the Newton method fails when f'(x)=0. Ex.1 shows an instance of this, because -sin(0)=0.
The second part of the reading describes the Secant method, which iterates through different secants until the answer is within the tolerance. This is described as slightly slower than the Newton method.

CHALLENGES
I've used Newton's method before, so it wasn't quite a challenge. However, the diagram on p. 64 was a little unclear— about how Newton's method actually works graphically. The secant method /was/ confusing to me, because I don't think the actual description of how it works was included in the reading assignment. This is another thing where fitting the illustration to the formula would be helpful for understanding the concept itself (rather than just the equation).

REFLECTION
I learned the Newton method in high school, where we learned how to use the graphing calculator to make it go quickly. This was my first exposure to numerical methods! This seems like one of the optimal methods scientifically, if you are given explicit formulas, however it is not guaranteed to produce results. I'm wondering how a computer might be able to tell if it is producing results. Plus, in order to carry it out, the computer would have to be able to estimate derivatives— and what if it used the Newton method to do that— and ended up in an infinite cycle of using derivatives to estimate derivatives...