Wed 1/30 Taylor Series

Main Points
This chapter is simply a list of calculus theorems with examples of these theorems being used. The 4 pages given state Taylor's theorem, and then show how to solve for low-order Taylor polynomials for f(x)=cosx, approximating cos(0.01), and then approximating a simple definite integral of cosx. Finding the polynomials is a simple matter of subsituting into the formula. Approximating cox(0.01) is another matter of substitution, and the truncation error is calculated by using the knowledge that Xi(x) lies in the interval [-1,1] (trivially), and then in the interval [0,0.1] (using Taylor's theorem). The integral approximation is performed by simply integrating the polynomial produced by simplifying the third-order Taylor polynomial for f(x)=cosx. The error is then calculated by integrating the Taylor polynomial remainder term within the same bounds.

Challenges
I was confused about why the example first uses [-1,1] as the error bound for Xi(x), even though we know by Taylor's theorem that Xi(x) is between 0 and 0.01. I did not find this reading challenging, but I feel like the logic tying the Taylor theorem to the actual approximation of the function could be difficult, because the Taylor's theorem does not explicitly state that it is for approximation with known error bounds. Also, the text did not really point out how error grows as x increases in distance from x0.

Reflections
I thought it was really elegant how the Taylor polynomial can be integrated to approximate the integral of the actual function, and even cooler how you can extend this to a calculation of the error bound. I see this theorem as playing an important part in numerical approximation of functions and integrals, although I am uncertain how we would deal with finding all of the derivatives using a computer.