Comment on Activity 2

As you calculate successive derivatives of `f text[(] x text[)]`, you should find a repeating pattern: `cos x, -sin x, -cos x, sin x`. Since the fourth derivative is `f text[(] x text[)]` itself, you now know all the derivatives. When you evaluate these functions at `x=0` (starting with the `0`th derivative, `sin x`), you get a repeating pattern of numbers: `0, 1, 0, -1, 0, 1`, and so on. Thus the first six coefficients are `0, 1, 0, -1//3!, 0`, and `1//5!`. This gives us the following six polynomials:

Notice that, because the even-degree coefficients are all zero, `P_2` is the same as `P_1`, `P_4` is the same as `P_3`, and so on. Counting the constant function `P_0`, there are four distinct functions among the first six Taylor approximations.