Problem:
a) Find the Taylor polynomial $T_6(x)$ for $f(x) = \sin(x)$ about $x=0$.
I found this to be $x-\frac{x^3}{6} + \frac{x^5}{120} + O(x^6)$.
b) Use this to estimate $\int_0^1\sin(x^2)dx$ with an error less than $0.00002$.
This is where I hit the wall.
Question:
How can I use the Taylor polynomial for a (slightly) different function, and how can I tell when I've come within a certain range of the analytical answer?
Update:
Since $f(x)$ and all its derivatives are continuous on the interval $[0,1]$, I may use that the error $|R_6(x^2)| \leq \frac M{7!}\int_0^1x^6dx = \frac{M}{7\cdot7!}$, if I haven't made any mistakes. Now I need to find an $M$ which satisfies $M\geq |f^{(7)}(x)$ on the interval. Do I need to find the 7th derivative of $\sin(x^2)$ for this, or may I use the 7th derivative of $\sin(x)$ somehow?