Use the Taylor polynomial of degree up to 3 for y = cos x along with Taylor's inequality to estimate (integral from 0 to 1) of cos(x^2) and give a bound on the error that your estimate makes.
I'm not quite sure how to do this.
Use the Taylor polynomial of degree up to 3 for y = cos x along with Taylor's inequality to estimate (integral from 0 to 1) of cos(x^2) and give a bound on the error that your estimate makes.
I'm not quite sure how to do this.
$\displaystyle \int_0^1 \cos(x^2)\approx\int_0^1 (1-\frac{x^4}{2}) = \int_0^1 1-\int_0^1 \frac{x^4}{2}$
Also, you need to find the remainder term to find the bound on error. You know... $\displaystyle R_n(x)= \frac{f^{n+1}(x)}{n!}$
Yes, this is correct. Note that if you want $\displaystyle equality$ to $\displaystyle cos(x^2),$ use the $\displaystyle \infty ^{th}$ order expansion, which is what I implied with $\displaystyle +...$
I've edited the post to show we are finding an $\displaystyle approximation.$