Let .
Find the MacLaurin polynomial of degree 7 for .
Use this polynomial to estimate the value of
I don't really understand what it means by degree 7..
$\displaystyle \sin{u} = u - \frac{u^3}{3!} + \frac{u^5}{5!} - \frac{u^7}{7!} + ...$
so ...
$\displaystyle \sin(8t^2) = 8t^2 - \frac{(8t^2)^3}{3!} + \frac{(8t^2)^5}{5!} - \frac{(8t^2)^7}{7!} + ...$
$\displaystyle \sin(8t^2) = 8t^2 - \frac{8^3t^6}{3!} + \frac{8^5t^{10}}{5!} - \frac{8^7t^{14}}{7!} + ...$
since you only need a 7th degree approximation for $\displaystyle F(x)$ ...
$\displaystyle F(x) \approx \int_0^x 8t^2 - \frac{8^3t^6}{3!} \, dt
$
$\displaystyle F(x) \approx \frac{8x^3}{3} - \frac{8^3x^7}{7 \cdot 3!}$
finish up by determining the approximation for $\displaystyle F(.67)$