ghtly off. Let's now illustrate how we
    can use polynomials to give a first, second, and third order
    approximation of the volume if the radius is 3.02m instead of
    3m. We start with $V=\frac{4}{3} \pi r^3$ and then compute the
    derivatives $$V'=4\pi r^2, V''=8\pi r, \text{ and }
    V'''=8\pi.$$ Because we are approximating the increase in
    volume from $r=3$ to something new, we'll create our polynomial
    approximations centered at $r=3$. We'll consider the polynomial
    $$P(r)=a_0+a_1(r-