Moderate# Taylor's Theorem

SVCALC-YXL7PQ

Let $f(x)=e^x$ and let $T_2(x)=c_0+c_1x+c_2x^2$ be its second degree Taylor polynomial at $x=0$.

To 3 decimal places, find the number $c$ between $0$ and $2$ guaranteed to exist by Taylor's theorem that satisfies the equation:

$$f(2)-T_2(2)=\frac{f^{(3)}(c)}{3!}2^3$$

*This problem requires a calculator.*