514. Calculate approximately $3.002^{4}$.
Solution. Consider the function $f(x)=x^{4}$. Here $x_{1}=3.002, x_{0}=3$, $\Delta x=0.002$ and, therefore, $f\left(x_{1}\right)=f(3.002)=3.002^{4}$. We find $f\left(x_{0}\right)=$ $=f(3)=3^{4}=81, f^{\prime}\left(x_{0}\right) \Delta x=4 \cdot 3^{3} \cdot 0.002=4 \cdot 27 \cdot 0.002=0.216$. Thus, $3.002^{4} \approx$ $\approx 81+0.216=81.216$.

Verification shows that the exact value of the number $3.002^{4}$ is 81.216116096016. The relative error is quite small and amounts to $0.00014 \%$.