SOLUTION 25: The cube has an edge length of $30$ cm and it's frosting is $0.5$ cm thick. Recall that the volume of a cube of edge length $x$ is $$V = x^3$$

and choose
$$x-values: 30 \rightarrow 30 +(0.5 + 0.5)= 31$$
so that
$$\Delta x = 31-30 = 1$$

The derivative of $V$ is
$$V'= 3 x^2$$
$\ \ \ \$ a.) The exact volume of the frosting is the exact change of $V-$values and is
$$\Delta V = V(31) - V(30)$$ $$= (31)^3 - (30)^3$$ $$= 29,791 - 27,000$$ $$= 2791 \ cm^3$$
$\ \ \ \$ b.) An estimate for volume of the frosting is the Differential of $V$ (since $\Delta V \approx dV)$ and is
$$dV = V'(30) \ \Delta x$$ $$= 3 (30)^2 \cdot (1)$$ $$= 2700 \ cm^3$$