Find the slope of the curve at the indicated point.
-A cubic salt crystal expands by accumulation on all sides. As it expands outward find the rate of change of its volume with respect to the length of an edge when the edge is $0.33 \mathrm {~mm}$.
A) $0.11 \mathrm {~mm} ^ { 3 } / \mathrm { mm }$
B) $32.67 \mathrm {~mm} ^ { 3 } / \mathrm { mm }$
C) $0.3267 \mathrm {~mm} ^ { 3 } / \mathrm { mm }$
D) $3.27 \mathrm {~mm} ^ { 3 } / \mathrm { mm }$

Question

Quizplus · Accepted Answer

Let the length of the edge be $x$. Then the volume $V$ of the cube is given by $V=x^3$. Taking the derivative with respect to $x$, we get $\frac{dV}{dx}=3x^2$, which represents the rate of change of volume with respect to the edge length. Evaluating at $x=0.33	ext{ mm}$ gives $\frac{dV}{dx}\bigg|_{x=0.33} = 3(0.33)^2 \approx 0.3267	ext{ mm}^3/	ext{mm}$. Thus, the correct answer is $\boxed{	extbf{(C)}}$.

Find the Slope of the Curve at the Indicated Point 0.33 mm0.33 \mathrm {~mm}0.33 mm

Find the Slope of the Curve at the Indicated Point $0.33 \mathrm {~mm}$