Interpretations of Partial Derivatives — Question 8

Question 8

Cylinder volume is $V(r,h)=\pi r^2h$.

Tasks

1. Interpret $V_r,V_h$ geometrically.

2. Evaluate at $(3,10)$ cm.

3. Estimate separate effects of $0.1$ cm increases.

Question 8 – Solution

Strategy. Connect derivatives to familiar areas.

Step 1: Geometry $V_r=2\pi rh$ is circumference times height; $V_h=\pi r^2$ is base area.

Step 2: Values $\boxed{V_r=60\pi\ \mathrm{cm^2}}$, $\boxed{V_h=9\pi\ \mathrm{cm^2}}$.

Step 3: Estimates $\Delta V_r\approx\boxed{6\pi\ \mathrm{cm^3}}$ and $\Delta V_h\approx\boxed{0.9\pi\ \mathrm{cm^3}}$. The radial change has the larger first-order effect.