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Calculus IIIChapter 2: Partial DerivativesSection 2-3

Interpretations of Partial Derivatives — Question 8

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Question 8

Cylinder volume is V(r,h)=πr2hV(r,h)=\pi r^2h.

Tasks

  1. Interpret Vr,VhV_r,V_h geometrically.

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

  3. Estimate separate effects of 0.10.1 cm increases.

Original worksheet page 1: question and worked solution for 2-3-008
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Question 8 – Solution

Strategy. Connect derivatives to familiar areas.

Step 1: Geometry Vr=2πrhV_r=2\pi rh is circumference times height; Vh=πr2V_h=\pi r^2 is base area.

Step 2: Values Vr=60πcm2\boxed{V_r=60\pi\ \mathrm{cm^2}}, Vh=9πcm2\boxed{V_h=9\pi\ \mathrm{cm^2}}.

Step 3: Estimates ΔVr≈6πcm3\Delta V_r\approx\boxed{6\pi\ \mathrm{cm^3}} and ΔVh≈0.9πcm3\Delta V_h\approx\boxed{0.9\pi\ \mathrm{cm^3}}. The radial change has the larger first-order effect.

Original worksheet page 2: question and worked solution for 2-3-008

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