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Calculus IIIChapter 2: Partial DerivativesSection 2-3

Interpretations of Partial Derivatives — Question 3

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Question 3

Revenue is R(p,a)=1200p−30p2+50a−2a2+4paR(p,a)=1200p-30p^2+50a-2a^2+4pa, where pp is dollars and aa is hundreds of advertising dollars.

Tasks

  1. Compute Rp(10,5),Ra(10,5)R_p(10,5),R_a(10,5).

  2. State units.

  3. Interpret both values.

Original worksheet page 1: question and worked solution for 2-3-003
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Question 3 – Solution

Strategy. Attach output units per input unit.

Step 1: Compute Rp=1200−60p+4aR_p=1200-60p+4a and Ra=50−4a+4pR_a=50-4a+4p, so Rp(10,5)=620\boxed{R_p(10,5)=620} and Ra(10,5)=70\boxed{R_a(10,5)=70}.

Step 2: Units These are revenue dollars per price dollar and revenue dollars per hundred advertising dollars.

Step 3: Meaning Locally, a $1 price rise predicts $620 more revenue; an added $100 advertising predicts $70 more, with the other input fixed.

Original worksheet page 2: question and worked solution for 2-3-003

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