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Calculus IIIChapter 2: Partial DerivativesSection 2-3

Interpretations of Partial Derivatives — Question 10

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Question 10

At the same input, models satisfy Ax=5,Ay=−2A_x=5,A_y=-2 and Bx=−1,By=−2B_x=-1,B_y=-2.

Tasks

  1. Compare positive-xx responses.

  2. Compare positive-yy responses.

  3. State what cannot be concluded from these data.

Original worksheet page 1: question and worked solution for 2-3-010
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Question 10 – Solution

Strategy. Partial derivatives give local coordinate-specific rates.

Step 1: xx AA locally rises at 55 per unit; BB falls at 11 per unit. Their signs oppose and AA has greater absolute sensitivity.

Step 2: yy Both locally fall at 22 per positive yy unit.

Step 3: Limits We cannot infer A>BA>B, global monotonicity, or exact finite changes without more information. The data justify local rate comparisons only.\boxed{\text{The data justify local rate comparisons only.}}

Original worksheet page 2: question and worked solution for 2-3-010

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