Interpretations of Partial Derivatives — Question 10

Question 10

At the same input, models satisfy $A_x=5,A_y=-2$ and $B_x=-1,B_y=-2$.

Tasks

1. Compare positive-$x$ responses.

2. Compare positive-$y$ responses.

3. State what cannot be concluded from these data.

Question 10 – Solution

Strategy. Partial derivatives give local coordinate-specific rates.

Step 1: $x$ $A$ locally rises at $5$ per unit; $B$ falls at $1$ per unit. Their signs oppose and $A$ has greater absolute sensitivity.

Step 2: $y$ Both locally fall at $2$ per positive $y$ unit.

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