Interpretations of Partial Derivatives — Question 6

Question 6

Near $(2,3)$, the columns give $x$ and the rows give $y$: $$\begin{array}{c|ccc}&1.9&2.0&2.1\\\hline 2.9&7.62&7.80&7.98\\3.0&7.71&7.90&8.09\\3.1&7.80&8.00&8.20\end{array}$$ Tasks

1. Estimate $f_x(2,3)$ centrally.

2. Estimate $f_y(2,3)$ centrally.

3. Interpret both.

Question 6 – Solution

Strategy. Use the center row and column.

Step 1: $x$ rate $f_x\approx(8.09-7.71)/.2=\boxed{1.90}$.

Step 2: $y$ rate $f_y\approx(8.00-7.80)/.2=\boxed{1.00}$.

Step 3: Meaning Output rises about $1.90$ per $x$ unit with $y$ fixed and $1.00$ per $y$ unit with $x$ fixed. Symmetric differences reduce one-sided bias.