Interpretations of Partial Derivatives — Question 2

Question 2

Temperature is $T(x,y)=80-2x^2-y^2$ degrees Celsius, with distance in centimeters.

Tasks

1. Find $T_x(3,4),T_y(3,4)$ with units.

2. Interpret signs and magnitudes.

3. Estimate separate changes for $0.05$ cm moves in $+x$ and $+y$.

Question 2 – Solution

Strategy. Read each partial as temperature change per centimeter with the other coordinate fixed.

Step 1: Rates $T_x=-4x$, $T_y=-2y$, so $\boxed{T_x=-12\ ^\circ\mathrm C/\mathrm{cm}}$ and $\boxed{T_y=-8\ ^\circ\mathrm C/\mathrm{cm}}$ at $(3,4)$.

Step 2: Meaning Both directions cool; the $x$ trace is locally steeper.

Step 3: Estimates $\Delta T_x\approx-12(.05)=\boxed{-0.60^\circ\mathrm C}$ and $\Delta T_y\approx-8(.05)=\boxed{-0.40^\circ\mathrm C}$.