Interpretations of Partial Derivatives — Question 5

Question 5 – Solution

Strategy. A trace slope is the tangent of its signed inclination.

Step 1: Rise/run A one-meter positive $x$ move predicts $-0.25$ m elevation change; a positive $y$ move predicts $0.60$ m, separately.

Step 2: Angles $\boxed{\alpha_x=\arctan(-.25)\approx-14.0^\circ}$ and $\boxed{\alpha_y=\arctan(.60)\approx 31.0^\circ}$.

Step 3: Compare Since $.60>.25$, the $y$ trace is steeper; its positive sign indicates ascent.