Interpretations of Partial Derivatives — Question 1

Question 1

For $f(x,y)=x^2+3xy-y^2$ at $P=(1,2,3)$:

Tasks

1. Find $f_x,f_y$ at $(1,2)$.

2. Interpret them as trace slopes.

3. Write both trace tangent lines.

Question 1 – Solution

Strategy. Hold one input fixed at a time.

Step 1: Rates $f_x=2x+3y$, $f_y=3x-2y$, hence $\boxed{f_x(1,2)=8}$ and $\boxed{f_y(1,2)=-1}$.

Step 2: Meaning With $y=2$, height rises $8$ units per $x$ unit; with $x=1$, it falls $1$ unit per $y$ unit.

Step 3: Lines $\boxed{y=2,\ z-3=8(x-1)}$ and $\boxed{x=1,\ z-3=-(y-2)}$.