Section 14.3
Worksheet by Jim Brown
In this worksheet we give some graphs illustrating the partial derivatives of a function of two variables $z = f(x,y)$.
Example 1: Consider the function $ z = f(x,y) = x^2  2 y^2$. We have the point $(1,1,3)$ on the graph and consider the partial derivatives of $f(x,y)$ with respect to $x$ and $y$ at this point. The point is in black on the graph.
(x, y) (x, y) 
To calculate the partial derivative of $f(x,y)$ with respect to $x$ at $(1,1,3)$, we fix $y = 1$ and consider the slope of the tangent line at the point $(1,1,3)$ to the curve $z = f(x,1)$. The curve is given in yellow and the tangent line is given in red. Thus, we have $\frac{\partial f}{\partial x}(1,1,3) = 2$ is the slope of the red line.

We now add the curve $z = f(1,y)$ and the tangent line to this curve at $(1,1,3)$. The curve is given in green and the tangent line in purple. We have the quantity $\frac{\partial f}{\partial y}(1,1,3)$ is the slope of the purple line.
