Functions of Two Variables

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Objectives:

1.: Understand what a function of two variables is.
2.: Be able to identify the domain (and possibly range) of a function of two variables.
3.: Be able to find the level curves of a function of two variables.
4.: Be able to identify graphs of paraboloids, cones, spheres, planes, cylinders.

Recap Video

The following video recaps the main ideas of the section.

Example: Level Curves

The following video works out an example on level curves:

Sketch a contour map for the function $f(x,y) = \ln (y-x^2)$ . Include at least four level curves.

Problems

Which of the following is the whole domain of the function $f(x,y) = \displaystyle e^{y/x}$ ?

All points $(x,y)$ in $\mathbb {R}^2$ All points $(x,y)$ in $\mathbb {R}^2$ with $x \neq 0$ All points $(x,y)$ in $\mathbb {R}^2$ except $(0,0)$

The level curves of the function $f(x,y) = \sqrt {\frac {y}{x}}$ are portions of:

Lines Parabolas Exponentials

Which of the following is a contour map for the function $f(x,y) = \ln (xy)$ ?

The graph of $f(x,y) = \sqrt {x^2+y^2}$ is a:

Cone Paraboloid Sphere Plane Cylinder

The graph of $f(x,y) = 2- x^2-y^2$ is a:

Cone Paraboloid Sphere Plane Cylinder

The graph of $y^2+z^2 = 9$ is a:

Cone Paraboloid Sphere Plane Cylinder

The graph of $2x+3y+z=1$ is a:

Cone Paraboloid Sphere Plane Cylinder