Free Practice Quiz Question List

02 Multivariable Functions Online Quiz Questions

Use this free practice quiz with 20 questions to review 02 Multivariable Functions, test your knowledge, and prepare for your next test or exam.

20 questions

Choose one

1 point

What is the domain of $g(x,y)=\frac{\ln(x+y)}{x^2-y}$ ?

A
$x+y\ge 0$ and $x^2-y\ne 0$
B
$x+y>0$ and $x^2-y\ne 0$
C
$x+y>0$ and $x^2-y=0$
D
All points in $\mathbb{R}^2$

Choose one

1 point

For the surface $z=x^2+y^2$ , what is the vertical trace obtained by setting $y=0$ ?

A
A horizontal line in the $xz$ -plane
B
A circle in the $xz$ -plane
C
A parabola in the $xz$ -plane
D
A hyperbola in the $xz$ -plane

Choose one

1 point

Evaluate $\lim_{(x,y)\to(1,2)}\left(x^2+3xy+y^2\right)$ .

A
$11$
B
$9$
C
$7$
D
$13$

Choose all

1 point

Select all conditions that must hold for $k(x,y)=\frac{\sqrt{x-y}}{x+y}$ to be defined.

A
$x-y\ge 0$
B
$x+y=0$
C
$x+y\ne 0$
D
$x-y<0$

Choose all

1 point

Select all correct statements about the level curves of $f(x,y)=x^2+y^2$ .

A
For every real $c$ , the level curve is a line.
B
For $c>0$ , the level curve is a circle centered at the origin.
C
For $c<0$ , the level curve is a circle of radius $\sqrt{-c}$ .
D
For $c=0$ , the level curve is only the point $(0,0)$ .

True or false

1 point

True or false: The graph of a function $z=f(x,y)$ is generally a surface in three-dimensional space.

A
True
B
False

True or false

1 point

If a function approaches the same value along two different paths toward a point, that is sufficient to prove that the two-variable limit exists.

A
True
B
False

Written response

1 point

For $f(x,y)=x^2+3xy-y^2$ , enter the value of $f(1,2)$ .

Written response

1 point

For $f(x,y)=x^2+y^2$ , what is the radius of the level curve corresponding to $c=4$ ?

Fill in the blank

1 point

The set of allowed input values is the , and the set of resulting output values is the .

Fill in the blank

1 point

For a function of two variables, the set satisfying $f(x,y)=c$ is a , and a collection of such curves is a .

Open ended

1 point

Determine whether $\lim_{(x,y)\to(0,0)}\frac{x^2-y^2}{x^2+y^2}$ exists. Justify your conclusion using two paths of approach.

Choose one

1 point

For $f(x,y,z)=x^2+y^2+z^2$ , what is the level surface corresponding to a constant $c>0$ ?

A
A plane with normal vector $(1,1,1)$
B
A sphere centered at the origin with radius $\sqrt{c}$
C
A cylinder of radius $c$ around the $z$ -axis
D
A paraboloid opening in the positive $z$ -direction

Choose one

1 point

Which geometric region is the domain of $h(x,y)=\ln(x+y)$ ?

A
The open half-plane $x+y>0$
B
The closed half-plane $x+y\ge 0$
C
The line $x+y=0$ only
D
The disk $x^2+y^2\le 1$

Choose one

1 point

For the function $f(x,y)=x^2+3xy-y^2$ , what is the value of $f(1,2)$ ?

True or false

1 point

True or false: The graph of a function of two variables, $z=f(x,y)$ , is generally a surface in three-dimensional space.

A
True
B
False

Choose one

1 point

Which set is the domain of $h(x,y)=\ln(x+y)$ ?

A
The region $x+y\ge 0$
B
The half-plane $x+y>0$
C
The half-plane $x+y<0$
D
All of $\mathbb{R}^2$

Written response

1 point

Find the limit $\lim_{(x,y)\to(1,2)}\left(x^2+3xy+y^2\right)$ . Enter the numerical value.

Choose one

1 point

For $f(x,y)=2x-y$ , which equation describes the level curve corresponding to $c=3$ ?

A
$y=2x-3$
B
$y=3x-2$
C
$y=-2x+3$
D
$y=2x+3$

Written response

1 point

What is the standard name of the surface $(z=x^2+y^2)$ ? Enter the name in words.