Free Practice Quiz Question List

12 Applications of Multivariable Calculus Online Quiz Questions

Use this free practice quiz with 20 questions to review 12 Applications of Multivariable Calculus, test your knowledge, and prepare for your next test or exam.

20 questions

Choose one

1 point

Which formula gives the surface area of $z=f(x,y)$ over a region $R$ in the $xy$ -plane?

A
$A=\iint_R f(x,y)\,dA$
B
$A=\iint_R \sqrt{1+f_x^2+f_y^2}\,dA$
C
$A=\iint_R (f_x+f_y)\,dA$
D
$A=\iint_R \frac{1}{\sqrt{1+f_x^2+f_y^2}}\,dA$

True or false

1 point

True or false: If $\mathbf r(t)$ is the position of a particle, then its speed at time $t$ is $\|\mathbf r'(t)\|$ .

A
True
B
False

Written response

1 point

What is the name of the vector $\nabla f=\langle f_x,f_y\rangle$ , which is perpendicular to a level curve $f(x,y)=c$ ?

Fill in the blank

1 point

For a thin plate occupying a planar region, integrating over the region gives the plate's .

Choose one

1 point

A region with circular symmetry is described using polar coordinates. Which Jacobian factor must appear in the area element?

A
$\rho^2\sin\phi$
B
$1$
C
$r$
D
$r^2$

Choose all

1 point

Which three statements about divergence and curl are correct? Select all correct choices.

A
Positive divergence can indicate a local source.
B
Curl measures local tendency to rotate.
C
Negative divergence can indicate local compression.
D
Divergence is defined only for scalar fields.

Written response

1 point

For $\mathbf r(t)=\langle 3t,4t,12t\rangle$ , enter the particle's constant speed.

True or false

1 point

True or false: In a conservative vector field, the work between two fixed endpoints is independent of the path taken.

A
True
B
False

Fill in the blank

1 point

For a lamina with constant density, the center of mass is the , and its location depends only on the region's .

Choose all

1 point

At a critical point, which statements correctly describe the second-derivative test? Select all correct choices.

A
If $D>0$ and $f_{xx}>0$ , the point is a local minimum.
B
If $D>0$ and $f_{xx}<0$ , the point is a local maximum.
C
If $D=0$ , the point is always a saddle point.
D
If $D<0$ , the point is a saddle point.

Choose one

1 point

Which system should be solved to find candidates for optimizing $f(x,y,z)$ subject to $g(x,y,z)=c$ ?

A
$\nabla f=\nabla g$ , with no constraint equation
B
$\nabla f=\lambda\nabla g$ and $g=c$
C
$\nabla f=\mathbf 0$ , with $f=c$
D
$\nabla g=\lambda\nabla f$ and $f=c$

Open ended

1 point

Explain the divergence theorem and describe one reason it can simplify a flux calculation.

Choose one

1 point

A constant force $\mathbf F=\langle 2,3\rangle$ moves an object from $A=(1,1)$ to $B=(4,5)$ . What work does the force do?

A
$18$
B
$10$
C
$14$
D
$24$

Choose one

1 point

For a differentiable function $f(x,y)$ , what is the geometric meaning of $\nabla f$ at a point on the level curve $f(x,y)=c$ where $\nabla f\ne\mathbf 0$ ?

A
It is tangent to the level curve.
B
It is perpendicular to the level curve.
C
It always points toward the origin.
D
It gives the curvature of the level curve.

Choose one

1 point

A particle has constant velocity $\mathbf v(t)=\langle 3,4,12\rangle$ . What is its speed?

A
5
B
12
C
13
D
19

Choose one

1 point

When evaluating a double integral in polar coordinates, which area element should be used?

A
$dA=dr\,d\theta$
B
$dA=r\,dr\,d\theta$
C
$dA=r^2\,dr\,d\theta$
D
$dA=\sin\theta\,dr\,d\theta$

Choose one

1 point

A lamina has surface density $\rho(x,y)$ . Which integral gives its moment about the $y$ -axis?

A
$M_x=\iint_R x\rho\,dA$
B
$M_x=\iint_R y\rho\,dA$
C
$M_y=\iint_R x\rho\,dA$
D
$M_y=\iint_R y\rho\,dA$

True or false

1 point

The divergence theorem relates the outward flux through a closed surface to the volume integral of the divergence over the region enclosed by that surface.

A
True
B
False

Written response

1 point

A particle has position $\mathbf r(t)=\langle 2t,t^2,2t^3\rangle$ . What is its speed at $t=1$ ? Enter the value as a nonnegative number; answers within an absolute tolerance of 0.00001 are accepted.

Written response

1 point

In spherical coordinates, the volume element is written as $dV=J\,d\rho\,d\phi\,d\theta$ . What is the Jacobian factor $J$ ?