Master Multivariable Calculus: Challenging Practice Test

Question 1

Let $\mathbf{F}(x,y,z) = (2xy + z^2)\mathbf{i} + (x^2 + 2yz)\mathbf{j} + (2xz + y^2)\mathbf{k}$. Which of the following is true?

Accepted Answer

$\mathbf{F}$ is conservative and $
abla 	imes \mathbf{F} = \mathbf{0}$

Question 2

Evaluate $\iint_S \mathbf{F} \cdot d\mathbf{S}$ where $\mathbf{F} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$ and $S$ is the sphere $x^2 + y^2 + z^2 = 4$ with outward orientation.

Accepted Answer

$32\pi$

Question 3

Find the directional derivative of $f(x,y,z) = x^2yz + 4xz^2$ at the point $(1, -2, 1)$ in the direction of $\mathbf{v} = \langle 2, -1, 2 \rangle$.

Accepted Answer

$\frac{8}{3}$

Question 4

Let $C$ be the curve of intersection of the cylinder $x^2 + y^2 = 1$ and the plane $z = x$. Evaluate $\oint_C y^2 \, dx + z \, dy + x \, dz$ where $C$ is oriented counterclockwise when viewed from above.

Accepted Answer

$-\pi$

Question 5

Find the absolute maximum value of $f(x,y) = x^2 + y^2 - 2x$ on the closed disk $x^2 + y^2 \leq 4$.

Accepted Answer

$8$

Question 6

Evaluate $\int_0^1 \int_0^1 \int_0^1 |x - y| \, dz \, dy \, dx$.

Accepted Answer

$\frac{1}{3}$

Question 7

Find the volume of the solid bounded by the paraboloid $z = x^2 + y^2$ and the plane $z = 4$.

Accepted Answer

$8\pi$

Question 8

If $\mathbf{r}(t) = \langle t^2, \ln t, t \rangle$ for $t > 0$, find the curvature $\kappa$ at $t = 1$.

Accepted Answer

$\frac{\sqrt{6}}{6}$

Question 9

Use Lagrange multipliers to find the maximum value of $f(x,y,z) = xyz$ subject to $x + y + z = 3$ and $x, y, z > 0$.

Accepted Answer

$1$

Question 10

Evaluate $\int_C \mathbf{F} \cdot d\mathbf{r}$ where $\mathbf{F} = \langle -y, x \rangle$ and $C$ is the ellipse $\frac{x^2}{4} + y^2 = 1$ traversed counterclockwise.

Accepted Answer

$4\pi$

Question 11

Evaluate $\iiint_E z \, dV$ where $E$ is the solid bounded by $z = 0$, $z = y$, and $y = 4 - x^2$. Round your answer to 2 decimal places.

Accepted Answer

8.53

Question 12

Find the surface area of the part of the paraboloid $z = 9 - x^2 - y^2$ that lies above the $xy$-plane. Round to 2 decimal places.

Accepted Answer

117.32

Question 13

Evaluate $\int_0^{\pi/2} \int_0^{\pi/2} \int_0^1 ho^4 \sin^3\phi \, dho \, d\phi \, d	heta$. Round to 4 decimal places.

Accepted Answer

0.2094

Question 14

Find the work done by the force field $\mathbf{F}(x,y,z) = \langle yz, xz, xy \rangle$ in moving a particle from $(1,0,0)$ to $(2,1,4)$ along any smooth path. Round to 1 decimal place.

Accepted Answer

8.0

Question 15

Find the flux of $\mathbf{F} = \langle x^3, y^3, z^3 \rangle$ outward through the surface of the cube $[0,1]^3$. Round to 2 decimal places.

Accepted Answer

1.50

Question 16

Let $f(x,y) = e^{x^2 + y^2}$. Find and classify all critical points of $f$.

Question 17

Use the Chain Rule to find $\frac{\partial w}{\partial s}$ and $\frac{\partial w}{\partial t}$ where $w = xy + yz + zx$, $x = s + t$, $y = st$, $z = s - t$.

Question 18

Verify Stokes' Theorem for $\mathbf{F} = \langle z, x, y \rangle$ where $S$ is the hemisphere $z = \sqrt{1-x^2-y^2}$ with upward orientation, by computing both the line integral and the surface integral.

Question 19

A particle moves along the curve $\mathbf{r}(t) = \langle t, t^2, t^3 \rangle$ for $t \geq 0$.

(a) Find the unit tangent vector $\mathbf{T}(t)$ and unit normal vector $\mathbf{N}(t)$.
(b) Find the curvature $\kappa(t)$.
(c) Find the tangential and normal components of acceleration, $a_T$ and $a_N$.
(d) At $t = 1$, find the osculating plane.

Question 20

Let $E$ be the solid region bounded by the cone $z = \sqrt{x^2 + y^2}$ and the sphere $x^2 + y^2 + z^2 = 2$.

(a) Set up (but do not evaluate) a triple integral in cylindrical coordinates for the volume of $E$.
(b) Set up (but do not evaluate) a triple integral in spherical coordinates for the volume of $E$.
(c) Evaluate the integral from part (b) to find the volume.
(d) Find the centroid $(\bar{x}, \bar{y}, \bar{z})$ of the solid $E$.