Q: Which of the following is the correct setup for the volume of the solid bounded above by $z = 4 - x^2 - y^2$ and below by $z = 0$ using polar coordinates?

$\int_0^{2\pi} \int_0^2 (4 - r^2) r \, dr \, d\theta$

Q: Which of the following is the correct statement of the Divergence Theorem for a solid region $E$ with boundary surface $S$?

$\iint_S \vec{F} \cdot d\vec{S} = \iiint_E \nabla \cdot \vec{F} \, dV$

Q: A surface $S$ is given by $z = x^2 + y^2$ for $0 \leq x^2 + y^2 \leq 1$. What is the surface area of $S$?

$\int_0^{2\pi} \int_0^1 r \sqrt{1 + 4r^2} \, dr \, d\theta$

Question 1

Which of the following is the correct formula for the dot product of two vectors $\vec{a} = \langle a_1, a_2, a_3 \rangle$ and $\vec{b} = \langle b_1, b_2, b_3 \rangle$ in $\mathbb{R}^3$?

Accepted Answer

$a_1 b_1 + a_2 b_2 + a_3 b_3$

Question 2

Given the vectors $\vec{u} = \langle 2, -1, 3 angle$ and $\vec{v} = \langle 0, 4, -2 angle$, what is their cross product $\vec{u} 	imes \vec{v}$?

Accepted Answer

$\langle -10, 4, 8 \rangle$

Question 3

Which of the following equations represents a plane in $\mathbb{R}^3$ passing through the point $(1,2,3)$ with normal vector $\vec{n} = \langle 2, -1, 4 \rangle$?

Accepted Answer

$2(x-1) - 1(y-2) + 4(z-3) = 0$

Question 4

A particle moves along the curve $\vec{r}(t) = \langle t, t^2, t^3 \rangle$. What is the length of the curve from $t=0$ to $t=1$?

Accepted Answer

$\int_0^1 \sqrt{1 + 4t^2 + 9t^4} \, dt$

Question 5

Let $f(x, y) = x^2 y + y^3$. What is the directional derivative of $f$ at the point $(1, 2)$ in the direction of the vector $\vec{v} = \langle 3, 4 \rangle$?

Accepted Answer

$36$

Question 6

Which of the following statements is TRUE about the gradient vector $
abla f$ of a function $f(x, y)$ at a point?

Accepted Answer

It is always perpendicular to the level curve of $f$ at that point.

Question 7

Suppose $f(x, y) = x^2 + y^2$ is defined on the closed disk $x^2 + y^2 \leq 4$. What are the absolute maximum and minimum values of $f$ on this region?

Accepted Answer

Maximum: $8$, Minimum: $0$

Question 8

A company wants to minimize the cost $C(x, y) = x^2 + y^2$ subject to the constraint $x + y = 10$. Using Lagrange multipliers, what are the values of $x$ and $y$ at the minimum?

Accepted Answer

$x = 5, y = 5$

Question 9

Evaluate the double integral $\iint_R x y \, dA$, where $R$ is the region bounded by $x=0$, $y=0$, and $x+y=1$.

Accepted Answer

$\frac{1}{12}$

Question 10

Which of the following is the correct setup for the volume of the solid bounded above by $z = 4 - x^2 - y^2$ and below by $z = 0$ using polar coordinates?

Accepted Answer

$\int_0^{2\pi} \int_0^2 (4 - r^2) r \, dr \, d	heta$

Question 11

A thin plate occupies the region $0 \leq x \leq 2$, $0 \leq y \leq 1$ and has density $\delta(x, y) = x + y$. What is the mass of the plate?

Accepted Answer

$4$

Question 12

Which of the following is a necessary and sufficient condition for a vector field $\vec{F} = \langle P(x, y), Q(x, y) \rangle$ to be conservative in a simply connected region?

Accepted Answer

$\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$

Question 13

Evaluate the line integral $\int_C (2x \, dx + 3y \, dy)$ where $C$ is the straight line from $(0,0)$ to $(1,2)$.

Accepted Answer

$7$

Question 14

Which of the following is a correct statement of Green's Theorem for a region $R$ with positively oriented boundary $C$?

Accepted Answer

$\oint_C P \, dx + Q \, dy = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA$

Question 15

A vector field $\vec{F}(x, y, z) = \langle y, z, x angle$ is given. What is the divergence $
abla \cdot \vec{F}$?

Accepted Answer

$0$

Question 16

Which of the following is the correct statement of the Divergence Theorem for a solid region $E$ with boundary surface $S$?

Accepted Answer

$\iint_S \vec{F} \cdot d\vec{S} = \iiint_E 
abla \cdot \vec{F} \, dV$

Question 17

Suppose a vector field $\vec{F}$ is conservative. Which of the following is always true?

Accepted Answer

The line integral of $\vec{F}$ over any closed curve is zero.

Question 18

A surface $S$ is given by $z = x^2 + y^2$ for $0 \leq x^2 + y^2 \leq 1$. What is the surface area of $S$?

Accepted Answer

$\int_0^{2\pi} \int_0^1 r \sqrt{1 + 4r^2} \, dr \, d	heta$

Question 19

Which of the following best describes the geometric meaning of the cross product $\vec{a} 	imes \vec{b}$ in $\mathbb{R}^3$?

Accepted Answer

It gives a vector perpendicular to both $\vec{a}$ and $\vec{b}$ with magnitude equal to the area of the parallelogram they span.

Calculus III Syllabus and Topical Outline: Multivariable Calculus and Vector Calculus

Study Guide - Practice Questions

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