Q: Let $$\vec{a} = \langle 2, -1, 3 \rangle$$ and $$\vec{b} = \langle 1, 4, -2 \rangle. $$What is the dot product $$\vec{a} \cdot \vec{b}$$?

$$2 \times 1 + (-1) \times 4 + 3 \times (-2) = 2 - 4 - 6 = -8$$

Q: Which of the following expressions gives the curvature $$\kappa of a $$smooth curve $$\vec{r}(t)$$?

$$\kappa = \dfrac{\|\vec{r}'(t) \times \vec{r}''(t)\|}{\|\vec{r}'(t)\|^3}$$

Q: Which of the following is the correct setup for the area inside the circle $$x^2$$ + $$y^2 = 4$ using polar coordinates?

$$\$$int_0$$^{2\pi} \$$int_0$$^2 r \, dr \, d\theta$$

Question 1

Let $$\vec{a} = \langle 2, -1, 3 \rangle$$ and $$\vec{b} = \langle 1, 4, -2 \rangle. $$What is the dot product $$\vec{a} \cdot \vec{b}$$?

Accepted Answer

$$2 	imes 1 + (-1) 	imes 4 + 3 	imes (-2) = 2 - 4 - 6 = -8$$

Question 2

Which of the following is the correct formula for the cross product $$\vec{a} 	imes \vec{b}$$ for vectors $$\vec{a} = \langle a_1$, a_2$, a_3$ angle$$ and $$\vec{b} = \langle b_1$, b_2$, b_3$ angle$$?

Accepted Answer

$$\langle a_2$ b_3$ - a_3$ b_2$, a_3$ b_1$ - a_1$ b_3$, a_1$ b_2$ - a_2$ b_1$ \rangle$$

Question 3

A plane passes through the point $$(1,2,3)$$ and has normal vector $$\vec{n} = \langle 2, -1, 4 \rangle. $$Which of the following is the equation of the plane?

Accepted Answer

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

Question 4

Which of the following is a quadric surface?

Accepted Answer

$$x^2$$ + $$y^2$$ - $$z^2 = 1$

Question 5

Given the vector-valued function $$\vec{r}(t) = \langle t, t^2$, t^3$ \rangle$$, what is $$\vec{r}'(t)$$?

Accepted Answer

$$\langle 1, 2t, 3t^2$ \rangle$$

Question 6

A particle moves in space with position $$\vec{r}(t) = \langle \cos t, \sin t, t \rangle. $$What is the speed of the particle at time $$t$$?

Accepted Answer

$$\sqrt{(-\sin t)^2$ + (\cos t)^2$ + 1^2$} = \sqrt{1 + 1} = \sqrt{2}$$

Question 7

The length of the curve $$\vec{r}(t) = \langle t, t^2$ \rangle$$ for $$t in$$ $$[0,1] is $$given by which integral?

Accepted Answer

$$\$$int_0$$^1 \sqrt{1 + (2t)^2$} \, dt$$

Question 8

Which of the following expressions gives the curvature $$\kappa of a $$smooth curve $$\vec{r}(t)$$?

Accepted Answer

$$\kappa = \dfrac{\|\vec{r}'(t) 	imes \vec{r}''(t)\|}{\|\vec{r}'(t)\|^3}$$

Question 9

Suppose $$f(x, y) = x^2$ y + y^3$. What is the partial derivative $$\dfrac{\partial f}{\partial y}$$?

Accepted Answer

$$x^2$$ + $$3y^2$$

Question 10

Evaluate the limit $$\lim_{(x, y) 	o (0, 0)} \dfrac{x^2$ y}{x^2$ + y^2$}$$, or determine if it does not exist.

Accepted Answer

The limit is $$0$$

Question 11

If $$z = f(x, y)$$ where $$x = r \cos 	heta$$, $$y = r \sin 	heta$$, which formula correctly expresses $$\dfrac{\partial z}{\partial r}$$ using the chain rule?

Accepted Answer

$$\dfrac{\partial z}{\partial r} = \dfrac{\partial f}{\partial x} \cos 	heta + \dfrac{\partial f}{\partial y} \sin 	heta$$

Question 12

The gradient vector $$
abla f(x, y)$$ points in which direction?

Accepted Answer

The direction of greatest increase of $$f$$

Question 13

A function $$f(x, y)$$ has a critical point at $$(1, 2). $$The second partial derivatives are $$f_{xx} = 4$$, $$f_{yy} = 5$$, $$f_{xy} = 0 at $$that point. What type of critical point is it?

Accepted Answer

Local minimum

Question 14

Evaluate the double integral $$\iint_R x y \, dA$$ over the rectangle $$R = [0,1] 	imes [0,2].$$

Accepted Answer

$$\dfrac{1}{2}$$

Question 15

Which of the following is the correct setup for the area inside the circle $$x^2$$ + $$y^2 = 4$ using polar coordinates?

Accepted Answer

$$\$$int_0$$^{2\pi} \$$int_0$$^2 r \, dr \, d	heta$$

Question 16

Which of the following is the correct formula for the divergence of a vector field $$\vec{F} = \langle P, Q, R \rangle$$?

Accepted Answer

$$
abla \cdot \vec{F} = \dfrac{\partial P}{\partial x} + \dfrac{\partial Q}{\partial y} + \dfrac{\partial R}{\partial z}$$

Question 17

Suppose $$\vec{F} = \langle y, -x \rangle. $$Compute the line integral $$\int_C \vec{F} \cdot d\vec{r}$$ where $$C is $$the unit circle $$x^2$$ + $$y^2 = 1$ traversed counterclockwise.

Accepted Answer

$$2\pi$$

Question 18

Which of the following statements about conservative vector fields is TRUE?

Accepted Answer

A vector field $$\vec{F} is $$conservative if there exists a scalar function $$f$$ such that $$\vec{F} = 
abla f$$

Question 19

Which of the following is Green’s Theorem?

Accepted Answer

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

Multivariable Calculus Course Syllabus and Topic Overview

Study Guide - Practice Questions

Study Guide - Flashcards