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 $$S$$?

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

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

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