Q: Given two nonzero vectors $$\vec{a}$$ and $$\vec{b}$$, the angle $$\theta$$ between them is given by:

$$\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}$$

Q: A parallelepiped in 3D formed by vectors $$\vec{a}$$, $$\vec{b}$$, and $$\vec{c}$$ has volume given by:

$$|\vec{a} \cdot (\vec{b} \times \vec{c})|$$

Question 1

Given the vector equation for a line in $$\mathbb{R}^3$$ passing through point $$(x_0$, y_0$, z_0$)$$ and parallel to vector $$\vec{a} = \langle a_1$, a_2$, a_3$ \rangle$$, which of the following is the correct parametric equation for the line?

Accepted Answer

$$x = x_0$ + a_1$ t,\ y = y_0$ + a_2$ t,\ z = z_0$ + a_3$ t$$

Question 2

Suppose two lines in $$\mathbb{R}^3$$ are given by $$\vec{r}_1 = \vec{a}_1 + t\vec{b}_1$$ and $$\vec{r}_2 = \vec{a}_2 + s\vec{b}_2. $$Which condition guarantees that the lines are parallel?

Accepted Answer

$$\vec{b}_1$$ and $$\vec{b}_2$$ are scalar multiples of each other

Question 3

The equation of a plane passing through point $$(x_0$, y_0$, z_0$)$$ and perpendicular to vector $$\vec{n} = \langle a, b, c \rangle is$$:

Accepted Answer

$$a(x - x_0$) + b(y - y_0$) + c(z - z_0$) = 0$$

Question 4

A particle moves along a path given by $$\vec{r}(t) = \langle t^2$, \sin t, e^t \rangle. $$What is the velocity vector $$\vec{v}(t)$$?

Accepted Answer

$$\langle 2t, \cos t, e^t \rangle$$

Question 5

Given $$f(x, y) = x^2$ + y^2$, what is the gradient $$
abla f$$ at the point (1$$, 2)$$?

Accepted Answer

$$\langle 2, 4 \rangle$$

Question 6

Which of the following series converges absolutely?

Accepted Answer

$$\sum_{n=1}^\infty \frac{(-1)^n}{n^2$}$$

Question 7

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

Accepted Answer

$$\frac{1}{\sqrt{2}}(4 + 13)$$

Question 8

The Taylor series for $$f(x)$$ about $$x = a is $$given by:

Accepted Answer

$$f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2$ + \cdots$$

Question 9

If $$\vec{a} = \langle 1, 2, 3 \rangle$$ and $$\vec{b} = \langle 4, 5, 6 \rangle$$, what is the dot product $$\vec{a} \cdot \vec{b}$$?

Accepted Answer

$$32$$

Question 10

Which of the following statements about the cross product $$\vec{a} 	imes \vec{b} is $$true?

Accepted Answer

It is perpendicular to both $$\vec{a}$$ and $$\vec{b}$$

Question 11

Given the series $$\sum_{n=1}^\infty \frac{1}{n^p}$$, for which values of $$p$$ does the series converge?

Accepted Answer

$$p \u003e 1$$

Question 12

The area of the parallelogram formed by vectors $$\vec{a}$$ and $$\vec{b} is $$given by:

Accepted Answer

$$|\vec{a} 	imes \vec{b}|$$

Question 13

Suppose $$f(x, y)$$ has continuous second partial derivatives and $$D = f_{xx}(a, b)f_{yy}(a, b) - [f_{xy}(a, b)]^2. If$$ $$D \u003e 0$$ and $$f_{xx}(a, b) \u003c 0$$, what can be concluded about $$f at$$ $$(a, b)$$?

Accepted Answer

$$(a, b) is a $$local maximum

Question 14

Which of the following is the correct formula for the distance between two points $$(x_1$, y_1$, z_1$)$$ and $$(x_2$, y_2$, z_2$) in$$ $$\mathbb{R}^3$$?

Accepted Answer

$$\sqrt{(x_2$ - x_1$)^2$$ + ($$y_2$$ - $$y_1)^2$ + (z_2$ - z_1$)^2$$}$

Question 15

If $$\vec{r}(t) is a $$position vector of a particle, which of the following best describes the acceleration vector $$\vec{a}(t)$$?

Accepted Answer

$$\vec{a}(t) = \vec{r}''(t)$$

Question 16

Which test can be used to determine the convergence of the series $$\sum_{n=1}^\infty (-1)^n a_n$$ where $$a_n is $$decreasing and $$\lim_{n 	o \infty} a_n = 0$$?

Accepted Answer

Alternating Series Test

Question 17

Given two nonzero vectors $$\vec{a}$$ and $$\vec{b}$$, the angle $$	heta$$ between them is given by:

Accepted Answer

$$\cos 	heta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}$$

Question 18

A parallelepiped in 3D formed by vectors $$\vec{a}$$, $$\vec{b}$$, and $$\vec{c}$$ has volume given by:

Accepted Answer

$$|\vec{a} \cdot (\vec{b} 	imes \vec{c})|$$

Question 19

If $$f(x, y) is $$differentiable at $$(a, b)$$, which of the following statements is true?

Accepted Answer

The gradient $$
abla f(a, b)$$ points in the direction of greatest increase of $$f$$

Calculus II & III Study Guide: Vectors, Multivariable Functions, Sequences, Series, and Taylor Expansions

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