Q: Which of the following is a necessary condition for the continuity of a vector-valued function $$\vec{r}(t) = (f(t), g(t), h(t)) at$$ $$t = a$$?

$$\lim_{t \to a} \vec{r}(t)$$ exists and equals $$\vec{r}(a)$$

Question 1

Which of the following best describes the geometric figure in $$\mathbb{R}^2$$ given by the equation $$y = x^2$?

Accepted Answer

A parabola opening upwards

Question 2

A cable company charges $$100$$ for a monthly subscription to its services and $$5$$ for each special feature movie that a subscriber chooses to watch. If $$n is $$the number of subscribers and $$m is $$the total number of special feature movies watched by all subscribers, which formula gives the total monthly revenue $$R$$?

Accepted Answer

$$R = 100n + 5m$$

Question 3

Given the points $$A = (0,1,0)$$, $$B = (1,1,0)$$, and $$C = (2,2,0)$$, what is the vector $$\overrightarrow{AB} + \overrightarrow{BC}$$?

Accepted Answer

$$(2,1,0)$$

Question 4

Let $$\vec{a} = 2\vec{i} - 3\vec{j} + 4\vec{k}$$ and $$\vec{b} = 4\vec{i} - 5\vec{k}. $$What is $$\|\vec{a} - 3\vec{b}\|$$?

Accepted Answer

$$\sqrt{(-10)^2$ + (-3)^2$ + (19)^2$}$$

Question 5

Which of the following statements about the dot product of two vectors $$\vec{a}$$ and $$\vec{b} is $$correct?

Accepted Answer

The dot product is zero if and only if $$\vec{a}$$ and $$\vec{b}$$ are orthogonal

Question 6

Given $$\vec{a} = \vec{i} + 2\vec{j} + \vec{k}$$ and $$\vec{b} = \vec{j} + \vec{k}$$, what is the angle $$	heta$$ between $$\vec{a}$$ and $$\vec{b}$$?

Accepted Answer

Question 7

Find the determinant of the matrix $$\begin{pmatrix} 1 & 2 & 3 \ 0 & 1 & 4 \ 6 & 7 & 8 \end{pmatrix}.$$

Accepted Answer

$$1(1 \cdot 8 - 4 \cdot 7) - 2(0 \cdot 8 - 4 \cdot 6) + 3(0 \cdot 7 - 1 \cdot 6) = 1(-20) - 2(-24) + 3(-6) = -20 + 48 - 18 = 10$$

Question 8

Given vectors $$\vec{a} = (1,2,3)$$ and $$\vec{b} = (1,-1,4)$$, what is the value of $$k$$ such that $$\vec{a}$$ and $$k\vec{b}$$ are orthogonal?

Accepted Answer

$$k = -\dfrac{11}{14}$$

Question 9

Which of the following is the correct formula for the cross product $$\vec{v}_1 	imes \vec{v}_2 of $$vectors $$\vec{v}_1 = (a_1$, b_1$, c_1$)$$ and $$\vec{v}_2 = (a_2$, b_2$, c_2$)$$?

Accepted Answer

Question 10

Find the area of the triangle with vertices $$A(0,0,0)$$, $$B(1,0,1)$$, and $$C(2,1,0).$$

Accepted Answer

$$\dfrac{\sqrt{14}}{2}$$

Question 11

Which of the following is the equation of the plane passing through the point $$(1,0,1)$$ and parallel to the plane $$x + y - 2z = 3$$?

Accepted Answer

Question 12

Let $$\vec{r}(t) = (\cos t, \sin t, t^2$). $$What is the unit tangent vector $$\vec{T}(t) at$$ $$t = 0$$?

Accepted Answer

Question 13

Suppose a moving particle has initial position $$\vec{r}_0 = (1,0,0)$$ and initial velocity $$\vec{v}_0 = (1,0,1). If$$ $$\vec{a}(t) = (2t, 4t, 6t) is $$its acceleration vector, what is its velocity at time $$t$$?

Accepted Answer

Question 14

Which of the following best describes the indefinite integral of a continuous vector-valued function $$\vec{r}(t) = (f(t), g(t), h(t))$$?

Accepted Answer

$$\int \vec{r}(t) dt = \left( \int f(t) dt, \int g(t) dt, \int h(t) dt \right)$$

Question 15

Given the vector-valued function $$\vec{r}(t) = (t^2$, t^3$, t^4$)$$, what is the speed $$|\vec{v}(t)| at$$ $$t = 1$$?

Accepted Answer

$$\sqrt{2^2$ + 3^2$ + 4^2$}$$

Question 16

Suppose the position vector of a particle is $$\vec{r}(t) = (e^t, \sin t, \ln t). $$What is the velocity vector $$\vec{v}(t)$$?

Accepted Answer

$$(e^t, \cos t, \dfrac{1}{t})$$

Question 17

Which of the following is a necessary condition for the continuity of a vector-valued function $$\vec{r}(t) = (f(t), g(t), h(t)) at$$ $$t = a$$?

Accepted Answer

$$\lim_{t 	o a} \vec{r}(t)$$ exists and equals $$\vec{r}(a)$$

Question 18

Let $$\vec{r}(t) = (t, t^2$, t^3$). At $$what value(s) of $$t$$ does the speed $$|\vec{v}(t)|$$ equal $$\sqrt{14}$$?

Accepted Answer

$$t = 2$$

Study Notes: Vectors, Matrices, and Vector-Valued Functions in Calculus (Math 200)

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