Q: Which of the following best describes the derivative of a vector function $\vec{r}(t)$?

It is the limit $\displaystyle \lim_{h \to 0} \frac{\vec{r}(t+h) - \vec{r}(t)}{h}$ if this limit exists.

Question 1

Which of the following best describes the derivative of a vector function $\vec{r}(t)$?

Accepted Answer

It is the limit $\displaystyle \lim_{h 	o 0} \frac{\vec{r}(t+h) - \vec{r}(t)}{h}$ if this limit exists.

Question 2

Given $\vec{r}(t) = f_1(t)\hat{i} + f_2(t)\hat{j} + f_3(t)\hat{k}$, what is $\frac{d}{dt}\vec{r}(t)$?

Accepted Answer

$f_1'(t)\hat{i} + f_2'(t)\hat{j} + f_3'(t)\hat{k}$

Question 3

If $\vec{r}(t) = t^2\hat{i} + e^t\hat{j} + \sin(2t)\hat{k}$, what is $\frac{d}{dt}\vec{r}(t)$?

Accepted Answer

$2t\hat{i} + e^t\hat{j} + 2\cos(2t)\hat{k}$

Question 4

For a vector function $\vec{r}(t)$, the unit tangent vector $\vec{T}(t)$ is defined as:

Accepted Answer

$\vec{T}(t) = \frac{\vec{r}'(t)}{\|\vec{r}'(t)\|}$

Question 5

Suppose $\vec{r}(t) = \sqrt{t}\hat{i} + (2-t)\hat{j}$ represents the position of a particle. What is the velocity vector $\vec{r}'(t)$?

Accepted Answer

$\frac{1}{2\sqrt{t}}\hat{i} - \hat{j}$

Question 6

If $\vec{r}(t) = c\vec{a}$, where $c$ is a constant and $\vec{a}$ is a constant vector, what is $\vec{r}'(t)$?

Accepted Answer

$\vec{0}$

Question 7

If $\vec{r}(t) = f(t)\vec{a}$, where $\vec{a}$ is a constant vector, what is $\frac{d}{dt}\vec{r}(t)$?

Accepted Answer

$f'(t)\vec{a}$

Question 8

Which of the following statements is TRUE about the definite integral of a continuous vector function $\vec{r}(t)$ from $a$ to $b$?

Accepted Answer

$\displaystyle \int_a^b \vec{r}(t)\,dt = \vec{R}(b) - \vec{R}(a)$, where $\vec{R}$ is any antiderivative of $\vec{r}(t)$.

Question 9

Given $\vec{r}(t) = t\hat{i} + t^2\hat{j}$, what is the indefinite integral $\int \vec{r}(t)\,dt$?

Accepted Answer

$\frac{1}{2}t^2\hat{i} + \frac{1}{3}t^3\hat{j} + \vec{C}$

Question 10

If $\vec{r}(t) = f_1(t)\hat{i} + f_2(t)\hat{j} + f_3(t)\hat{k}$, which of the following is the correct expression for $\int_a^b \vec{r}(t)\,dt$?

Accepted Answer

$\displaystyle \int_a^b f_1(t)\,dt\,\hat{i} + \int_a^b f_2(t)\,dt\,\hat{j} + \int_a^b f_3(t)\,dt\,\hat{k}$

Question 11

A particle moves along a path described by $\vec{r}(t) = t\hat{i} + t^2\hat{j}$. What is the unit tangent vector $\vec{T}(t)$ at $t=1$?

Accepted Answer

$\frac{1}{\sqrt{5}}\hat{i} + \frac{2}{\sqrt{5}}\hat{j}$

Question 12

If $\vec{r}(t) = \cos(t)\hat{i} + \sin(t)\hat{j}$, which statement is TRUE about $\vec{r}'(t)$ and $\vec{r}(t)$?

Accepted Answer

$\vec{r}'(t)$ is always orthogonal to $\vec{r}(t)$ for all $t$.

Question 13

Suppose $\vec{r}(t) = t^3\hat{i} + t^2\hat{j} + t\hat{k}$. What is $\vec{r}'(t)$ at $t=2$?

Accepted Answer

$12\hat{i} + 4\hat{j} + 1\hat{k}$

Question 14

A drone's position is given by $\vec{r}(t) = (2t)\hat{i} + (3t^2)\hat{j} + (4t^3)\hat{k}$. What is the total displacement from $t=0$ to $t=1$?

Accepted Answer

$2\hat{i} + 3\hat{j} + 4\hat{k}$

Question 15

If $\vec{r}(t) = t\hat{i} + t^2\hat{j}$, what is the definite integral $\int_0^2 \vec{r}(t)\,dt$?

Accepted Answer

$2\hat{i} + \frac{8}{3}\hat{j}$

Question 16

A car moves along a path $\vec{r}(t) = t^2\hat{i} + 3t\hat{j}$. What is the velocity at $t=3$?

Accepted Answer

$6\hat{i} + 3\hat{j}$

Question 17

If $\vec{r}(t) = e^{2t}\hat{i} + \ln(t)\hat{j}$, what is $\vec{r}'(t)$?

Accepted Answer

$2e^{2t}\hat{i} + \frac{1}{t}\hat{j}$

Question 18

A rocket's acceleration is given by $\vec{a}(t) = 3t^2\hat{i} + 2t\hat{j}$. If its initial velocity is $\vec{v}(0) = \vec{0}$, what is its velocity at $t=2$?

Accepted Answer

$8\hat{i} + 4\hat{j}$

Section 3.2 - Derivatives and Integrals of Vector Functions

Study Guide - Practice Questions

Study Guide - Flashcards