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

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