Calculus Early Transcendentals

Howard Anton, Irl Bivens, Stephen Davis

Chapter 12

VECTOR-VALUED FUNCTIONS - all with Video Answers

Educators

Section 1

Introduction to Vector-Valued Functions

Problem 1

Find the domain of $\mathbf{r}(t)$ and the value of $\mathbf{r}\left(t_{0}\right)$
$$
\mathbf{r}(t)=\cos t \mathbf{i}-3 t \mathbf{j} ; \quad t_{0}=\pi
$$

Thomas Pauly

Thomas Pauly

Numerade Educator

Problem 2

Find the domain of $\mathbf{r}(t)$ and the value of $\mathbf{r}\left(t_{0}\right)$
$$
\mathbf{r}(t)=\left\langle\sqrt{3 t+1}, t^{2}\right\rangle ; t_{0}=1
$$

Thomas Pauly

Numerade Educator

Problem 3

Find the domain of $\mathbf{r}(t)$ and the value of $\mathbf{r}\left(t_{0}\right)$
$$
\mathbf{r}(t)=\cos \pi t \mathbf{i}-\ln t \mathbf{j}+\sqrt{t-2} \mathbf{k} ; t_{0}=3
$$

Thomas Pauly

Numerade Educator

Problem 4

Find the domain of $\mathbf{r}(t)$ and the value of $\mathbf{r}\left(t_{0}\right)$
$$
\mathbf{r}(t)=\left\langle 2 e^{-t}, \sin ^{-1} t, \ln (1-t)\right\rangle ; t_{0}=0
$$

Thomas Pauly

Numerade Educator

Problem 5

Express the parametric equations as a single vector equation of the form
$$
\mathbf{r}=x(t) \mathbf{i}+y(t) \mathbf{j} \quad \text { or } \quad \mathbf{r}=x(t) \mathbf{i}+y(t) \mathbf{j}+z(t) \mathbf{k}
$$
$$
x=3 \cos t, y=t+\sin t
$$

Thomas Pauly

Numerade Educator

Problem 6

Express the parametric equations as a single vector equation of the form
$$
\mathbf{r}=x(t) \mathbf{i}+y(t) \mathbf{j} \quad \text { or } \quad \mathbf{r}=x(t) \mathbf{i}+y(t) \mathbf{j}+z(t) \mathbf{k}
$$
$$
x=2 t, \quad y=2 \sin 3 t, \quad z=5 \cos 3 t
$$

Thomas Pauly

Numerade Educator

Problem 7

Find the parametric equations that correspond to the given vector equation.
$$
\mathbf{r}=3 t^{2} \mathbf{i}-2 \mathbf{j}
$$

Thomas Pauly

Numerade Educator

Problem 8

Find the parametric equations that correspond to the given vector equation.
$$
\mathbf{r}=(2 t-1) \mathbf{i}-3 \sqrt{t} \mathbf{j}+\sin 3 t \mathbf{k}
$$

Thomas Pauly

Numerade Educator

Problem 9

Describe the graph of the equation.
$$
\mathbf{r}=(3-2 t) \mathbf{i}+5 t \mathbf{j}
$$

Thomas Pauly

Numerade Educator

Problem 10

Describe the graph of the equation.
$$
\mathbf{r}=2 \sin 3 t \mathbf{i}-2 \cos 3 t \mathbf{j}
$$

Thomas Pauly

Numerade Educator

Problem 11

Describe the graph of the equation.
$$
\mathbf{r}=2 t \mathbf{i}-3 \mathbf{j}+(1+3 t) \mathbf{k}
$$

Thomas Pauly

Numerade Educator

Problem 12

Describe the graph of the equation.
$$
\mathbf{r}=3 \mathbf{i}+2 \cos t \mathbf{j}+2 \sin t \mathbf{k}
$$

Thomas Pauly

Numerade Educator

Problem 13

Describe the graph of the equation.
$$
\mathbf{r}=2 \cos t \mathbf{i}-3 \sin t \mathbf{j}+\mathbf{k}
$$

Thomas Pauly

Numerade Educator

Problem 14

Describe the graph of the equation.
$$
\mathbf{r}=-3 \mathbf{i}+\left(1-t^{2}\right) \mathbf{j}+t \mathbf{k}
$$

Thomas Pauly

Numerade Educator

Problem 15

(a) Find the slope of the line in 2 -space that is represented
by the vector equation $\mathbf{r}=(1-2 t) \mathbf{i}-(2-3 t) \mathbf{j}$.
(b) Find the coordinates of the point where the line
$$
\mathbf{r}=(2+t) \mathbf{i}+(1-2 t) \mathbf{j}+3 t \mathbf{k}
$$
intersects the $x z$ -plane.

Thomas Pauly

Numerade Educator

Problem 16

(a) Find the $y$ -intercept of the line in 2 -space that is represented by the vector equation $\mathbf{r}=(3+2 t) \mathbf{i}+5 t \mathbf{j}$.
(b) Find the coordinates of the point where the line
$$
\mathbf{r}=t \mathbf{i}+(1+2 t) \mathbf{j}-3 t \mathbf{k}
$$
intersects the plane $3 x-y-z=2$

Thomas Pauly

Numerade Educator

Problem 17

Sketch the line segment represented by each vector
equation.
$$
\begin{array}{l}{\text { (a) } \mathbf{r}=(1-t) \mathbf{i}+t \mathbf{j} ; 0 \leq t \leq 1} \\ {\text { (b) } \mathbf{r}=(1-t)(\mathbf{i}+\mathbf{j})+t(\mathbf{i}-\mathbf{j}) ; 0 \leq t \leq 1}\end{array}
$$

Thomas Pauly

Numerade Educator

Problem 18

Sketch the line segment represented by each vector
equation.
$$
\begin{array}{l}{\text { (a) } \mathbf{r}=(1-t)(\mathbf{i}+\mathbf{j})+t \mathbf{k} ; 0 \leq t \leq 1} \\ {\text { (b) } \mathbf{r}=(1-t)(\mathbf{i}+\mathbf{j}+\mathbf{k})+t(\mathbf{i}+\mathbf{j}) ; 0 \leq t \leq 1}\end{array}
$$

Carl David Cepeda

Carl David Cepeda

Numerade Educator

Problem 19

Write a vector equation for the line segment from P to Q.
(Figure cant copy)

Thomas Pauly

Numerade Educator

Problem 20

Write a vector equation for the line segment from P to Q.
(Figure cant copy)

Thomas Pauly

Numerade Educator

Problem 21

Sketch the graph of r(t) and show the direction of increasing t.
$$
\mathbf{r}(t)=2 \mathbf{i}+t \mathbf{j}
$$

Thomas Pauly

Numerade Educator

Problem 22

Sketch the graph of r(t) and show the direction of increasing t.
$$
\mathbf{r}(t)=\langle 3 t-4,6 t+2\rangle
$$

Thomas Pauly

Numerade Educator

Problem 23

Sketch the graph of r(t) and show the direction of increasing t.
$$
\mathbf{r}(t)=(1+\cos t) \mathbf{i}+(3-\sin t) \mathbf{j} ; \quad 0 \leq t \leq 2 \pi
$$

Thomas Pauly

Numerade Educator

Problem 24

Sketch the graph of r(t) and show the direction of increasing t.
$$
\mathbf{r}(t)=\langle 2 \cos t, 5 \sin t\rangle ; 0 \leq t \leq 2 \pi
$$

Thomas Pauly

Numerade Educator

Problem 25

Sketch the graph of r(t) and show the direction of increasing t.
$$
\mathbf{r}(t)=\cosh t \mathbf{i}+\sinh t \mathbf{j} \quad
$$

Tanishq Gupta

Numerade Educator

Problem 26

Sketch the graph of r(t) and show the direction of increasing t.
$$
\mathbf{r}(t)=\sqrt{t} \mathbf{i}+(2 t+4) \mathbf{j}
$$

Thomas Pauly

Numerade Educator

Problem 27

Sketch the graph of r(t) and show the direction of increasing t.
$$
\mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}+t \mathbf{k}
$$

Tanishq Gupta

Numerade Educator

Problem 28

Sketch the graph of r(t) and show the direction of increasing t.
$$
\mathbf{r}(t)=9 \cos t \mathbf{i}+4 \sin t \mathbf{j}+t \mathbf{k}
$$

Carl David Cepeda

Carl David Cepeda

Numerade Educator

Problem 29

Sketch the graph of r(t) and show the direction of increasing t.
$$
\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+2 \mathbf{k}
$$

Carl David Cepeda

Carl David Cepeda

Numerade Educator

Problem 30

Sketch the graph of r(t) and show the direction of increasing t.
$$
\mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+\sin t \mathbf{k} ; \quad 0 \leq t \leq 2 \pi
$$

Carl David Cepeda

Carl David Cepeda

Numerade Educator

Problem 31

True–False Determine whether the statement is true or
false. Explain your answer.
The natural domain of a vector-valued function is the union
of the domains of its component functions.

Thomas Pauly

Numerade Educator

Problem 32

True–False Determine whether the statement is true or
false. Explain your answer.
If $\mathbf{r}(t)=\langle x(t), y(t)\rangle$ is a vector-valued function in 2 -space,
then the graph of $\mathbf{r}(t)$ is a surface in 3 -space.

Tanishq Gupta

Numerade Educator

Problem 33

True–False Determine whether the statement is true or
false. Explain your answer.
If $\mathbf{r}_{0}$ and $\mathbf{r}_{1}$ are vectors in 3 -space, then the graph of the
vector-valued function
$$
\mathbf{r}(t)=(1-t) \mathbf{r}_{0}+t \mathbf{r}_{1} \quad(0 \leq t \leq 1)
$$
is the straight line segment joining the terminal points of $\mathbf{r}_{0}$
and $\mathbf{r}_{1} .$

Tanishq Gupta

Numerade Educator

Problem 34

True–False Determine whether the statement is true or
false. Explain your answer.
The graph of $\mathbf{r}(t)=\langle 2 \cos t, 2 \sin t, t\rangle$ is a circular helix.

Tanishq Gupta

Numerade Educator

Problem 35

Sketch the curve of intersection of the surfaces, and find
parametric equations for the intersection in terms of parameter
x = t. Check your work with a graphing utility by generating
the parametric curve over the interval ?1 ? t ? 1.
$$
z=x^{2}+y^{2}, x-y=0
$$

Thomas Pauly

Numerade Educator

Problem 36

Sketch the curve of intersection of the surfaces, and find
parametric equations for the intersection in terms of parameter
x = t. Check your work with a graphing utility by generating
the parametric curve over the interval ?1 ? t ? 1.
$$
y+x=0, z=\sqrt{2-x^{2}-y^{2}}
$$

Thomas Pauly

Numerade Educator

Problem 37

Sketch the curve of intersection of the surfaces, and find
a vector equation for the curve in terms of the parameter x = t.
$$
9 x^{2}+y^{2}+9 z^{2}=81, y=x^{2} \quad(z>0)
$$

Thomas Pauly

Numerade Educator

Problem 38

Sketch the curve of intersection of the surfaces, and find
a vector equation for the curve in terms of the parameter x = t.
$$
y=x, x+y+z=1
$$

Thomas Pauly

Numerade Educator

Problem 39

Show that the graph of
$$
\mathbf{r}=t \sin t \mathbf{i}+t \cos t \mathbf{j}+t^{2} \mathbf{k}
$$
lies on the paraboloid $z=x^{2}+y^{2}$

Carl David Cepeda

Carl David Cepeda

Numerade Educator

Problem 40

Show that the graph of
$$
\mathbf{r}=t \mathbf{i}+\frac{1+t}{t} \mathbf{j}+\frac{1-t^{2}}{t} \mathbf{k}, \quad t>0
$$
lies in the plane $x-y+z+1=0$

Carl David Cepeda

Carl David Cepeda

Numerade Educator

Problem 41

Show that the graph of

$$
\mathbf{r}=\sin t \mathbf{i}+2 \cos t \mathbf{j}+\sqrt{3} \sin t \mathbf{k}
$$
is a circle, and find its center and radius. [Hint: Show
that the curve lies on both a sphere and a plane. $]$

Carl David Cepeda

Carl David Cepeda

Numerade Educator

Problem 42

Show that the graph of
$$
\mathbf{r}=3 \cos t \mathbf{i}+3 \sin t \mathbf{j}+3 \sin t \mathbf{k}
$$
is an ellipse, and find the lengths of the major and minor
axes. [Hint: Show that the graph lies on both a circular
cylinder and a plane and use the result in Exercise 42 of
Section $10.4 .]$

Carl David Cepeda

Carl David Cepeda

Numerade Educator

Problem 43

For the helix $\mathbf{r}=a \cos t \mathbf{i}+a \sin t \mathbf{j}+c t \mathbf{k},$ find the
value of $c(c>0)$ so that the helix will make one com-
plete turn in a distance of 3 units measured along the
$z$ -axis.

Carl David Cepeda

Carl David Cepeda

Numerade Educator

Problem 44

How many revolutions will the circular helix
$\mathbf{r}=a \cos t \mathbf{i}+a \sin t \mathbf{j}+0.2 t \mathbf{k}$
make in a distance of 10 units measured along the $z$ -axis?

Carl David Cepeda

Carl David Cepeda

Numerade Educator

Problem 45

Show that the curve $\mathbf{r}=t \cos t \mathbf{i}+t \sin t \mathbf{j}+t \mathbf{k}, t \geq 0$
lies on the cone $z=\sqrt{x^{2}+y^{2}} .$ Describe the curve.

Carl David Cepeda

Carl David Cepeda

Numerade Educator

Problem 46

Describe the curve $\mathbf{r}=a \cos t \mathbf{i}+b \sin t \mathbf{j}+c t \mathbf{k},$ where
$a, b,$ and $c$ are positive constants such that $a \neq b$

Carl David Cepeda

Carl David Cepeda

Numerade Educator

Problem 47

In each part, match the vector equation with one of the
accompanying graphs, and explain your reasoning.
$$
\begin{array}{l}{\text { (a) } \mathbf{r}=t \mathbf{i}-t \mathbf{j}+\sqrt{2-t^{2}} \mathbf{k}} \\ {\text { (b) } \mathbf{r}=\sin \pi t \mathbf{i}-t \mathbf{j}+t \mathbf{k}} \\ {\text { (c) } \mathbf{r}=\sin t \mathbf{i}+\cos t \mathbf{j}+\sin 2 t \mathbf{k}} \\ {\text { (d) } \mathbf{r}=\frac{1}{2} t \mathbf{i}+\cos 3 t \mathbf{j}+\sin 3 t \mathbf{k}}\end{array}
$$

Carson Merrill

Numerade Educator

Problem 48

Check your conclusions in Exercise 47 by generating the
curves with a graphing utility. [Note: Your graphing util-
ity may look at the curve from a different viewpoint. Read
the documentation for your graphing utility to determine
how to control the viewpoint, and see if you can generate a
reasonable facsimile of the graphs shown in the figure by ad-
justing the viewpoint and choosing the interval of $t$ -values
appropriately. $]$

Carl David Cepeda

Carl David Cepeda

Numerade Educator

Problem 49

(a) Find parametric equations for the curve of intersection
of the circular cylinder $x^{2}+y^{2}=9$ and the parabolic
cylinder $z=x^{2}$ in terms of a parameter $t$ for which $x=3 \cos t$
(b) Use a graphing utility to generate the curve of intersection in part (a).

Linh Vu

Numerade Educator

Problem 50

(a) Sketch the graph of
$$
\mathbf{r}(t)=\left\langle 2 t, \frac{2}{1+t^{2}}\right\rangle
$$
(b) Prove that the curve in part (a) is also the graph of the
function
$y=\frac{8}{4+x^{2}}$
[The graphs of $y=a^{3} /\left(a^{2}+x^{2}\right),$
[The graphs of y = a3/(a2 + x2), where a denotes a

constant, were first studied by the French mathemati-
cian Pierre de Fermat, and later by the Italian mathe-
maticians Guido Grandi and Maria Agnesi. Any such

curve is now known as a “witch of Agnesi.” There are
a number of theories for the origin of this name. Some
suggest there was a mistranslation by either Grandi or
Agnesi of some less colorful Latin name into Italian.
Others lay the blame on a translation into English of
Agnesi’s 1748 treatise, Analytical Institutions.]

Linh Vu

Numerade Educator

Problem 51

Writing Consider the curve $C$ of intersection of the cone
$z=\sqrt{x^{2}+y^{2}}$ and the plane $z=y+2 .$ Sketch and iden-
tify the curve $C,$ and describe a procedure for finding a
vector-valued function $\mathbf{r}(t)$ whose graph is $C$.

WZ

Wen Zheng

Numerade Educator

Problem 52

Writing Suppose that $\mathbf{r}_{1}(t)$ and $\mathbf{r}_{2}(t)$ are vector-valued
functions in 2 -space. Explain why solving the equation
$\mathbf{r}_{1}(t)=\mathbf{r}_{2}(t)$ may not produce all of the points where the
graphs of these functions intersect.

Bobby Barnes

University of North Texas