Calculus Early Transcendental Functions

Ron Larson, Bruce Edwards

Chapter 12

Vector-Valued Functions - all with Video Answers

Educators

Section 1

Vector-Valued Functions

Problem 1

Find the domain of the vector-valued function.
$$\mathbf{r}(t)=\frac{1}{t+1} \mathbf{i}+\frac{t}{2} \mathbf{j}-3 t \mathbf{k}$$

Monica Miller

Monica Miller

Numerade Educator

Problem 2

Find the domain of the vector-valued function.
$$\mathbf{r}(t)=\sqrt{4-t^{2}} \mathbf{i}+t^{2} \mathbf{j}-6 t \mathbf{k}$$

Monica Miller

Numerade Educator

Problem 3

Find the domain of the vector-valued function.
$$\mathbf{r}(t)=\ln t \mathbf{i}-e^{t} \mathbf{j}-t \mathbf{k}$$

Monica Miller

Numerade Educator

Problem 4

Find the domain of the vector-valued function.
$$\mathbf{r}(t)=\sin t \mathbf{i}+4 \cos t \mathbf{j}+t \mathbf{k}$$

Monica Miller

Numerade Educator

Problem 5

Find the domain of the vector-valued function.
$$\begin{aligned}
&\mathbf{r}(t)=\mathbf{F}(t)+\mathbf{G}(t), \text { where }\\
&\mathbf{F}(t)=\cos t \mathbf{i}-\sin t \mathbf{j}+\sqrt{t} \mathbf{k}, \quad \mathbf{G}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}
\end{aligned}$$

Monica Miller

Numerade Educator

Problem 6

Find the domain of the vector-valued function.
$$\begin{aligned}&\mathbf{r}(t)=\mathbf{F}(t)-\mathbf{G}(t), \text { where }\\&\mathbf{F}(t)=\ln t \mathbf{i}+5 t \mathbf{j}-3 t^{2} \mathbf{k}, \quad \mathbf{G}(t)=\mathbf{i}+4 t \mathbf{j}-3 t^{2}\mathbf{k}\end{aligned}$$

Monica Miller

Numerade Educator

Problem 7

Find the domain of the vector-valued function.
$$\begin{aligned}&\mathbf{r}(t)=\mathbf{F}(t) \times \mathbf{G}(t), \text { where }\\&\mathbf{F}(t)=\sin t \mathbf{i}+\cos t \mathbf{j}, \quad \mathbf{G}(t)=\sin t \mathbf{j}+\cos t \mathbf{k}\end{aligned}$$

Monica Miller

Numerade Educator

Problem 8

Find the domain of the vector-valued function.
$$\begin{aligned}&\mathbf{r}(t)=\mathbf{F}(t) \times \mathbf{G}(t), \text { where }\\&\mathbf{F}(t)=t^{3} \mathbf{i}-t \mathbf{j}+t \mathbf{k}, \quad \mathbf{G}(t)=\sqrt[3]{t} \mathbf{i}+\frac{1}{t+1} \mathbf{j}+(t+2) \mathbf{k}\end{aligned}$$

Monica Miller

Numerade Educator

Problem 9

Evaluate (if possible) the vector-valued function at each given value of $t$.
$\mathbf{r}(t)=\frac{1}{2} t^{2} \mathbf{i}-(t-1) \mathbf{j}$
(a) $\mathbf{r}(1)$
(b) $\mathbf{r}(0)$
(c) $\mathbf{r}(s+1)$
(d) $\mathbf{r}(2+\Delta t)-\mathbf{r}(2)$

Monica Miller

Numerade Educator

Problem 10

Evaluate (if possible) the vector-valued function at each given value of $t$.
$\mathbf{r}(t)=\cos t \mathbf{i}+2 \sin t \mathbf{j}$
(a) $\mathbf{r}(0)$
(b) $\mathbf{r}(\pi / 4)$
(c) $\mathbf{r}(\theta-\pi)$
(d) $\mathbf{r}(\pi / 6+\Delta t)-\mathbf{r}(\pi / 6)$

Monica Miller

Numerade Educator

Problem 11

Evaluate (if possible) the vector-valued function at each given value of $t$.
$\mathbf{r}(t)=\ln t \mathbf{i}+\frac{1}{t} \mathbf{j}+3 t \mathbf{k}$
(a) $\mathbf{r}(2)$
(b) $r(-3)$
(c) $\mathbf{r}(t-4)$
(d) $\mathbf{r}(1+\Delta t)-\mathbf{r}(1)$

Monica Miller

Numerade Educator

Problem 12

Evaluate (if possible) the vector-valued function at each given value of $t$.
$\mathbf{r}(t)=\sqrt{t} \mathbf{i}+t^{3 / 2} \mathbf{j}+e^{-t / 4} \mathbf{k}$
(a) $\mathbf{r}(0)$
(b) $\mathbf{r}(4)$
(c) $\mathbf{r}(c+2)$
(d) $\mathbf{r}(9+\Delta t)-\mathbf{r}(9)$

Monica Miller

Numerade Educator

Problem 13

Represent the line segment from $P$ to $Q$ by a vector-valued function and by a set of parametric equations.
$$P(0,0,0), Q(3,1,2)$$

Monica Miller

Numerade Educator

Problem 14

Represent the line segment from $P$ to $Q$ by a vector-valued function and by a set of parametric equations.
$$P(0,2,-1), Q(4,7,2)$$

Monica Miller

Numerade Educator

Problem 15

Represent the line segment from $P$ to $Q$ by a vector-valued function and by a set of parametric equations.
$$P(-2,5,-3), Q(-1,4,9)$$

Monica Miller

Numerade Educator

Problem 16

Represent the line segment from $P$ to $Q$ by a vector-valued function and by a set of parametric equations.
$$P(1,-6,8), Q(-3,-2,5)$$

Monica Miller

Numerade Educator

Problem 17

Find $r(t) \cdot u(t) .$ Is the result a vector-valued function? Explain.
$$\mathbf{r}(t)=(3 t-1) \mathbf{i}+\frac{1}{4} t^{3} \mathbf{j}+4 \mathbf{k}, \quad \mathbf{u}(t)=t^{2} \mathbf{i}-8 \mathbf{j}+t^{3} \mathbf{k}$$

Monica Miller

Numerade Educator

Problem 18

Find $r(t) \cdot u(t) .$ Is the result a vector-valued function? Explain.
$$\mathbf{r}(t)=\langle 3 \cos t, 2 \sin t, t-2\rangle, \quad \mathbf{u}(t)=\left\langle 4 \sin t,-6 \cos t, t^{2}\right\rangle$$

Monica Miller

Numerade Educator

Problem 19

Match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]
(Check your book to see graph)
$$\mathbf{r}(t)=t \mathbf{i}+2 t \mathbf{j}+t^{2} \mathbf{k}, \quad-2 \leq t \leq 2$$

Monica Miller

Numerade Educator

Problem 20

Match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]
(Check your book to see graph)
$$\mathbf{r}(t)=\cos (\pi t) \mathbf{i}+\sin (\pi t) \mathbf{j}+t^{2} \mathbf{k}, \quad-1 \leq t \leq 1$$

Monica Miller

Numerade Educator

Problem 21

Match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]
(Check your book to see graph)
$$\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+e^{0.75 t} \mathbf{k}, \quad-2 \leq t \leq 2$$

Monica Miller

Numerade Educator

Problem 22

Match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]
(Check your book to see graph)
$$\mathbf{r}(t)=t \mathbf{i}+\ln t \mathbf{j}+\frac{2 t}{3} \mathbf{k}, \quad 0.1 \leq t \leq 5$$

Monica Miller

Numerade Educator

Problem 23

Sketch the curve represented by the vector-valued function and give the orientation of the curve.
$$\mathbf{r}(t)=\frac{t}{4} \mathbf{i}+(t-1) \mathbf{j}$$

Monica Miller

Numerade Educator

Problem 24

Sketch the curve represented by the vector-valued function and give the orientation of the curve.
$$\mathbf{r}(t)=\frac{t}{4} \mathbf{i}+(t-1) \mathbf{j}$$

Monica Miller

Numerade Educator

Problem 25

Sketch the curve represented by the vector-valued function and give the orientation of the curve.
$$\mathbf{r}(t)=t^{3} \mathbf{i}+t^{2} \mathbf{j}$$

Monica Miller

Numerade Educator

Problem 26

Sketch the curve represented by the vector-valued function and give the orientation of the curve.
$$\mathbf{r}(t)=\left(t^{2}+t\right) \mathbf{i}+\left(t^{2}-t\right) \mathbf{j}$$

Monica Miller

Numerade Educator

Problem 27

Sketch the curve represented by the vector-valued function and give the orientation of the curve.
$$\mathbf{r}(\theta)=\cos \theta \mathbf{i}+3 \sin \theta \mathbf{j}$$

Monica Miller

Numerade Educator

Problem 28

Sketch the curve represented by the vector-valued function and give the orientation of the curve.
$$\mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}$$

Monica Miller

Numerade Educator

Problem 29

Sketch the curve represented by the vector-valued function and give the orientation of the curve.
$$\mathbf{r}(\theta)=3 \sec \theta \mathbf{i}+2 \tan \theta \mathbf{j}$$

Monica Miller

Numerade Educator

Problem 30

Sketch the curve represented by the vector-valued function and give the orientation of the curve.
$$\mathbf{r}(t)=2 \cos ^{3} t \mathbf{i}+2 \sin ^{3} t \mathbf{j}$$

Monica Miller

Numerade Educator

Problem 31

Sketch the curve represented by the vector-valued function and give the orientation of the curve.
$$\mathbf{r}(t)=(-t+1) \mathbf{i}+(4 t+2) \mathbf{j}+(2 t+3) \mathbf{k}$$

Monica Miller

Numerade Educator

Problem 32

Sketch the curve represented by the vector-valued function and give the orientation of the curve.
$$\mathbf{r}(t)=t \mathbf{i}+(2 t-5) \mathbf{j}+3 t \mathbf{k}$$

Monica Miller

Numerade Educator

Problem 33

Sketch the curve represented by the vector-valued function and give the orientation of the curve.
$$\mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}+t \mathbf{k}$$

Monica Miller

Numerade Educator

Problem 34

Sketch the curve represented by the vector-valued function and give the orientation of the curve.
$$\mathbf{r}(t)=t \mathbf{i}+3 \cos t \mathbf{j}+3 \sin t \mathbf{k}$$

Monica Miller

Numerade Educator

Problem 35

Sketch the curve represented by the vector-valued function and give the orientation of the curve.
$$\mathbf{r}(t)=2 \sin t \mathbf{i}+2 \cos t \mathbf{j}+e^{-t} \mathbf{k}$$

Monica Miller

Numerade Educator

Problem 36

Sketch the curve represented by the vector-valued function and give the orientation of the curve.
$$\mathbf{r}(t)=t^{2} \mathbf{i}+2 t \mathbf{j}+\frac{3}{2} t \mathbf{k}$$

Monica Miller

Numerade Educator

Problem 37

Sketch the curve represented by the vector-valued function and give the orientation of the curve.
$$\mathbf{r}(t)=\left\langle t, t^{2}, \frac{2}{3} t^{3}\right\rangle$$

Monica Miller

Numerade Educator

Problem 38

Sketch the curve represented by the vector-valued function and give the orientation of the curve.
$$\mathbf{r}(t)=\langle\cos t+t \sin t, \sin t-t \cos t, t\rangle$$

Monica Miller

Numerade Educator

Problem 39

Use a computer algebra system to graph the vector-valued function and identify the common curve.
$$\mathbf{r}(t)=-\frac{1}{2} t^{2} \mathbf{i}+t \mathbf{j}-\frac{\sqrt{3}}{2} t^{2} \mathbf{k}$$

Monica Miller

Numerade Educator

Problem 40

Use a computer algebra system to graph the vector-valued function and identify the common curve.
$$\mathbf{r}(t)=t \mathbf{i}-\frac{\sqrt{3}}{2} t^{2} \mathbf{j}+\frac{1}{2} t^{2} \mathbf{k}$$

Monica Miller

Numerade Educator

Problem 41

Use a computer algebra system to graph the vector-valued function and identify the common curve.
$$\mathbf{r}(t)=\sin t \mathbf{i}+\left(\frac{\sqrt{3}}{2} \cos t-\frac{1}{2} t\right) \mathbf{j}+\left(\frac{1}{2} \cos t+\frac{\sqrt{3}}{2}\right) \mathbf{k}$$

Monica Miller

Numerade Educator

Problem 42

Use a computer algebra system to graph the vector-valued function and identify the common curve.
$$\mathbf{r}(t)=-\sqrt{2} \sin t \mathbf{i}+2 \cos t \mathbf{j}+\sqrt{2} \sin t \mathbf{k}$$

Monica Miller

Numerade Educator

Problem 43

Use a computer algebra system to graph the vector-valued function $\mathbf{r}(t) .$ For each $\mathbf{u}(t),$ make a conjecture about the transformation (if any) of the graph of $r(t) .$ Use a computer algebra system to verify your conjecture.
$\mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}+\frac{1}{2} t \mathbf{k}$
(a) $\mathbf{u}(t)=2(\cos t-1) \mathbf{i}+2 \sin t \mathbf{j}+\frac{1}{2} t \mathbf{k}$
(b) $\mathbf{u}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}+2 t \mathbf{k}$
(c) $\mathbf{u}(t)=2 \cos (-t) \mathbf{i}+2 \sin (-t) \mathbf{j}+\frac{1}{2}(-t) \mathbf{k}$
(d) $\mathbf{u}(t)=\frac{1}{2} t \mathbf{i}+2 \sin t \mathbf{j}+2 \cos t \mathbf{k}$
(e) $\mathbf{u}(t)=6 \cos t \mathbf{i}+6 \sin t \mathbf{j}+\frac{1}{2} t \mathbf{k}$

Monica Miller

Numerade Educator

Problem 44

Use a computer algebra system to graph the vector-valued function $\mathbf{r}(t) .$ For each $\mathbf{u}(t),$ make a conjecture about the transformation (if any) of the graph of $r(t) .$ Use a computer algebra system to verify your conjecture.
$\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+\frac{1}{2} t^{3} \mathbf{k}$
(a) $\mathbf{u}(t)=t \mathbf{i}+\left(t^{2}-2\right) \mathbf{j}+\frac{1}{2} t^{3} \mathbf{k}$
(b) $\mathbf{u}(t)=t^{2} \mathbf{i}+t \mathbf{j}+\frac{1}{2} t^{3} \mathbf{k}$
(c) $\mathbf{u}(t)=t \mathbf{i}+t^{2} \mathbf{j}+\left(\frac{1}{2} t^{3}+4\right) \mathbf{k}$
(d) $\mathbf{u}(t)=t \mathbf{i}+t^{2} \mathbf{j}+\frac{1}{8} t^{3} \mathbf{k}$
(e) $\mathbf{u}(t)=(-t) \mathbf{i}+(-t)^{2} \mathbf{j}+\frac{1}{2}(-t)^{3} \mathbf{k}$

Monica Miller

Numerade Educator

Problem 45

Represent the plane curve by a vector-valued function. (There are many correct answers.)
$$y=x+5$$

Monica Miller

Numerade Educator

Problem 46

Represent the plane curve by a vector-valued function. (There are many correct answers.)
$$2 x-3 y+5=0$$

Monica Miller

Numerade Educator

Problem 47

Represent the plane curve by a vector-valued function. (There are many correct answers.)
$$y=(x-2)^{2}$$

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Numerade Educator

Problem 48

Represent the plane curve by a vector-valued function. (There are many correct answers.)
$$y=4-x^{2}$$

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Numerade Educator

Problem 49

Represent the plane curve by a vector-valued function. (There are many correct answers.)
$$x^{2}+y^{2}=25$$

Monica Miller

Numerade Educator

Problem 50

Represent the plane curve by a vector-valued function. (There are many correct answers.)
$$(x-2)^{2}+y^{2}=4$$

Monica Miller

Numerade Educator

Problem 51

Represent the plane curve by a vector-valued function. (There are many correct answers.)
$$\frac{x^{2}}{16}-\frac{y^{2}}{4}=1$$

Monica Miller

Numerade Educator

Problem 52

Represent the plane curve by a vector-valued function. (There are many correct answers.)
$$\frac{x^{2}}{9}+\frac{y^{2}}{16}=1$$

Monica Miller

Numerade Educator

Problem 53

Sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter.
Surfaces $\quad$ Parameter
$z=x^{2}+y^{2}, \quad x+y=0 \quad x=t$

Monica Miller

Numerade Educator

Problem 54

Sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter.
Surfaces $\quad$ Parameter
$z=x^{2}+y^{2}, z=4 \quad x=2 \cos t$

Monica Miller

Numerade Educator

Problem 55

Sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter.
Surfaces $\quad$ Parameter
$x^{2}+y^{2}=4, \quad z=x^{2} \quad \quad x=2 \sin t$

Monica Miller

Numerade Educator

Problem 56

Sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter.
Surfaces $\quad$ Parameter
$4 x^{2}+4 y^{2}+z^{2}=16, \quad x=z^{2} \quad z=t$

Monica Miller

Numerade Educator

Problem 57

Sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter.
Surfaces $\quad$ Parameter
$x^{2}+y^{2}+z^{2}=4, \quad x+z=2 \quad x=1+\sin t$

Monica Miller

Numerade Educator

Problem 58

Sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter.
Surfaces $\quad$ Parameter
$x^{2}+y^{2}+z^{2}=10, \quad x+y=4 \quad x=2+\sin t$

Monica Miller

Numerade Educator

Problem 59

Sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter.
Surfaces $\quad$ Parameter
$x^{2}+z^{2}=4, \quad y^{2}+z^{2}=4 \quad x=t$ (first octant)

Monica Miller

Numerade Educator

Problem 60

Sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter.
Surfaces $\quad$ Parameter
$x^{2}+z^{2}=4, \quad y^{2}+z^{2}=4 \quad x=t$ (first octant)

Monica Miller

Numerade Educator

Problem 61

Show that the vector-valued function $\mathbf{r}(t)=t \mathbf{i}+2 t \cos t \mathbf{j}+2 t \sin t \mathbf{k}$ lies on the cone $4 x^{2}=y^{2}+z^{2}$ Sketch the curve.

Monica Miller

Numerade Educator

Problem 62

Show that the vector-valued function $\mathbf{r}(t)=e^{-t} \cos t \mathbf{i}+e^{-t} \sin t \mathbf{j}+e^{-t} \mathbf{k}$ lies on the cone $z^{2}=x^{2}+y^{2} .$ Sketch the curve.

Monica Miller

Numerade Educator

Problem 63

Find the limit (if it exists).
$$\lim _{t \rightarrow \pi}(t \mathbf{i}+\cos t \mathbf{j}+\sin t \mathbf{k})$$

Monica Miller

Numerade Educator

Problem 64

Find the limit (if it exists).
$$\lim _{t \rightarrow 2}\left(3 t \mathbf{i}+\frac{2}{t^{2}-1} \mathbf{j}+\frac{1}{t} \mathbf{k}\right)$$

Monica Miller

Numerade Educator

Problem 65

Find the limit (if it exists).
$$\lim _{t \rightarrow 0}\left(t^{2} \mathbf{i}+3 t \mathbf{j}+\frac{1-\cos t}{t} \mathbf{k}\right)$$

Monica Miller

Numerade Educator

Problem 66

Find the limit (if it exists).
$$\lim _{t \rightarrow 1}\left(\sqrt{t} \mathbf{i}+\frac{\ln t}{t^{2}-1} \mathbf{j}+\frac{1}{t-1} \mathbf{k}\right)$$

Monica Miller

Numerade Educator

Problem 67

Find the limit (if it exists).
$$\lim _{t \rightarrow 0}\left(e^{t} \mathbf{i}+\frac{\sin t}{t} \mathbf{j}+e^{-t} \mathbf{k}\right)$$

Monica Miller

Numerade Educator

Problem 68

Find the limit (if it exists).
$$\lim _{t \rightarrow \infty}\left(e^{-t} \mathbf{i}+\frac{1}{t} \mathbf{j}+\frac{t}{t^{2}+1} \mathbf{k}\right)$$

Monica Miller

Numerade Educator

Problem 69

Determine the interval(s) on which the vector-valued function is continuous.
$$\mathbf{r}(t)=t \mathbf{i}+\frac{1}{t} \mathbf{j}$$

Monica Miller

Numerade Educator

Problem 70

Determine the interval(s) on which the vector-valued function is continuous.
$$\mathbf{r}(t)=\sqrt{t} \mathbf{i}+\sqrt{t-1} \mathbf{j}$$

Monica Miller

Numerade Educator

Problem 71

Determine the interval(s) on which the vector-valued function is continuous.
$$\mathbf{r}(t)=t \mathbf{i}+\arcsin t \mathbf{j}+(t-1) \mathbf{k}$$

Monica Miller

Numerade Educator

Problem 72

Determine the interval(s) on which the vector-valued function is continuous.
$$\mathbf{r}(t)=2 e^{-t} \mathbf{i}+e^{-t} \mathbf{j}+\ln (t-1) \mathbf{k}$$

Monica Miller

Numerade Educator

Problem 73

Determine the interval(s) on which the vector-valued function is continuous.
$$\mathbf{r}(t)=\left\langle e^{-t}, t^{2}, \tan t\right\rangle$$

Monica Miller

Numerade Educator

Problem 74

Determine the interval(s) on which the vector-valued function is continuous.
$$\mathbf{r}(t)=\langle 8, \sqrt{t}, \sqrt[3]{t}\rangle$$

Monica Miller

Numerade Educator

Problem 75

Consider the vector-valued function
$\mathbf{r}(t)=t^{2} \mathbf{i}+(t-3) \mathbf{j}+t \mathbf{k}$
Write a vector-valued function $s(t)$ that is the specified transformation of $\mathbf{r}.$

A vertical translation three units upward

Monica Miller

Numerade Educator

Problem 76

Consider the vector-valued function
$\mathbf{r}(t)=t^{2} \mathbf{i}+(t-3) \mathbf{j}+t \mathbf{k}$
Write a vector-valued function $s(t)$ that is the specified transformation of $\mathbf{r}.$

A vertical translation four units downward

Monica Miller

Numerade Educator

Problem 77

Consider the vector-valued function
$\mathbf{r}(t)=t^{2} \mathbf{i}+(t-3) \mathbf{j}+t \mathbf{k}$
Write a vector-valued function $s(t)$ that is the specified transformation of $\mathbf{r}.$

A horizontal translation two units in the direction of the negative $x$-axis

Monica Miller

Numerade Educator

Problem 78

Consider the vector-valued function
$\mathbf{r}(t)=t^{2} \mathbf{i}+(t-3) \mathbf{j}+t \mathbf{k}$
Write a vector-valued function $s(t)$ that is the specified transformation of $\mathbf{r}.$

A horizontal translation five units in the direction of the positive $y$-axis

Monica Miller

Numerade Educator

Problem 79

State the definition of continuity of a vector-valued function. Give an example of a vector-valued function that is defined but not continuous at $t=2$.

Monica Miller

Numerade Educator

Problem 80

Which of the following vectorvalued functions represent the same graph?
(a) $\mathbf{r}(t)=(-3 \cos t+1) \mathbf{i}+(5 \sin t+2) \mathbf{j}+4 \mathbf{k}$
(b) $\mathbf{r}(t)=4 \mathbf{i}+(-3 \cos t+1) \mathbf{j}+(5 \sin t+2) \mathbf{k}$
(c) $\mathbf{r}(t)=(3 \cos t-1) \mathbf{i}+(-5 \sin t-2) \mathbf{j}+4 \mathbf{k}$
(d) $\mathbf{r}(t)=(-3 \cos 2 t+1) \mathbf{i}+(5 \sin 2 t+2) \mathbf{j}+4 \mathbf{k}$

Monica Miller

Numerade Educator

Problem 81

The outer edge of a playground slide is in the shape of a helix of radius 1.5 meters. The slide has a height of 2 meters and makes one complete revolution from top to bottom. Find a vector-valued
function for the helix. Use a computer algebra system to graph your function. (There are many correct answers.)

Monica Miller

Numerade Educator

Problem 82

The four figures below are graphs of the vector-valued function $\mathbf{r}(t)=4 \cos t \mathbf{i}+4 \sin t \mathbf{j}+(t / 4) \mathbf{k} .$ Match each of the four graphs with the point in space from which the helix is viewed. The four points are (0,0,20) (20,0,0),(-20,0,0), and (10,20,10).
(Check your book to see graph)

Monica Miller

Numerade Educator

Problem 83

Let $\mathbf{r}(t)$ and $\mathbf{u}(t)$ be vector-valued functions whose limits exist as $t \rightarrow c .$ Prove that
$$\lim _{t \rightarrow c}[\mathbf{r}(t) \times \mathbf{u}(t)]=\lim _{t \rightarrow c} \mathbf{r}(t) \times \lim _{t \rightarrow c} \mathbf{u}(t)$$

Monica Miller

Numerade Educator

Problem 84

Let $\mathbf{r}(t)$ and $\mathbf{u}(t)$ be vector-valued functions whose limits exist as $t \rightarrow c .$ Prove that
$$\lim _{t \rightarrow c}[\mathbf{r}(t) \cdot \mathbf{u}(t)]=\lim _{t \rightarrow c} \mathbf{r}(t) \cdot \lim _{t \rightarrow c} \mathbf{u}(t)$$

Monica Miller

Numerade Educator

Problem 85

Prove that if $\mathbf{r}$ is a vector-valued function that is continuous at $c,$ then $\|\mathbf{r}\|$ is continuous at $c.$

Monica Miller

Numerade Educator

Problem 86

Verify that the converse of Exercise 85 is not true by finding a vector-valued function $\mathbf{r}$ such that $\|\mathbf{r}\|$ is continuous at $c$ but $\mathbf{r}$ is not continuous at $c.$

Monica Miller

Numerade Educator

Problem 87

Two particles travel along the space curves $r(t)$ and $u(t) .$ A collision will occur at the point of intersection $P$ when both particles are at $P$ at the same time. Do the particles collide? Do their paths intersect?
$$\begin{array}{l}\mathbf{r}(t)=t^{2} \mathbf{i}+(9 t-20) \mathbf{j}+t^{2} \mathbf{k} \\ \mathbf{u}(t)=(3 t+4) \mathbf{i}+t^{2} \mathbf{j}+(5 t-4) \mathbf{k}\end{array}$$

Monica Miller

Numerade Educator

Problem 88

Two particles travel along the space curves $r(t)$ and $u(t) .$ A collision will occur at the point of intersection $P$ when both particles are at $P$ at the same time. Do the particles collide? Do their paths intersect?
$$\begin{aligned}&\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k}\\&\mathbf{u}(t)=(-2 t+3) \mathbf{i}+8 t \mathbf{j}+(12 t+2) \mathbf{k}\end{aligned}$$

Monica Miller

Numerade Educator

Problem 89

Two particles travel along the space curves $\mathbf{r}(t)$ and $\mathbf{u}(t)$.
If $\mathbf{r}(t)$ and $\mathbf{u}(t)$ intersect, will the particles collide?

Monica Miller

Numerade Educator

Problem 90

Two particles travel along the space curves $\mathbf{r}(t)$ and $\mathbf{u}(t)$.
If the particles collide, do their paths $\mathbf{r}(t)$ and $\mathbf{u}(t)$ intersect?

Monica Miller

Numerade Educator

Problem 91

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $f, g,$ and $h$ are first-degree polynomial functions, then the curve given by $x=f(t), y=g(t),$ and $z=h(t)$ is a line.

Monica Miller

Numerade Educator

Problem 92

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If the curve given by $x=f(t), y=g(t),$ and $z=h(t)$ is a line, then $f, g,$ and $h$ are first-degree polynomial functions of $t$.

Monica Miller

Numerade Educator

Problem 93

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
Two particles travel along the space curves $\mathbf{r}(t)$ and $\mathbf{u}(t) .$ The intersection of their paths depends only on the curves traced out by $\mathbf{r}(t)$ and $\mathbf{u}(t),$ while collision depends on the parametrizations.

Monica Miller

Numerade Educator

Problem 94

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
The vector-valued function $\mathbf{r}(t)=t^{2} \mathbf{i}+t \sin t \mathbf{j}+t \cos t \mathbf{k}$ lies on the paraboloid $x=y^{2}+z^{2}$.

Monica Miller

Numerade Educator