Calculus

James Stewart

Chapter 13

Vector Functions - all with Video Answers

Educators

Section 1

13.1 Exercises

Problem 1

Find the domain of the vector function.
$\mathbf{r}(t)=\left\langle\sqrt{4-t^2}, e^{-3 t}, \ln (t+1)\right\rangle$

Carson Merrill

Carson Merrill

Numerade Educator

Problem 2

Find the domain of the vector function.
$\mathbf{r}(t)=\frac{t-2}{t+2} \mathbf{i}+\sin t \mathbf{j}+\ln \left(9-t^2\right) \mathbf{k}$

James Kiss

Numerade Educator

Problem 3

Find the limit.
$\lim _{t \rightarrow 1}\left(\frac{t^2-t}{t-1} \mathbf{i}+\sqrt{t+8} \mathbf{j}+\frac{\sin \pi t}{\ln t} \mathbf{k}\right)$

WZ

Wen Zheng

Numerade Educator

Problem 3

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

William Semus

Numerade Educator

Problem 5

Find the limit.
$\lim _{t \rightarrow \infty}\left\langle\frac{1+t^2}{1-t^2}, \tan ^{-1} t, \frac{1-e^{-2 t}}{t}\right\rangle$

WZ

Wen Zheng

Numerade Educator

Problem 6

Find the limit.
$\lim _{t \rightarrow \infty}\left\langle t e^{-t}, \frac{t^3+t}{2 t^3-1}, t \sin \frac{1}{t}\right\rangle$

James Kiss

Numerade Educator

Problem 7

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $t$ increases.
$\mathbf{r}(t)=\langle\sin t, t\rangle$

WZ

Wen Zheng

Numerade Educator

Problem 8

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $t$ increases.
$\mathbf{r}(t)=\left\langle t^3, t^2\right\rangle$

Carson Merrill

Numerade Educator

Problem 9

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $t$ increases.
$\mathbf{r}(t)=\langle t, 2-t, 2 t\rangle$

Carson Merrill

Numerade Educator

Problem 10

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $t$ increases.
$\mathbf{r}(t)=\langle\sin \pi t, t, \cos \pi t\rangle$

Carson Merrill

Numerade Educator

Problem 11

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $t$ increases.
$\mathbf{r}(t)=\langle 1, \cos t, 2 \sin t\rangle$

Carson Merrill

Numerade Educator

Problem 12

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $t$ increases.
$\mathbf{r}(t)=t^2 \mathbf{i}+t \mathbf{j}+2 \mathbf{k}$

Carson Merrill

Numerade Educator

Problem 13

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $t$ increases.
$\mathbf{r}(t)=t^2 \mathbf{i}+t^4 \mathbf{j}+t^6 \mathbf{k}$

James Kiss

Numerade Educator

Problem 14

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $t$ increases.
$\mathbf{r}(t)=\cos t \mathbf{i}-\cos t \mathbf{j}+\sin t \mathbf{k}$

Check back soon!

Problem 15

Draw the projections of the curve on the three coordinate planes. Use these projections to help sketch the curve.
$\mathbf{r}(t)=\langle t, \sin t, 2 \cos t\rangle$

Nick Johnson

Numerade Educator

Problem 16

Draw the projections of the curve on the three coordinate planes. Use these projections to help sketch the curve.
$\mathbf{r}(t)=\left\langle t, t, t^2\right\rangle$

Nick Johnson

Numerade Educator

Problem 17

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$.
$P(2,0,0), \quad Q(6,2,-2)$

William Semus

Numerade Educator

Problem 18

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$.
$P(-1,2,-2), \quad Q(-3,5,1)$

Bobby Barnes

University of North Texas

Problem 19

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$.
$P(0,-1,1), \quad Q\left(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\right)$

Bobby Barnes

University of North Texas

Problem 20

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$.
$P(a, b, c), \quad Q(u, v, w)$

Bobby Barnes

University of North Texas

Problem 21

Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices.
$x=t \cos t, \quad y=t, \quad z=t \sin t, \quad t \geqslant 0$

James Kiss

Numerade Educator

Problem 22

Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices.
$x=\cos t, \quad y=\sin t, \quad z=1 /\left(1+t^2\right)$

James Kiss

Numerade Educator

Problem 23

Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices.
$x=t, \quad y=1 /\left(1+t^2\right), \quad z=t^2$

Carson Merrill

Numerade Educator

Problem 24

Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices.
$x=\cos t, \quad y=\sin t, \quad z=\cos 2 t$

James Kiss

Numerade Educator

Problem 25

Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices.
$x=\cos 8 t, \quad y=\sin 8 t, \quad z=e^{0.8 t}, \quad t \geqslant 0$

James Kiss

Numerade Educator

Problem 26

Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices.
$x=\cos ^2 t, \quad y=\sin ^2 t, \quad z=t$

James Kiss

Numerade Educator

Problem 27

Show that the curve with parametric equations $x=t \cos t$, $y=t \sin t, z=t$ lies on the cone $z^2=x^2+y^2$, and use this fact to help sketch the curve.

WZ

Wen Zheng

Numerade Educator

Problem 28

Show that the curve with parametric equations $x=\sin t$, $y=\cos t, z=\sin ^2 t$ is the curve of intersection of the surfaces $z=x^2$ and $x^2+y^2=1$. Use this fact to help sketch the curve.

WZ

Wen Zheng

Numerade Educator

Problem 29

At what points does the curve $\mathbf{r}(t)=t \mathbf{i}+\left(2 t-t^2\right) \mathbf{k}$ intersect the paraboloid $z=x^2+y^2$ ?

Bobby Barnes

University of North Texas

Problem 30

At what points does the helix $\mathbf{r}(t)=\langle\sin t, \cos t, t\rangle$ intersect the sphere $x^2+y^2+z^2=5$ ?

WZ

Wen Zheng

Numerade Educator

Problem 31

Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.
$\mathbf{r}(t)=\langle\cos t \sin 2 t, \sin t \sin 2 t, \cos 2 t\rangle$

Carson Merrill

Numerade Educator

Problem 32

Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.
$\mathbf{r}(t)=\left\langle t^2, \ln t, t\right\rangle$

Check back soon!

Problem 33

Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.
$\mathbf{r}(t)=\langle t, t \sin t, t \cos t\rangle$

Carson Merrill

Numerade Educator

Problem 34

Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.
$\mathbf{r}(t)=\left\langle t, e^t, \cos t\right\rangle$

Carson Merrill

Numerade Educator

Problem 35

Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.
$\mathbf{r}(t)=\langle\cos 2 t, \cos 3 t, \cos 4 t\rangle$

Carson Merrill

Numerade Educator

Problem 36

Graph the curve with parametric equations $x=\sin t, y=\sin 2 t$, $z=\cos 4 t$. Explain its shape by graphing its projections onto the three coordinate planes.

WZ

Wen Zheng

Numerade Educator

Problem 37

Graph the curve with parametric equations

$$
\begin{aligned}
& x=(1+\cos 16 t) \cos t \\
& y=(1+\cos 16 t) \sin t \\
& z=1+\cos 16 t
\end{aligned}
$$

Explain the appearance of the graph by showing that it lies on a cone.

Carson Merrill

Numerade Educator

Problem 38

Graph the curve with parametric equations

$$
\begin{aligned}
& x=\sqrt{1-0.25 \cos ^2 10 t} \cos t \\
& y=\sqrt{1-0.25 \cos ^2 10 t} \sin t \\
& z=0.5 \cos 10 t
\end{aligned}
$$

Explain the appearance of the graph by showing that it lies on a sphere.

Carson Merrill

Numerade Educator

Problem 39

Show that the curve with parametric equations $x=t^2$, $y=1-3 t, z=1+t^3$ passes through the points $(1,4,0)$ and $(9,-8,28)$ but not through the point $(4,7,-6)$.

Aman Gupta

Numerade Educator

Problem 40

Find a vector function that represents the curve of intersection of the two surfaces.
The cylinder $x^2+y^2=4$ and the surface $z=x y$

Carson Merrill

Numerade Educator

Problem 41

Find a vector function that represents the curve of intersection of the two surfaces.
The cone $z=\sqrt{x^2+y^2}$ and the plane $z=1+y$

WZ

Wen Zheng

Numerade Educator

Problem 42

Find a vector function that represents the curve of intersection of the two surfaces.
The paraboloid $z=4 x^2+y^2$ and the parabolic cylinder $y=x^2$

WZ

Wen Zheng

Numerade Educator

Problem 43

Find a vector function that represents the curve of intersection of the two surfaces.
The hyperboloid $z=x^2-y^2$ and the cylinder $x^2+y^2=1$

James Kiss

Numerade Educator

Problem 44

Find a vector function that represents the curve of intersection of the two surfaces.
The semiellipsoid $x^2+y^2+4 z^2=4, y \geqslant 0$, and the cylinder $x^2+z^2=1$

Carson Merrill

Numerade Educator

Problem 45

Try to sketch by hand the curve of intersection of the circular cylinder $x^2+y^2=4$ and the parabolic cylinder $z=x^2$. Then find parametric equations for this curve and use these equations and a computer to graph the curve.

WZ

Wen Zheng

Numerade Educator

Problem 46

Try to sketch by hand the curve of intersection of the parabolic cylinder $y=x^2$ and the top half of the ellipsoid $x^2+4 y^2+4 z^2=16$. Then find parametric equations for this curve and use these equations and a computer to graph the curve.

WZ

Wen Zheng

Numerade Educator

Problem 47

If two objects travel through space along two different curves, it's often important to know whether they will collide. (Will a missile hit its moving target? Will two aircraft collide?) The curves might intersect, but we need to know whether the objects are in the same position at the same time. Suppose the trajectories of two particles are given by the vector functions

$$
\mathbf{r}_1(t)=\left\langle t^2, 7 t-12, t^2\right\rangle \quad \mathbf{r}_2(t)=\left\langle 4 t-3, t^2, 5 t-6\right\rangle
$$

for $t \geqslant 0$. Do the particles collide?

Bobby Barnes

University of North Texas

Problem 48

Two particles travel along the space curves

$$
\mathbf{r}_1(t)=\left\langle t, t^2, t^3\right\rangle \quad \mathbf{r}_2(t)=\langle 1+2 t, 1+6 t, 1+14 t\rangle
$$

Do the particles collide? Do their paths intersect?

WZ

Wen Zheng

Numerade Educator

Problem 49

Suppose $\mathbf{u}$ and $\mathbf{v}$ are vector functions that possess limits as $t \rightarrow a$ and let $c$ be a constant. Prove the following properties of limits.
(a) $\lim _{t \rightarrow a}[\mathbf{u}(t)+\mathbf{v}(t)]=\lim _{t \rightarrow a} \mathbf{u}(t)+\lim _{t \rightarrow a} \mathbf{v}(t)$
(b) $\lim _{t \rightarrow a} c \mathbf{u}(t)=c \lim _{t \rightarrow a} \mathbf{u}(t)$
(c) $\lim _{t \rightarrow a}[\mathbf{u}(t) \cdot \mathbf{v}(t)]=\lim _{t \rightarrow a} \mathbf{u}(t) \cdot \lim _{t \rightarrow a} \mathbf{v}(t)$
(d) $\lim _{t \rightarrow a}[\mathbf{u}(t) \times \mathbf{v}(t)]=\lim _{t \rightarrow a} \mathbf{u}(t) \times \lim _{t \rightarrow a} \mathbf{v}(t)$

Carson Merrill

Numerade Educator

Problem 50

The view of the trefoil knot shown in Figure 8 is accurate, but it doesn't reveal the whole story. Use the parametric equations

$$
\begin{aligned}
& x=(2+\cos 1.5 t) \cos t \\
& y=(2+\cos 1.5 t) \sin t \\
& z=\sin 1.5 t
\end{aligned}
$$

to sketch the curve by hand as viewed from above, with gaps indicating where the curve passes over itself. Start by showing that the projection of the curve onto the $x y$-plane has polar coordinates $r=2+\cos 1.5 t$ and $\theta=t$, so $r$ varies between $I$ and 3. Then show that $z$ has maximum and minimum values when the projection is halfway between $r=1$ and $r=3$.

When you have finished your sketch, use a computer to draw the curve with viewpoint directly above and compare with your sketch. Then use the computer to draw the curve from several other viewpoints. You can get a better impression of the curve if you plot a tube with radius 0.2 around the curve. (Use the tubeplot command in Maple or the tubecurve or Tube command in Mathematica.)

Carson Merrill

Numerade Educator

Problem 51

Show that $\lim _{t \rightarrow a} \mathbf{r}(t)=\mathbf{b}$ if and only if for every $\varepsilon>0$ there is a number $\delta>0$ such that

$$
\text { if } 0<|t-a|<\delta \text { then }|\mathbf{r}(t)-\mathbf{b}|<\varepsilon
$$

Carson Merrill

Numerade Educator