Advanced Engineering Mathematics

Dennis G. Zill, Warren S. Wright

Chapter 9

Vector Calculus - all with Video Answers

Educators

Section 1

Vector Functions

Problem 1

In Problems, graph the curve traced by the given vector function.
$\mathbf{r}(t)=2 \sin t \mathbf{i}+4 \cos t \mathbf{j}+t \mathbf{k} ; t \geq 0$

Amany Waheeb

Amany Waheeb

Numerade Educator

Problem 2

In Problems, graph the curve traced by the given vector function.
$\mathbf{r}(t)=\cos t \hat{\mathbf{i}}+t \mathbf{j}+\sin t \mathbf{k} ; t \geq 0$

Amany Waheeb

Numerade Educator

Problem 3

In Problems, graph the curve traced by the given vector function.
$\mathbf{r}(t)=t \mathbf{i}+2 t \mathbf{j}+\cos t \mathbf{k} ; t \geq 0$

Wendi Zhao

Numerade Educator

Problem 4

In Problems, graph the curve traced by the given vector function.
$\mathbf{r}(t)=4 \mathbf{i}+2 \cos t \mathbf{j}+3 \sin t \mathbf{k}$

Wendi Zhao

Numerade Educator

Problem 5

In Problems, graph the curve traced by the given vector function.
$\mathbf{r}(t)=\left\langle e^{t}, e^{2 t}\right\rangle$

Wendi Zhao

Numerade Educator

Problem 6

In Problems, graph the curve traced by the given vector function.
$\mathbf{r}(t)=\cosh t \mathbf{i}+3 \sinh t \mathbf{j}$

Wendi Zhao

Numerade Educator

Problem 7

In Problems, graph the curve traced by the given vector function.
$\mathbf{r}(t)=\langle\sqrt{2} \sin t, \sqrt{2} \sin t, 2 \cos t\rangle ; 0 \leq t \leq \pi / 2$

Amany Waheeb

Numerade Educator

Problem 8

In Problems, graph the curve traced by the given vector function.
$\mathbf{r}(t)=t \mathbf{i}+t^{3} \mathbf{j}+t \mathbf{k}$

Wendi Zhao

Numerade Educator

Problem 9

In Problems, graph the curve traced by the given vector function.
$\mathbf{r}(t)=e^{t} \cos t \mathbf{i}+e^{t} \sin t \mathbf{j}+e^{t} \mathbf{k}$

Wendi Zhao

Numerade Educator

Problem 10

In Problems, graph the curve traced by the given vector function.
$\mathbf{r}(t)=\left\langle t \cos t, t \sin t, t^{2}\right\rangle$

Wendi Zhao

Numerade Educator

Problem 11

In Problems, find the vector function that describes the curve $C$ of intersection between the given surfaces. Sketch the curve $C$. Use the indicated parameter.
$$
z=x^{2}+y^{2}, y=x ; x=t
$$

Thomas Pauly

Numerade Educator

Problem 12

In Problems, find the vector function that describes the curve $C$ of intersection between the given surfaces. Sketch the curve $C$. Use the indicated parameter.
$$
x^{2}+y^{2}-z^{2}=1, y=2 x ; x=t
$$

Thomas Pauly

Numerade Educator

Problem 13

In Problems, find the vector function that describes the curve $C$ of intersection between the given surfaces. Sketch the curve $C$. Use the indicated parameter.
$$
x^{2}+y^{2}=9, z=9-x^{2} ; x=3 \cos t
$$

Thomas Pauly

Numerade Educator

Problem 14

In Problems, find the vector function that describes the curve $C$ of intersection between the given surfaces. Sketch the curve $C$. Use the indicated parameter.
$$
z=x^{2}+y^{2}, z=1 ; x=\sin t
$$

Monica Miller

Numerade Educator

Problem 15

Given that $\mathbf{r}(t)=\frac{\sin 2 t}{t} \mathbf{i}+(t-2)^{5} \mathbf{j}+t \ln t \mathbf{k}$, find
$\lim _{t \rightarrow 0^{+}} \mathbf{r}(t)$

Hast Aggarwal

Numerade Educator

Problem 16

Given that $\lim _{t \rightarrow a} r_{1}(t)=\mathbf{i}-2 \mathbf{j}+\mathbf{k}$ and $\lim _{t \rightarrow a} \mathbf{r}_{2}(t)=2 \mathbf{i}+$
$5 \mathbf{j}+7 \mathbf{k}$, find:
(a) $\lim _{t \rightarrow a}\left[-4 \mathbf{r}_{1}(t)+3 \mathbf{r}_{2}(t)\right]$
(b) $\lim _{t \rightarrow a} \mathbf{r}_{1}(t) \cdot \mathbf{r}_{2}(t) .$

Hast Aggarwal

Numerade Educator

Problem 17

In Problems, find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$ for the given vector function.
$\mathbf{r}(t)=\ln t \mathbf{i}+\mathbf{j}, t>0$

Hast Aggarwal

Numerade Educator

Problem 18

In Problems, find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$ for the given vector function.
$\mathbf{r}(t)=\langle t \cos t-\sin t, t+\cos t\rangle$

Hast Aggarwal

Numerade Educator

Problem 19

In Problems, find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$ for the given vector function.
$\mathbf{r}(t)=\left\langle t e^{2 t}, t^{3}, 4 t^{2}-t\right\rangle$

Hast Aggarwal

Numerade Educator

Problem 20

In Problems, find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$ for the given vector function.
$\mathbf{r}(t)=t^{2} \mathbf{i}+t^{3} \mathbf{j}+\tan ^{-1} t \mathbf{k}$

Hast Aggarwal

Numerade Educator

Problem 21

In Problems, graph the curve $C$ that is described by $\mathbf{r}$ and $\operatorname{graph} \mathbf{r}^{\prime}$ at the indicated value of $t$.
$$
\mathbf{r}(t)=2 \cos t \mathbf{i}+6 \sin t \mathbf{j} ; t=\pi / 6
$$

Monica Miller

Numerade Educator

Problem 22

In Problems, graph the curve $C$ that is described by $\mathbf{r}$ and $\operatorname{graph} \mathbf{r}^{\prime}$ at the indicated value of $t$.
$$
\mathbf{r}(t)=t^{3} \mathbf{i}+t^{2} \mathbf{j} ; t=-1
$$

Linh Vu

Numerade Educator

Problem 23

In Problems, graph the curve $C$ that is described by $\mathbf{r}$ and $\operatorname{graph} \mathbf{r}^{\prime}$ at the indicated value of $t$.
$$
\mathbf{r}(t)=2 \mathbf{i}+t \mathbf{j}+\frac{4}{1+t^{2}} \mathbf{k} ; t=1
$$

Linh Vu

Numerade Educator

Problem 24

In Problems, graph the curve $C$ that is described by $\mathbf{r}$ and $\operatorname{graph} \mathbf{r}^{\prime}$ at the indicated value of $t$.
$$
\mathbf{r}(t)=3 \cos t \mathbf{i}+3 \sin t \mathbf{j}+2 t \mathbf{k} ; t=\pi / 4
$$

Wendi Zhao

Numerade Educator

Problem 25

In Problems, find parametric equations of the tangent line to the given curve at the indicated value of $t$.
$$
x=t, y=\frac{1}{2} t^{2}, z=\frac{1}{3} t^{3} ; t=2
$$

Hast Aggarwal

Numerade Educator

Problem 26

In Problems, find parametric equations of the tangent line to the given curve at the indicated value of $t$.
$$
x=t^{3}-t, y=\frac{6 t}{t+1}, z=(2 t+1)^{2} ; t=1
$$

Hast Aggarwal

Numerade Educator

Problem 27

In Problems, find the indicated derivative. Assume that all vector functions are differentiable.
$$
\frac{d}{d t}\left[\mathbf{r}(t) \times \mathbf{r}^{\prime}(t)\right]
$$

Hast Aggarwal

Numerade Educator

Problem 28

In Problems, find the indicated derivative. Assume that all vector functions are differentiable.
$$
\frac{d}{d t}[\mathbf{r}(t) \cdot(t \mathbf{r}(t))]
$$

Wendi Zhao

Numerade Educator

Problem 29

In Problems, find the indicated derivative. Assume that all vector functions are differentiable.
$$
\frac{d}{d t}\left[\mathbf{r}(t) \cdot\left(\mathbf{r}^{\prime}(t) \times \mathbf{r}^{\prime \prime}(t)\right)\right]
$$

Wendi Zhao

Numerade Educator

Problem 30

In Problems, find the indicated derivative. Assume that all vector functions are differentiable.
$$
\frac{d}{d t}\left[\mathbf{r}_{1}(t) \times\left(\mathbf{r}_{2}(t) \times \mathbf{r}_{3}(t)\right)\right]
$$

Wendi Zhao

Numerade Educator

Problem 31

In Problems, find the indicated derivative. Assume that all vector functions are differentiable.
$$
\frac{d}{d t}\left[\mathbf{r}_{1}(2 t)+\mathbf{r}_{2}\left(\frac{1}{t}\right)\right]
$$

Hast Aggarwal

Numerade Educator

Problem 32

In Problems, find the indicated derivative. Assume that all vector functions are differentiable.
$$
\frac{d}{d t}\left[t^{3} r\left(t^{2}\right)\right]
$$

Hast Aggarwal

Numerade Educator

Problem 33

In Problems, evaluate the given integral.
$$
\int_{-1}^{2}\left(t \mathrm{i}+3 t^{2} \mathrm{j}+4 t^{3} \mathbf{k}\right) d t
$$

Hast Aggarwal

Numerade Educator

Problem 34

In Problems, evaluate the given integral.
$$
\int_{0}^{4}(\sqrt{2 t+1} \mathbf{i}-\sqrt{t} \mathbf{j}+\sin \pi t \mathbf{k}) d t
$$

Wendi Zhao

Numerade Educator

Problem 35

In Problems, evaluate the given integral.
$$
\int\left(t e^{t} \mathbf{i}-e^{-2 t} \mathbf{j}+t e^{t^{2}} \mathbf{k}\right) d t
$$

Wendi Zhao

Numerade Educator

Problem 36

In Problems, evaluate the given integral.
$$
\int \frac{1}{1+t^{2}}\left(\mathbf{i}+t \mathbf{j}+t^{2} \mathbf{k}\right) d t
$$

Wendi Zhao

Numerade Educator

Problem 37

In Problems, find a vector function $\mathbf{r}$ that satisfies the indicated conditions.
$$
\mathbf{r}^{\prime}(t)=6 \mathbf{i}+6 t \mathbf{j}+3 t^{2} \mathbf{k} ; \mathbf{r}(0)=\mathbf{i}-2 \mathbf{j}+\mathbf{k}
$$

Wendi Zhao

Numerade Educator

Problem 38

In Problems, find a vector function $\mathbf{r}$ that satisfies the indicated conditions.
$$
\mathbf{r}^{\prime}(t)=t \sin t^{2} \mathbf{i}-\cos 2 t \mathbf{j} ; \mathbf{r}(0)=\frac{3}{2} \mathbf{i}
$$

Wendi Zhao

Numerade Educator

Problem 39

In Problems, find a vector function $\mathbf{r}$ that satisfies the indicated conditions.
$$
\mathbf{r}^{\prime \prime}(t)=12 t \mathbf{i}-3 t^{-1 / 2} \mathbf{j}+2 \mathbf{k} ; \mathbf{r}^{\prime}(1)=\mathbf{j}, \mathbf{r}(1)=2 \mathbf{i}-\mathbf{k}
$$

Wendi Zhao

Numerade Educator

Problem 40

In Problems, find a vector function $\mathbf{r}$ that satisfies the indicated conditions.
$$
\begin{aligned}
&\mathbf{r}^{\prime \prime}(t)=\sec ^{2} t \mathbf{i}+\cos t \mathbf{j}-\sin t \mathbf{k} ; \\
&\mathbf{r}^{\prime}(0)=\mathbf{i}+\mathbf{j}+\mathbf{k}, \mathbf{r}(0)=-\mathbf{j}+5 \mathbf{k}
\end{aligned}
$$

Wendi Zhao

Numerade Educator

Problem 41

In Problems, find the length of the curve traced by the given vector function on the indicated interval.
$$
\mathbf{r}(t)=a \cos t \mathbf{i}+a \sin t \mathbf{j}+\text { ct } \mathbf{k} ; 0 \leq t \leq 2 \pi
$$

Wendi Zhao

Numerade Educator

Problem 42

In Problems, find the length of the curve traced by the given vector function on the indicated interval.
$$
\mathbf{r}(t)=t \mathbf{i}+t \cos t \mathbf{j}+t \sin t \mathbf{k} ; 0 \leq t \leq \pi
$$

Wendi Zhao

Numerade Educator

Problem 43

In Problems, find the length of the curve traced by the given vector function on the indicated interval.
$$
\mathbf{r}(t)=e^{t} \cos 2 t \mathbf{i}+e^{t} \sin 2 t \mathbf{j}+e^{t} \mathbf{k} ; 0 \leq t \leq 3 \pi
$$

Wendi Zhao

Numerade Educator

Problem 44

In Problems, find the length of the curve traced by the given vector function on the indicated interval.
$$
\mathbf{r}(t)=3 t \mathbf{i}+\sqrt{3} t^{2} \mathbf{j}+\frac{2}{3} t^{3} \mathbf{k} ; 0 \leq t \leq 1
$$

Wendi Zhao

Numerade Educator

Problem 45

Express the vector equation of a circle $\mathbf{r}(t)=a \cos t \mathbf{i}+$ $a \sin t \mathbf{j}$ as a function of arc length $s$. Verify that $\mathbf{r}^{\prime}(s)$ is a unit vector.

Wendi Zhao

Numerade Educator

Problem 46

If $\mathbf{r}(s)$ is the vector function given in (4), verify that $\mathbf{r}^{\prime}(s)$ is a unit vector.

Wendi Zhao

Numerade Educator

Problem 47

Suppose $\mathbf{r}$ is a differentiable vector function for which $\|\mathbf{r}(t)\|=c$ for all $t$. Show that the tangent vector $\mathbf{r}^{\prime}(t)$ is perpendicular to the position vector $\mathbf{r}(t)$ for all $t$.

Hast Aggarwal

Numerade Educator

Problem 48

In Problem 47 , describe geometrically the kind of curve $C$ for which $\|\mathbf{r}(t)\|=c$

Hast Aggarwal

Numerade Educator

Problem 49

Prove Theorem 9.1.4(ii).

Wendi Zhao

Numerade Educator

Problem 50

Prove Theorem 9.1.4(iii).

Wendi Zhao

Numerade Educator

Problem 51

Prove Theorem 9.1.4(iv).

Wendi Zhao

Numerade Educator

Problem 52

If $\mathbf{v}$ is a constant vector and $\mathbf{r}$ is integrable on $[a, b]$, prove that $\int_{a}^{b} \mathbf{v} \cdot \mathbf{r}(t) d t=\mathbf{v} \cdot \int_{a}^{b} \mathbf{r}(t) d t$

Wendi Zhao

Numerade Educator