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Calculus, Early Transcendentals

Michael Sullivan, Kathleen Miranda

Chapter 11

Vectors; Lines, Planes, and Quadric Surfaces in Space - all with Video Answers

Educators

Section 1

Rectangular Coordinates in Space

01:01

Problem 1

The domain of $\mathbf{r}(t)=(4 t+1) \mathbf{i}+\sqrt{4-t} \mathbf{j}+\mathbf{k}$ is _____.

Monica Miller
Monica Miller
Numerade Educator
00:28

Problem 2

True or False A vector function is continuous at a real number $t_{0}$ if at least one of its component functions is continuous at $t_{0}$.

Monica Miller
Monica Miller
Numerade Educator
00:20

Problem 3

True or False To find the derivative of a vector function $\mathbf{r}=\mathbf{r}(t),$ find the derivative of each of its component functions.

Monica Miller
Monica Miller
Numerade Educator
00:26

Problem 4

True or False The derivative of a dot product is a scalar function.

Monica Miller
Monica Miller
Numerade Educator
00:53

Problem 5

True or False If $y=f(t)$ is a differentiable real (scalar) function and $\mathbf{r}=\mathbf{r}(t)$ is a differentiable vector function, then $\frac{d}{d t}[f(t) \mathbf{r}(t)]=f^{\prime}(t) \frac{d}{d t} \mathbf{r}(t)$.

Monica Miller
Monica Miller
Numerade Educator
00:33

Problem 6

If $\mathbf{u}=\mathbf{u}(t)$ and $\mathbf{v}=\mathbf{v}(t)$ are vector functions in space that are differentiable, then $\frac{d}{d t}(\mathbf{u} \times \mathbf{v})=$ ______.

Monica Miller
Monica Miller
Numerade Educator
00:37

Problem 7

In Problems $7-14,$ find the value of each vector function at $t$.
$$
\mathbf{r}(t)=t^{2} \mathbf{i}-2 t \mathbf{j} \text { at } t=1
$$

Monica Miller
Monica Miller
Numerade Educator
00:26

Problem 8

Find the value of each vector function at $t$.
$$
\mathbf{r}(t)=t^{3} \mathbf{i}+2 t \mathbf{j} \text { at } t=2
$$

Monica Miller
Monica Miller
Numerade Educator
00:42

Problem 9

Find the value of each vector function at $t$.
$$
\mathbf{v}(t)=\sin t \mathbf{i}-\cos t \mathbf{j} \text { at } t=\frac{\pi}{4}
$$

Monica Miller
Monica Miller
Numerade Educator
00:33

Problem 10

Find the value of each vector function at $t$.
$$
\mathbf{v}(t)=\tan t \mathbf{i}+\cos (2 t) \mathbf{j} \text { at } t=0
$$

Monica Miller
Monica Miller
Numerade Educator
00:29

Problem 11

Find the value of each vector function at $t$.
$$
\mathbf{u}(t)=e^{2 t} \mathbf{i}+t^{2} \mathbf{j}-2 \mathbf{k} \text { at } t=0
$$

Monica Miller
Monica Miller
Numerade Educator
00:33

Problem 12

Find the value of each vector function at $t$.
$$
\mathbf{u}(t)=\ln t \mathbf{i}-\mathbf{j}+t^{3} \mathbf{k} \text { at } t=1
$$

Monica Miller
Monica Miller
Numerade Educator
00:38

Problem 13

Find the value of each vector function at $t$.
$$
\mathbf{g}(t)=t \mathbf{i}-\cos \left(\frac{\pi t}{4}\right) \mathbf{j}+2 t \mathbf{k} \text { at } t=4
$$

Monica Miller
Monica Miller
Numerade Educator
00:44

Problem 14

Find the value of each vector function at $t$.
$$
\mathbf{f}(t)=\sin \left(\frac{3 \pi t}{4}\right) \mathbf{i}+3 \mathbf{j}-3 t^{2} \mathbf{k} \text { at } t=1
$$

Monica Miller
Monica Miller
Numerade Educator
00:27

Problem 15

In Problems 15-22, find the domain of each vector function.
$$
\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+2 \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
00:26

Problem 16

Find the domain of each vector function.
$$
\mathbf{r}(t)=\cos (2 t) \mathbf{i}+\sin t \mathbf{j}+2 t \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
00:35

Problem 17

Find the domain of each vector function.
$$
\mathbf{f}(t)=\sqrt{t} \mathbf{i}+t \mathbf{j}-\mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
00:45

Problem 18

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

Monica Miller
Monica Miller
Numerade Educator
00:33

Problem 19

Find the domain of each vector function.
$$
\mathbf{u}(t)=\frac{\left(t^{2}+1\right)}{t} \mathbf{i}+3 t \mathbf{j}-\frac{2}{t} \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
00:30

Problem 20

Find the domain of each vector function.
$$
\mathbf{v}(t)=e^{t} \mathbf{i}+(5+3 t) \mathbf{j}-\frac{2}{t} \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
00:32

Problem 21

Find the domain of each vector function.
$$
\mathbf{v}(t)=\ln t \mathbf{i}+(t+1) \mathbf{j}+4 t \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
00:33

Problem 22

Find the domain of each vector function.
$$
\mathbf{v}(t)=\sqrt{t} \mathbf{i}+\ln t \mathbf{j}+t \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
01:37

Problem 23

In Problems $23-36,$ graph the curve $C$ traced out by each vector function and show its orientation.
$$
\mathbf{r}(t)=2 t \mathbf{i}+t^{2} \mathbf{j}, \quad t \geq 0
$$

Monica Miller
Monica Miller
Numerade Educator
01:30

Problem 24

Graph the curve $C$ traced out by each vector function and show its orientation.
$$
\mathbf{r}(t)=t^{2} \mathbf{i}-4 t \mathbf{j}, \quad t \leq 0
$$

Monica Miller
Monica Miller
Numerade Educator
00:46

Problem 25

Graph the curve $C$ traced out by each vector function and show its orientation.
$$
\mathbf{r}(t)=t \mathbf{i}, \quad-1 \leq t \leq 1
$$

Monica Miller
Monica Miller
Numerade Educator
00:36

Problem 26

Graph the curve $C$ traced out by each vector function and show its orientation.
$$
\mathbf{r}(t)=t \mathbf{j}, \quad-1 \leq t \leq 1
$$

Monica Miller
Monica Miller
Numerade Educator
00:34

Problem 27

Graph the curve $C$ traced out by each vector function and show its orientation.
$$
\mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
00:55

Problem 28

Graph the curve $C$ traced out by each vector function and show its orientation.
$$
\mathbf{r}(t)=3 t \mathbf{i}+2 t \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
00:50

Problem 29

Graph the curve $C$ traced out by each vector function and show its orientation.
$$
\mathbf{r}(t)=3 t \mathbf{i}-2 t \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
01:07

Problem 30

Graph the curve $C$ traced out by each vector function and show its orientation.
$$
\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
01:23

Problem 31

Graph the curve $C$ traced out by each vector function and show its orientation.
$$
\mathbf{r}(t)=\cos t \mathbf{i}-\sin t \mathbf{j}, \quad 0 \leq t \leq \frac{\pi}{2}
$$

Monica Miller
Monica Miller
Numerade Educator
01:11

Problem 32

Graph the curve $C$ traced out by each vector function and show its orientation.
$$
\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}, \quad 0 \leq t \leq \frac{\pi}{2}
$$

Monica Miller
Monica Miller
Numerade Educator
01:25

Problem 33

Graph the curve $C$ traced out by each vector function and show its orientation.
$$
\mathbf{r}(t)=\sin ^{2} t \mathbf{i}+\cos ^{2} t \mathbf{j}, \quad 0 \leq t \leq \frac{\pi}{2}
$$

Monica Miller
Monica Miller
Numerade Educator
01:10

Problem 34

Graph the curve $C$ traced out by each vector function and show its orientation.
$$
\mathbf{r}(t)=\sin ^{2} t \mathbf{i}-\cos ^{2} t \mathbf{j}, \quad 0 \leq t \leq \frac{\pi}{2}
$$

Monica Miller
Monica Miller
Numerade Educator
01:19

Problem 35

Graph the curve $C$ traced out by each vector function and show its orientation.
$$
\mathbf{r}(t)=\sin t \mathbf{i}+t^{2} \mathbf{j}, \quad 0 \leq t \leq 4 \pi
$$

Monica Miller
Monica Miller
Numerade Educator
01:29

Problem 36

Graph the curve $C$ traced out by each vector function and show its orientation.
$$
\mathbf{r}(t)=\sin ^{3} t \mathbf{i}+\cos ^{3} t \mathbf{j}, \quad 0 \leq t \leq 2 \pi
$$

Monica Miller
Monica Miller
Numerade Educator
00:55

Problem 37

In Problems $37-40,$ match each vector function to its plane curve.
$$
\mathbf{r}(t)=t^{2} \mathbf{i}+\cos t \mathbf{j}, \quad 0 \leq t \leq 2 \pi
$$

Monica Miller
Monica Miller
Numerade Educator
00:41

Problem 38

Match each vector function to its plane curve.
$$
\mathbf{r}(t)=\cos t \mathbf{i}+t^{2} \mathbf{j}, \quad 0 \leq t \leq 2 \pi
$$

Monica Miller
Monica Miller
Numerade Educator
00:47

Problem 39

Match each vector function to its plane curve.
$$
\mathbf{r}(t)=\sqrt{t} \mathbf{i}+\cos t \mathbf{j}, \quad 0 \leq t \leq 2 \pi
$$

Monica Miller
Monica Miller
Numerade Educator
00:40

Problem 40

Match each vector function to its plane curve.
$$
\mathbf{r}(t)=\cos t \mathbf{i}-t \mathbf{j}, \quad 0 \leq t \leq 2 \pi
$$

Monica Miller
Monica Miller
Numerade Educator
01:55

Problem 41

In Problems $41-48$, graph the curve $C$ traced out by each vector function.
$$
\mathbf{r}(t)=(2 t+1) \mathbf{i}+\frac{t}{3} \mathbf{j}-(t+1) \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
01:36

Problem 42

Graph the curve $C$ traced out by each vector function.
$$
\mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}-(t+1) \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
02:02

Problem 43

Graph the curve $C$ traced out by each vector function.
$$
\mathbf{r}(t)=2 \cos t \mathbf{i}+t \mathbf{j}+2 \sin t \mathbf{k}, \quad-2 \pi \leq t \leq 2 \pi
$$

Monica Miller
Monica Miller
Numerade Educator
01:21

Problem 44

Graph the curve $C$ traced out by each vector function.
$$
\mathbf{r}(t)=\frac{t}{2} \mathbf{i}+3 \sin t \mathbf{j}+2 \cos t \mathbf{k}, \quad-2 \pi \leq t \leq 2 \pi
$$

Monica Miller
Monica Miller
Numerade Educator
02:08

Problem 45

Graph the curve $C$ traced out by each vector function.
$$
\mathbf{r}(t)=4 \cos t \mathbf{i}+\sin t \mathbf{j}+e^{t} \mathbf{k}, \quad-\pi \leq t \leq 2 \pi
$$

Monica Miller
Monica Miller
Numerade Educator
02:34

Problem 46

Graph the curve $C$ traced out by each vector function.
$$
\mathbf{r}(t)=t \mathbf{i}+\sin t \mathbf{j}+e^{t} \mathbf{k}, \quad-5 \leq t \leq 5
$$

Monica Miller
Monica Miller
Numerade Educator
02:05

Problem 47

Graph the curve $C$ traced out by each vector function.
$$
\mathbf{r}(t)=3 \cos t \mathbf{i}+3 \sin t \mathbf{j}+\sin (2 t) \mathbf{k}, \quad 0 \leq t \leq 2 \pi
$$

Monica Miller
Monica Miller
Numerade Educator
00:52

Problem 48

Graph the curve $C$ traced out by each vector function.
$$
\mathbf{r}(t)=t \mathbf{i}+t \cos (3 t) \mathbf{j}+t \sin (3 t) \mathbf{k}, \quad 0 \leq t \leq 2 \pi
$$

Monica Miller
Monica Miller
Numerade Educator
00:50

Problem 49

In Problems 49-56, determine where each vector function is continuous.
$$
\mathbf{r}(t)=t^{2} \mathbf{i}+\frac{2}{1-t^{2}} \mathbf{j}+\sin t \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
00:29

Problem 50

Determine where each vector function is continuous.
$$
\mathbf{r}(t)=\ln t \mathbf{i}+e^{-t} \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
01:14

Problem 51

Determine where each vector function is continuous.
$$
\mathbf{r}(t)=\sec t \mathbf{i}-\cos t \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
01:25

Problem 52

Determine where each vector function is continuous.
$$
\mathbf{r}(t)=\sqrt{t+1} \mathbf{i}+\tan t \mathbf{j}-\frac{\sin t-1}{t} \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
00:30

Problem 53

Determine where each vector function is continuous.
$$
\mathbf{r}(t)=\frac{t}{t+1} \mathbf{i}-t \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
00:42

Problem 54

Determine where each vector function is continuous.
$$
\mathbf{r}(t)=\frac{t}{t^{2}+1} \mathbf{i}+t \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
01:12

Problem 55

Determine where each vector function is continuous.
$$
\mathbf{r}(t)=\sin t \mathbf{i}-\tan t \mathbf{j}+\mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
01:48

Problem 56

Determine where each vector function is continuous.
$$
\mathbf{r}(t)=\sec t \mathbf{i}+\csc t \mathbf{j}+\mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
00:38

Problem 57

In Problems 57-72, find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$
$$
\mathbf{r}(t)=4 t^{2} \mathbf{i}-2 t^{3} \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
00:41

Problem 58

Find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$
$$
\mathbf{r}(t)=8 t \mathbf{i}+4 t^{3} \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
01:11

Problem 59

Find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$
$$
\mathbf{r}(t)=4 \sqrt{t} \mathbf{i}+2 e^{t} \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
01:21

Problem 60

Find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$
$$
\mathbf{r}(t)=e^{3 t} \mathbf{i}+\sqrt[3]{t} \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
00:46

Problem 61

Find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$
$$
\mathbf{r}(t)=\mathbf{i}+t \mathbf{j}+t^{2} \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
00:31

Problem 62

Find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$
$$
\mathbf{r}(t)=\mathbf{i}-\mathbf{j}+t \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
00:42

Problem 63

Find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$
$$
\mathbf{r}(t)=t^{2} \mathbf{i}+t^{3} \mathbf{j}-t \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
00:38

Problem 64

Find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$
$$
\mathbf{r}(t)=(1+t) \mathbf{i}-3 t^{2} \mathbf{j}+t \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
01:33

Problem 65

Find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$
$$
\mathbf{r}(t)=\sin ^{2} t \mathbf{i}-\cos ^{2} t \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
00:55

Problem 66

Find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$
$$
\mathbf{r}(t)=\sin (2 t) \mathbf{i}-\cos (2 t) \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
01:57

Problem 67

Find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$
$$
\mathbf{r}(t)=5 e^{t^{2}+t} \mathbf{i}-\ln e^{t^{2}+t} \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
03:01

Problem 68

Find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$
$$
\mathbf{r}(t)=\sin \left(3 t^{3}-t\right) \mathbf{i}-\cos ^{2}\left(3 t^{3}-t\right) \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
03:08

Problem 69

Find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$
$$
\mathbf{r}(t)=e^{t} \cos t \mathbf{i}+e^{t} \sin t \mathbf{j}+t \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
03:09

Problem 70

Find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$
$$
\mathbf{r}(t)=e^{-t} \cos t \mathbf{i}+e^{-t} \sin t \mathbf{j}-t \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
00:42

Problem 71

Find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$
$$
\mathbf{r}(t)=\left(t-t^{3}\right) \mathbf{i}+\left(t+t^{3}\right) \mathbf{j}-t \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
00:40

Problem 72

Find $\mathbf{r}^{\prime}(t)$ and $\mathbf{r}^{\prime \prime}(t)$
$$
\mathbf{r}(t)=\left(t^{2}-t\right) \mathbf{i}+\left(t^{2}+t\right) \mathbf{j}+t \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
01:56

Problem 73

In Problems 73-78, find $\frac{d}{d t}[\mathbf{u}(t) \cdot \mathbf{v}(t)]$
$$
\mathbf{u}(t)=t^{2} \mathbf{i}-t \mathbf{j} \quad \text { and } \quad \mathbf{v}(t)=t \mathbf{i}+t^{2} \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
01:59

Problem 74

Find $\frac{d}{d t}[\mathbf{u}(t) \cdot \mathbf{v}(t)]$
$$
\mathbf{u}(t)=t^{3} \mathbf{i}+t \mathbf{j} \quad \text { and } \quad \mathbf{v}(t)=t \mathbf{i}-2 t \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
01:24

Problem 75

Find $\frac{d}{d t}[\mathbf{u}(t) \cdot \mathbf{v}(t)]$
$$
\mathbf{u}(t)=e^{t} \mathbf{i}+e^{-t} \mathbf{j} \quad \text { and } \quad \mathbf{v}(t)=t \mathbf{i}-t^{2} \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
01:11

Problem 76

$$
\mathbf{u}(t)=t \mathbf{i}+4 \sqrt{t} \mathbf{j} \quad \text { and } \quad \mathbf{v}(t)=2 t \mathbf{i}-t^{2} \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
00:45

Problem 77

Find $\frac{d}{d t}[\mathbf{u}(t) \cdot \mathbf{v}(t)]$
$$
\mathbf{u}(t)=\sin (\omega t) \mathbf{i}+\cos (\omega t) \mathbf{j} \quad \text { and } \quad \mathbf{v}(t)=\mathbf{i}+\mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
01:20

Problem 78

Find $\frac{d}{d t}[\mathbf{u}(t) \cdot \mathbf{v}(t)]$
$$
\mathbf{u}(t)=\sin ^{2} t \mathbf{i}-\cos ^{2} t \mathbf{j} \quad \text { and } \quad \mathbf{v}(t)=\mathbf{i}-\mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
02:07

Problem 79

In Problems 79-82, find $\frac{d}{d t}[\mathbf{u}(t) \cdot \mathbf{v}(t)]$ and $\frac{d}{d t}[\mathbf{u}(t) \times \mathbf{v}(t)]$
$$
\mathbf{u}(t)=2 t \mathbf{i}+t^{2} \mathbf{j}-5 \mathbf{k} \quad \text { and } \quad \mathbf{v}(t)=t^{2} \mathbf{i}+2 t \mathbf{j}+\mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
02:10

Problem 80

Find $\frac{d}{d t}[\mathbf{u}(t) \cdot \mathbf{v}(t)]$ and $\frac{d}{d t}[\mathbf{u}(t) \times \mathbf{v}(t)]$
$$
\mathbf{u}(t)=t^{3} \mathbf{i}-t^{2} \mathbf{j}+t \mathbf{k} \quad \text { and } \quad \mathbf{v}(t)=t \mathbf{i}-t^{2} \mathbf{j}+t^{3} \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
06:07

Problem 81

Find $\frac{d}{d t}[\mathbf{u}(t) \cdot \mathbf{v}(t)]$ and $\frac{d}{d t}[\mathbf{u}(t) \times \mathbf{v}(t)]$
$$
\mathbf{u}(t)=\cos (2 t) \mathbf{i}+\sin (2 t) \mathbf{j}+\mathbf{k} \quad \text { and } \quad \mathbf{v}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+\mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
02:10

Problem 82

Find $\frac{d}{d t}[\mathbf{u}(t) \cdot \mathbf{v}(t)]$ and $\frac{d}{d t}[\mathbf{u}(t) \times \mathbf{v}(t)]$
$$
\mathbf{u}(t)=e^{2 t} \mathbf{i}+e^{-2 t} \mathbf{j}+t \mathbf{k} \quad \text { and } \quad \mathbf{v}(t)=e^{-t} \mathbf{i}+e^{-2 t} \mathbf{j}-t \mathbf{k}
$$

Monica Miller
Monica Miller
Numerade Educator
01:14

Problem 83

Given the vector function $\mathbf{u}(t)=\cos (\omega t) \mathbf{i}+\sin (\omega t) \mathbf{j}$ find $\frac{d \mathbf{u}}{d t}$ and $\left\|\frac{d \mathbf{u}}{d t}\right\| .$

Monica Miller
Monica Miller
Numerade Educator
01:05

Problem 84

Given the vector function $\mathbf{v}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k},$ find $\frac{d^{2} \mathbf{v}}{d t^{2}}$ and $\left\|\frac{d^{2} \mathbf{v}}{d t^{2}}\right\|$.

Monica Miller
Monica Miller
Numerade Educator
01:08

Problem 85

For $\mathbf{f}(t)=\sin t \mathbf{i}-\cos t \mathbf{j}$, show that $\mathbf{f}(t)$ and $\mathbf{f}^{\prime \prime}(t)$ are parallel.

Monica Miller
Monica Miller
Numerade Educator
01:09

Problem 86

For $\mathbf{f}(t)=e^{3 t} \mathbf{i}+e^{-3 t} \mathbf{j},$ show that $\mathbf{f}(t)$ and $\mathbf{f}^{\prime \prime}(t)$ are parallel.

Monica Miller
Monica Miller
Numerade Educator
01:57

Problem 87

In Problems 87-94, a vector function $\mathbf{r}=\mathbf{r}(t)$ defining a curve in the plane is given. Graph each curve, indicating its orientation. Include in the graph the vector $\mathbf{r}(0)$ and $\mathbf{r}^{\prime}(0) .$ Draw $\mathbf{r}^{\prime}(0)$ so its initial point is at the terminal point of $\mathbf{r}(0)$.
$$
\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
02:17

Problem 88

A vector function $\mathbf{r}=\mathbf{r}(t)$ defining a curve in the plane is given. Graph each curve, indicating its orientation. Include in the graph the vector $\mathbf{r}(0)$ and $\mathbf{r}^{\prime}(0) .$ Draw $\mathbf{r}^{\prime}(0)$ so its initial point is at the terminal point of $\mathbf{r}(0)$.
$$
\mathbf{r}(t)=2 t^{2} \mathbf{i}-t \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
02:31

Problem 89

A vector function $\mathbf{r}=\mathbf{r}(t)$ defining a curve in the plane is given. Graph each curve, indicating its orientation. Include in the graph the vector $\mathbf{r}(0)$ and $\mathbf{r}^{\prime}(0) .$ Draw $\mathbf{r}^{\prime}(0)$ so its initial point is at the terminal point of $\mathbf{r}(0)$.
$$
\mathbf{r}(t)=t \mathbf{i}+e^{t} \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
01:55

Problem 90

A vector function $\mathbf{r}=\mathbf{r}(t)$ defining a curve in the plane is given. Graph each curve, indicating its orientation. Include in the graph the vector $\mathbf{r}(0)$ and $\mathbf{r}^{\prime}(0) .$ Draw $\mathbf{r}^{\prime}(0)$ so its initial point is at the terminal point of $\mathbf{r}(0)$.
$$
\mathbf{r}(t)=t \mathbf{i}+\ln (1+t) \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
01:37

Problem 91

A vector function $\mathbf{r}=\mathbf{r}(t)$ defining a curve in the plane is given. Graph each curve, indicating its orientation. Include in the graph the vector $\mathbf{r}(0)$ and $\mathbf{r}^{\prime}(0) .$ Draw $\mathbf{r}^{\prime}(0)$ so its initial point is at the terminal point of $\mathbf{r}(0)$.
$$
\mathbf{r}(t)=3 \sin t \mathbf{i}-3 \cos t \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
02:10

Problem 92

A vector function $\mathbf{r}=\mathbf{r}(t)$ defining a curve in the plane is given. Graph each curve, indicating its orientation. Include in the graph the vector $\mathbf{r}(0)$ and $\mathbf{r}^{\prime}(0) .$ Draw $\mathbf{r}^{\prime}(0)$ so its initial point is at the terminal point of $\mathbf{r}(0)$.
$$
\mathbf{r}(t)=4 \sin t \mathbf{i}+4 \cos t \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
01:59

Problem 93

A vector function $\mathbf{r}=\mathbf{r}(t)$ defining a curve in the plane is given. Graph each curve, indicating its orientation. Include in the graph the vector $\mathbf{r}(0)$ and $\mathbf{r}^{\prime}(0) .$ Draw $\mathbf{r}^{\prime}(0)$ so its initial point is at the terminal point of $\mathbf{r}(0)$.
$$
\mathbf{r}(t)=2 \cos t \mathbf{i}-3 \sin t \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
01:26

Problem 94

A vector function $\mathbf{r}=\mathbf{r}(t)$ defining a curve in the plane is given. Graph each curve, indicating its orientation. Include in the graph the vector $\mathbf{r}(0)$ and $\mathbf{r}^{\prime}(0) .$ Draw $\mathbf{r}^{\prime}(0)$ so its initial point is at the terminal point of $\mathbf{r}(0)$.
$$
\mathbf{r}(t)=-\cos t \mathbf{i}+2 \sin t \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
01:31

Problem 95

In Problems 95-98, the position of an object at time $t$ is given.
(a) Eliminate the parameter $t$ to find $y$ as a function of $x$.
(b) Graph $\mathbf{r}=\mathbf{r}(t)$ and indicate the orientation.
$$
\mathbf{r}(t)=\left(1-t^{2}\right) \mathbf{i}+t^{2} \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
02:10

Problem 96

The position of an object at time $t$ is given.
(a) Eliminate the parameter $t$ to find $y$ as a function of $x$.
(b) Graph $\mathbf{r}=\mathbf{r}(t)$ and indicate the orientation.
$$
\mathbf{r}(t)=t^{2} \mathbf{i}+\left(4 t^{2}-t^{4}\right) \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
03:04

Problem 97

The position of an object at time $t$ is given.
(a) Eliminate the parameter $t$ to find $y$ as a function of $x$.
(b) Graph $\mathbf{r}=\mathbf{r}(t)$ and indicate the orientation.
$$
\mathbf{r}(t)=\sin ^{2} t \mathbf{i}+\tan t \mathbf{j}, \quad 0<t<\frac{\pi}{2}
$$

Monica Miller
Monica Miller
Numerade Educator
00:53

Problem 98

The position of an object at time $t$ is given.
(a) Eliminate the parameter $t$ to find $y$ as a function of $x$.
(b) Graph $\mathbf{r}=\mathbf{r}(t)$ and indicate the orientation.
$$
\mathbf{r}(t)=e^{t} \mathbf{i}+e^{-t} \mathbf{j}
$$

Monica Miller
Monica Miller
Numerade Educator
01:54

Problem 99

Proof of the Sum Formula If the vector functions $\mathbf{u}=\mathbf{u}(t)$ and $\mathbf{v}=\mathbf{v}(t)$ are differentiable, show that
$$
[\mathbf{u}(t)+\mathbf{v}(t)]^{\prime}=\mathbf{u}^{\prime}(t)+\mathbf{v}^{\prime}(t)
$$

Monica Miller
Monica Miller
Numerade Educator
02:17

Problem 100

Proof of the Dot Product Formula If the vector functions
$\mathbf{u}=\mathbf{u}(t)$ and $\mathbf{v}=\mathbf{v}(t)$ are differentiable, show that
$$
[\mathbf{u}(t) \cdot \mathbf{v}(t)]^{\prime}=\mathbf{u}^{\prime}(t) \cdot \mathbf{v}(t)+\mathbf{u}(t) \cdot \mathbf{v}^{\prime}(t)
$$

Monica Miller
Monica Miller
Numerade Educator
03:58

Problem 101

Proof of the Cross Product Formula
$\mathbf{u}=\mathbf{u}(t)$ and $\mathbf{v}=\mathbf{v}(t)$ in space are differentiable, show that
$$
[\mathbf{u}(t) \times \mathbf{v}(t)]^{\prime}=\mathbf{u}^{\prime}(t) \times \mathbf{v}(t)+\mathbf{u}(t) \times \mathbf{v}^{\prime}(t)
$$

Monica Miller
Monica Miller
Numerade Educator
02:22

Problem 102

Proof of the Chain Rule If the vector function $\mathbf{v}=\mathbf{v}(t)$ is differentiable and the real function $y=f(t)$ is differentiable, show that
$$
\frac{d}{d t} \mathbf{v}(f(t))=\mathbf{v}^{\prime}(f(t)) f^{\prime}(t)
$$

Monica Miller
Monica Miller
Numerade Educator
02:41

Problem 103

The derivative of a cross product is not commutative. Give an example of two vector functions that demonstrate this fact.

Monica Miller
Monica Miller
Numerade Educator
01:02

Problem 104

Suppose the vector function $\mathbf{r}=\mathbf{r}(t)$ is twice differentiable; show that $\left[\mathbf{r}(t) \times \mathbf{r}^{\prime (t)\right]^{\prime}=\mathbf{r}(t) \times \mathbf{r}^{\prime \prime}(t)$

Raj Bala
Raj Bala
Numerade Educator
View

Problem 105

Show that $\lim _{t \rightarrow t_{0}} \mathbf{r}(t)=\mathbf{L}$ if and only if $\lim _{t \rightarrow t_{0}}\|\mathbf{r}(t)-\mathbf{L}\|=0 .$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:00

Problem 106

If the vector function $\mathbf{r}=\mathbf{r}(t)$ is differentiable, show that $\frac{d}{d t}\|\mathbf{r}(t)\|=\frac{\mathbf{r}(t) \cdot \mathbf{r}^{\prime}(t)}{\|\mathbf{r}(t)\|} .$ Use this result to show that $\|\mathbf{r}(t)\|$ is constant if and only if $\mathbf{r}(t) \cdot \mathbf{r}^{\prime}(t)=0$ for all $t$

Amy Jiang
Amy Jiang
Numerade Educator
00:53

Problem 107

If the vector function $\mathbf{r}$ is differentiable, show that
$\frac{d}{d t} \frac{\mathbf{r}(t)}{\|\mathbf{r}(t)\|}=\frac{\mathbf{r}^{\prime}(t)}{\|\mathbf{r}(t)\|}-\frac{\mathbf{r}(t) \cdot \mathbf{r}^{\prime}(t)}{\|\mathbf{r}(t)\|^{3}} \mathbf{r}(t)
$$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:04

Problem 108

(a) Discuss the curve $C$ traced out by the vector function $\mathbf{r}(t)=e^{t} \sin t \mathbf{i}+e^{t} \cos t \mathbf{j}+e^{t} \mathbf{k}$
(b) Name the surface on which the curve $C$ lies.

Howard Francis
Howard Francis
Numerade Educator