Deck 13: Vector Functions

Question 1

Find the velocity, acceleration, and speed of an object with position function for Sketch the path of the object and its velocity and acceleration vectors.

Answer 1

not answered

Question 2

The following table gives coordinates of a particle moving through space along a smooth curve. $$\begin{array} { | c | c | c | c | } \hline \mathbf { t } & x & y & z \\\hline 0.5 & 5.8 & 9.1 & 4.3 \\\hline 1 & 12.6 & 14.9 & 16.8 \\\hline 1.5 & 25.6 & 21.2 & 29.4 \\\hline 2 & 39.2 & 39.5 & 37.9 \\\hline 2.5 & 42.4 & 42.4 & 43 \\\hline\end{array}$$ Find the average velocity over the time interval $[ 2.5,1.5 ]$ .

A) $\mathbf { v } = 13.6 \mathbf { i } + 21.12 \mathbf { j } + 16.8 \mathbf { k }$
B) $v = 13.6 \mathbf { i } + 16.8 \mathbf { j } + 16.8 \mathbf { k }$
C) $$\mathbf { v } = 13.6 \mathbf { i } + 16.8 \mathbf { j } + 13.6 \mathbf { k }$$
D) $\mathbf { v } = 16.8 \mathbf { i } + 21.12 \mathbf { j } + 13.6 \mathbf { k }$
E) $\mathbf { v } = 21.12 \mathbf { i } + 21.12 \mathbf { j } + 13.6 \mathbf { k }$

Answer 2

$\mathbf { v } = 16.8 \mathbf { i } + 21.12 \mathbf { j } + 13.6 \mathbf { k }$

Question 3

What force is required so that a particle of mass $m$ has the following position function?. $r ( t ) = 5 t ^ { 3 } \mathbf { i } + 10 t ^ { 2 } \mathbf { j } + 7 t ^ { 3 } \mathbf { k }$

A) $\mathrm { F } ( t ) = 30 \mathrm {~m} t \mathbf { i } + 20 \mathrm {~m} \mathbf { j } + 42 \mathrm {~m} t \mathbf { k }$
B) $\mathrm { F } ( t ) = 30 m t ^ { 2 } \mathbf { i } + 20 m \mathbf { j } + 42 m t \mathbf { k }$
C) $\mathrm { F } ( t ) = 42 m t \mathbf { i } + 42 m \mathbf { j } + 20 m t \mathbf { k }$
D) $\mathrm { F } ( t ) = m t ^ { 2 } \mathbf { i } + 5 m t \mathbf { j } + 10 m t ^ { 2 } \mathbf { k }$
E) $\mathrm { F } ( t ) = 30 m t \mathbf { i } + 20 m \mathbf { j } + t \mathbf { k }$

Answer 3

$\mathrm { F } ( t ) = 30 \mathrm {~m} t \mathbf { i } + 20 \mathrm {~m} \mathbf { j } + 42 \mathrm {~m} t \mathbf { k }$

Question 4

Find the acceleration of a particle with the following position function. $\mathbf { r } ( t ) = \left\{ 2 t ^ { 2 } - 2,4 t \right\}$

A) $\mathbf { a } ( t ) = 4 \mathbf { i }$
B) $\mathbf { a } ( t ) = 2 t \mathbf { i } + 2 \mathbf { j }$
C) $\mathbf { a } ( t ) = ( 2 + 4 t ) \mathbf { i } - 2 \mathbf { j }$
D) $\mathbf { a } ( t ) = 2 \mathbf { i } - \mathbf { j }$
E) $$\mathbf { a } ( t ) = 4 t \mathbf { i } + 2 \mathbf { j }$$

Question 5

A force with magnitude $12$ N acts directly upward from the xy-plane on an object with mass $2$ kg. The object starts at the origin with initial velocity $\mathbf { v } ( 0 ) = 3 \mathbf { i } - 2 \mathbf { j }$ . Find its position function.

A) $\mathbf { r } ( t ) = 2 t ^ { 2 } \mathbf { i } - 3 t ^ { 2 } \mathbf { j } + 12 t ^ { 3 } \mathbf { k }$
B) $\mathbf { r } ( t ) = 2 t \mathbf { i } - 4 t \mathbf { j } + t ^ { 2 } \mathbf { k }$
C) $\mathbf { r } ( t ) = 3 t \mathbf { i } - 2 t \mathbf { j } +$ $3$ $\text { k }$
D) $\mathbf { r } ( t ) = 3 t \mathbf { i } - 2 t \mathbf { j }$
E) $\mathbf { r } ( t ) = 5 t ^ { 3 } \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }$

Question 6

A projectile is fired with an initial speed of $700 \mathrm {~m} / \mathrm { s }$ and angle of elevation $60 ^ { \circ }$ . Find the range of the projectile.

A) $d \approx 43.3$ km
B) $d \approx 350$ km
C) $d \approx 63.3$ km
D) $d \approx 433$ km
E) $$d \approx 53.3$$ km

Question 7

Find the velocity of a particle with the given position function. $\mathbf { r } ( t ) = 11 e ^ { 9 t } \mathbf { i } + 9 e ^ { - 13 t } \mathbf { j }$

A) $\mathbf { v } ( t ) = e ^ { 9 t } \mathbf { i } + 117 e ^ { - 13 t } \mathbf { j }$
B) $\mathbf { v } ( t ) = 99 e ^ { 9 t } \mathbf { i } - 117 e ^ { - 13 t } \mathbf { j }$
C) $\mathbf { v } ( t ) = 11 e ^ { 9 t } \mathbf { i } + e ^ { - 13 t } \mathbf { j }$
D) $\mathbf { v } ( t ) = 11 e ^ { 9 t } \mathbf { i } - 117 e ^ { - 13 t } \mathbf { j }$
E) $\mathbf { v } ( t ) = 99 e ^ { 9 t } \mathbf { i } + 117 e ^ { - 13 t } \mathbf { j }$

Question 8

A particle moves with position function $\mathbf { r } ( t ) = \left( 21 t - 7 t ^ { 3 } - 5 \right) \mathbf { i } + 21 t ^ { 2 } \mathbf { j }$ . Find the tangential component of the acceleration vector.

A) $a _ { T } = 5 t$
B) $a _ { T } = 542 t$
C) $a _ { T } = 42 t + 5$
D) $a _ { T } = - 55 t$
E) $a _ { T } = 42 t$

Question 9

A particle moves with position function . Find the acceleration of the particle.

Question 10

Find the position vector of a particle that has the given acceleration and the given initial velocity and position.

Question 11

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = e ^ { t } ( \cos 8 t , \sin 8 t , 8 )$

A) $a _ { \mathbf { T } } = \sqrt { 65 } e ^ { t }$ , $a _ { \mathrm { N } } = 8 \sqrt { 65 } e ^ { t }$
B) $a _ { \mathbf { T } } = 8 \sqrt { 65 } e ^ { t }$ , $a _ { \mathrm { N } } = \sqrt { 65 } e ^ { t }$
C) $a _ { \mathbf { T } } = 65 e ^ { t }$ , $a _ { \mathrm { N } } ( t ) = 8 e ^ { t }$
D) $a _ { \mathbf { T } } = 8 e ^ { t }$ , $a _ { \mathrm { N } } = 65 e ^ { t }$

Question 12

Find the acceleration of a particle with the given position function. $\mathbf { r } ( t ) = 9 \sin t \mathbf { i } + 10 t \mathbf { j } - 8 \cos t \mathbf { k }$

A) $\mathbf { a } ( t ) = - 9 \sin t \mathbf { i } - 9 \cos t \mathbf { j }$
B) $\mathbf { a } ( t ) = - 9 \sin t \mathbf { i } + 9 \cos t \mathbf { j }$
C) $\mathbf { a } ( t ) = - 9 \sin t \mathbf { i } + 10 \cos t \mathbf { k }$
D) $\mathbf { a } ( t ) = - 9 \sin t \mathbf { i } + 8 \cos t \mathbf { k }$
E) $\mathbf { a } ( t ) = 9 \sin t \mathbf { i } - 10 t \mathbf { k }$

Question 13

A mortar shell is fired with a muzzle speed of 325 ft/sec. Find the angle of elevation of the mortar if the shell strikes a target located 1500 ft away. Round your answer to 2 decimal places.

A) $12.22 ^ { \circ }$
B) $0.64 ^ { \circ }$
C) $13.51 ^ { \circ }$
D) $0.24 ^ { \circ }$

Question 14

Find the speed of a particle with the given position function. $r ( t ) = t \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 3 t ^ { 6 } \mathbf { k }$

A) $| \mathbf { v } ( t ) | = \sqrt { 1 + 100 t ^ { 2 } + 100 t ^ { 10 } }$
B) $| \mathbf { v } ( t ) | = \sqrt { 1 + 100 t + 324 t ^ { 9 } }$
C) $| \mathbf { v } ( t ) | = 1 + 100 t ^ { 2 } + 324 t ^ { 10 }$
D) $| \mathbf { v } ( t ) | = \sqrt { 1 + 100 t ^ { 2 } + 324 t ^ { 10 } }$
E) $| \mathbf { v } ( t ) | = \sqrt { 1 + 100 t + t ^ { 9 } }$

Question 15

A particle moves with position function .
Find the normal component of the acceleration vector.

Question 16

Find the speed of a particle with the given position function. $\mathbf { r } ( t ) = 5 \sqrt { 2 } t \mathbf { i } + e ^ { 5 t } \mathbf { j } - e ^ { - 5 t } \mathbf { k }$

A) $| v ( t ) | = 5 \left( e ^ { 5 t } + e ^ { - 5 t } \right)$
B) $| v ( t ) | = \left( e ^ { 5 t } + e ^ { - 5 t } \right)$
C) $| v ( t ) | = \sqrt { 5 + 5 e ^ { t } + 5 e ^ { - t } }$
D) $| v ( t ) | = 5 \left( e ^ { t } + e ^ { - t } \right)$
E) $| v ( t ) | = \sqrt { 5 + e ^ { 5 t } + e ^ { - 50 t } }$

Question 17

A ball is thrown at an angle of $45 ^ { \circ }$ to the ground. If the ball lands $30$ m away, what was the initial speed of the ball? Let $g = 9.82 \mathrm {~m} / \mathrm { s }$ .

A) $v _ { 0 } \approx 17 \mathrm {~m} / \mathrm { s }$
B) $v _ { 0 } \approx 22 \mathrm {~m} / \mathrm { s }$ .
C) $v _ { 0 } \approx 42 \mathrm {~m} / \mathrm { s }$ .
D) $v _ { 0 } \approx 27 \mathrm {~m} / \mathrm { s }$
E) $v _ { 0 } \approx 27 \mathrm {~m} / \mathrm { s }$ .

Question 18

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = 3 \sin t \mathbf { i } + 3 \cos t \mathbf { j } + 5 t \mathbf { k }$

A) $a _ { \mathbf { T } } = 0$ $a _ { \mathrm { N } } = 3$
B) $a _ { \mathbf { T } } = 5$ $a _ { \mathrm { N } } = 3$
C) $a _ { \mathbf { T } } = 0$ $a _ { \mathrm { N } } = 3 \sqrt { 26 }$
D) $a _ { \mathbf { T } } = 5$ $a _ { \mathrm { N } } = 3 \sqrt { 26 }$

Question 19

The position function of a particle is given by $\mathbf { r } ( t ) = \left\langle 5 t ^ { 2 } , 5 t , 5 t ^ { 2 } - 100 t \right\rangle$ When is the speed a minimum?

A) $$t = 5$$
B) $t = 30$
C) $t = 20$
D) $t = 0$
E) $t = 10$

Question 20

Find the velocity of a particle that has the given acceleration and the given initial velocity. $\mathbf { a } ( t ) = 3 \mathbf { k } , \mathbf { v } ( 0 ) = 12 \mathbf { i } - 7 \mathbf { j }$

A) $\mathbf { v } ( t ) = 12 \mathbf { i } - \mathbf { j } + 3 t \mathbf { k }$
B) $\mathbf { v } ( t ) = 12 \mathbf { i } - 7 \mathbf { j } + t \mathbf { k }$
C) $\mathbf { v } ( t ) = ( 12 t - 9 ) \mathbf { i } + 7 t \mathbf { k }$
D) $\mathbf { v } ( t ) = \mathbf { i } - 7 \mathbf { j } + 3 t \mathbf { k }$
E) $\mathbf { v } ( t ) = 12 \mathbf { i } - 7 \mathbf { j } + 3 t \mathbf { k }$

Question 21

Find the curvature of the curve $\mathbf { r } ( t ) = 3 \sin 4 t \mathbf { i } + 3 \cos 4 t \mathbf { j } + 3 t \mathbf { k }$ .

A) $\frac { 4 } { 3 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 51 } { 16 }$
D) $\frac { 16 } { 51 }$

Question 22

Find the scalar tangential and normal components of acceleration of a particle with position vector

Question 23

Find the curvature of $y = x ^ { 4 }$ .

A) $\frac { 12 | x | ^ { 2 } } { \left( 1 - 16 x ^ { 6 } \right) ^ { 3 / 2 } }$
B) $\frac { 12 | x | ^ { 2 } } { \left( 1 + 16 x ^ { 6 } \right) ^ { 1 / 2 } }$
C) $\frac { | x | ^ { 2 } } { \left( 1 + x ^ { 6 } \right) ^ { 3 / 2 } }$
D) $\frac { | x | ^ { 2 } } { \left( 1 + 16 x ^ { 6 } \right) ^ { 1 / 2 } }$
E) $\frac { 12 | x | ^ { 2 } } { \left( 1 + 16 x ^ { 6 } \right) ^ { 3 / 2 } }$

Question 24

Find the velocity, acceleration, and speed of an object with position function for Sketch the path of the object and its velocity and acceleration vectors.

Question 25

A projectile is fired from a height of 400 ft with an initial speed of 200 ft/sec and an angle of elevation of .
a. What are the scalar tangential and normal components of acceleration of the projectile?
b. What are the scalar tangential and normal components of acceleration of the projectile when the projectile is at its maximum height?

Question 26

Find the velocity and position vectors of an object with acceleration initial velocity and initial position

Question 27

Find the length of the curve $\mathbf { r } ( t ) = 2 t \mathbf { i } + t ^ { 2 } \mathbf { j } + \ln t \mathbf { k }$ $1 \leq t \leq e ^ { 3 }$

A) $e ^ { 6 }$
B) $e ^ { 3 }$
C) $e ^ { 6 } + 2$
D) $e ^ { 3 } + 2$

Question 28

For the curve given by $\mathbf { r } ( t ) = (4 \sin t , 5 t , 4 \mathrm { cos } t )$ , find the unit normal vector. $\mathbf { a } ( t ) = 2 \mathbf { k } , \mathbf { v } ( 0 ) = 10 \mathbf { i } - 9 \mathbf { j }$

A) $( 2 \sin t , 5 , - 2 \cos t)$
B) $(- 2 \sin t , 0 , - 2 \cos t )$
C) $\left( - \frac { \sin t } { \sqrt { 29 } } , 0 , - \frac { \cos t } { \sqrt { 29 } } \right)$
D) $( \sqrt { 29 } \sin t , 0 , - \sqrt { 29 } \mathrm { cost } )$
E) None of these

Question 29

Find the length of the curve $\mathbf { r } ( t ) = 8 t \mathbf { i } + 3 \cos t \mathbf { j } + 3 \sin t \mathbf { k }$ $0 \leq t \leq 2 \pi$

A) $2 \sqrt { 73 }$ $\pi$
B) $\sqrt { 73 }$ $\pi$
C) $\sqrt { 11 }$ $\pi$
D) $2 \sqrt { 11 }$ $\pi$

Question 30

Find the unit tangent and unit normal vectors and for the curve C defined by

Question 31

Let C be a smooth curve defined by $\mathbf { r } ( t ) = 2 \mathbf { i } + 3 t \mathbf { j } + 2 t ^ { 2 } \mathbf { k }$ , and let $\mathbf { T } ( t )$ and $\mathrm { N } ( t )$ be the unit tangent vector and unit normal vector to C corresponding to t. The plane determined by T and N is called the osculating plane. Find an equation of the osculating plane of the curve described by $\mathbf { r } ( t )$ at $t = 1$

A) $z = 2$
B) $y = 3$
C) $2 x + 3 y + 2 z = 1$
D) $x = 2$

Question 32

Reparametrize the curve with respect to arc length measured from the point where $t = 0$ in the direction of increasing $t$ . $\mathbf { r } ( t ) = ( 5 + 3 t ) \mathbf { i } + ( 8 + 9 t ) \mathbf { j } - ( 6 t ) \mathbf { k }$

A) $\mathbf { r } ( t ( s ) ) = \left( 5 - \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - \left( \frac { 6 s } { \sqrt { 126 } } \right) \mathbf { k }$
B) $\mathbf { r } ( t ( s ) ) = \left( 5 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 - \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - \left( \frac { 6 s } { \sqrt { 126 } } \right) \mathbf { k }$
C) $\mathbf { r } ( t ( s ) ) = \left( 5 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } + ( 6 s ) \mathbf { k }$
D) $\mathbf { r } ( t ( s ) ) = \left( 5 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - \left( \frac { 6 s } { \sqrt { 126 } } \right) \mathbf { k }$
E) $\mathbf { r } ( t ( s ) ) = \left( 5 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - ( 6 s ) \mathbf { k }$

Question 33

Find the velocity, acceleration, and speed of an object with position vector .

Question 34

Find the velocity, acceleration, and speed of an object with position function for Sketch the path of the object and its velocity and acceleration vectors.

Question 35

Find the unit tangent and unit normal vectors and for the curve C defined by Sketch the graph of C, and show and for

Question 36

Find the velocity, acceleration, and speed of an object with position vector .

Question 37

A projectile is fired from ground level with an initial speed of 1100 ft/sec and an angle of elevation of

A) Find the range of the projectile.
B) What is the maximm height attained by the projectile?
C) What is the speed of the projectile at impact?
Round your answers to the nearest integer.

Question 38

Find the length of the curve $\mathbf { r } ( t ) = - 2 t \mathbf { i } - t \mathbf { j } + t \mathbf { k }$ $- 2 \leq t \leq 1 .$

A) $3 \sqrt { 6 }$
B) $\sqrt { 6 }$
C) $2 \sqrt { 6 }$
D) $6 \sqrt { 6 }$

Question 39

Find the velocity, acceleration, and speed of an object with position function for Sketch the path of the object and its velocity and acceleration vectors.

Question 40

Find the curvature of the curve $\mathbf { r } ( t ) = 2 t \mathbf { i } + 6 t \mathbf { j } + 9 \mathbf { k }$ .

A) 1
B) $2 \sqrt { 10 }$
C) 11
D) 0

Question 41

Find parametric equations for the tangent line to the curve with parametric equations $x = 2 t$ $y = 7 t ^ { 2 }$ $z = 4 t ^ { 3 }$ at the point with $t = 1$

A) $x = 2 + 2 t$ $y = 7 + 14 t$ $z = 4 + 12 t$
B) $x = 2 + 2 t$ $y = 7 + 7 t$ $z = 4 + 4 t$
C) $x = 1 + 2 t$ $y = 1 + 7 t$ $z = 1 + 4 t$
D) $x = 1 + 2 t$ $y = 1 + 14 t$ $z = 1 + 12 t$

Question 42

Evaluate the integral. $\int \left( e ^ { 9 t } \mathbf { i } + 14 t \mathbf { j } + \ln t \mathbf { k } \right) d t$

A) $\frac { e ^ { 9 t } } { 9 } \mathbf { i } + 7 t ^ { 2 } \mathbf { j } + t ( \ln t - 1 ) \mathbf { k } + C$
B) $e ^ { 9 t } \mathbf { i } + 7 t ^ { 2 } \mathbf { j } + ( \ln t - 1 ) \mathbf { k } + \mathrm { C }$
C) $\frac { e ^ { 9 t } } { 9 } \mathbf { i } + 7 t ^ { 2 } \mathbf { j } + t ( \ln t + 9 ) \mathbf { k } + C$
D) $\frac { e ^ { 9 t } } { 9 } \mathbf { i } - 7 t ^ { 2 } \mathbf { j } + t ( \ln t + 1 ) \mathbf { k } + C$
E) $e ^ { 9 t } \mathbf { i } - 7 t ^ { 2 } \mathbf { j } + ( \ln t - 1 ) \mathbf { k } + \mathrm { C }$

Question 43

Find $\mathbf { r } ( t )$ satisfying the conditions for $\mathbf { r } ^ { \prime } ( t ) = 5 \mathbf { i } + 8 t \mathbf { j } - 6 t ^ { 2 } \mathbf { k }$ $\mathbf { r } ( 0 ) = \mathbf { i } + \mathbf { j }$

A) $( 5 t + 1 ) \mathbf { i } + \left( 4 t ^ { 2 } + 1 \right) \mathbf { j } - 2 t ^ { 3 } \mathbf { k }$
B) $( 5 t + 1 ) \mathbf { i } + \left( 8 t ^ { 2 } + 1 \right) \mathbf { j } - 6 t ^ { 3 } \mathbf { k }$
C) $5 t \mathbf { i } + 4 t ^ { 2 } \mathbf { j } - 2 t ^ { 3 } \mathbf { k }$
D) $5 t \mathbf { i } + 8 t ^ { 2 } \mathbf { j } - 6 t ^ { 3 } \mathbf { k }$

Question 44

The figure shows a curve $C$ given by a vector function $\mathbf { r } ( t )$ . Choose the correct expression for $\mathbf { r } ^ { \prime } ( 4 )$ .

A) $\lim _ { h \rightarrow 0 } \frac { r ( 4 - h ) + r ( 4 ) } { h }$
B) $\lim _ { h \rightarrow 0 } \frac { r ( 4 + h ) - r ( 4 ) } { h }$
C) $\lim _ { h \rightarrow 0 } \frac { r ( 4 + h ) - r ( h ) } { h }$
D) $\lim _ { h \rightarrow 0 } \frac { r ( 4 - h ) - r ( 4 ) } { h }$
E) $\lim _ { h \rightarrow 0 } \frac { r ( 4 + h ) + r ( 4 ) } { h }$

Question 45

If
$\mathbf { r } ( t ) = \left\langle t , t ^ { 9 } , t ^ { 11 } \right\rangle$ , find $\mathbf { r } ^ { \prime \prime } ( t )$ .

A) $\left\langle 0,110 t ^ { 9 } , 72 t ^ { 7 } \right\rangle$
B) $\left\langle 0,72 t ^ { 6 } , 110 t ^ { 8 } \right\rangle$
C) $\left\langle0,72 t ^ { 7 } , 110 t ^ { 9 } \right\rangle$
D) $\left\langle0,110 t ^ { 8 } , 72 t ^ { 6 } \right\rangle$
E) $\left\langle 1,9 t ^ { 6 } , 11 t ^ { 8 } \right\rangle$

Question 46

Find equations of the normal plane to at the point (2, 4, 8).

Question 47

Find $\mathbf { r } ( t )$ satisfying the conditions for $\mathbf { r } ^ { t } ( t ) = 9 e ^ { 9 t } \mathbf { i } + 9 e ^ { - t } \mathbf { j } + e ^ { t } \mathbf { k }$ $\mathbf { r } ( 0 ) = \mathbf { i } - \mathbf { j } + 9 \mathbf { k }$

A) $\left( e ^ { 9 t } + 1 \right) \mathbf { i } - \left( 9 e ^ { - t } + 1 \right) \mathbf { j } + \left( e ^ { t } + 9 \right) \mathbf { k }$
B) $9 e ^ { 9 t } \mathbf { i } - \left( 9 e ^ { - t } + 8 \right) \mathbf { j } + \left( e ^ { t } + 8 \right) \mathbf { k }$
C) $e ^ { 9 t } \mathbf { i } - \left( 9 e ^ { - t } - 8 \right) \mathbf { j } + \left( e ^ { t } + 8 \right) \mathbf { k }$
D) $\left( 9 e ^ { 9 t } + 1 \right) \mathbf { i } - \left( 9 e ^ { - t } - 1 \right) \mathbf { j } + \left( e ^ { t } + 9 \right) \mathbf { k }$

Question 48

The torsion of a curve defined by $\mathbf { r } ( t )$ is given by $\tau = \frac { \left( \mathbf { r } ^ { t } \times \mathbf { r } ^ { \prime \prime } \right) \cdot \mathbf { r } ^ { m \prime } } { \left| \mathbf { r } ^ { t } \times \mathbf { r } ^ { \prime t } \right| ^ { 2 } }$ Find the torsion of the curve defined by $\mathbf { r } ( t ) = \cos 2 t \mathbf { i } + \sin 2 t \mathbf { j } + 5 t \mathbf { k }$ .

A) $\frac { 27 } { 29 }$
B) $\frac { 10 } { 29 }$
C) $\frac { 50 } { 29 }$
D) $\frac { 7 } { 29 }$

Question 49

Find the integral $\int \left( 2 t \mathbf { i } + 9 t ^ { 2 } \mathbf { j } + 7 \mathbf { k } \right) d t$

A) $2 t ^ { 2 } \mathbf { i } + 9 t ^ { 3 } \mathbf { j } + 7 t \mathbf { k } + \mathbf { C }$
B) $t ^ { 2 } \mathbf { i } + 3 t ^ { 3 } \mathbf { j } + 7 t \mathbf { k } + \mathbf { C }$
C) $2 t \mathbf { i } + 9 t ^ { 2 } \mathbf { j } + 7 \mathbf { k } + \mathbf { C }$
D) $2 \mathbf { i } + 18 t \mathbf { j } + \mathbf { C }$

Question 50

Find parametric equations for the tangent line to the curve with parametric equations $x = 3 t$ $y = 7 t ^ { 2 }$ $z = 8 t ^ { 3 }$ at the point with $t = 1$

A) $x = 3 + 3 t$ $y = 7 + 14 t$ $z = 8 + 24 t$
B) $x = 1 + 3 t$ $y = 1 + 14 t$ $z = 1 + 24 t$
C) $x = 1 + 3 t$ $y = 1 + 7 t$ $z = 1 + 8 t$
D) $x = 3 + 3 t$ $y = 7 + 7 t$ $z = 8 + 8 t$

Question 51

If $\mathbf { r } ( t ) = \mathbf { i } + t \cos \pi t \mathbf { j } + 2 \sin \pi t\mathbf { k }$ , evaluate $\int _ { 0 } ^ { 1 } r ( t ) d t$ .

A) $\mathbf { i } - \frac { 2 } { \pi ^ { 2 } } \mathbf { j } - \frac { 1 } { \pi } \mathbf { k }$
B) $\mathbf { i } + \frac { 2 } { \pi ^ { 2 } } \mathbf { j } + \frac { 4 } { \pi } \mathbf { k }$
C) $$\mathbf { i } + \frac { 2 } { \pi ^ { 2 } } \mathbf { j } - \frac { 4 } { \pi } \mathbf { k }$$
D) $\mathbf { i } - \frac { 2 } { \pi ^ { 2 } } \mathbf { j } + \frac { 1 } { \pi } \mathbf { k }$
E) $\mathbf { i } - \frac { 2 } { \pi ^ { 2 } } \mathbf { j } + \frac { 4 } { \pi } \mathbf { k }$

Question 52

Use Simpson's Rule with n = 4 to estimate the length of the arc of the curve with equations $x = \sqrt { t } , y = \frac { 4 } { t } , z = t ^ { 2 } + 1$ , from $( 1,4,2 )$ to $( 2,1,17 )$ . Round your answer to four decimal places.

A) $14.824$
B) $7.2041$
C) $7.4106$
D) $6.5706$
E) None of these

Question 53

Find $\mathbf { r } ( t )$ if $\mathbf { r } ^ { \prime } ( t ) = \sin t \mathbf { i } - \cos t \mathbf { j } + 6 t \mathbf { k }$ and $r ( 0 ) = i + j + 5 k$ .

A) $( - \cos t + 2 ) \mathbf { i } + ( \sin t - 1 ) \mathbf { j } + \left( 3 t ^ { 2 } + 5 \right) \mathbf { k }$
B) $( \cos t ) \mathbf { i } + ( \sin t - 1 ) \mathbf { j } + \left( 3 t ^ { 2 } + 5 \right) \mathbf { k }$
C) $( \cos t ) \mathbf { i } + ( \sin t + 1 ) \mathbf { j } + \left( 3 t ^ { 2 } + 5 \right) \mathbf { k }$
D) $( - \cos t + 2 ) \mathbf { i } - ( \sin t - 1 ) \mathbf { j } + \left( 3 t ^ { 2 } + 5 \right) \mathbf { k }$
E) $( - \cos t + 1 ) \mathbf { i } - ( \sin t - 1 ) \mathbf { j } + \left( 3 t ^ { 2 } + 5 \right) \mathbf { k }$

Question 54

Find the arc length function for the curve defined by for Then use this result to find a parametrization of C in terms of s.

Question 55

The helix $\mathbf { r } _ { 1 } ( t ) = 8 \cos t \mathbf { i } + \sin t \mathbf { j } + t \mathbf { k }$ intersects the curve $\mathbf { r } _ { 2 } ( t ) = ( 8 + t ) \mathbf { i } + 10 t ^ { 2 } \mathbf { j } + 9 t ^ { 3 } \mathbf { k }$ at the point $( 8,0,0 )$ . Find the angle of intersection.

A) $\frac { \pi } { 3 }$
B) $\frac { \pi } { 4 }$
C) $\frac { \pi } { 2 }$
D) $0$
E) None of these

Question 56

Find the unit tangent vector $\mathbf { T } ( t )$ for $\mathbf { r } ( t ) = 2 t \mathbf { i } + 6 t \mathbf { j } + 3 t \mathbf { k }$ at $t = - 1$

A) $- 2 \mathbf { i } - 6 \mathbf { j } - 3 \mathbf { k }$
B) $\frac { 2 } { 7 }$ i + $\frac { 6 } { 7 }$ j + $\frac { 3 } { 7 }$ k
C) $- \frac { 2 } { 49 }$ i $- \frac { 6 } { 49 }$ j $- \frac { 3 } { 49 }$ k
D) $2 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }$

Question 57

Find the point(s) on the graph of at which the curvature is zero.

Question 58

Find the unit tangent vector $T ( t )$ . $\mathbf { r } ( t ) = \langle 2 \sin t , 4 t , 2 \cos t \rangle$

A) $\left\langle\frac { \cos t } { 2 \sqrt { 5 } } , \frac { 2 } { \sqrt { 5 } } , - \frac { \sin t } { 2 \sqrt { 5 } } \right\rangle$
B) $\langle3 \cos t , 6 , - 3 \sin t \rangle$
C) $\left\langle\frac { \cos t } { \sqrt { 5 } } , \frac { 2 } { \sqrt { 5 } } , - \frac { \sin t } { \sqrt { 5 } } \right\rangle$
D) $\langle3 \sqrt { 5 } \cos t , 6 , - 3 \sqrt { 5 } \sin t \rangle$
E) $\left\langle- \frac { \cos t } { 2 \sqrt { 5 } } , \frac { 4 } { 2 \sqrt { 5 } } , \frac { 3 \sin t } { 2 \sqrt { 5 } } \right\rangle$

Question 59

At what point on the curve $x = t ^ { 3 } , y = 9 t , z = t ^ { 4 }$ is the normal plane parallel to the plane $3 x + 9 y - 4 z = 4$ ?

A) $( - 1 , - 3,9 )$
B) $( 9,18 , - 2 )$
C) $( - 1 , - 9,1 )$
D) $( - 18,9,1 )$
E) $( - 9,1,1 )$

Question 60

The curvature of the curve given by the vector function $r$ is $k ( t ) = \frac { \left| \mathbf { r } ^ { \prime } ( t ) \times \mathbf { r } ^ { \prime \prime } ( t ) \right| } { \left| \mathbf { r } ^ { \prime } ( t ) \right| ^ { 3 } }$ Use the formula to find the curvature of $\mathbf { r } ( t ) = \left\langle \sqrt { 13 } t , e ^ { t } , e ^ { - t } \right\rangle$
at the point $( 0,1,1 )$ .

A) $\sqrt { 15 }$
B) $\frac { \sqrt { 2 } } { 15 }$
C) $15 \sqrt { 2 }$
D) $\frac { 15 } { \sqrt { 2 } }$
E) $\frac { \sqrt { 15 } } { 15 }$

Question 61

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $x = t ^ { 11 } , y = t ^ { 3 } , z = t ^ { 6 } ; ( 4,4,4 )$

A) $x = 4 - 11 t , y = 4 + 3 t , z = 4 + 6 t$
B) $x = 4 + 11 t , y = 4 + 3 t , z = 4 + 6 t$
C) $x = 4 + 11 t , y = 4 + 3 t , z = 4 - 6 t$
D) $x = 11 t , y = 4 + 3 t , z = 4 + 6 t$
E) $x = 4 + 11 t , y = 4 - 3 t , z = 4 + 6 t$

Question 62

Given
a. Find and .
b. Sketch the curve defined by r and the vectors and on the same set of axes.

Question 63

Find the domain of the vector function $\mathbf { r } ( t ) = 8 t \mathbf { i } + \frac { 1 } { t - 4 } \mathbf { j }$ .

A) $( - \infty , - 4 ) \cup ( - 4 , \infty )$
B) $( - \infty , 8 ) \cup ( 8 , \infty )$
C) $( - \infty , - 8 ) \cup ( - 8 , \infty )$
D) $( - \infty , 4 ) \cup ( 4 , \infty )$

Question 64

Find the unit tangent vector for the curve given by $\mathbf { r } ( t ) = \left\langle \frac { 1 } { 7 } t ^ { 7 } , \frac { 1 } { 3 } t ^ { 3 } , t \right\rangle$ .

A) $\frac { \left\langle t ^ { 6 } , t ^ { 2 } , 1 \right\rangle } { \sqrt { t ^ { 10 } + t ^ { 4 } } }$
B) $\frac { \left\langle t ^ { 6 } , t ^ { 2 } , 1 \right\rangle } { \sqrt { 6 t ^ { 12 } + 4 t ^ { 4 } + 1 } }$
C) $\frac { \left\langle t ^ { 6 } , t ^ { 2 } , 1 \right\rangle } { \sqrt { t ^ { 12 } + t ^ { 4 } } }$
D) $\frac { \left\langle t ^ { 6 } , t ^ { 2 } , 1 \right\rangle } { \sqrt { t ^ { 12 } + t ^ { 4 } + 1 } }$
E) None of these

Question 65

Find the derivative of the vector function.

Question 66

If and , find .

Question 67

Find and for

Question 68

Find the point of intersection of the tangent lines to the curve , at the points where and .

Question 69

The curves and intersects at the origin. Find their angle of intersection correct to the nearest degree.

Question 70

Let $\mathbf { r } ( t ) = \left\langle \sqrt { 6 - t } , \frac { \left( e ^ { t } - 2 \right) } { t } , \ln ( t + 1 ) \right\rangle$ . Find the domain of $r$ .

A) $( - 2,6 ]$
B) $( - 2,0 ) \cup ( 0,6 ]$
C) $( 6 , \infty ]$
D) $( - \infty , - 2 )$
E) $[ 6,0 ) \cup ( 0,2 )$

Question 71

Find an expression for .

Question 72

Find if and .

Question 73

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $x = \cos t , y = 4 e ^ { 6 t } , z = 4 e ^ { - 6 t } ; ( 1,5,5 )$

A) $x = 0 , y = 5 - 24 t , z = 5 + 24 t$
B) $x = 1 , y = 5 + 24 t , z = 4 - 24 t$
C) $x = t , y = 5 + 24 t , z = 5 - 24 t$
D) $x = t , y = 5 - 24 t , z = 5 + 24 t$
E) $x = 1 , y = 5 + 24 t , z = 5 - 24 t$

Question 74

Find and for

Question 75

Find the domain of the vector function $\mathbf { r } ( t ) = \left\langle 9 \sqrt { t } , \frac { 1 } { t - 3 } , \ln t \right\rangle$ .

A) $[ 0,3 ) \cup ( 3,9 ) \cup ( 9 , \infty )$
B) $( 0,3 ) \cup ( 3 , \infty )$
C) $( 0,3 ) \cup ( 3,9 ) \cup ( 9 , \infty )$
D) $( 0,9 ) \cup ( 9 , \infty )$

Question 76

Find the limit. $\lim _ { t \rightarrow 0 ^ { + } } \langle10 \cos t , 30 \sin t , 5 t \ln t\rangle$

A) $\mathbf { r } ( t ) = 10 \mathbf { k }$
B) $\mathbf { r } ( t ) = 10 \mathbf { j }$
C) $\mathbf { r } ( t ) = 10 \mathbf { i } - 5 \mathbf { k }$
D) $\mathbf { r } ( t ) = 10 \mathbf { i } + 30 \mathbf { j } + 5 \mathbf { k }$
E) $\mathbf { r } ( t ) = 10 \mathbf { i }$

Question 77

Find the derivative

Question 78

Find $\mathbf { r } ^ { \prime \prime } ( t )$ for the function given. $\mathbf { r } ( t ) = 4 \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k }$

A) $r ^ { \prime \prime } ( t ) = \sin t j - \cos t \mathrm { k }$
B) $r ^ { \prime \prime } ( t ) = - 4 \sin t \mathrm { j } - \cos t \mathrm { k }$
C) $r ^ { \prime \prime } ( t ) = - 4 \cos t j - \sin t \mathrm { k }$
D) $r ^ { \prime \prime } ( t ) = \cos t j - \sin t \mathrm { k }$
E) $r ^ { \prime \prime } ( t ) = - \sin t \mathrm { j } + 4 \cos t \mathrm { k }$

Question 79

Find the integral

Question 80

Find for the function given.