Deck 14: Vector-Valued Functions

Question 1

Find a function r(t) that describes the line or line segment.

-The line through P(4, 9, 3) and Q(1, 6, 7)

A) r(t) =
B) r(t) =
C) r(t) =
D) r(t) =

Answer 1

r(t) =

Question 2

Find a function r(t) that describes the line or line segment.

-The line segment from P(2, 7, 3) to Q(3, 1, 1)

A) r(t) = ; 1 $\le$ t $\le$ 2
B) r(t) = ; 1 $\le$ t $\le$ 2
C) r(t) = ; 0 $\le$ t $\le$ 1
D) r(t) = ; 0 $\le$ t $\le$ 1

Answer 2

r(t) = ; 0 $\le$ t $\le$ 1

Question 3

Graph the curve described by the function, indicating the positive orientation.

-r(t) = , for 0 $\le$ t $\le$ 2 $\pi$

A)

B)

C)

D)

Answer 3

Question 4

Graph the curve described by the function, indicating the positive orientation.

-r(t) = 2cos t i + 3j + 2 sin t k, for 0 $\le$ t $\le$ 2 $\pi$

A)

B)

C)

D)

Question 5

Graph the curve described by the function. Use analysis to anticipate the shape of the curve before using a graphing utility.

-r(t) = 3 cos t i + 2 sin t j + cos 5t k, for 0 $\le$ t $\le$ 2 $\pi$

A)

B)

C)

D)

Question 6

Graph the curve described by the function. Use analysis to anticipate the shape of the curve before using a graphing utility.

-r(t) = cos 2t sin t i + sin 2t sin t j + k, for 0 $\le$ t $\le$ 16

A)

B)

C)

D)

Question 7

Evaluate the limit.

- = ( 7 cos ti+ 6 sin tj)

A) 7i
B) -6j
C) 6j
D) 7i - 6j

Question 8

Evaluate the limit.

-

A)

B)

C)

D)

Question 9

Find the domain of the vector-valued function.

-r(t) = i + j

A) t $\ge$ 5
B) $\ge$ 5
C) < 5
D) t > 5

Question 10

Find the domain of the vector-valued function.

-r(t) = sin 3t i + j

A) t $\ge$
B) t $\ge$ 0
C) t > 3
D)All real numbers

Question 11

Find a function r(t) that describes the curve where the surfaces intersect.

-z = 16; z = +

A) r(t) =
B) r(t) =
C) r(t) =
D) r(t) =

Question 12

Find a function r(t) that describes the curve where the surfaces intersect.

- + = 16; z = 2x + 3y

A) r(t) =
B) r(t) =
C) r(t) =
D) r(t) =

Question 13

Verify that the curve r(t) lies on the surface. Give the name of the surface.

-r(t) = (2t cos t)i + (2t sin t)j + 2t k; + =

A)

B)

C)

D)

Question 14

Verify that the curve r(t) lies on the surface. Give the name of the surface.

-r(t) = ; z= +

A)

B)

C)

D)

Question 15

Differentiate the function.

-r(t) = ( -7 - 6)i + j

A)

B)

C)

D)

Question 16

Differentiate the function.

-r(t) = (cot t)i + (csc t)j

A)

B)

C)

D)

Question 17

Find the unit tangent vector of the given curve.

-r(t) = 3 i - 12 j + 4 k

A)

B)

C)

D)

Question 18

Find the unit tangent vector of the given curve.

-r(t) = ( 6 - 2t)i + (2t - 9)j + ( 9 + t)k

A)

B)

C)

D)

Question 19

Find the unit tangent vector of the given curve.

-r(t) = ( 6 + 10 )i + ( 9 + 11 )j + ( 1 + 2 )k

A)

B)

C) T = 10i + 11j + 2k
D)

Question 20

Find the unit tangent vector of the given curve.

-r(t) = i + j - 12tk

A)

B)

C)

D)

Question 21

Find the unit tangent vector of the given curve.

-r(t) = ( 6 2t)i + ( 6 2t)j

A) T = ( 6 sin 2t)i - ( 6 cos 2t)j
B) T = ( 36 sin 2t)i -( 36 cos 2t)j
C) T = (sin 2t)i - (cos 2t)j
D) T = ( 6 cos 2t)i - ( 6 sin 2t)j

Question 22

Find the unit tangent vector of the given curve.

-r(t) = ( 8t cos t - 8 sin t)j + ( 8t sin t + 8 cos t)k

A) T = (-8 sin t)j + ( 8 cos t)k
B) T = ( 8 cos t)j - ( 8 sin t)k
C) T = (-sin t)j + (cos t)k
D) T = - (sin t)j + (cos t)k

Question 23

Compute r''(t).

-r(t) = ( 2 cos t)i + ( 4 sin t)j

A) r''(t) = (-2 cos t)i + (-4 sin t)j
B) r''(t) = ( 2 cos t)i + ( 4 sin t)j
C) r''(t) = ( 2 sin t)i + ( 4 cos t)j
D) r''(t) = (-2 sin t)i + (-4 cos t)j

Question 24

Compute r''(t).

-r(t) = (cos 2t)i + ( 3 sin t)j

A) r''(t) = (-2 cos 2t)i + ( 3 sin t)j
B) r''(t) = (-4 cos 2t)i + (-3 sin t)j
C) r''(t) = (-4 cos 2t)i + (-9 sin t)j
D) r''(t) = ( 4 cos 2t)i + (-3 sin t)j

Question 25

Compute r''(t).

-r(t) = ( 3 ln( 6t))i + ( 2 )j

A)

B)

C)

D)

Question 26

Evaluate the integral.

-

A)

B)

C)

D)

Question 27

Evaluate the integral.

-

A)

B)

C)

D)

Question 28

Evaluate the integral.

-

A) 11i + 4 ln 2j + (1 - )k
B) 5i + 4 ln 2j + (1 - )k
C) 11i + 4 ln 2j + k
D) 11i + 4 ln 2j + (1 + )k

Question 29

Evaluate the integral.

-

A) +4i + j + 24k
B) -4i - j - 24k
C) -4i - j + 24k
D) -4i + j + 24k

Question 30

Evaluate the integral.

-

A)

B)

C)

D)

Question 31

Evaluate the integral.

-

A) 3i + 3j - k
B) 3i - 3j - 11ee9522_3540_09d3_bdb6_ddd7428c17b9_TB9662_11 k
C) 6i - 6j + 3k
D) 6i + 6j - 3k

Question 32

Evaluate the integral.

-

A) 7 i
B) 0
C) 7 i + 3 j
D) 3 i

Question 33

Evaluate the integral.

-

A)

B) 7 j
C)

D)

Question 34

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.

-Find the velocity vector.

A)

B)

C)

D)

Question 35

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.

-Find the acceleration vector. r(t) = ( 6 cos t)i + ( 8 sin t)j

A) a = (-6 sin t)i + (-8 cos t)j
B) a = ( 6 sin t)i + ( 8 cos t)j
C) a = ( 6 cos t)i + ( 8 sin t)j
D) a = (-6 cos t)i + (-8 sin t)j

Question 36

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.

-Find the velocity vector. r(t) = (cot t)i + (csc t)j

A) v = ( t)i + (cot t csc t)j
B) v = ( t)i + (tan t sec t)j
C) v = (- t)i - (cot t csc t)j
D) v = (- t)i - (tan t sec t)j

Question 37

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.

-Find the acceleration vector. r(t) = ( 7 ln( 5t))i + ( 3 )j

A)

B)

C)

D)

Question 38

The position vector of a particle is r(t). Find the requested vector.

-

A) v( 3) = 58i - 135j - 6k
B) v( 3) = 31i - 45j - 3k
C) v( 3) = 58i + 135j + 6k
D) v( 3) = 50i - 135j - 6k

Question 39

The position vector of a particle is r(t). Find the requested vector.

-

A)

B)

C)

D)

Question 40

The position vector of a particle is r(t). Find the requested vector.

-The velocity at t = 0 for r(t) = cos( 2t)i + 7ln(t - 3)j - k

A) v(0) = j
B) v(0) = -2i - j
C) v(0) = - j
D) v(0) = 2i - j

Question 41

The position vector of a particle is r(t). Find the requested vector.

-The velocity at t = 3 for r(t) = ( 8 - 4 )i + ( 6t + 7)j - k

A)v( 3) = -24i + 6j + 5 k
B) v( 3) = -24i + 6j - 5 k
C) v( 3) = -12i +6j + 5 k
D) v( 3) = 24i + 6j + 5 k

Question 42

The position vector of a particle is r(t). Find the requested vector.

-The velocity at t = 0 for r(t) = ln( - 5 + 3)i - j - 5cos(t)k

A) v( 0) = i - j + 5k
B) v( 0) = j
C) v( 0) = i - j
D) v( 0) = - j

Question 43

The position vector of a particle is r(t). Find the requested vector.

-The acceleration at t = for r(t) = ( 9 sin 5t)i - ( 10 cos 5t)j + ( 2 csc 5t)k

A) a = -225i - 50k
B) a = 225i + 50k
C) a = -225i + 50k
D) a = 250j + 50k

Question 44

The position vector of a particle is r(t). Find the requested vector.

-The acceleration at t = 2 for r(t) = ( 7t - 3 )i + ( 10 - t)j + ( 2 - 3t)k

A) a( 2) = -144i - j + 4k
B) a( 2) = 144i + 4k
C) a( 2) = -144i + 4k
D) a( 2) = -36i + 4k

Question 45

The position vector of a particle is r(t). Find the requested vector.

-The acceleration at t = 1 for r(t) = i + 2ln j + k

A) a(1) = 20i + 2j + 12k
B) a(1) = 20i + j - 12k
C) a(1) = 20i - 2j - 12k
D) a(1) = 20i + j + 12k

Question 46

The position vector of a particle is r(t). Find the requested vector.

-The acceleration at t = for r(t) = )i + 2tan( 3t)j + k

A)

B)

C)

D)

Question 47

The position vector of a particle is r(t). Find the requested vector.

-The acceleration at t = 0 for r(t) = i + ( 10 - 2)j + k

A) a(0) = 2i - 2k
B) a(0) = 2i - k
C)a(0) = 2i - k
D) a(0) = 2i +

k

Question 48

Solve the problem. Assume the x-axis is horizontal, the positive y-axis is vertical (opposite g), the ground is horizontal, and only the gravitational force acts on the objects

-A projectile is launched from the origin at an angle of $\alpha$ radians to the horizontal and an initial speed of Find the position function r(t) for this projectile.

A) r(t) = ( 75t cos $\alpha$ - 32 )i + ( 75t sin $\alpha$ )j
B) r(t) = ( 75t sin $\alpha$ )i + ( 75t cos $\alpha$ - 16 )j
C) r(t) = ( 75t cos $\alpha$ )i + ( 75t sin $\alpha$ - 16 )j
D) r(t) = ( 75t sin $\alpha$ - 16 )i + ( 75t cos $\alpha$ )j

Question 49

Solve the problem. Assume the x-axis is horizontal, the positive y-axis is vertical (opposite g), the ground is horizontal, and only the gravitational force acts on the objects

-A projectile is fired at a speed of 800 m/sec at an angle of 34°. How long will it take to get 20 km downrange? Round your answer to the nearest whole number.

A) 32 sec
B) It will never get that far downrange.
C) 30 sec
D) 28 sec

Question 50

Solve the problem. Assume the x-axis is horizontal, the positive y-axis is vertical (opposite g), the ground is horizontal, and only the gravitational force acts on the objects

-A projectile is fired with an initial speed of 585 m/sec at an angle of 45°. What is the greatest height reached by the projectile? Round your answer to the nearest tenth.

A) 85,556.3 m
B) 8730.2 m
C) 34,920.9 m
D) 84.4 m

Question 51

Find the length of the indicated portion of the trajectory.

-r(t) = ( 2cos t)i + ( 2sin t)j + 5tk, 0 $\le$ t $\le$ $\pi$ /2

A) $\pi$
B) $\pi$
C) $\pi$
D) $\pi$

Question 52

Find the length of the indicated portion of the trajectory.

-r(t) = ( cos t)i + ( sin t)j + k, -ln 2 $\le$ t $\le$ 0

A)
B)
C)
D)

Question 53

Find the length of the indicated portion of the trajectory.

-r(t) = (1 + 5t)i + (1 + 8t)j + ( 2 - 2t)k, -1 $\le$ t $\le$ 0

A)
B) 2
C)
D)

Question 54

Find the length of the indicated portion of the trajectory.

-r(t) = ( 2 + 2t)i + ( 3 + 3t)j + ( 3 - 6t)k, -1 $\le$ t $\le$ 0

A) 8
B) 5
C) 9
D) 7

Question 55

Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response
-Show that the arc length of one petal of the rose is given by
2 and use this formula to help make a conjecture about the limit of such arc lengths as

Question 56

Find the curvature of the curve r(t).

-r(t) = ( 8 + ln(sec t))i + ( 3 + t)k, - $\pi$ /2 < t < $\pi$ /2

A) = cos t
B) 11ee983b_f671_a800_a6de_a1c79ef1661c_TB9662_11= sin t
C) 11ee983b_f671_a800_a6de_a1c79ef1661c_TB9662_11 = 1-cos t
D) 11ee983b_f671_a800_a6de_a1c79ef1661c_TB9662_11= -cos t

Question 57

Find the curvature of the curve r(t).

-r(t) = ( 10 + 6 cos 9t) i - ( 2 + 6 sin 9t)j + 2k

A) = 6
B) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 =
C) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 =
D) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 =

Question 58

Find the curvature of the curve r(t).

-r(t) = ( 7 + cos 8t - sin 8t)i + ( 5 + sin 8t + cos 8t)j + 6k

A) =
B) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 = 4
C) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11= 4
D) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 =

Question 59

Find the curvature of the curve r(t).

-r(t) = ( 7t + 3)i - 7j + ( 4 - )k

A) =
B) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 = 7
C)11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 = 7
D) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 =

Question 60

Find the curvature of the space curve.

-r(t) = 12ti + j + k

A) =
B) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 =
C) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 =
D) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 =

Question 61

Find the curvature of the space curve.

-r(t) = (t + 5)i + 8j + (ln(sec t) + 1)k

A) = cos t
B) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 = sec t
C) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 = sin t
D) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11= csc t

Question 62

Find the curvature of the space curve.

-r(t) = -10i + (t + 5)j +(ln(cos t) + 4)k

A) = sin t
B) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 = csc t
C) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 = sec t
D) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 = cos t

Question 63

Find the curvature of the space curve.

-r(t) = -10i + ( 6 + 2t)j + ( + 2)k

A) = -
B) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11=
C) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 =
D) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 =

Question 64

Find the curvature of the space curve.

-r(t) = ( 9 t sin t + 9 cos t)i + 9j + ( 9t cos t - 9 sin t)k

A) = -
B) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 = 9t
C) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 =
D) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 =

Question 65

Find the curvature of the space curve.

-r(t) = ti + (sinh t)j + (cosh t)k

A) = t
B) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 =
C) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 = t
D) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 =

Question 66

Find the curvature of the space curve.

-

A) =
B) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 =
C) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 =
D) 11ee983c_18fb_aab1_a6de_e7ae6ccbb311_TB9662_11 =

Question 67

Find the unit tangent vector T and the principal unit normal vector N.

-r(t) = i + j + 3tk

A)
B)
C)
D)

Question 68

Find the unit tangent vector T and the principal unit normal vector N.

-r(t) = ( 3 + t)i + ( 7 + ln(sec t))j - 3k, - $\pi$ /2 < t < $\pi$ /2

A) T = (-cos t)i - (sin t)j; N = (-cos t)i + (sin t)j
B) T = (cos t)i + (sin t)j; N = (- sin t)i + (cos t)j
C)T = (cos t)i - (sin t)j; N = (-sin t)i - (cos t)j
D) T = (-cos t)i - (sin t)j; N = (sin t)i - (cos t)j

Question 69

Find the unit tangent vector T and the principal unit normal vector N.

-r(t) = (ln(cos t) + 9)i + 9j + ( 10 + t )k, - $\pi$ /2 < t < $\pi$ /2

A) T = (sin t)i - (cos t)k; N = (cos t)i - (sin t)k;
B) T = (sin t)i + (cos t)k; N = (cos t)i - (sin t)k
C) T = (-sin t)i + (cos t)k; N = (-cos t)i + (sin t)k
D) T = (-sin t)i + (cos t)k; N = (-cos t)i - (sin t)k

Question 70

Find the unit tangent vector T and the principal unit normal vector N.

-r(t) = ( - 8)i + (2t - 6)j + 5k

A)
B)
C)
D)

Question 71

Find the unit tangent vector T and the principal unit normal vector N.

-r(t) = ( 7t sin t + 7cos t)i + ( 7t cos t - 7 sin t)j - 4k

A) T = (cos t)i + (sin t)j; N = (-sin t)i + (cos t)j
B) T = (-cos t)i - (sin t)j; N = (sin t)i - (cos t)j
C) T = (cos t)i - (sin t)j; N = (-sin t)i - (cos t)j
D) T = (-cos t)i - (sin t)j; N = (sin t)i + (cos t)j

Question 72

Find the unit tangent vector T and the principal unit normal vector N.

-r(t) = (cosh t)i + (sinh t)j + tk

A)
B)
C)
D)

Question 73

Find the unit tangent vector T and the principal unit normal vector N.

-

A)
B)
C)
D)

Question 74

FInd the tangential and normal components of the acceleration.

-r(t) = i + j + 12tk

A) a = N
B) a = 25T + 25N
C) a = T + 25N
D) a = T

Question 75

FInd the tangential and normal components of the acceleration.

-r(t) = (t + 6)i + (ln(sec t) - 1)j + 7k, - $\pi$ /2 < t < $\pi$ /2

A) a = (sec t tan t)T + (sec t)N
B) a = (csc t)T + (sec t)N
C) a = ( t)T + (cos t)N
D) a = (cos t)T + (cos t)N

Question 76

FInd the tangential and normal components of the acceleration.

-r(t) = ( - 3)i + ( 2t - 4)j + 9k

A)

B)

C)

D)

Question 77

FInd the tangential and normal components of the acceleration.

-r(t) = ( 8t sin t + 8 cos t)i + ( 8t cos t - 8 sin t)j + 8k

A) a = N
B) a = 8T + 8tN
C) a = 8T + N
D) a = 8tN

Question 78

FInd the tangential and normal components of the acceleration.

-r(t) = (cosh t)i + (sinh t)j + tk

A) a = ( sinh t)T + N
B) a = (- sinh t)T + N
C) a = (sinh t)T + N
D) a = (-sinh t)T + N

Question 79

FInd the tangential and normal components of the acceleration.

-r(t) = 4 i + 4 j + 3tk

A) a = T + 3 N
B) a = 3 N
C) a = N
D) a = 3 T

Question 80

Compute the unit binormal vector and torsion of the curve.

-r(t) =

A) B(t) = , = 0
B) B(t) = , 11ee983d_146e_64f2_a6de_f3be5c6bdb70_TB9662_11 = 0
C) B(t) = , 11ee983d_146e_64f2_a6de_f3be5c6bdb70_TB9662_11 = 1
D) B(t) = ,11ee983d_146e_64f2_a6de_f3be5c6bdb70_TB9662_11= 1