Question 1

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

Accepted Answer

The answer of Find the unit tangent vector of the...

Question 2

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

Accepted Answer

The answer of Find the unit tangent vector of the...

Question 3

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

Accepted Answer

The second derivative of $r(t) = (2 \cos t)i + (4 \sin t)j$ with respect to $t$ is $r''(t) = (-2 \cos t)i + (-4 \sin t)j$, because the second derivative of $\cos t$ is $-\cos t$ and the second derivative of $\sin t$ is $-\sin t$.

Question 4

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

Accepted Answer

To find r''(t), first find r'(t): r'(t) = (-2 sin 2t)i + (3 cos t)j. Then differentiate again: r''(t) = (-4 cos 2t)i + (-3 sin t)j.

Question 5

Compute r''(t).
-r(t) = ( 3 ln( 6t))i + ( 2  )j
A)

B)

C)

D)

Accepted Answer

The answer of Compute r''(t).
-r(t) = ( 3 ln( 6t))i...

Question 6

Evaluate the integral.
-
A)

B)

C)

D)

Accepted Answer

The answer of Evaluate the integral.
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A)

B)

C)

D) 
...

Question 7

Evaluate the integral.
-
A)

B)

C)

D)

Accepted Answer

The answer of Evaluate the integral.
-
A)

B)

C)

D) 
...

Question 8

Evaluate the integral.
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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

Accepted Answer

The answer of Evaluate the integral.
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A) 11i + 4 ln...

Question 9

Evaluate the integral.
-
A) +4i +  j + 24k
B) -4i -  j - 24k
C) -4i -  j + 24k
D) -4i +  j + 24k

Accepted Answer

The answer of Evaluate the integral.
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A) +4i +  j...

Question 10

Evaluate the integral.
-
A)

B)

C)

D)

Accepted Answer

The answer of Evaluate the integral.
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A)

B)

C)

D) 
...

Question 11

Evaluate the integral.
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A) 3i + 3j -  k
B) 3i - 3j -  k
C) 6i - 6j + 3k
D) 6i + 6j - 3k

Accepted Answer

The answer of Evaluate the integral.
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A) 3i + 3j -...

Question 12

Evaluate the integral.
-
A) 7  i
B) 0
C) 7  i + 3  j
D) 3  i

Accepted Answer

The answer of Evaluate the integral.
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A) 7  i
B) 0
C)...

Question 13

Evaluate the integral.
-
A)

B) 7  j
C)

D)

Accepted Answer

The answer of Evaluate the integral.
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A)

B) 7  j
C)...

Question 14

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)

Accepted Answer

The answer of If r(t) is the position vector of...

Question 15

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

Accepted Answer

The answer of If r(t) is the position vector of...

Question 16

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

Accepted Answer

The answer of If r(t) is the position vector of...

Question 17

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)

Accepted Answer

The answer of If r(t) is the position vector of...

Question 18

The position vector of a particle is r(t). Find the requested vector.
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A) v( 3) = 58i - 135j - 6k
B) v( 3) = 31i - 45j - 3k
C) v( 3) = 58i + 135j + 6k
D) v( 3) = 50i - 135j - 6k

Accepted Answer

The answer of The position vector of a particle is...

Question 19

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

B)

C)

D)

Accepted Answer

The answer of The position vector of a particle is...

Question 20

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

Accepted Answer

The answer of The position vector of a particle is...

Quiz 14: Vector-Valued Functions

Find the unit tangent vector of the given curve. -r(t) = ( 6 2t) i + ( 6 2t) j

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

Compute r''(t) . -r(t) = ( 2 cos t) i + ( 4 sin t) j

Compute r''(t) . -r(t) = (cos 2t) i + ( 3 sin t) j

Compute r''(t) . -r(t) = ( 3 ln( 6t) ) i + ( 2 ) j

Evaluate the integral. -

Evaluate the integral. -

Evaluate the integral. -

Evaluate the integral. -

Evaluate the integral. -

Evaluate the integral. -

Evaluate the integral. -

Evaluate the integral. -

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

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

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

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

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

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

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