Question 1

Find parametric equations for the tangent line to the curve with parametric equations $x = 3 t$, $y = 5 t ^ { 2 } , z = 4 t ^ { 3 }$ at the point with $t = 1$.

Accepted Answer

A)  $x = 1 + 3 t , y = 1 + 10 t , z = 1 + 12 t$ 
B)  $x = 3 + 3 t , y = 5 + 10 t , z = 4 + 12 t$ 
C)  $x = 1 + 3 t , y = 1 + 5 t , z = 1 + 4 t$ 
D)  $x = 3 + 3 t , y = 5 + 5 t , z = 4 + 4 t$ 
A)  $x = 1 + 3 t , y = 1 + 10 t , z = 1 + 12 t$ 
B)  $x = 3 + 3 t , y = 5 + 10 t , z = 4 + 12 t$ 
C)  $x = 1 + 3 t , y = 1 + 5 t , z = 1 + 4 t$ 
D)  $x = 3 + 3 t , y = 5 + 5 t , z = 4 + 4 t$ B

Question 2

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.

Accepted Answer

$$d \approx 43.3 \mathrm {~km}$$

Question 3

Find a vector function that represents the curve of intersection of the two surfaces:
The circular cylinder $x ^ { 2 } + y ^ { 2 } = 4$ and the parabolic cylinder $z = x y$.

Accepted Answer

A)  $\mathbf { r } ( t ) = 4 \cos t \mathbf { i } + 4 t \mathbf { j } + 4 \cos ^ { 2 } t \mathbf { k }$ 
B)  $\mathbf { r } ( t ) = 2 \cos ( t ) \mathbf { i } + 2 \sin ( t ) \mathbf { j } - \sin ( t ) \cos ( t ) \mathbf { k }$ 
C)  $\mathbf { r } ( t ) = \cos t \mathbf { i } + \sin t \mathbf { j } - 4 \cos 2 t \mathbf { k }$ 
D)  $\mathbf { r } ( t ) = \cos ( t ) \mathbf { i } + \sin ( t ) \mathbf { j } + 4 \sin ( t ) \cos ( t ) \mathbf { k }$ 
E)  $\mathbf { r } ( t ) = 2 \cos ( t ) \mathbf { i } + 2 \sin ( t ) \mathbf { j } + 4 \sin ( t ) \cos ( t ) \mathbf { k }$ 
A)  $\mathbf { r } ( t ) = 4 \cos t \mathbf { i } + 4 t \mathbf { j } + 4 \cos ^ { 2 } t \mathbf { k }$ 
B)  $\mathbf { r } ( t ) = 2 \cos ( t ) \mathbf { i } + 2 \sin ( t ) \mathbf { j } - \sin ( t ) \cos ( t ) \mathbf { k }$ 
C)  $\mathbf { r } ( t ) = \cos t \mathbf { i } + \sin t \mathbf { j } - 4 \cos 2 t \mathbf { k }$ 
D)  $\mathbf { r } ( t ) = \cos ( t ) \mathbf { i } + \sin ( t ) \mathbf { j } + 4 \sin ( t ) \cos ( t ) \mathbf { k }$ 
E)  $\mathbf { r } ( t ) = 2 \cos ( t ) \mathbf { i } + 2 \sin ( t ) \mathbf { j } + 4 \sin ( t ) \cos ( t ) \mathbf { k }$ E

Question 4

Sketch the curve of the vector function $\mathbf { r } ( t ) = \left\langle t ^ { 2 } , t ^ { 3 } , t \right\rangle , t \geq 0$, and indicate the orientation of the curve.

Accepted Answer

The answer of Sketch the curve of the vector function...

Question 5

The curves $\mathbf { r } _ { 1 } ( t ) = \left\langle t , t ^ { 3 } , t ^ { 9 } \right\}$ and $\mathbf { r } _ { 2 } ( t ) = \langle \sin t , \sin 5 t , t \rangle$ intersects at the origin. Find their angle of intersection correct to the nearest degree.

Accepted Answer

The answer of The curves \(\mathbf { r } _...

Question 6

$$\text { Find } \mathbf { r } ^ { t } ( t ) \text { and } \mathbf { r } ^ {tt } ( t ) \text { for } \mathbf { r } ( t ) = 10 t \mathbf { i } + 7 t ^ { 2 } \mathbf { j } + 5 t ^ { 3 } \mathbf { k }$$

Accepted Answer

The answer of \[\text { Find } \mathbf { r...

Question 7

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = 4 \sin t \mathbf { i } + 4 \cos t \mathbf { j } + 3 t \mathbf { k }$

Accepted Answer

A)  $a _ { \mathbf { T } } = 0 , a _ { \mathrm { N } } = 4 \sqrt { 10 }$ 
B)  $a _ { \mathrm { T } } = 3 , a _ { \mathrm { N } } = 4 \sqrt { 10 }$ 
C)  $a _ { \mathrm { T } } = 3 , a _ { \mathrm { N } } = 4$ 
D)  $a _ { \mathbf { T } } = 0 , a _ { \mathrm { N } } = 4$ 
A)  $a _ { \mathbf { T } } = 0 , a _ { \mathrm { N } } = 4 \sqrt { 10 }$ 
B)  $a _ { \mathrm { T } } = 3 , a _ { \mathrm { N } } = 4 \sqrt { 10 }$ 
C)  $a _ { \mathrm { T } } = 3 , a _ { \mathrm { N } } = 4$ 
D)  $a _ { \mathbf { T } } = 0 , a _ { \mathrm { N } } = 4$

Question 8

A particle moves with position function $\mathbf { r } ( t ) = \left( 21 t - 7 t ^ { 3 } - 6 \right) \mathbf { i } + 21 t ^ { 2 } \mathbf { j }$
Find the tangential component of the acceleration vector. Select the correct answer.

Accepted Answer

A)  $a _ { T } = 6 t$ 
B)  $a _ { T } = 42 t$ 
C)  $a _ { T } = 42 t + 6$ 
D)  $a _ { T } = 642 t$ 
E)  $a _ { T } = - 66 t$ 
A)  $a _ { T } = 6 t$ 
B)  $a _ { T } = 42 t$ 
C)  $a _ { T } = 42 t + 6$ 
D)  $a _ { T } = 642 t$ 
E)  $a _ { T } = - 66 t$

Question 9

$$\text { If } \mathbf { r } ( t ) = 7 \mathbf { i } + t \cos \pi t \mathbf { j } + 4 \sin \pi t \mathbf { k } \text {, evaluate } \int _ { 0 } ^ { 1 } r ( t ) d t$$

Accepted Answer

The answer of \[\text { If } \mathbf { r...

Question 10

Find a vector function that represents the curve of intersection of the two surfaces: the top half of the ellipsoid $x ^ { 2 } + 6 y ^ { 2 } + 6 z ^ { 2 } = 36$ and the parabolic cylinder $y = x ^ { 2 }$.

Accepted Answer

The answer of Find a vector function that represents the...

Question 11

Find the scalar tangential and normal components of acceleration of a particle with position vector
$$\mathbf { r } ( t ) = e ^ { t } \{ \cos 4 t , \sin 4 t , 0 \}$$

Accepted Answer

The answer of Find the scalar tangential and normal components...

Question 12

$$\text { Find the unit tangent vector } \mathrm { T } ( t ) \text { for } \mathbf { r } ( t ) = 4 t \mathbf { i } + 2 t \mathbf { j } + 4 t \mathbf { k } \text { at } t = 2 \text {. }$$

Accepted Answer

The answer of \[\text { Find the unit tangent vector...

Question 13

Given $\mathbf { r } ( t ) = 7 t ^ { 2 } \mathbf { i } + 4 t ^ { 3 } \mathbf { j }$.
a. Find $\mathbf { r } ( - 1 )$ and $\mathbf { r } ^ { \prime } ( - 1 )$.
b. Sketch the curve defined by $\mathbf { r }$ and the vectors $\mathbf { r } ( - 1 )$ and $\mathbf { r } ^ { t } ( - 1 )$ on the same set of axes.

Accepted Answer

The answer of Given \(\mathbf { r } ( t...

Question 14

Find a vector function that represents the curve of intersection of the two surfaces: the top half of the ellipsoid $x ^ { 2 } + 7 y ^ { 2 } + 7 z ^ { 2 } = 49$ and the parabolic cylinder $y = x ^ { 2 }$.

Accepted Answer

A)  $\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 4 } \mathbf { j } + \sqrt { \frac { 49 - t ^ { 2 } + 7 t } { 7 } } \mathbf { k }$ 
B)  $\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + \sqrt { \frac { 49 + t ^ { 2 } - 7 t ^ { 4 } } { 7 } } \mathbf { k }$ 
C)  $r ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + \sqrt { \frac { 49 - t ^ { 2 } - 7 t } { 7 } } \mathbf { k }$ 
D)  $\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + \sqrt { \frac { 7 - t ^ { 2 } - 7 t ^ { 4 } } { 7 } } \mathbf { k }$ 
E)  $\mathbf { r } ( t ) = t \mathbf { i } - t ^ { 2 } \mathbf { j } - \sqrt { \frac { 49 - t ^ { 2 } - 7 t } { 7 } } \mathbf { k }$ 
A)  $\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 4 } \mathbf { j } + \sqrt { \frac { 49 - t ^ { 2 } + 7 t } { 7 } } \mathbf { k }$ 
B)  $\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + \sqrt { \frac { 49 + t ^ { 2 } - 7 t ^ { 4 } } { 7 } } \mathbf { k }$ 
C)  $r ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + \sqrt { \frac { 49 - t ^ { 2 } - 7 t } { 7 } } \mathbf { k }$ 
D)  $\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + \sqrt { \frac { 7 - t ^ { 2 } - 7 t ^ { 4 } } { 7 } } \mathbf { k }$ 
E)  $\mathbf { r } ( t ) = t \mathbf { i } - t ^ { 2 } \mathbf { j } - \sqrt { \frac { 49 - t ^ { 2 } - 7 t } { 7 } } \mathbf { k }$

Question 15

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

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

$$\text { Find } r ( t ) \text { if } \mathbf { r } ^ { t } ( t ) = t ^ { 10 } \mathbf { i } + 5 t ^ { 4 } \mathbf { j } - t ^ { 7 } \mathbf { k } \text { and } \mathrm { r } ( 0 ) = \mathrm { j }$$

Accepted Answer

The answer of \[\text { Find } r ( t...

Question 18

The torsion of a curve defined by $\mathbf { r } ( t )$ is given by
$$\tau = \frac { \left( \mathbf { r } ^ { t } \times \mathbf { r } ^ { tt } \right) \cdot \mathbf { r } ^ {ttt } } { \left| \mathbf { r } ^ { t } \times \mathbf { r } ^ { t t } \right| ^ { 2 } }$$ Find the torsion of the curve defined by $r ( t ) = \cos 5 t \mathbf { i } + \sin 5 t \mathbf { j } + 4 t \mathbf { k }$.

Accepted Answer

A)  $\frac { 21 } { 41 }$ 
B)  $\frac { 9 } { 41 }$ 
C)  $\frac { 20 } { 41 }$ 
D)  $\frac { 80 } { 41 }$ 
A)  $\frac { 21 } { 41 }$ 
B)  $\frac { 9 } { 41 }$ 
C)  $\frac { 20 } { 41 }$ 
D)  $\frac { 80 } { 41 }$

Question 19

Use Simpson's Rule with $\mathrm { n } = 4$ 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.

Accepted Answer

A)  $14.824$ 
B)  $7.2041$ 
C)  $15.8306$ 
D)  $6.5706$ 
E)  None of these 
A)  $14.824$ 
B)  $7.2041$ 
C)  $15.8306$ 
D)  $6.5706$ 
E)  None of these

Question 20

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

Accepted Answer

The answer of \[\text { A particle moves with position...

Find parametric equations for the tangent line to the curve with parametric equations $x = 3 t$ , $y = 5 t ^ { 2 } , z = 4 t ^ { 3 }$ at the point with $t = 1$ .

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.

Find a vector function that represents the curve of intersection of the two surfaces: The circular cylinder $x ^ { 2 } + y ^ { 2 } = 4$ and the parabolic cylinder $z = x y$ .

Sketch the curve of the vector function $\mathbf { r } ( t ) = \left\langle t ^ { 2 } , t ^ { 3 } , t \right\rangle , t \geq 0$ , and indicate the orientation of the curve.

The curves $\mathbf { r } _ { 1 } ( t ) = \left\langle t , t ^ { 3 } , t ^ { 9 } \right\}$ and $\mathbf { r } _ { 2 } ( t ) = \langle \sin t , \sin 5 t , t \rangle$ intersects at the origin. Find their angle of intersection correct to the nearest degree.

$\text { Find } \mathbf { r } ^ { t } ( t ) \text { and } \mathbf { r } ^ {tt } ( t ) \text { for } \mathbf { r } ( t ) = 10 t \mathbf { i } + 7 t ^ { 2 } \mathbf { j } + 5 t ^ { 3 } \mathbf { k }$

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = 4 \sin t \mathbf { i } + 4 \cos t \mathbf { j } + 3 t \mathbf { k }$

A particle moves with position function $\mathbf { r } ( t ) = \left( 21 t - 7 t ^ { 3 } - 6 \right) \mathbf { i } + 21 t ^ { 2 } \mathbf { j }$ Find the tangential component of the acceleration vector. Select the correct answer.

$\text { If } \mathbf { r } ( t ) = 7 \mathbf { i } + t \cos \pi t \mathbf { j } + 4 \sin \pi t \mathbf { k } \text {, evaluate } \int _ { 0 } ^ { 1 } r ( t ) d t$

Find a vector function that represents the curve of intersection of the two surfaces: the top half of the ellipsoid $x ^ { 2 } + 6 y ^ { 2 } + 6 z ^ { 2 } = 36$ and the parabolic cylinder $y = x ^ { 2 }$ .

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = e ^ { t } \{ \cos 4 t , \sin 4 t , 0 \}$

$\text { Find the unit tangent vector } \mathrm { T } ( t ) \text { for } \mathbf { r } ( t ) = 4 t \mathbf { i } + 2 t \mathbf { j } + 4 t \mathbf { k } \text { at } t = 2 \text {. }$

Find a vector function that represents the curve of intersection of the two surfaces: the top half of the ellipsoid $x ^ { 2 } + 7 y ^ { 2 } + 7 z ^ { 2 } = 49$ and the parabolic cylinder $y = x ^ { 2 }$ .

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

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

$\text { Find } r ( t ) \text { if } \mathbf { r } ^ { t } ( t ) = t ^ { 10 } \mathbf { i } + 5 t ^ { 4 } \mathbf { j } - t ^ { 7 } \mathbf { k } \text { and } \mathrm { r } ( 0 ) = \mathrm { j }$

Use Simpson's Rule with $\mathrm { n } = 4$ 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.

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

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Exam 13: Vector Functions

Find parametric equations for the tangent line to the curve with parametric equations x=3tx = 3 tx=3t , y=5t2,z=4t3y = 5 t ^ { 2 } , z = 4 t ^ { 3 }y=5t2,z=4t3 at the point with t=1t = 1t=1 .

A projectile is fired with an initial speed of 700 m/s700 \mathrm {~m} / \mathrm { s }700 m/s and angle of elevation 60∘60 ^ { \circ }60∘ . Find the range of the projectile.

Find a vector function that represents the curve of intersection of the two surfaces: The circular cylinder x2+y2=4x ^ { 2 } + y ^ { 2 } = 4x2+y2=4 and the parabolic cylinder z=xyz = x yz=xy .

Sketch the curve of the vector function r(t)=⟨t2,t3,t⟩,t≥0\mathbf { r } ( t ) = \left\langle t ^ { 2 } , t ^ { 3 } , t \right\rangle , t \geq 0r(t)=⟨t2,t3,t⟩,t≥0 , and indicate the orientation of the curve.

Find rt(t) and rtt(t) for r(t)=10ti+7t2j+5t3k\text { Find } \mathbf { r } ^ { t } ( t ) \text { and } \mathbf { r } ^ {tt } ( t ) \text { for } \mathbf { r } ( t ) = 10 t \mathbf { i } + 7 t ^ { 2 } \mathbf { j } + 5 t ^ { 3 } \mathbf { k } Find rt(t) and rtt(t) for r(t)=10ti+7t2j+5t3k

Find the scalar tangential and normal components of acceleration of a particle with position vector r(t)=4sin⁡ti+4cos⁡tj+3tk\mathbf { r } ( t ) = 4 \sin t \mathbf { i } + 4 \cos t \mathbf { j } + 3 t \mathbf { k }r(t)=4sinti+4costj+3tk

A particle moves with position function r(t)=(21t−7t3−6)i+21t2j\mathbf { r } ( t ) = \left( 21 t - 7 t ^ { 3 } - 6 \right) \mathbf { i } + 21 t ^ { 2 } \mathbf { j }r(t)=(21t−7t3−6)i+21t2j Find the tangential component of the acceleration vector. Select the correct answer.

If r(t)=7i+tcos⁡πtj+4sin⁡πtk, evaluate ∫01r(t)dt\text { If } \mathbf { r } ( t ) = 7 \mathbf { i } + t \cos \pi t \mathbf { j } + 4 \sin \pi t \mathbf { k } \text {, evaluate } \int _ { 0 } ^ { 1 } r ( t ) d t If r(t)=7i+tcosπtj+4sinπtk, evaluate ∫01​r(t)dt

Find a vector function that represents the curve of intersection of the two surfaces: the top half of the ellipsoid x2+6y2+6z2=36x ^ { 2 } + 6 y ^ { 2 } + 6 z ^ { 2 } = 36x2+6y2+6z2=36 and the parabolic cylinder y=x2y = x ^ { 2 }y=x2 .

Find the scalar tangential and normal components of acceleration of a particle with position vector r(t)=et{cos⁡4t,sin⁡4t,0}\mathbf { r } ( t ) = e ^ { t } \{ \cos 4 t , \sin 4 t , 0 \}r(t)=et{cos4t,sin4t,0}

Find a vector function that represents the curve of intersection of the two surfaces: the top half of the ellipsoid x2+7y2+7z2=49x ^ { 2 } + 7 y ^ { 2 } + 7 z ^ { 2 } = 49x2+7y2+7z2=49 and the parabolic cylinder y=x2y = x ^ { 2 }y=x2 .

Find the length of the curve r(t)=2ti+t2j+ln⁡tk,1≤t≤e3\mathbf { r } ( t ) = 2 t \mathbf { i } + t ^ { 2 } \mathbf { j } + \ln t \mathbf { k } , 1 \leq t \leq e ^ { 3 }r(t)=2ti+t2j+lntk,1≤t≤e3 .

If r(t)=⟨t,t9,t11}\mathbf { r } ( t ) = \left\langle t , t ^ { 9 } , t ^ { 11 } \right\}r(t)=⟨t,t9,t11} , find rtt(t)\mathbf { r } ^ { tt } ( t )rtt(t)

Find r(t) if rt(t)=t10i+5t4j−t7k and r(0)=j\text { Find } r ( t ) \text { if } \mathbf { r } ^ { t } ( t ) = t ^ { 10 } \mathbf { i } + 5 t ^ { 4 } \mathbf { j } - t ^ { 7 } \mathbf { k } \text { and } \mathrm { r } ( 0 ) = \mathrm { j } Find r(t) if rt(t)=t10i+5t4j−t7k and r(0)=j

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Find parametric equations for the tangent line to the curve with parametric equations $x = 3 t$ , $y = 5 t ^ { 2 } , z = 4 t ^ { 3 }$ at the point with $t = 1$ .

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.

Find a vector function that represents the curve of intersection of the two surfaces: The circular cylinder $x ^ { 2 } + y ^ { 2 } = 4$ and the parabolic cylinder $z = x y$ .

Sketch the curve of the vector function $\mathbf { r } ( t ) = \left\langle t ^ { 2 } , t ^ { 3 } , t \right\rangle , t \geq 0$ , and indicate the orientation of the curve.

$\text { Find } \mathbf { r } ^ { t } ( t ) \text { and } \mathbf { r } ^ {tt } ( t ) \text { for } \mathbf { r } ( t ) = 10 t \mathbf { i } + 7 t ^ { 2 } \mathbf { j } + 5 t ^ { 3 } \mathbf { k }$

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = 4 \sin t \mathbf { i } + 4 \cos t \mathbf { j } + 3 t \mathbf { k }$

A particle moves with position function $\mathbf { r } ( t ) = \left( 21 t - 7 t ^ { 3 } - 6 \right) \mathbf { i } + 21 t ^ { 2 } \mathbf { j }$ Find the tangential component of the acceleration vector. Select the correct answer.

$\text { If } \mathbf { r } ( t ) = 7 \mathbf { i } + t \cos \pi t \mathbf { j } + 4 \sin \pi t \mathbf { k } \text {, evaluate } \int _ { 0 } ^ { 1 } r ( t ) d t$

Find a vector function that represents the curve of intersection of the two surfaces: the top half of the ellipsoid $x ^ { 2 } + 6 y ^ { 2 } + 6 z ^ { 2 } = 36$ and the parabolic cylinder $y = x ^ { 2 }$ .

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = e ^ { t } \{ \cos 4 t , \sin 4 t , 0 \}$

Find a vector function that represents the curve of intersection of the two surfaces: the top half of the ellipsoid $x ^ { 2 } + 7 y ^ { 2 } + 7 z ^ { 2 } = 49$ and the parabolic cylinder $y = x ^ { 2 }$ .

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

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

$\text { Find } r ( t ) \text { if } \mathbf { r } ^ { t } ( t ) = t ^ { 10 } \mathbf { i } + 5 t ^ { 4 } \mathbf { j } - t ^ { 7 } \mathbf { k } \text { and } \mathrm { r } ( 0 ) = \mathrm { j }$