Question 1

$$\text { Find the limit } \lim _ { t \rightarrow \infty } \left( e ^ { - 4 t } , \frac { 1 } { t } , \frac { 7 t ^ { 2 } } { t ^ { 2 } + 1 } \right)$$

Accepted Answer

The answer of \[\text { Find the limit } \lim...

Question 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

A)  $7 \mathbf { i } - \frac { 2 } { \pi ^ { 2 } } \mathbf { j } - \frac { 1 } { \pi } \mathbf { k }$ 
B)  $7 \mathbf { i } - \frac { 2 } { \pi ^ { 2 } } \mathbf { j } + \frac { 8 } { \pi } \mathbf { k }$ 
C)  $\mathbf { i } + \frac { 2 } { \pi ^ { 2 } } \mathbf { j } - \frac { 8 } { \pi } \mathbf { k }$ 
D)  $\mathbf { i } - \frac { 2 } { \pi ^ { 2 } } \mathbf { j } + \frac { 1 } { \pi } \mathbf { k }$ 
E)  $7 \mathbf { i } + \frac { 2 } { \pi ^ { 2 } } \mathbf { j } + \frac { 8 } { \pi } \mathbf { k }$ 
A)  $7 \mathbf { i } - \frac { 2 } { \pi ^ { 2 } } \mathbf { j } - \frac { 1 } { \pi } \mathbf { k }$ 
B)  $7 \mathbf { i } - \frac { 2 } { \pi ^ { 2 } } \mathbf { j } + \frac { 8 } { \pi } \mathbf { k }$ 
C)  $\mathbf { i } + \frac { 2 } { \pi ^ { 2 } } \mathbf { j } - \frac { 8 } { \pi } \mathbf { k }$ 
D)  $\mathbf { i } - \frac { 2 } { \pi ^ { 2 } } \mathbf { j } + \frac { 1 } { \pi } \mathbf { k }$ 
E)  $7 \mathbf { i } + \frac { 2 } { \pi ^ { 2 } } \mathbf { j } + \frac { 8 } { \pi } \mathbf { k }$

Question 4

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

Accepted Answer

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

Question 5

Find the following limit. $$\lim _ { t \rightarrow \infty } \left( \arctan t , \mathrm { e } ^ { - 7 t } , \frac { \ln 4 t } { t } \right)$$

Accepted Answer

The answer of Find the following limit. \[\lim _ {...

Question 6

Find the velocity and position vectors of an object with acceleration $\mathbf { a } ( t ) = 4 \mathbf { i } - 72 \mathbf { j } + ( 72 t + 4 ) \mathbf { k }$, initial velocity $\mathbf { v } ( 0 ) = \mathbf { i } + \mathbf { k }$, and initial position $r ( 0 ) = \mathbf { j } + 3 \mathbf { k }$.

Accepted Answer

The answer of Find the velocity and position vectors of...

Question 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

A)  $2 \sqrt { 14 }$ 
B)  $\sqrt { 14 }$ 
C)  $6 \sqrt { 14 }$ 
D)  $3 \sqrt { 14 }$ 
A)  $2 \sqrt { 14 }$ 
B)  $\sqrt { 14 }$ 
C)  $6 \sqrt { 14 }$ 
D)  $3 \sqrt { 14 }$

Question 9

Find the point of intersection of the tangent lines to the curve $\mathbf { r } ( t ) = \langle \sin \pi t , 7 \sin \pi t , \cos \pi t \rangle$, at the points where $t = 0$ and $t = 0.5$.

Accepted Answer

The answer of Find the point of intersection of the...

Question 10

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 11

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

Accepted Answer

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)  $v _ { 0 } \approx 27 \mathrm {~m} / \mathrm { s }$. 
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 12

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

Accepted Answer

A)  $( - 1,0 ) \cup ( 0,2 ]$ 
B)  $( 2 , \infty ]$ 
C)  $[ 2,0 ) \cup ( 0,2 )$ 
D)  $( - \infty , - 1 )$ 
E)  $( - 1,2 ]$ 
A)  $( - 1,0 ) \cup ( 0,2 ]$ 
B)  $( 2 , \infty ]$ 
C)  $[ 2,0 ) \cup ( 0,2 )$ 
D)  $( - \infty , - 1 )$ 
E)  $( - 1,2 ]$

Question 13

Find $\mathbf { r } ( t )$ satisfying the conditions for $\mathbf { r } ^ { \prime } ( t ) = 6 e ^ { 6 t } \mathbf { i } + 7 e ^ { - t } \mathbf { j } + e ^ { t } \mathbf { k } , \quad \mathbf { r } ( 0 ) = \mathbf { i } - \mathbf { j } + 5 \mathbf { k }$

Accepted Answer

A)  $6 e ^ { 6 t } \mathbf { i } - \left( 7 e ^ { - t } + 6 \right) \mathbf { j } + \left( e ^ { t } + 4 \right) \mathbf { k }$ 
B)  $e ^ { 6 t } \mathbf { i } - \left( 7 e ^ { - t } - 6 \right) \mathbf { j } + \left( e ^ { t } + 4 \right) \mathbf { k }$ 
C)  $\left( 6 e ^ { 6 t } + 1 \right) \mathbf { i } - \left( 7 e ^ { - t } - 1 \right) \mathbf { j } + \left( e ^ { t } + 5 \right) \mathbf { k }$ 
D)  $\left( e ^ { 6 t } + 1 \right) \mathbf { i } - \left( 7 e ^ { - t } + 1 \right) \mathbf { j } + \left( e ^ { t } + 5 \right) \mathbf { k }$ 
A)  $6 e ^ { 6 t } \mathbf { i } - \left( 7 e ^ { - t } + 6 \right) \mathbf { j } + \left( e ^ { t } + 4 \right) \mathbf { k }$ 
B)  $e ^ { 6 t } \mathbf { i } - \left( 7 e ^ { - t } - 6 \right) \mathbf { j } + \left( e ^ { t } + 4 \right) \mathbf { k }$ 
C)  $\left( 6 e ^ { 6 t } + 1 \right) \mathbf { i } - \left( 7 e ^ { - t } - 1 \right) \mathbf { j } + \left( e ^ { t } + 5 \right) \mathbf { k }$ 
D)  $\left( e ^ { 6 t } + 1 \right) \mathbf { i } - \left( 7 e ^ { - t } + 1 \right) \mathbf { j } + \left( e ^ { t } + 5 \right) \mathbf { k }$

Question 14

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)  $\mathbf { r } ( t ) = 2 \cos ( t ) \mathbf { i } + 2 \sin ( t ) \mathbf { j } + 4 \sin ( t ) \cos ( t ) \mathbf { k }$

Question 15

A particle moves with position function $\mathbf { r } ( t ) = 4 \sqrt { 2 } t \mathbf { i } + e ^ { 4 t } \mathbf { j } + e ^ { - 4 t } \mathbf { k }$. Find the acceleration of the particle.

Accepted Answer

The answer of A particle moves with position function \(\mathbf...

Question 16

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

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 17

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

Accepted Answer

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)  $| \mathbf { v } ( t ) | = \sqrt { 1 + 100 t + t ^ { 9 } }$ 
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 18

Find the velocity of a particle with the given position function. $$\mathbf { r } ( t ) = 19 e ^ { 5 t } \mathbf { i } + 5 e ^ { - 21 t } \mathbf { j }$$

Accepted Answer

A)  $\mathbf { v } ( t ) = 19 e ^ { 5 t } \mathbf { i } + e ^ { - 21 t } \mathbf { j }$ 
B)  $\mathbf { v } ( t ) = 95 e ^ { 5 t } \mathbf { i } - 105 e ^ { - 21 t } \mathbf { j }$ 
C)  $\mathbf { v } ( t ) = 19 e ^ { 5 t } \mathbf { i } - 105 e ^ { - 21 t } \mathbf { j }$ 
D)  $\mathbf { v } ( t ) = e ^ { 5 t } \mathbf { i } + 105 e ^ { - 21 t } \mathbf { j }$ 
E)  $\mathbf { v } ( t ) = 95 e ^ { 5 t } \mathbf { i } + 105 e ^ { - 21 t } \mathbf { j }$ 
A)  $\mathbf { v } ( t ) = 19 e ^ { 5 t } \mathbf { i } + e ^ { - 21 t } \mathbf { j }$ 
B)  $\mathbf { v } ( t ) = 95 e ^ { 5 t } \mathbf { i } - 105 e ^ { - 21 t } \mathbf { j }$ 
C)  $\mathbf { v } ( t ) = 19 e ^ { 5 t } \mathbf { i } - 105 e ^ { - 21 t } \mathbf { j }$ 
D)  $\mathbf { v } ( t ) = e ^ { 5 t } \mathbf { i } + 105 e ^ { - 21 t } \mathbf { j }$ 
E)  $\mathbf { v } ( t ) = 95 e ^ { 5 t } \mathbf { i } + 105 e ^ { - 21 t } \mathbf { j }$

$\text { Find the limit } \lim _ { t \rightarrow \infty } \left( e ^ { - 4 t } , \frac { 1 } { t } , \frac { 7 t ^ { 2 } } { t ^ { 2 } + 1 } \right)$

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

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

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

Find the following limit. $\lim _ { t \rightarrow \infty } \left( \arctan t , \mathrm { e } ^ { - 7 t } , \frac { \ln 4 t } { t } \right)$

Find the velocity and position vectors of an object with acceleration $\mathbf { a } ( t ) = 4 \mathbf { i } - 72 \mathbf { j } + ( 72 t + 4 ) \mathbf { k }$ , initial velocity $\mathbf { v } ( 0 ) = \mathbf { i } + \mathbf { k }$ , and initial position $r ( 0 ) = \mathbf { j } + 3 \mathbf { k }$ .

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

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

Find the point of intersection of the tangent lines to the curve $\mathbf { r } ( t ) = \langle \sin \pi t , 7 \sin \pi t , \cos \pi t \rangle$ , at the points where $t = 0$ and $t = 0.5$ .

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

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

Find $\mathbf { r } ( t )$ satisfying the conditions for $\mathbf { r } ^ { \prime } ( t ) = 6 e ^ { 6 t } \mathbf { i } + 7 e ^ { - t } \mathbf { j } + e ^ { t } \mathbf { k } , \quad \mathbf { r } ( 0 ) = \mathbf { i } - \mathbf { j } + 5 \mathbf { k }$

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 $z = x y$ .

A particle moves with position function $\mathbf { r } ( t ) = 4 \sqrt { 2 } t \mathbf { i } + e ^ { 4 t } \mathbf { j } + e ^ { - 4 t } \mathbf { k }$ . Find the acceleration of the particle.

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

Find the velocity of a particle with the given position function. $\mathbf { r } ( t ) = 19 e ^ { 5 t } \mathbf { i } + 5 e ^ { - 21 t } \mathbf { j }$

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Exam 13: Vector Functions

Find the limit lim⁡t→∞(e−4t,1t,7t2t2+1)\text { Find the limit } \lim _ { t \rightarrow \infty } \left( e ^ { - 4 t } , \frac { 1 } { t } , \frac { 7 t ^ { 2 } } { t ^ { 2 } + 1 } \right) Find the limit limt→∞​(e−4t,t1​,t2+17t2​)

If r(t)=⟨t,t9,t11⟩\mathbf { r } ( t ) = \left\langle t , t ^ { 9 } , t ^ { 11 } \right\rangler(t)=⟨t,t9,t11⟩ , find r′′(t)\mathbf { r } ^ { \prime \prime } ( t )r′′(t)

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

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 ) = \mathbf { j } Find r(t) if rt(t)=t10i+5t4j−t7k and r(0)=j

Find the following limit. lim⁡t→∞(arctan⁡t,e−7t,ln⁡4tt)\lim _ { t \rightarrow \infty } \left( \arctan t , \mathrm { e } ^ { - 7 t } , \frac { \ln 4 t } { t } \right)limt→∞​(arctant,e−7t,tln4t​)

Find the length of the curve r(t)=−3ti+2tj−tk,−2≤t≤1\mathbf { r } ( t ) = - 3 t \mathbf { i } + 2 t \mathbf { j } - t \mathbf { k } , - 2 \leq t \leq 1r(t)=−3ti+2tj−tk,−2≤t≤1 .

Find the point of intersection of the tangent lines to the curve r(t)=⟨sin⁡πt,7sin⁡πt,cos⁡πt⟩\mathbf { r } ( t ) = \langle \sin \pi t , 7 \sin \pi t , \cos \pi t \rangler(t)=⟨sinπt,7sinπt,cosπt⟩ , at the points where t=0t = 0t=0 and t=0.5t = 0.5t=0.5 .

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 .

A ball is thrown at an angle of 45∘45 ^ { \circ }45∘ to the ground. If the ball lands 30 m30 \mathrm {~m}30 m away, what was the initial speed of the ball? Let g=9.82 m/sg = 9.82 \mathrm {~m} / \mathrm { s }g=9.82 m/s .

Let r(t)={2−t,(et−2)t,ln⁡(t+1)}\mathbf { r } ( t ) = \left\{ \sqrt { 2 - t } , \frac { \left( e ^ { t } - 2 \right) } { t } , \ln ( t + 1 ) \right\}r(t)={2−t​,t(et−2)​,ln(t+1)} Find the domain of rrr .

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 .

A particle moves with position function r(t)=42ti+e4tj+e−4tk\mathbf { r } ( t ) = 4 \sqrt { 2 } t \mathbf { i } + e ^ { 4 t } \mathbf { j } + e ^ { - 4 t } \mathbf { k }r(t)=42​ti+e4tj+e−4tk . Find the acceleration of the particle.

Find the speed of a particle with the given position function. r(t)=ti+5t2j+3t6k\mathrm { r } ( t ) = t \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 3 t ^ { 6 } \mathbf { k }r(t)=ti+5t2j+3t6k

Find the velocity of a particle with the given position function. r(t)=19e5ti+5e−21tj\mathbf { r } ( t ) = 19 e ^ { 5 t } \mathbf { i } + 5 e ^ { - 21 t } \mathbf { j }r(t)=19e5ti+5e−21tj

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$\text { Find the limit } \lim _ { t \rightarrow \infty } \left( e ^ { - 4 t } , \frac { 1 } { t } , \frac { 7 t ^ { 2 } } { t ^ { 2 } + 1 } \right)$

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

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

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

Find the following limit. $\lim _ { t \rightarrow \infty } \left( \arctan t , \mathrm { e } ^ { - 7 t } , \frac { \ln 4 t } { t } \right)$

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

Find the point of intersection of the tangent lines to the curve $\mathbf { r } ( t ) = \langle \sin \pi t , 7 \sin \pi t , \cos \pi t \rangle$ , at the points where $t = 0$ and $t = 0.5$ .

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

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

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 $z = x y$ .

A particle moves with position function $\mathbf { r } ( t ) = 4 \sqrt { 2 } t \mathbf { i } + e ^ { 4 t } \mathbf { j } + e ^ { - 4 t } \mathbf { k }$ . Find the acceleration of the particle.

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

Find the velocity of a particle with the given position function. $\mathbf { r } ( t ) = 19 e ^ { 5 t } \mathbf { i } + 5 e ^ { - 21 t } \mathbf { j }$