Question 1

Find the integral $\int \left( 4 t \mathbf { i } + 6 t ^ { 2 } \mathbf { j } + 4 \mathbf { k } \right) d t$

Accepted Answer

A)  $4 \mathbf { i } + 12 t \mathbf { j } + \mathbf { C }$ 
B)  $4 t \mathbf { i } + 6 t ^ { 2 } \mathbf { j } + 4 \mathbf { k } + \mathbf { C }$ 
C)  $2 t ^ { 2 } \mathbf { i } + 2 t ^ { 3 } \mathbf { j } + 4 t \mathbf { k } + \mathbf { C }$ 
D)  $4 t ^ { 2 } \mathbf { i } + 6 t ^ { 3 } \mathbf { j } + 4 t \mathbf { k } + \mathbf { C }$ 
A)  $4 \mathbf { i } + 12 t \mathbf { j } + \mathbf { C }$ 
B)  $4 t \mathbf { i } + 6 t ^ { 2 } \mathbf { j } + 4 \mathbf { k } + \mathbf { C }$ 
C)  $2 t ^ { 2 } \mathbf { i } + 2 t ^ { 3 } \mathbf { j } + 4 t \mathbf { k } + \mathbf { C }$ 
D)  $4 t ^ { 2 } \mathbf { i } + 6 t ^ { 3 } \mathbf { j } + 4 t \mathbf { k } + \mathbf { C }$

Question 2

Sketch the curve of the vector function $\mathbf { r } ( t ) = 3 t \mathbf { i } + ( 3 t + 8 ) \mathbf { j } , - 1 \leq t \leq 2$, and indicate the orientation of the curve.

Accepted Answer

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

Question 3

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 [1, 2].

Accepted Answer

A)  $\mathbf { v } = 21.1 \mathbf { i } + 24.6 \mathbf { j } + 26.6 \mathbf { k }$ 
B)  $\quad \mathbf { v } = 21.1 \mathbf { i } + 26.6 \mathbf { j } + 21.1 \mathbf { k }$ 
C)  $\quad \mathbf { v } = 24.6 \mathbf { i } + 24.6 \mathbf { j } + 21.1 \mathbf { k }$ 
D)  $\mathbf { v } = 26.6 \mathbf { i } + 24.6 \mathbf { j } + 21.1 \mathbf { k }$ 
E)  $\mathbf { v } = 21.1 \mathbf { i } + 26.6 \mathbf { j } + 26.6 \mathbf { k }$ 
A)  $\mathbf { v } = 21.1 \mathbf { i } + 24.6 \mathbf { j } + 26.6 \mathbf { k }$ 
B)  $\quad \mathbf { v } = 21.1 \mathbf { i } + 26.6 \mathbf { j } + 21.1 \mathbf { k }$ 
C)  $\quad \mathbf { v } = 24.6 \mathbf { i } + 24.6 \mathbf { j } + 21.1 \mathbf { k }$ 
D)  $\mathbf { v } = 26.6 \mathbf { i } + 24.6 \mathbf { j } + 21.1 \mathbf { k }$ 
E)  $\mathbf { v } = 21.1 \mathbf { i } + 26.6 \mathbf { j } + 26.6 \mathbf { k }$

Question 4

The helix $\mathbf { r } _ { 1 } ( t ) = 4 \cos t \mathbf { i } + \sin t \mathbf { j } + t \mathbf { k }$ intersects the curve $\mathbf { r } _ { 2 } ( t ) = ( 4 + t ) \mathbf { i } + 6 t ^ { 2 } \mathbf { j } + 5 t ^ { 3 } \mathbf { k }$ at the point $( 4,0,0 )$. Find the angle of intersection.

Accepted Answer

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

Question 5

Find the velocity of a particle that has the given acceleration and the given initial velocity.
$$\mathbf { a } ( t ) = 3 \mathrm { k } , \mathbf { v } ( 0 ) = 18 \mathbf { i } - 7 \mathbf { j }$$

Accepted Answer

The answer of Find the velocity of a particle that...

Question 6

Sketch the curve of the vector function $\mathbf { r } ( t ) = 5 \sin t \mathbf { i } + 4 \cos t \mathbf { j }$, and indicate the orientation of the curve.

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

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

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 $\mathbf { r } ( t ) = \cos 2 t \mathbf { i } + \sin 2 t \mathbf { j } + 5 t \mathbf { k }$.

Accepted Answer

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

Question 10

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 $\mathbf { r } ( 0 ) = \mathbf { j } + 3 \mathbf { k }$.

Accepted Answer

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

Question 11

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.

Accepted Answer

A)  $a _ { T } = 5 t$ 
B)  $a _ { T } = 542 t$ 
C)  $a _ { T } = 42 t + 5$ 
D)  $a _ { T } = - 55 t$ 
E)  $a _ { T } = 42 t$ 
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 12

A ball is thrown at an angle of $45 ^ { \circ }$ to the ground. If the ball lands $90 \mathrm {~m}$ away, what was the initial speed of the ball? Let $g = 9.82 \mathrm {~m} / \mathrm { s }$.

Accepted Answer

The answer of A ball is thrown at an angle...

Question 13

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

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

$$\text { Find an expression for } \frac { d } { d t } [ x ( t ) . ( y ( t ) \times z ( t ) ) ] \text {. }$$

Accepted Answer

The answer of \[\text { Find an expression for }...

Question 15

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) $\frac { 2 } { \pi ^ { 2 } } \mathbf { j } + \frac { 1 } { \pi } \mathbf { k }$ 
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) $\frac { 2 } { \pi ^ { 2 } } \mathbf { j } + \frac { 1 } { \pi } \mathbf { k }$ 
E)  $7 \mathbf { i } + \frac { 2 } { \pi ^ { 2 } } \mathbf { j } + \frac { 8 } { \pi } \mathbf { k }$

Question 16

The torsion of a curve defined by $r ( t )$ 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 $\mathbf { r } ( t ) = \cos 7 t \mathbf { i } + \sin 7 t \mathbf { j } + 4 t \mathbf { k }$.

Accepted Answer

The answer of The torsion of a curve defined by...

Question 17

Find $\mathbf { r } ^ { \prime \prime } ( t )$ for the function given.
$$\mathbf { r } ( t ) = 8 \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k }$$

Accepted Answer

The answer of Find \(\mathbf { r } ^ {...

Question 18

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

Accepted Answer

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

Question 19

$$\text { Find } \mathbf { r } ^ { t } ( t ) \text { and } \mathbf { r } ^ {tt } ( t ) \text { for } \mathbf { r } ( t ) = \langle t \cos 9 t - \sin 9 t , t \sin 10 t + \cos 10 t \rangle$$

Accepted Answer

\[\begin{array} { l } 
\mathbf { r } ^ {

Question 20

Find the derivative of the vector function.
$$\mathbf { r } ( t ) = a + t b + t ^ { 2 } c$$

Accepted Answer

The answer of Find the derivative of the vector function.
\[\mathbf...

Find the integral $\int \left( 4 t \mathbf { i } + 6 t ^ { 2 } \mathbf { j } + 4 \mathbf { k } \right) d t$

Sketch the curve of the vector function $\mathbf { r } ( t ) = 3 t \mathbf { i } + ( 3 t + 8 ) \mathbf { j } , - 1 \leq t \leq 2$ , and indicate the orientation of the curve.

The following table gives coordinates of a particle moving through space along a smooth curve. x y z 0.5 5.8 9.1 4.3 1 12.6 14.9 16.8 1.5 25.6 21.2 29.4 2 39.2 39.5 37.9 2.5 42.4 42.4 43 Find the average velocity over the time interval [1, 2].

The helix $\mathbf { r } _ { 1 } ( t ) = 4 \cos t \mathbf { i } + \sin t \mathbf { j } + t \mathbf { k }$ intersects the curve $\mathbf { r } _ { 2 } ( t ) = ( 4 + t ) \mathbf { i } + 6 t ^ { 2 } \mathbf { j } + 5 t ^ { 3 } \mathbf { k }$ at the point $( 4,0,0 )$ . Find the angle of intersection.

Find the velocity of a particle that has the given acceleration and the given initial velocity. $\mathbf { a } ( t ) = 3 \mathrm { k } , \mathbf { v } ( 0 ) = 18 \mathbf { i } - 7 \mathbf { j }$

Sketch the curve of the vector function $\mathbf { r } ( t ) = 5 \sin t \mathbf { i } + 4 \cos t \mathbf { j }$ , and indicate the orientation of the curve.

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

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 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 ball is thrown at an angle of $45 ^ { \circ }$ to the ground. If the ball lands $90 \mathrm {~m}$ away, what was the initial speed of the ball? Let $g = 9.82 \mathrm {~m} / \mathrm { s }$ .

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

$\text { Find an expression for } \frac { d } { d t } [ x ( t ) . ( y ( t ) \times z ( t ) ) ] \text {. }$

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

Find $\mathbf { r } ^ { \prime \prime } ( t )$ for the function given. $\mathbf { r } ( t ) = 8 \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k }$

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

$\text { Find } \mathbf { r } ^ { t } ( t ) \text { and } \mathbf { r } ^ {tt } ( t ) \text { for } \mathbf { r } ( t ) = \langle t \cos 9 t - \sin 9 t , t \sin 10 t + \cos 10 t \rangle$

Find the derivative of the vector function. $\mathbf { r } ( t ) = a + t b + t ^ { 2 } c$

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Exam 13: Vector Functions

Find the integral ∫(4ti+6t2j+4k)dt\int \left( 4 t \mathbf { i } + 6 t ^ { 2 } \mathbf { j } + 4 \mathbf { k } \right) d t∫(4ti+6t2j+4k)dt

Sketch the curve of the vector function r(t)=3ti+(3t+8)j,−1≤t≤2\mathbf { r } ( t ) = 3 t \mathbf { i } + ( 3 t + 8 ) \mathbf { j } , - 1 \leq t \leq 2r(t)=3ti+(3t+8)j,−1≤t≤2 , and indicate the orientation of the curve.

The following table gives coordinates of a particle moving through space along a smooth curve. x y z 0.5 5.8 9.1 4.3 1 12.6 14.9 16.8 1.5 25.6 21.2 29.4 2 39.2 39.5 37.9 2.5 42.4 42.4 43 Find the average velocity over the time interval [1, 2].

Find the velocity of a particle that has the given acceleration and the given initial velocity. a(t)=3k,v(0)=18i−7j\mathbf { a } ( t ) = 3 \mathrm { k } , \mathbf { v } ( 0 ) = 18 \mathbf { i } - 7 \mathbf { j }a(t)=3k,v(0)=18i−7j

Sketch the curve of the vector function r(t)=5sin⁡ti+4cos⁡tj\mathbf { r } ( t ) = 5 \sin t \mathbf { i } + 4 \cos t \mathbf { j }r(t)=5sinti+4costj , and indicate the orientation of the curve.

Find the curvature of the curve r(t)=3sin⁡4ti+3cos⁡4tj+3tkr ( t ) = 3 \sin 4 t \mathbf { i } + 3 \cos 4 t \mathbf { j } + 3 t \mathbf { k }r(t)=3sin4ti+3cos4tj+3tk .

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

A particle moves with position function r(t)=(21t−7t3−5)i+21t2j\mathbf { r } ( t ) = \left( 21 t - 7 t ^ { 3 } - 5 \right) \mathbf { i } + 21 t ^ { 2 } \mathbf { j }r(t)=(21t−7t3−5)i+21t2j . Find the tangential component of the acceleration vector.

A ball is thrown at an angle of 45∘45 ^ { \circ }45∘ to the ground. If the ball lands 90 m90 \mathrm {~m}90 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 .

Find an expression for ddt[x(t).(y(t)×z(t))]. \text { Find an expression for } \frac { d } { d t } [ x ( t ) . ( y ( t ) \times z ( t ) ) ] \text {. } Find an expression for dtd​[x(t).(y(t)×z(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)\mathbf { r } ^ { \prime \prime } ( t )r′′(t) for the function given. r(t)=8i+sin⁡tj+cos⁡tk\mathbf { r } ( t ) = 8 \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k }r(t)=8i+sintj+costk

The position function of a particle is given by r(t)=⟨5t2,5t,5t2−100t}\mathbf { r } ( t ) = \left\langle 5 t ^ { 2 } , 5 t , 5 t ^ { 2 } - 100 t \right\}r(t)=⟨5t2,5t,5t2−100t} When is the speed a minimum?

Find the derivative of the vector function. r(t)=a+tb+t2c\mathbf { r } ( t ) = a + t b + t ^ { 2 } cr(t)=a+tb+t2c

Functions and Models

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Find the integral $\int \left( 4 t \mathbf { i } + 6 t ^ { 2 } \mathbf { j } + 4 \mathbf { k } \right) d t$

Sketch the curve of the vector function $\mathbf { r } ( t ) = 3 t \mathbf { i } + ( 3 t + 8 ) \mathbf { j } , - 1 \leq t \leq 2$ , and indicate the orientation of the curve.

Find the velocity of a particle that has the given acceleration and the given initial velocity. $\mathbf { a } ( t ) = 3 \mathrm { k } , \mathbf { v } ( 0 ) = 18 \mathbf { i } - 7 \mathbf { j }$

Sketch the curve of the vector function $\mathbf { r } ( t ) = 5 \sin t \mathbf { i } + 4 \cos t \mathbf { j }$ , and indicate the orientation of the curve.

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

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 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 ball is thrown at an angle of $45 ^ { \circ }$ to the ground. If the ball lands $90 \mathrm {~m}$ away, what was the initial speed of the ball? Let $g = 9.82 \mathrm {~m} / \mathrm { s }$ .

$\text { Find an expression for } \frac { d } { d t } [ x ( t ) . ( y ( t ) \times z ( t ) ) ] \text {. }$

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

Find $\mathbf { r } ^ { \prime \prime } ( t )$ for the function given. $\mathbf { r } ( t ) = 8 \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k }$

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

Find the derivative of the vector function. $\mathbf { r } ( t ) = a + t b + t ^ { 2 } c$