Question 1

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

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 2

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

Question 3

$$\text { Find equations of the normal plane to } x = t , y = t ^ { 2 } , z = t ^ { 3 } \text { at the point } ( 2,4,8 ) \text {. }$$

Accepted Answer

The answer of \[\text { Find equations of the normal...

Question 4

The curves $\mathbf { r } _ { 1 } ( t ) = \left\langle t , t ^ { 3 } , t ^ { 9 } \right\rangle$ 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 5

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....

Question 6

Find the position vector of a particle that has the given acceleration and the given initial velocity and position. $$\mathbf { a } ( t ) = - 4 \mathbf { k } , \mathbf { v } ( 0 ) = \mathbf { i } + \mathbf { j } - 20 \mathbf { k } , \mathbf { r } ( 0 ) = 4 \mathbf { i } + 9 \mathbf { j }$$

Accepted Answer

The answer of Find the position vector of a particle...

Question 7

The curvature of the curve given by the vector function $r$ is
$\mathrm { k } ( t ) = \frac { \left| \mathbf { r } ^ { t } ( t ) \times \mathbf { r } ^ { tt } ( t ) \right| } { \left| \mathbf { r } ^ { t } ( t ) \right| ^ { 3 } }$
Use the formula to find the curvature of $\mathbf { r } ( t ) = \left( \sqrt { 19 } t , e ^ { t } , e ^ { - t } \right\}$ at the point $( 0,1,1 )$.

Accepted Answer

A)  $\sqrt { 21 }$ 
B)  $\frac { \sqrt { 2 } } { 21 }$ 
C)  $21 \sqrt { 2 }$ 
D)  $\frac { 21 } { \sqrt { 2 } }$ 
E)  $\frac { \sqrt { 21 } } { 21 }$ 
A)  $\sqrt { 21 }$ 
B)  $\frac { \sqrt { 2 } } { 21 }$ 
C)  $21 \sqrt { 2 }$ 
D)  $\frac { 21 } { \sqrt { 2 } }$ 
E)  $\frac { \sqrt { 21 } } { 21 }$

Question 8

At what point on the curve $x = t ^ { 3 } , y = 9 t , z = t ^ { 4 }$ is the normal plane parallel to the plane $3 x + 9 y - 4 z = 4 ?$

Accepted Answer

A)  $( - 1 , - 3,9 )$ 
B)  $( 9,18 , - 2 )$ 
C)  $( - 1 , - 9,1 )$ 
D)  $( - 18,9,1 )$ 
E)  $( - 9,1,1 )$ 
A)  $( - 1 , - 3,9 )$ 
B)  $( 9,18 , - 2 )$ 
C)  $( - 1 , - 9,1 )$ 
D)  $( - 18,9,1 )$ 
E)  $( - 9,1,1 )$

Question 9

Find the unit tangent and unit normal vectors $\mathrm { T } ( t )$ and $\mathrm { N } ( t )$ for the curve $C$ defined by $\mathbf { r } ( t ) = t \mathbf { i } + 2 t ^ { 2 } \mathbf { j }$. Sketch the graph of $C$, and show $\mathrm { T } ( t )$ and $\mathrm { N } ( t )$ for $t = 1$.

Accepted Answer

The answer of Find the unit tangent and unit normal...

Question 10

Find the limit $\lim _ { t \rightarrow 0 } \left[ \left( t ^ { 2 } + 3 \right) \mathbf { i } + \cos 5 t \mathbf { j } - 6 \mathbf { k } \right]$

Accepted Answer

A)  $3 i - 6 \mathbf { k }$ 
B)  $4 \mathbf { i } + \mathbf { j }$ 
C)  $3 i + j - 6 \mathbf { k }$ 
D)  $4 \mathbf { i }$ 
A)  $3 i - 6 \mathbf { k }$ 
B)  $4 \mathbf { i } + \mathbf { j }$ 
C)  $3 i + j - 6 \mathbf { k }$ 
D)  $4 \mathbf { i }$

Question 11

If $\mathbf { r } ( t ) = \left( t , t ^ { 5 } , t ^ { \imath } \right\rangle$, find $\mathbf { r } ^ { tt } ( t )$

Accepted Answer

A)  $\left( 0,42 t ^ { 4 } , 20 t ^ { 2 } \right\rangle$ 
B)  $\left( 0,42 t ^ { 5 } , 20 t ^ { 3 } \right)$ 
C)  $\left( 0,20 t ^ { 3 } , 42 t ^ { 5 } \right)$ 
D)  $\left\langle 1,5 t ^ { 6 } , 7 t ^ { 8 } \right\}$ 
E)  $\left( 0,20 t ^ { 2 } , 42 t ^ { 4 } \right)$ 
A)  $\left( 0,42 t ^ { 4 } , 20 t ^ { 2 } \right\rangle$ 
B)  $\left( 0,42 t ^ { 5 } , 20 t ^ { 3 } \right)$ 
C)  $\left( 0,20 t ^ { 3 } , 42 t ^ { 5 } \right)$ 
D)  $\left\langle 1,5 t ^ { 6 } , 7 t ^ { 8 } \right\}$ 
E)  $\left( 0,20 t ^ { 2 } , 42 t ^ { 4 } \right)$

Question 12

Find the domain of the vector function $\mathbf { r } ( t ) = \left\langle 8 \sqrt { t } , \frac { 1 } { t - 9 } , \ln t \right\rangle$.

Accepted Answer

A)  $( 0,9 ) \cup ( 9 , \infty )$ 
B)  $[ 0,8 ) \cup ( 8,9 ) \cup ( 9 , \infty )$ 
C)  $( 0,8 ) \cup ( 8 , \infty )$ 
D)  $( 0,8 ) \cup ( 8,9 ) \cup ( 9 , \infty )$ 
A)  $( 0,9 ) \cup ( 9 , \infty )$ 
B)  $[ 0,8 ) \cup ( 8,9 ) \cup ( 9 , \infty )$ 
C)  $( 0,8 ) \cup ( 8 , \infty )$ 
D)  $( 0,8 ) \cup ( 8,9 ) \cup ( 9 , \infty )$

Question 13

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 14

$$\text { Find the integral } \int \left( \sin 7 t \mathbf { i } + \cos 7 t \mathbf { j } + e ^ { - t / 5 } \mathbf { k } \right) d t \text {. }$$

Accepted Answer

The answer of \[\text { Find the integral } \int...

Question 15

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) 
$\mathrm { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + \sqrt { \frac { 49 - t ^ { 2 } - 7 t } { 7 } } \mathbf { k }$ 
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) 
$\mathrm { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + \sqrt { \frac { 49 - t ^ { 2 } - 7 t } { 7 } } \mathbf { k }$ 
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 16

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)  $x = 3 + 3 t , y = 5 + 5 t , z = 4 + 4 t$

Question 17

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

Accepted Answer

A)  $\mathrm { r } ^ { tt } ( t ) = - 8 \sin t \mathrm { j } - \cos t \mathrm { k }$ 
B)  $\mathrm { r } ^ { tt} ( t ) = - \sin t \mathrm { j } - \cos t \mathrm { k }$ 
C)  $\mathrm { r } ^ { tt } ( t ) = \cos t \mathrm { j } - \sin t \mathrm { k }$ 
D)  $\mathrm { r } ^ { tt } ( t ) = - \sin t \mathrm { j } + 8 \cos t \mathrm { k }$ 
E)  $\mathrm { r } ^ { tt } ( t ) = - 8 \cos t \mathrm { j } - \sin t \mathrm { k }$ 
A)  $\mathrm { r } ^ { tt } ( t ) = - 8 \sin t \mathrm { j } - \cos t \mathrm { k }$ 
B)  $\mathrm { r } ^ { tt} ( t ) = - \sin t \mathrm { j } - \cos t \mathrm { k }$ 
C)  $\mathrm { r } ^ { tt } ( t ) = \cos t \mathrm { j } - \sin t \mathrm { k }$ 
D)  $\mathrm { r } ^ { tt } ( t ) = - \sin t \mathrm { j } + 8 \cos t \mathrm { k }$ 
E)  $\mathrm { r } ^ { tt } ( t ) = - 8 \cos t \mathrm { j } - \sin t \mathrm { k }$

Question 18

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 19

Find the unit tangent vector for the curve given by $\mathbf { r } ( t ) = \left\langle \frac { 1 } { 5 } t ^ { 5 } , \frac { 1 } { 3 } t ^ { 3 } , t \right\rangle$.

Accepted Answer

A)  $\frac { \left\langle t ^ { 4 } , t ^ { 2 } , 1 \right\} } { \sqrt { t ^ { 10 } + t ^ { 4 } } }$ 
B)  $\frac { \left\langle t ^ { 4 } , t ^ { 2 } , 1 \right) } { \sqrt { 4 t ^ { 8 } + 4 t ^ { 4 } + 1 } }$ 
C) 
$\frac { \left\langle t ^ { 4 } , t ^ { 2 } , 1 \right) } { \sqrt { t ^ { 8 } + t ^ { 4 } + 1 } }$ 
D) 
$\frac { \left\langle t ^ { 4 } , t ^ { 2 } , 1 \right\rangle } { \sqrt { t ^ { 8 } + t ^ { 4 } } }$ 
E)  None of these 
A)  $\frac { \left\langle t ^ { 4 } , t ^ { 2 } , 1 \right\} } { \sqrt { t ^ { 10 } + t ^ { 4 } } }$ 
B)  $\frac { \left\langle t ^ { 4 } , t ^ { 2 } , 1 \right) } { \sqrt { 4 t ^ { 8 } + 4 t ^ { 4 } + 1 } }$ 
C) 
$\frac { \left\langle t ^ { 4 } , t ^ { 2 } , 1 \right) } { \sqrt { t ^ { 8 } + t ^ { 4 } + 1 } }$ 
D) 
$\frac { \left\langle t ^ { 4 } , t ^ { 2 } , 1 \right\rangle } { \sqrt { t ^ { 8 } + t ^ { 4 } } }$ 
E)  None of these

Question 20

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

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

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

$\text { Find equations of the normal plane to } x = t , y = t ^ { 2 } , z = t ^ { 3 } \text { at the point } ( 2,4,8 ) \text {. }$

The curves $\mathbf { r } _ { 1 } ( t ) = \left\langle t , t ^ { 3 } , t ^ { 9 } \right\rangle$ 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.

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

Find the position vector of a particle that has the given acceleration and the given initial velocity and position. $\mathbf { a } ( t ) = - 4 \mathbf { k } , \mathbf { v } ( 0 ) = \mathbf { i } + \mathbf { j } - 20 \mathbf { k } , \mathbf { r } ( 0 ) = 4 \mathbf { i } + 9 \mathbf { j }$

At what point on the curve $x = t ^ { 3 } , y = 9 t , z = t ^ { 4 }$ is the normal plane parallel to the plane $3 x + 9 y - 4 z = 4 ?$

Find the unit tangent and unit normal vectors $\mathrm { T } ( t )$ and $\mathrm { N } ( t )$ for the curve $C$ defined by $\mathbf { r } ( t ) = t \mathbf { i } + 2 t ^ { 2 } \mathbf { j }$ . Sketch the graph of $C$ , and show $\mathrm { T } ( t )$ and $\mathrm { N } ( t )$ for $t = 1$ .

Find the limit $\lim _ { t \rightarrow 0 } \left[ \left( t ^ { 2 } + 3 \right) \mathbf { i } + \cos 5 t \mathbf { j } - 6 \mathbf { k } \right]$

If $\mathbf { r } ( t ) = \left( t , t ^ { 5 } , t ^ { \imath } \right\rangle$ , find $\mathbf { r } ^ { tt } ( t )$

Find the domain of the vector function $\mathbf { r } ( t ) = \left\langle 8 \sqrt { t } , \frac { 1 } { t - 9 } , \ln t \right\rangle$ .

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.

$\text { Find the integral } \int \left( \sin 7 t \mathbf { i } + \cos 7 t \mathbf { j } + e ^ { - t / 5 } \mathbf { k } \right) d t \text {. }$

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 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 $t = 1$ .

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

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

Find the unit tangent vector for the curve given by $\mathbf { r } ( t ) = \left\langle \frac { 1 } { 5 } t ^ { 5 } , \frac { 1 } { 3 } t ^ { 3 } , t \right\rangle$ .

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 $[ 1,2 ]$ .

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Exam 13: Vector Functions

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

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

Find equations of the normal plane to x=t,y=t2,z=t3 at the point (2,4,8). \text { Find equations of the normal plane to } x = t , y = t ^ { 2 } , z = t ^ { 3 } \text { at the point } ( 2,4,8 ) \text {. } Find equations of the normal plane to x=t,y=t2,z=t3 at the point (2,4,8).

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

At what point on the curve x=t3,y=9t,z=t4x = t ^ { 3 } , y = 9 t , z = t ^ { 4 }x=t3,y=9t,z=t4 is the normal plane parallel to the plane 3x+9y−4z=4?3 x + 9 y - 4 z = 4 ?3x+9y−4z=4?

Find the limit lim⁡t→0[(t2+3)i+cos⁡5tj−6k]\lim _ { t \rightarrow 0 } \left[ \left( t ^ { 2 } + 3 \right) \mathbf { i } + \cos 5 t \mathbf { j } - 6 \mathbf { k } \right]limt→0​[(t2+3)i+cos5tj−6k]

If r(t)=(t,t5,tı⟩\mathbf { r } ( t ) = \left( t , t ^ { 5 } , t ^ { \imath } \right\rangler(t)=(t,t5,t⟩ , find rtt(t)\mathbf { r } ^ { tt } ( t )rtt(t)

Find the domain of the vector function r(t)=⟨8t,1t−9,ln⁡t⟩\mathbf { r } ( t ) = \left\langle 8 \sqrt { t } , \frac { 1 } { t - 9 } , \ln t \right\rangler(t)=⟨8t​,t−91​,lnt⟩ .

Find the integral ∫(sin⁡7ti+cos⁡7tj+e−t/5k)dt. \text { Find the integral } \int \left( \sin 7 t \mathbf { i } + \cos 7 t \mathbf { j } + e ^ { - t / 5 } \mathbf { k } \right) d t \text {. } Find the integral ∫(sin7ti+cos7tj+e−t/5k)dt.

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 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 .

Find r(t)\mathbf { r } ^ { \tt } ( 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

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 .

Find the unit tangent vector for the curve given by r(t)=⟨15t5,13t3,t⟩\mathbf { r } ( t ) = \left\langle \frac { 1 } { 5 } t ^ { 5 } , \frac { 1 } { 3 } t ^ { 3 } , t \right\rangler(t)=⟨51​t5,31​t3,t⟩ .

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][ 1,2 ][1,2] .

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

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

$\text { Find equations of the normal plane to } x = t , y = t ^ { 2 } , z = t ^ { 3 } \text { at the point } ( 2,4,8 ) \text {. }$

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

At what point on the curve $x = t ^ { 3 } , y = 9 t , z = t ^ { 4 }$ is the normal plane parallel to the plane $3 x + 9 y - 4 z = 4 ?$

Find the limit $\lim _ { t \rightarrow 0 } \left[ \left( t ^ { 2 } + 3 \right) \mathbf { i } + \cos 5 t \mathbf { j } - 6 \mathbf { k } \right]$

If $\mathbf { r } ( t ) = \left( t , t ^ { 5 } , t ^ { \imath } \right\rangle$ , find $\mathbf { r } ^ { tt } ( t )$

Find the domain of the vector function $\mathbf { r } ( t ) = \left\langle 8 \sqrt { t } , \frac { 1 } { t - 9 } , \ln t \right\rangle$ .

$\text { Find the integral } \int \left( \sin 7 t \mathbf { i } + \cos 7 t \mathbf { j } + e ^ { - t / 5 } \mathbf { k } \right) d t \text {. }$

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 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 $t = 1$ .

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

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

Find the unit tangent vector for the curve given by $\mathbf { r } ( t ) = \left\langle \frac { 1 } { 5 } t ^ { 5 } , \frac { 1 } { 3 } t ^ { 3 } , t \right\rangle$ .

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 $[ 1,2 ]$ .