Question 1

Find the length of the curve $\mathbf { r } ( t ) = 2 t \mathbf { i } + t ^ { 2 } \mathbf { j } + \ln t \mathbf { k } , 1 \leq t \leq e ^ { 7 }$
Select the correct answer.

Accepted Answer

A)  $e ^ { 14 }$ 
B)  $e ^ { 14 } + 6$ 
C)  $e ^ { 7 }$ 
D)  $e ^ { 7 } + 6$ 
A)  $e ^ { 14 }$ 
B)  $e ^ { 14 } + 6$ 
C)  $e ^ { 7 }$ 
D)  $e ^ { 7 } + 6$

Question 2

$$\text { For the curve given by } \mathbf { r } ( t ) = \langle 2 \sin t , 5 t , 2 \operatorname { cost } \} \text {, find the unit normal vector. }$$

Accepted Answer

The answer of \[\text { For the curve given by...

Question 3

Find the curvature of the curve $\mathbf { r } ( t ) = 2 t \mathbf { i } + 6 t \mathbf { j } + 9 \mathbf { k }$
Select the correct answer.

Accepted Answer

A)  1 
B)  $2 \sqrt { 10 }$ 
C)  11 
D)  0 
A)  1 
B)  $2 \sqrt { 10 }$ 
C)  11 
D)  0

Question 4

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)  $| v ( t ) | = \sqrt { 1 + 100 t ^ { 2 } + 100 t ^ { 10 } }$ 
B)  $| \mathrm { v } ( t ) | = \sqrt { 1 + 100 t + 324 t ^ { 9 } }$ 
C)  $| \mathrm { v } ( t ) | = 1 + 100 t ^ { 2 } + 324 t ^ { 10 }$ 
D)  $| v ( t ) | = \sqrt { 1 + 100 t ^ { 2 } + 324 t ^ { 10 } }$ 
E)  $| \mathrm { v } ( t ) | = \sqrt { 1 + 100 t + t ^ { 9 } }$ 
A)  $| v ( t ) | = \sqrt { 1 + 100 t ^ { 2 } + 100 t ^ { 10 } }$ 
B)  $| \mathrm { v } ( t ) | = \sqrt { 1 + 100 t + 324 t ^ { 9 } }$ 
C)  $| \mathrm { v } ( t ) | = 1 + 100 t ^ { 2 } + 324 t ^ { 10 }$ 
D)  $| v ( t ) | = \sqrt { 1 + 100 t ^ { 2 } + 324 t ^ { 10 } }$ 
E)  $| \mathrm { v } ( t ) | = \sqrt { 1 + 100 t + t ^ { 9 } }$

Question 5

Find the limit.
$$\lim _ { t \rightarrow 0 ^ { + } } ( 10 \cos t , 30 \sin t , 5 t \ln t \}$$

Accepted Answer

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

Question 6

$$\text { If } \mathbf { r } ( t ) = 7 \mathbf { i } + 3 t \cos \pi \mathbf { j } + 4 \sin \pi k \mathbf { k } \text {, evaluate } \int _ { 0 } ^ { 1 } r ( t ) d t$$

Accepted Answer

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

Question 7

Find the curvature of the curve $\mathbf { r } ( t ) = 6 t \mathbf { i } + 6 t \mathbf { j } + 3 \mathbf { k }$. Select the correct answer.

Accepted Answer

A)  $6 \sqrt { 2 }$ 
B)  0 
C)  9 
D)  1 
A)  $6 \sqrt { 2 }$ 
B)  0 
C)  9 
D)  1

Question 8

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

Accepted Answer

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

Question 9

Find the curvature of the curve $\mathbf { r } ( t ) = 2 \sin 7 t \mathbf { i } + 2 \cos 7 t \mathbf { j } + 2 t \mathbf { k }$. Select the correct answer.

Accepted Answer

A)  $\frac { 7 } { 2 }$ 
B)  $\frac { 2 } { 7 }$ 
C)  $\frac { 49 } { 100 }$ 
D)  $\frac { 100 } { 49 }$ 
A)  $\frac { 7 } { 2 }$ 
B)  $\frac { 2 } { 7 }$ 
C)  $\frac { 49 } { 100 }$ 
D)  $\frac { 100 } { 49 }$

Question 10

Find the velocity of a particle that has the given acceleration and the given initial velocity.
Select the correct answer.
$$\mathbf { a } ( t ) = 3 k , \mathbf { v } ( 0 ) = 16 \mathbf { i } - 3 \mathbf { j }$$

Accepted Answer

A)  $\quad \mathrm { v } ( t ) = 16 \mathbf { i } - \mathbf { j } + 3 t \mathbf { k }$ 
B)  $\mathbf { v } ( t ) = \mathbf { i } - 3 \mathbf { j } + 3 t \mathbf { k }$ 
C)  $\mathbf { v } ( t ) = 16 \mathbf { i } - 3 \mathbf { j } + t \mathbf { k }$ 
D)  $\mathbf { v } ( t ) = 16 \mathbf { i } - 3 \mathbf { j } + 3 t \mathbf { k }$ 
E)  $\mathbf { v } ( t ) = ( 16 t - 9 ) \mathbf { i } + 3 t \mathbf { k }$ 
A)  $\quad \mathrm { v } ( t ) = 16 \mathbf { i } - \mathbf { j } + 3 t \mathbf { k }$ 
B)  $\mathbf { v } ( t ) = \mathbf { i } - 3 \mathbf { j } + 3 t \mathbf { k }$ 
C)  $\mathbf { v } ( t ) = 16 \mathbf { i } - 3 \mathbf { j } + t \mathbf { k }$ 
D)  $\mathbf { v } ( t ) = 16 \mathbf { i } - 3 \mathbf { j } + 3 t \mathbf { k }$ 
E)  $\mathbf { v } ( t ) = ( 16 t - 9 ) \mathbf { i } + 3 t \mathbf { k }$

Question 11

The helix $\mathbf { r } _ { 1 } ( t ) = 2 \cos t \mathbf { i } + \sin t \mathbf { j } + t \mathbf { k }$ intersects the curve $\mathbf { r } _ { 2 } ( t ) = ( 2 + t ) \mathbf { i } + 10 t ^ { 2 } \mathbf { j } + 9 t ^ { 3 } \mathbf { k }$ at the point $( 2,0,0 )$. Find the angle of intersection.

Accepted Answer

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

Question 12

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

Accepted Answer

A)  $\frac { \left\langle t ^ { 4 } , t ^ { 2 } , 1 \right) } { \sqrt { t ^ { 10 } + t ^ { 4 } } }$ 
B)  $\frac { \left( t ^ { 4 } , t ^ { 2 } , 1 \right) } { \sqrt { 4 t ^ { 8 } + 4 t ^ { 4 } + 1 } }$ 
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)  $\frac { \left( t ^ { 4 } , t ^ { 2 } , 1 \right) } { \sqrt { 4 t ^ { 8 } + 4 t ^ { 4 } + 1 } }$ 
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 13

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = 3 \sin t \mathbf { i } + 3 \cos t \mathbf { j } + 5 t \mathbf { k }$

Accepted Answer

A)  $\quad a _ { \mathbf { T } } = 0 , a _ { \mathrm { N } } = 3$ 
B)  $a _ { \mathbf { T } } = 5 , a _ { \mathbf { N } } = 3$ 
C)  $a _ { \mathbf { T } } = 0 , a _ { \mathrm { N } } = 3 \sqrt { 26 }$ 
D)  $a _ { \mathbf { T } } = 5 , a _ { \mathrm { N } } = 3 \sqrt { 26 }$ 
A)  $\quad a _ { \mathbf { T } } = 0 , a _ { \mathrm { N } } = 3$ 
B)  $a _ { \mathbf { T } } = 5 , a _ { \mathbf { N } } = 3$ 
C)  $a _ { \mathbf { T } } = 0 , a _ { \mathrm { N } } = 3 \sqrt { 26 }$ 
D)  $a _ { \mathbf { T } } = 5 , a _ { \mathrm { N } } = 3 \sqrt { 26 }$

Question 14

Find the arc length function $s ( t )$ for the curve defined by $\mathbf { r } ( t ) = e ^ { 2 t } \cos t \mathbf { i } + e ^ { 2 t } \sin t \mathbf { j } + e ^ { 2 t } \mathbf { k }$ for $t \geq 0$. Then use this result to find a parametrization of $C$ in terms of $s$.

Accepted Answer

\[\begin{array} { l } 
s ( t ) = \frac {

Question 15

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = e ^ { t } \{ \cos 6 t , \sin 6 t , 0 \}$

Accepted Answer

A)  $a _ { \mathbf { T } } = \sqrt { 37 } e ^ { t } , a _ { \mathrm { N } } = 6 \sqrt { 37 } e ^ { t }$ 
B)  $a _ { \mathbf { T } } = 6 e ^ { t } , a _ { \mathbf { N } } = 37 e ^ { t }$ 
C)  $a _ { \mathbf { T } } = 6 \sqrt { 37 } e ^ { t } , a _ { \mathrm { N } } = \sqrt { 37 } e ^ { t }$ 
D)  $a _ { \mathbf { T } } = 37 e ^ { t } , a _ { \mathrm { N } } = 6 e ^ { t }$ 
A)  $a _ { \mathbf { T } } = \sqrt { 37 } e ^ { t } , a _ { \mathrm { N } } = 6 \sqrt { 37 } e ^ { t }$ 
B)  $a _ { \mathbf { T } } = 6 e ^ { t } , a _ { \mathbf { N } } = 37 e ^ { t }$ 
C)  $a _ { \mathbf { T } } = 6 \sqrt { 37 } e ^ { t } , a _ { \mathrm { N } } = \sqrt { 37 } e ^ { t }$ 
D)  $a _ { \mathbf { T } } = 37 e ^ { t } , a _ { \mathrm { N } } = 6 e ^ { t }$

Question 16

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)  $v ( t ) = 95 e ^ { 5 t } \mathbf { i } + 105 e ^ { - 21 t } \mathbf { j }$ 
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)  $v ( t ) = 95 e ^ { 5 t } \mathbf { i } + 105 e ^ { - 21 t } \mathbf { j }$

Question 17

$$\text { Find the curvature of } y = x ^ { 4 } \text {. }$$

Accepted Answer

The answer of \[\text { Find the curvature of }...

Question 18

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

Accepted Answer

The answer of Find the speed of a particle with...

Question 19

A mortar shell is fired with a muzzle speed of $325 \mathrm { ft } / \mathrm { sec }$. Find the angle of elevation of the if the shell strikes a target located $1500 \mathrm { ft }$ away. Round your answer to 2 decimal places.
Select the correct answer.

Accepted Answer

A)  $12.22 ^ { \circ }$ 
B)  $0.64 ^ { \circ }$ 
C)  $13.51 ^ { \circ }$ 
D)  $0.24 ^ { \circ }$ ortar 
A)  $12.22 ^ { \circ }$ 
B)  $0.64 ^ { \circ }$ 
C)  $13.51 ^ { \circ }$ 
D)  $0.24 ^ { \circ }$ ortar

Question 20

$$\text { Find the domain of the vector function } \mathbf { r } ( t ) = 8 t \mathbf { i } + \frac { 1 } { t - 3 } \mathbf { j } \text {. }$$ Select the correct answer.

Accepted Answer

A)  $( - \infty , - 8 ) \cup ( - 8 , \infty )$ 
B)  $( - \infty , 3 ) \cup ( 3 , \infty )$ 
C)  $( - \infty , - 3 ) \cup ( - 3 , \infty )$ 
D)  $( - \infty , 8 ) \cup ( 8 , \infty )$ 
A)  $( - \infty , - 8 ) \cup ( - 8 , \infty )$ 
B)  $( - \infty , 3 ) \cup ( 3 , \infty )$ 
C)  $( - \infty , - 3 ) \cup ( - 3 , \infty )$ 
D)  $( - \infty , 8 ) \cup ( 8 , \infty )$

Find the length of the curve $\mathbf { r } ( t ) = 2 t \mathbf { i } + t ^ { 2 } \mathbf { j } + \ln t \mathbf { k } , 1 \leq t \leq e ^ { 7 }$ Select the correct answer.

$\text { For the curve given by } \mathbf { r } ( t ) = \langle 2 \sin t , 5 t , 2 \operatorname { cost } \} \text {, find the unit normal vector. }$

Find the curvature of the curve $\mathbf { r } ( t ) = 2 t \mathbf { i } + 6 t \mathbf { j } + 9 \mathbf { k }$ Select the correct answer.

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 limit. $\lim _ { t \rightarrow 0 ^ { + } } ( 10 \cos t , 30 \sin t , 5 t \ln t \}$

$\text { If } \mathbf { r } ( t ) = 7 \mathbf { i } + 3 t \cos \pi \mathbf { j } + 4 \sin \pi k \mathbf { k } \text {, evaluate } \int _ { 0 } ^ { 1 } r ( t ) d t$

Find the curvature of the curve $\mathbf { r } ( t ) = 6 t \mathbf { i } + 6 t \mathbf { j } + 3 \mathbf { k }$ . Select the correct answer.

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

Find the curvature of the curve $\mathbf { r } ( t ) = 2 \sin 7 t \mathbf { i } + 2 \cos 7 t \mathbf { j } + 2 t \mathbf { k }$ . Select the correct answer.

Find the velocity of a particle that has the given acceleration and the given initial velocity. Select the correct answer. $\mathbf { a } ( t ) = 3 k , \mathbf { v } ( 0 ) = 16 \mathbf { i } - 3 \mathbf { j }$

The helix $\mathbf { r } _ { 1 } ( t ) = 2 \cos t \mathbf { i } + \sin t \mathbf { j } + t \mathbf { k }$ intersects the curve $\mathbf { r } _ { 2 } ( t ) = ( 2 + t ) \mathbf { i } + 10 t ^ { 2 } \mathbf { j } + 9 t ^ { 3 } \mathbf { k }$ at the point $( 2,0,0 )$ . Find the angle of intersection.

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

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = 3 \sin t \mathbf { i } + 3 \cos t \mathbf { j } + 5 t \mathbf { k }$

Find the arc length function $s ( t )$ for the curve defined by $\mathbf { r } ( t ) = e ^ { 2 t } \cos t \mathbf { i } + e ^ { 2 t } \sin t \mathbf { j } + e ^ { 2 t } \mathbf { k }$ for $t \geq 0$ . Then use this result to find a parametrization of $C$ in terms of $s$ .

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = e ^ { t } \{ \cos 6 t , \sin 6 t , 0 \}$

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

$\text { Find the curvature of } y = x ^ { 4 } \text {. }$

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

A mortar shell is fired with a muzzle speed of $325 \mathrm { ft } / \mathrm { sec }$ . Find the angle of elevation of the if the shell strikes a target located $1500 \mathrm { ft }$ away. Round your answer to 2 decimal places. Select the correct answer.

$\text { Find the domain of the vector function } \mathbf { r } ( t ) = 8 t \mathbf { i } + \frac { 1 } { t - 3 } \mathbf { j } \text {. }$ Select the correct answer.

Functions and Models

Limits and Derivatives

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Applications of Differentiation

Integrals

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Exam 13: Vector Functions

Find the length of the curve r(t)=2ti+t2j+ln⁡tk,1≤t≤e7\mathbf { r } ( t ) = 2 t \mathbf { i } + t ^ { 2 } \mathbf { j } + \ln t \mathbf { k } , 1 \leq t \leq e ^ { 7 }r(t)=2ti+t2j+lntk,1≤t≤e7 Select the correct answer.

Find the curvature of the curve r(t)=2ti+6tj+9k\mathbf { r } ( t ) = 2 t \mathbf { i } + 6 t \mathbf { j } + 9 \mathbf { k }r(t)=2ti+6tj+9k Select the correct answer.

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 limit. lim⁡t→0+(10cos⁡t,30sin⁡t,5tln⁡t}\lim _ { t \rightarrow 0 ^ { + } } ( 10 \cos t , 30 \sin t , 5 t \ln t \}limt→0+​(10cost,30sint,5tlnt}

If r(t)=7i+3tcos⁡πj+4sin⁡πkk, evaluate ∫01r(t)dt\text { If } \mathbf { r } ( t ) = 7 \mathbf { i } + 3 t \cos \pi \mathbf { j } + 4 \sin \pi k \mathbf { k } \text {, evaluate } \int _ { 0 } ^ { 1 } r ( t ) d t If r(t)=7i+3tcosπj+4sinπkk, evaluate ∫01​r(t)dt

Find the curvature of the curve r(t)=6ti+6tj+3k\mathbf { r } ( t ) = 6 t \mathbf { i } + 6 t \mathbf { j } + 3 \mathbf { k }r(t)=6ti+6tj+3k . Select the correct answer.

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))].

Find the curvature of the curve r(t)=2sin⁡7ti+2cos⁡7tj+2tk\mathbf { r } ( t ) = 2 \sin 7 t \mathbf { i } + 2 \cos 7 t \mathbf { j } + 2 t \mathbf { k }r(t)=2sin7ti+2cos7tj+2tk . Select the correct answer.

Find the velocity of a particle that has the given acceleration and the given initial velocity. Select the correct answer. a(t)=3k,v(0)=16i−3j\mathbf { a } ( t ) = 3 k , \mathbf { v } ( 0 ) = 16 \mathbf { i } - 3 \mathbf { j }a(t)=3k,v(0)=16i−3j

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)r(t)=⟨51​t5,31​t3,t) .

Find the scalar tangential and normal components of acceleration of a particle with position vector r(t)=3sin⁡ti+3cos⁡tj+5tk\mathbf { r } ( t ) = 3 \sin t \mathbf { i } + 3 \cos t \mathbf { j } + 5 t \mathbf { k }r(t)=3sinti+3costj+5tk

Find the scalar tangential and normal components of acceleration of a particle with position vector r(t)=et{cos⁡6t,sin⁡6t,0}\mathbf { r } ( t ) = e ^ { t } \{ \cos 6 t , \sin 6 t , 0 \}r(t)=et{cos6t,sin6t,0}

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

Find the curvature of y=x4. \text { Find the curvature of } y = x ^ { 4 } \text {. } Find the curvature of y=x4.

Find the speed of a particle with the given position function. r(t)=52ti+e5tj−e−5tk\mathbf { r } ( t ) = 5 \sqrt { 2 } t \mathbf { i } + e ^ { 5 t } \mathbf { j } - e ^ { - 5 t } \mathbf { k }r(t)=52​ti+e5tj−e−5tk

A mortar shell is fired with a muzzle speed of 325ft/sec325 \mathrm { ft } / \mathrm { sec }325ft/sec . Find the angle of elevation of the if the shell strikes a target located 1500ft1500 \mathrm { ft }1500ft away. Round your answer to 2 decimal places. Select the correct answer.

Find the domain of the vector function r(t)=8ti+1t−3j. \text { Find the domain of the vector function } \mathbf { r } ( t ) = 8 t \mathbf { i } + \frac { 1 } { t - 3 } \mathbf { j } \text {. } Find the domain of the vector function r(t)=8ti+t−31​j. Select the correct answer.

Functions and Models

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Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Find the length of the curve $\mathbf { r } ( t ) = 2 t \mathbf { i } + t ^ { 2 } \mathbf { j } + \ln t \mathbf { k } , 1 \leq t \leq e ^ { 7 }$ Select the correct answer.

Find the curvature of the curve $\mathbf { r } ( t ) = 2 t \mathbf { i } + 6 t \mathbf { j } + 9 \mathbf { k }$ Select the correct answer.

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 limit. $\lim _ { t \rightarrow 0 ^ { + } } ( 10 \cos t , 30 \sin t , 5 t \ln t \}$

$\text { If } \mathbf { r } ( t ) = 7 \mathbf { i } + 3 t \cos \pi \mathbf { j } + 4 \sin \pi k \mathbf { k } \text {, evaluate } \int _ { 0 } ^ { 1 } r ( t ) d t$

Find the curvature of the curve $\mathbf { r } ( t ) = 6 t \mathbf { i } + 6 t \mathbf { j } + 3 \mathbf { k }$ . Select the correct answer.

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

Find the curvature of the curve $\mathbf { r } ( t ) = 2 \sin 7 t \mathbf { i } + 2 \cos 7 t \mathbf { j } + 2 t \mathbf { k }$ . Select the correct answer.

Find the velocity of a particle that has the given acceleration and the given initial velocity. Select the correct answer. $\mathbf { a } ( t ) = 3 k , \mathbf { v } ( 0 ) = 16 \mathbf { i } - 3 \mathbf { j }$

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

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = 3 \sin t \mathbf { i } + 3 \cos t \mathbf { j } + 5 t \mathbf { k }$

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = e ^ { t } \{ \cos 6 t , \sin 6 t , 0 \}$

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

$\text { Find the curvature of } y = x ^ { 4 } \text {. }$

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

A mortar shell is fired with a muzzle speed of $325 \mathrm { ft } / \mathrm { sec }$ . Find the angle of elevation of the if the shell strikes a target located $1500 \mathrm { ft }$ away. Round your answer to 2 decimal places. Select the correct answer.

$\text { Find the domain of the vector function } \mathbf { r } ( t ) = 8 t \mathbf { i } + \frac { 1 } { t - 3 } \mathbf { j } \text {. }$ Select the correct answer.