Question 1

Find the unit tangent vector for the curve given by $\mathbf { r } ( t ) = \left\langle \frac { 1 } { 7 } t ^ { 7 } , \frac { 1 } { 3 } t ^ { 3 } , t \right\rangle$ .

Accepted Answer

A)  $\frac { \left\langle t ^ { 6 } , t ^ { 2 } , 1 \right\rangle } { \sqrt { t ^ { 10 } + t ^ { 4 } } }$ 
B)  $\frac { \left\langle t ^ { 6 } , t ^ { 2 } , 1 \right\rangle } { \sqrt { 6 t ^ { 12 } + 4 t ^ { 4 } + 1 } }$ 
C)  $\frac { \left\langle t ^ { 6 } , t ^ { 2 } , 1 \right\rangle } { \sqrt { t ^ { 12 } + t ^ { 4 } } }$ 
D)  $\frac { \left\langle t ^ { 6 } , t ^ { 2 } , 1 \right\rangle } { \sqrt { t ^ { 12 } + t ^ { 4 } + 1 } }$ 
E)  None of these 
A)  $\frac { \left\langle t ^ { 6 } , t ^ { 2 } , 1 \right\rangle } { \sqrt { t ^ { 10 } + t ^ { 4 } } }$ 
B)  $\frac { \left\langle t ^ { 6 } , t ^ { 2 } , 1 \right\rangle } { \sqrt { 6 t ^ { 12 } + 4 t ^ { 4 } + 1 } }$ 
C)  $\frac { \left\langle t ^ { 6 } , t ^ { 2 } , 1 \right\rangle } { \sqrt { t ^ { 12 } + t ^ { 4 } } }$ 
D)  $\frac { \left\langle t ^ { 6 } , t ^ { 2 } , 1 \right\rangle } { \sqrt { t ^ { 12 } + t ^ { 4 } + 1 } }$ 
E)  None of these

Question 2

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

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 3

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

A)  $d \approx 43.3$ km 
B)  $d \approx 350$ km 
C)  $d \approx 63.3$ km 
D)  $d \approx 433$ km 
E)  $$d \approx 53.3$$ km 
A)  $d \approx 43.3$ km 
B)  $d \approx 350$ km 
C)  $d \approx 63.3$ km 
D)  $d \approx 433$ km 
E)  $$d \approx 53.3$$ km

Question 4

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = 5 t \mathbf { i } + \left( t ^ { 2 } + 5 \right) \mathbf { j }$

Accepted Answer

\[\begin{array} { l } 
a _ { \

Question 5

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 6

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

Accepted Answer

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

Question 7

Find $\mathbf { r } ( t )$ if $\mathbf { r } ^ { \prime } ( t ) = \sin t \mathbf { i } - \cos t \mathbf { j } + 6 t \mathbf { k }$ and $r ( 0 ) = i + j + 5 k$ .

Accepted Answer

A)  $( - \cos t + 2 ) \mathbf { i } + ( \sin t - 1 ) \mathbf { j } + \left( 3 t ^ { 2 } + 5 \right) \mathbf { k }$ 
B)  $( \cos t ) \mathbf { i } + ( \sin t - 1 ) \mathbf { j } + \left( 3 t ^ { 2 } + 5 \right) \mathbf { k }$ 
C)  $( \cos t ) \mathbf { i } + ( \sin t + 1 ) \mathbf { j } + \left( 3 t ^ { 2 } + 5 \right) \mathbf { k }$ 
D)  $( - \cos t + 2 ) \mathbf { i } - ( \sin t - 1 ) \mathbf { j } + \left( 3 t ^ { 2 } + 5 \right) \mathbf { k }$ 
E)  $( - \cos t + 1 ) \mathbf { i } - ( \sin t - 1 ) \mathbf { j } + \left( 3 t ^ { 2 } + 5 \right) \mathbf { k }$ 
A)  $( - \cos t + 2 ) \mathbf { i } + ( \sin t - 1 ) \mathbf { j } + \left( 3 t ^ { 2 } + 5 \right) \mathbf { k }$ 
B)  $( \cos t ) \mathbf { i } + ( \sin t - 1 ) \mathbf { j } + \left( 3 t ^ { 2 } + 5 \right) \mathbf { k }$ 
C)  $( \cos t ) \mathbf { i } + ( \sin t + 1 ) \mathbf { j } + \left( 3 t ^ { 2 } + 5 \right) \mathbf { k }$ 
D)  $( - \cos t + 2 ) \mathbf { i } - ( \sin t - 1 ) \mathbf { j } + \left( 3 t ^ { 2 } + 5 \right) \mathbf { k }$ 
E)  $( - \cos t + 1 ) \mathbf { i } - ( \sin t - 1 ) \mathbf { j } + \left( 3 t ^ { 2 } + 5 \right) \mathbf { k }$

Question 8

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

Accepted Answer

The answer of Find an expression for \(\frac { d...

Question 9

Find the velocity, acceleration, and speed of an object with position vector $\mathbf { r } ( t ) = e ^ { t } \{ \cos 2 t , \sin 2 t , 8 \}$ .

Accepted Answer

\[\begin{array} { l } 
\mathbf { v } ( t

Question 10

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 11

Find $\mathrm { r } ( t )$ if $\mathbf { r } ^ { \prime } ( t ) = t ^ { 10 } \mathbf { i } + 3 t ^ { 2 } \mathbf { j } - t ^ { 7 } \mathbf { k }$ and $r ( 0 ) = j$ .

Accepted Answer

The answer of Find \(\mathrm { r } ( t...

Question 12

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

Accepted Answer

The answer of Find the integral \(\int \left( \sin 7...

Question 13

Find the unit tangent vector $\mathbf { T } ( t )$ for $\mathbf { r } ( t ) = 2 t \mathbf { i } + 6 t \mathbf { j } + 3 t \mathbf { k }$ at $t = - 1$

Accepted Answer

A)  $- 2 \mathbf { i } - 6 \mathbf { j } - 3 \mathbf { k }$ 
B)  $\frac { 2 } { 7 }$ i + $\frac { 6 } { 7 }$ j + $\frac { 3 } { 7 }$ k 
C)  $- \frac { 2 } { 49 }$ i $- \frac { 6 } { 49 }$ j $- \frac { 3 } { 49 }$ k 
D)  $2 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }$ 
A)  $- 2 \mathbf { i } - 6 \mathbf { j } - 3 \mathbf { k }$ 
B)  $\frac { 2 } { 7 }$ i + $\frac { 6 } { 7 }$ j + $\frac { 3 } { 7 }$ k 
C)  $- \frac { 2 } { 49 }$ i $- \frac { 6 } { 49 }$ j $- \frac { 3 } { 49 }$ k 
D)  $2 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }$

Question 14

Let C be a smooth curve defined by $\mathbf { r } ( t ) = 2 \mathbf { i } + 3 t \mathbf { j } + 2 t ^ { 2 } \mathbf { k }$ , and let $\mathbf { T } ( t )$ and $\mathrm { N } ( t )$ be the unit tangent vector and unit normal vector to C corresponding to t. The plane determined by T and N is called the osculating plane. Find an equation of the osculating plane of the curve described by $\mathbf { r } ( t )$ at $t = 1$

Accepted Answer

A)  $z = 2$ 
B)  $y = 3$ 
C)  $2 x + 3 y + 2 z = 1$ 
D)  $x = 2$ 
A)  $z = 2$ 
B)  $y = 3$ 
C)  $2 x + 3 y + 2 z = 1$ 
D)  $x = 2$

Question 15

Find the velocity of a particle with the given position function. $\mathbf { r } ( t ) = 11 e ^ { 9 t } \mathbf { i } + 9 e ^ { - 13 t } \mathbf { j }$

Accepted Answer

A)  $\mathbf { v } ( t ) = e ^ { 9 t } \mathbf { i } + 117 e ^ { - 13 t } \mathbf { j }$ 
B)  $\mathbf { v } ( t ) = 99 e ^ { 9 t } \mathbf { i } - 117 e ^ { - 13 t } \mathbf { j }$ 
C)  $\mathbf { v } ( t ) = 11 e ^ { 9 t } \mathbf { i } + e ^ { - 13 t } \mathbf { j }$ 
D)  $\mathbf { v } ( t ) = 11 e ^ { 9 t } \mathbf { i } - 117 e ^ { - 13 t } \mathbf { j }$ 
E)  $\mathbf { v } ( t ) = 99 e ^ { 9 t } \mathbf { i } + 117 e ^ { - 13 t } \mathbf { j }$ 
A)  $\mathbf { v } ( t ) = e ^ { 9 t } \mathbf { i } + 117 e ^ { - 13 t } \mathbf { j }$ 
B)  $\mathbf { v } ( t ) = 99 e ^ { 9 t } \mathbf { i } - 117 e ^ { - 13 t } \mathbf { j }$ 
C)  $\mathbf { v } ( t ) = 11 e ^ { 9 t } \mathbf { i } + e ^ { - 13 t } \mathbf { j }$ 
D)  $\mathbf { v } ( t ) = 11 e ^ { 9 t } \mathbf { i } - 117 e ^ { - 13 t } \mathbf { j }$ 
E)  $\mathbf { v } ( t ) = 99 e ^ { 9 t } \mathbf { i } + 117 e ^ { - 13 t } \mathbf { j }$

Question 16

Find the velocity of a particle that has the given acceleration and the given initial velocity. $\mathbf { a } ( t ) = 3 \mathbf { k } , \mathbf { v } ( 0 ) = 12 \mathbf { i } - 7 \mathbf { j }$

Accepted Answer

A)  $\mathbf { v } ( t ) = 12 \mathbf { i } - \mathbf { j } + 3 t \mathbf { k }$ 
B)  $\mathbf { v } ( t ) = 12 \mathbf { i } - 7 \mathbf { j } + t \mathbf { k }$ 
C)  $\mathbf { v } ( t ) = ( 12 t - 9 ) \mathbf { i } + 7 t \mathbf { k }$ 
D)  $\mathbf { v } ( t ) = \mathbf { i } - 7 \mathbf { j } + 3 t \mathbf { k }$ 
E)  $\mathbf { v } ( t ) = 12 \mathbf { i } - 7 \mathbf { j } + 3 t \mathbf { k }$ 
A)  $\mathbf { v } ( t ) = 12 \mathbf { i } - \mathbf { j } + 3 t \mathbf { k }$ 
B)  $\mathbf { v } ( t ) = 12 \mathbf { i } - 7 \mathbf { j } + t \mathbf { k }$ 
C)  $\mathbf { v } ( t ) = ( 12 t - 9 ) \mathbf { i } + 7 t \mathbf { k }$ 
D)  $\mathbf { v } ( t ) = \mathbf { i } - 7 \mathbf { j } + 3 t \mathbf { k }$ 
E)  $\mathbf { v } ( t ) = 12 \mathbf { i } - 7 \mathbf { j } + 3 t \mathbf { k }$

Question 17

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $x = t ^ { 11 } , y = t ^ { 3 } , z = t ^ { 6 } ; ( 4,4,4 )$

Accepted Answer

A)  $x = 4 - 11 t , y = 4 + 3 t , z = 4 + 6 t$ 
B)  $x = 4 + 11 t , y = 4 + 3 t , z = 4 + 6 t$ 
C)  $x = 4 + 11 t , y = 4 + 3 t , z = 4 - 6 t$ 
D)  $x = 11 t , y = 4 + 3 t , z = 4 + 6 t$ 
E)  $x = 4 + 11 t , y = 4 - 3 t , z = 4 + 6 t$ 
A)  $x = 4 - 11 t , y = 4 + 3 t , z = 4 + 6 t$ 
B)  $x = 4 + 11 t , y = 4 + 3 t , z = 4 + 6 t$ 
C)  $x = 4 + 11 t , y = 4 + 3 t , z = 4 - 6 t$ 
D)  $x = 11 t , y = 4 + 3 t , z = 4 + 6 t$ 
E)  $x = 4 + 11 t , y = 4 - 3 t , z = 4 + 6 t$

Question 18

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

Accepted Answer

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

Question 19

Find parametric equations for the tangent line to the curve with parametric equations $x = 2 t$ $y = 7 t ^ { 2 }$ $z = 4 t ^ { 3 }$ at the point with $t = 1$

Accepted Answer

A)  $x = 2 + 2 t$ $y = 7 + 14 t$ $z = 4 + 12 t$ 
B)  $x = 2 + 2 t$ $y = 7 + 7 t$ $z = 4 + 4 t$ 
C)  $x = 1 + 2 t$ $y = 1 + 7 t$ $z = 1 + 4 t$ 
D)  $x = 1 + 2 t$ $y = 1 + 14 t$ $z = 1 + 12 t$ 
A)  $x = 2 + 2 t$ $y = 7 + 14 t$ $z = 4 + 12 t$ 
B)  $x = 2 + 2 t$ $y = 7 + 7 t$ $z = 4 + 4 t$ 
C)  $x = 1 + 2 t$ $y = 1 + 7 t$ $z = 1 + 4 t$ 
D)  $x = 1 + 2 t$ $y = 1 + 14 t$ $z = 1 + 12 t$

Question 20

Find a vector function that represents the curve of intersection of the two surfaces:
The paraboloid $z = 4 x ^ { 2 } + 9 y ^ { 2 }$ and the parabolic cylinder $y = x ^ { 2 }$ .

Accepted Answer

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

Find the unit tangent vector for the curve given by $\mathbf { r } ( t ) = \left\langle \frac { 1 } { 7 } t ^ { 7 } , \frac { 1 } { 3 } t ^ { 3 } , t \right\rangle$ .

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 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 the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = 5 t \mathbf { i } + \left( t ^ { 2 } + 5 \right) \mathbf { j }$

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.

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

Find $\mathbf { r } ( t )$ if $\mathbf { r } ^ { \prime } ( t ) = \sin t \mathbf { i } - \cos t \mathbf { j } + 6 t \mathbf { k }$ and $r ( 0 ) = i + j + 5 k$ .

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

Find the velocity, acceleration, and speed of an object with position vector $\mathbf { r } ( t ) = e ^ { t } \{ \cos 2 t , \sin 2 t , 8 \}$ .

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

Find $\mathrm { r } ( t )$ if $\mathbf { r } ^ { \prime } ( t ) = t ^ { 10 } \mathbf { i } + 3 t ^ { 2 } \mathbf { j } - t ^ { 7 } \mathbf { k }$ and $r ( 0 ) = j$ .

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

Find the unit tangent vector $\mathbf { T } ( t )$ for $\mathbf { r } ( t ) = 2 t \mathbf { i } + 6 t \mathbf { j } + 3 t \mathbf { k }$ at $t = - 1$

Find the velocity of a particle with the given position function. $\mathbf { r } ( t ) = 11 e ^ { 9 t } \mathbf { i } + 9 e ^ { - 13 t } \mathbf { j }$

Find the velocity of a particle that has the given acceleration and the given initial velocity. $\mathbf { a } ( t ) = 3 \mathbf { k } , \mathbf { v } ( 0 ) = 12 \mathbf { i } - 7 \mathbf { j }$

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $x = t ^ { 11 } , y = t ^ { 3 } , z = t ^ { 6 } ; ( 4,4,4 )$

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

Find parametric equations for the tangent line to the curve with parametric equations $x = 2 t$ $y = 7 t ^ { 2 }$ $z = 4 t ^ { 3 }$ at the point with $t = 1$

Find a vector function that represents the curve of intersection of the two surfaces: The paraboloid $z = 4 x ^ { 2 } + 9 y ^ { 2 }$ and the parabolic cylinder $y = x ^ { 2 }$ .

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

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

Final Exam

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Exam 13: Vector Functions

Find the unit tangent vector for the curve given by r(t)=⟨17t7,13t3,t⟩\mathbf { r } ( t ) = \left\langle \frac { 1 } { 7 } t ^ { 7 } , \frac { 1 } { 3 } t ^ { 3 } , t \right\rangler(t)=⟨71​t7,31​t3,t⟩ .

Find the length of the curve r(t)=2ti+t2j+ln⁡tk\mathbf { r } ( t ) = 2 t \mathbf { i } + t ^ { 2 } \mathbf { j } + \ln t \mathbf { k }r(t)=2ti+t2j+lntk 1≤t≤e31 \leq t \leq e ^ { 3 }1≤t≤e3

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 the scalar tangential and normal components of acceleration of a particle with position vector r(t)=5ti+(t2+5)j\mathbf { r } ( t ) = 5 t \mathbf { i } + \left( t ^ { 2 } + 5 \right) \mathbf { j }r(t)=5ti+(t2+5)j

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.

What force is required so that a particle of mass mmm has the following position function?. r(t)=5t3i+10t2j+7t3kr ( t ) = 5 t ^ { 3 } \mathbf { i } + 10 t ^ { 2 } \mathbf { j } + 7 t ^ { 3 } \mathbf { k }r(t)=5t3i+10t2j+7t3k

Find r(t)\mathbf { r } ( t )r(t) if r′(t)=sin⁡ti−cos⁡tj+6tk\mathbf { r } ^ { \prime } ( t ) = \sin t \mathbf { i } - \cos t \mathbf { j } + 6 t \mathbf { k }r′(t)=sinti−costj+6tk and r(0)=i+j+5kr ( 0 ) = i + j + 5 kr(0)=i+j+5k .

Find an expression for ddu[x(u)⋅(y(u)×z(u))]\frac { d } { d u } [ x ( u ) \cdot ( y ( u ) \times z ( u ) ) ]dud​[x(u)⋅(y(u)×z(u))] .

Find the velocity, acceleration, and speed of an object with position vector r(t)=et{cos⁡2t,sin⁡2t,8}\mathbf { r } ( t ) = e ^ { t } \{ \cos 2 t , \sin 2 t , 8 \}r(t)=et{cos2t,sin2t,8} .

Find r(t)\mathrm { r } ( t )r(t) if r′(t)=t10i+3t2j−t7k\mathbf { r } ^ { \prime } ( t ) = t ^ { 10 } \mathbf { i } + 3 t ^ { 2 } \mathbf { j } - t ^ { 7 } \mathbf { k }r′(t)=t10i+3t2j−t7k and r(0)=jr ( 0 ) = jr(0)=j .

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

Find the unit tangent vector T(t)\mathbf { T } ( t )T(t) for r(t)=2ti+6tj+3tk\mathbf { r } ( t ) = 2 t \mathbf { i } + 6 t \mathbf { j } + 3 t \mathbf { k }r(t)=2ti+6tj+3tk at t=−1t = - 1t=−1

Find the velocity of a particle with the given position function. r(t)=11e9ti+9e−13tj\mathbf { r } ( t ) = 11 e ^ { 9 t } \mathbf { i } + 9 e ^ { - 13 t } \mathbf { j }r(t)=11e9ti+9e−13tj

Find the velocity of a particle that has the given acceleration and the given initial velocity. a(t)=3k,v(0)=12i−7j\mathbf { a } ( t ) = 3 \mathbf { k } , \mathbf { v } ( 0 ) = 12 \mathbf { i } - 7 \mathbf { j }a(t)=3k,v(0)=12i−7j

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x=t11,y=t3,z=t6;(4,4,4)x = t ^ { 11 } , y = t ^ { 3 } , z = t ^ { 6 } ; ( 4,4,4 )x=t11,y=t3,z=t6;(4,4,4)

Find the following limit. lim⁡t→∞⟨arctan⁡t,e−γt,ln⁡4tt⟩\lim _ { t \rightarrow \infty } \left\langle\arctan t , \mathrm { e } ^ { - \gamma t } , \frac { \ln 4 t } { t } \right\ranglelimt→∞​⟨arctant,e−γt,tln4t​⟩

Find parametric equations for the tangent line to the curve with parametric equations x=2tx = 2 tx=2t y=7t2y = 7 t ^ { 2 }y=7t2 z=4t3z = 4 t ^ { 3 }z=4t3 at the point with t=1t = 1t=1

Find a vector function that represents the curve of intersection of the two surfaces: The paraboloid z=4x2+9y2z = 4 x ^ { 2 } + 9 y ^ { 2 }z=4x2+9y2 and the parabolic cylinder y=x2y = x ^ { 2 }y=x2 .

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

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

Final Exam

Filters

Find the unit tangent vector for the curve given by $\mathbf { r } ( t ) = \left\langle \frac { 1 } { 7 } t ^ { 7 } , \frac { 1 } { 3 } t ^ { 3 } , t \right\rangle$ .

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 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 the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = 5 t \mathbf { i } + \left( t ^ { 2 } + 5 \right) \mathbf { j }$

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.

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

Find $\mathbf { r } ( t )$ if $\mathbf { r } ^ { \prime } ( t ) = \sin t \mathbf { i } - \cos t \mathbf { j } + 6 t \mathbf { k }$ and $r ( 0 ) = i + j + 5 k$ .

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

Find the velocity, acceleration, and speed of an object with position vector $\mathbf { r } ( t ) = e ^ { t } \{ \cos 2 t , \sin 2 t , 8 \}$ .

Find $\mathrm { r } ( t )$ if $\mathbf { r } ^ { \prime } ( t ) = t ^ { 10 } \mathbf { i } + 3 t ^ { 2 } \mathbf { j } - t ^ { 7 } \mathbf { k }$ and $r ( 0 ) = j$ .

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

Find the unit tangent vector $\mathbf { T } ( t )$ for $\mathbf { r } ( t ) = 2 t \mathbf { i } + 6 t \mathbf { j } + 3 t \mathbf { k }$ at $t = - 1$

Find the velocity of a particle with the given position function. $\mathbf { r } ( t ) = 11 e ^ { 9 t } \mathbf { i } + 9 e ^ { - 13 t } \mathbf { j }$

Find the velocity of a particle that has the given acceleration and the given initial velocity. $\mathbf { a } ( t ) = 3 \mathbf { k } , \mathbf { v } ( 0 ) = 12 \mathbf { i } - 7 \mathbf { j }$

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $x = t ^ { 11 } , y = t ^ { 3 } , z = t ^ { 6 } ; ( 4,4,4 )$

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

Find parametric equations for the tangent line to the curve with parametric equations $x = 2 t$ $y = 7 t ^ { 2 }$ $z = 4 t ^ { 3 }$ at the point with $t = 1$

Find a vector function that represents the curve of intersection of the two surfaces: The paraboloid $z = 4 x ^ { 2 } + 9 y ^ { 2 }$ and the parabolic cylinder $y = x ^ { 2 }$ .