Question 1

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 1 + \cos ( 3 t ) , 1 + \sin ( 3 t ) \rangle$ over $[ 4,7 ]$

Accepted Answer

The answer of Find the arc length of the curve...

Question 2

Find an equation for the line tangent to the given curve at the specified point. $$\vec { r } ( t ) = \left\langle e ^ { t } , \cos ( t ) , t \right\rangle , ( 1,1,0 )$$

Accepted Answer

The answer of Find an equation for the line tangent...

Question 3

Find the unit tangent for $\vec { r } ( t ) = \left\langle \cos ( 3 t ) , \sin ( 3 t ) , e ^ { t } \right\rangle$

Accepted Answer

A)  $\vec { T } ( t ) = \frac { \left\langle - 3 \sin ( 3 t ) , 3 \cos ( 3 t ) , e ^ { t } \right\rangle } { \sqrt { 9 + e ^ { 2 t } } }$ 
B)  $\vec { T } ( t ) = \frac { \left\langle \cos ( 3 t ) , \sin ( 3 t ) , e ^ { t } \right\rangle } { \sqrt { 9 + e ^ { 2 t } } }$ 
C)  $\vec { T } ( t ) = \frac { \left\langle 3 \sin ( 3 t ) , - 3 \cos ( 3 t ) , e ^ { t } \right\rangle } { \sqrt { 9 + e ^ { 2 t } } }$ 
D)  $\vec { T } ( t ) = \left\langle \cos ( 3 t ) , \sin ( 3 t ) , e ^ { t } \right\rangle$ 
A)  $\vec { T } ( t ) = \frac { \left\langle - 3 \sin ( 3 t ) , 3 \cos ( 3 t ) , e ^ { t } \right\rangle } { \sqrt { 9 + e ^ { 2 t } } }$ 
B)  $\vec { T } ( t ) = \frac { \left\langle \cos ( 3 t ) , \sin ( 3 t ) , e ^ { t } \right\rangle } { \sqrt { 9 + e ^ { 2 t } } }$ 
C)  $\vec { T } ( t ) = \frac { \left\langle 3 \sin ( 3 t ) , - 3 \cos ( 3 t ) , e ^ { t } \right\rangle } { \sqrt { 9 + e ^ { 2 t } } }$ 
D)  $\vec { T } ( t ) = \left\langle \cos ( 3 t ) , \sin ( 3 t ) , e ^ { t } \right\rangle$

Question 4

Find the tangential and normal components for the position function $\vec { r } ( t ) = \left\langle t , e ^ { t } , t ^ { 2 } \right\rangle$

Accepted Answer

None

Question 5

Find the acceleration for the given position vector, $\vec { r } ( t ) = \left\langle 2 t , \left( 1 + t ^ { 2 } \right) ^ { 3 } , t \right\rangle$

Accepted Answer

A)  $\vec { a } ( t ) = \left\langle 2,6 t \left( 1 + t ^ { 2 } \right) ^ { 2 } , t \right\rangle$ 
B)  $\vec { a } ( t ) = \left\langle 0,6 \left( 1 + t ^ { 2 } \right) ^ { 2 } + 24 t ^ { 2 } \left( 1 + t ^ { 2 } \right) , 0 \right\rangle$ 
C)  $\vec { a } ( t ) = \left\langle 0,2 t \left( 1 + t ^ { 2 } \right) , 0 \right\rangle$ 
D)  $\vec { a } ( t ) = \left\langle 0,6 t ^ { 2 } ( 1 + 2 t ) ^ { 2 } , 0 \right\rangle$ 
A)  $\vec { a } ( t ) = \left\langle 2,6 t \left( 1 + t ^ { 2 } \right) ^ { 2 } , t \right\rangle$ 
B)  $\vec { a } ( t ) = \left\langle 0,6 \left( 1 + t ^ { 2 } \right) ^ { 2 } + 24 t ^ { 2 } \left( 1 + t ^ { 2 } \right) , 0 \right\rangle$ 
C)  $\vec { a } ( t ) = \left\langle 0,2 t \left( 1 + t ^ { 2 } \right) , 0 \right\rangle$ 
D)  $\vec { a } ( t ) = \left\langle 0,6 t ^ { 2 } ( 1 + 2 t ) ^ { 2 } , 0 \right\rangle$

Question 6

Evaluate and simplify the quantity. $$\langle t , 7 t , \cos ( t ) \rangle \times \left\langle 1 , t , t ^ { 2 } \right\rangle$$

Accepted Answer

A)  $\left\langle t \cos ( t ) - 7 t ^ { 2 } , t ^ { 3 } - \cos ( t ) , 7 t - t ^ { 3 } \right\rangle$ 
B)  $\left\langle 7 t ^ { 3 } - t \cos ( t ) , \cos ( t ) - t ^ { 3 } , t ^ { 2 } - 7 t \right\rangle$ 
C)  $\left\langle t , 7 t ^ { 3 } , t ^ { 2 } \cos ( t ) \right\rangle$ 
D)  $t + 7 t ^ { 3 } + t ^ { 2 } \cos ( t )$ 
A)  $\left\langle t \cos ( t ) - 7 t ^ { 2 } , t ^ { 3 } - \cos ( t ) , 7 t - t ^ { 3 } \right\rangle$ 
B)  $\left\langle 7 t ^ { 3 } - t \cos ( t ) , \cos ( t ) - t ^ { 3 } , t ^ { 2 } - 7 t \right\rangle$ 
C)  $\left\langle t , 7 t ^ { 3 } , t ^ { 2 } \cos ( t ) \right\rangle$ 
D)  $t + 7 t ^ { 3 } + t ^ { 2 } \cos ( t )$

Question 7

Find the parametric equations for the vector-valued function $\vec { r } ( t ) = \left\langle t , 2 t ^ { 2 } , 3 t ^ { 3 } \right\rangle$

Accepted Answer

The answer of Find the parametric equations for the vector-valued...

Question 8

For $\vec { r } ( t ) = \left\langle \sin \left( \frac { \pi } { 2 } t \right) , \cos \left( \frac { \pi } { 2 } t \right) , t \right\rangle$ find the displacement vector as $t$ goes from $t = - 1$ to $t = 1$

Accepted Answer

The answer of For \(\vec { r } ( t...

Question 9

Find the unit tangent for $\vec { r } ( t ) = \left\langle t , e ^ { t } \right\rangle$

Accepted Answer

The answer of Find the unit tangent for \(\vec {...

Question 10

Given the velocity vector $\vec { v } ( t ) = \left\langle \frac { 1 } { t + 1 } , e ^ { t } , t \right\rangle$ and $\vec { r } ( 0 ) = \langle 1,0,2 \rangle$ , find $\vec { r } ( t )$

Accepted Answer

The answer of Given the velocity vector \(\vec { v...

Question 11

Find the parametric equations for the vector-valued function $\vec { r } ( t ) = \left\langle 2 t , t ^ { 2 } \right\rangle$

Accepted Answer

The answer of Find the parametric equations for the vector-valued...

Question 12

Find the principle binormal unit vector for $\vec { r } ( t ) = \langle \sin ( t ) , \cos ( t ) , t \rangle$

Accepted Answer

The answer of Find the principle binormal unit vector for...

Question 13

Evaluate and simplify the quantity. $$\left\langle t , e ^ { t } , \cos ( t ) \right\rangle + \left\langle 4 , e ^ { t } , t ^ { 2 } \right\rangle$$

Accepted Answer

The answer of Evaluate and simplify the quantity. \[\left\langle t...

Question 14

Find the parametric equations for the vector-valued function $\vec { r } ( t ) = \left\langle \cos ( t ) , t ^ { 2 } - 2 \right\rangle$

Accepted Answer

A)  $x ( t ) = \cos ( t ) , y ( t ) = t ^ { 2 } - 2$ 
B)  $x ( t ) = t ^ { 2 } - 2 , y ( t ) = \cos ( t )$ 
C)  $x ( t ) = - \sin ( t ) , y ( t ) = 2 t$ 
D)  $x ( t ) = 2 t , y ( t ) = - \sin ( t )$ 
A)  $x ( t ) = \cos ( t ) , y ( t ) = t ^ { 2 } - 2$ 
B)  $x ( t ) = t ^ { 2 } - 2 , y ( t ) = \cos ( t )$ 
C)  $x ( t ) = - \sin ( t ) , y ( t ) = 2 t$ 
D)  $x ( t ) = 2 t , y ( t ) = - \sin ( t )$

Question 15

Find the parametric equations for the vector-valued function $\vec { r } ( t ) = \langle \sin ( t ) , 14 \rangle$

Accepted Answer

The answer of Find the parametric equations for the vector-valued...

Question 16

Give an arc length parameterization of $\vec { r } ( t ) = \left\langle e ^ { t } , 2 e ^ { t } \right\rangle$

Accepted Answer

The answer of Give an arc length parameterization of \(\vec...

Question 17

Find the curvature of the vector valued function: $$\vec { r } ( t ) = \langle \sin ( t ) , \cos ( t ) , t \rangle$$

Accepted Answer

The answer of Find the curvature of the vector valued...

Question 18

Evaluate and simplify the quantity. $$t $$

Accepted Answer

The answer of Evaluate and simplify the quantity. $$t $$...

Question 19

For $\vec { r } ( t ) = \langle \sin ( 3 t ) , \cos ( 3 t ) \rangle$ find the displacement vector as $t$ goes from $t = 0$ to $t = \pi$

Accepted Answer

A)  $\langle - 2,0 \rangle$ 
B)  $\langle - 1,0 \rangle$ 
C)  $\langle 1,0 \rangle$ 
D)  $\langle 1,1 \rangle$ 
A)  $\langle - 2,0 \rangle$ 
B)  $\langle - 1,0 \rangle$ 
C)  $\langle 1,0 \rangle$ 
D)  $\langle 1,1 \rangle$

Question 20

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 1 + \cos ( 3 t ) , 1 + \sin ( 3 t ) , 2 \rangle$ over $[ 1,5 ]$

Accepted Answer

The answer of Find the arc length of the curve...

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 1 + \cos ( 3 t ) , 1 + \sin ( 3 t ) \rangle$ over $[ 4,7 ]$

Find an equation for the line tangent to the given curve at the specified point. $\vec { r } ( t ) = \left\langle e ^ { t } , \cos ( t ) , t \right\rangle , ( 1,1,0 )$

Find the unit tangent for $\vec { r } ( t ) = \left\langle \cos ( 3 t ) , \sin ( 3 t ) , e ^ { t } \right\rangle$

Find the tangential and normal components for the position function $\vec { r } ( t ) = \left\langle t , e ^ { t } , t ^ { 2 } \right\rangle$

Find the acceleration for the given position vector, $\vec { r } ( t ) = \left\langle 2 t , \left( 1 + t ^ { 2 } \right) ^ { 3 } , t \right\rangle$

Evaluate and simplify the quantity. $\langle t , 7 t , \cos ( t ) \rangle \times \left\langle 1 , t , t ^ { 2 } \right\rangle$

Find the parametric equations for the vector-valued function $\vec { r } ( t ) = \left\langle t , 2 t ^ { 2 } , 3 t ^ { 3 } \right\rangle$

For $\vec { r } ( t ) = \left\langle \sin \left( \frac { \pi } { 2 } t \right) , \cos \left( \frac { \pi } { 2 } t \right) , t \right\rangle$ find the displacement vector as $t$ goes from $t = - 1$ to $t = 1$

Find the unit tangent for $\vec { r } ( t ) = \left\langle t , e ^ { t } \right\rangle$

Given the velocity vector $\vec { v } ( t ) = \left\langle \frac { 1 } { t + 1 } , e ^ { t } , t \right\rangle$ and $\vec { r } ( 0 ) = \langle 1,0,2 \rangle$ , find $\vec { r } ( t )$

Find the parametric equations for the vector-valued function $\vec { r } ( t ) = \left\langle 2 t , t ^ { 2 } \right\rangle$

Find the principle binormal unit vector for $\vec { r } ( t ) = \langle \sin ( t ) , \cos ( t ) , t \rangle$

Evaluate and simplify the quantity. $\left\langle t , e ^ { t } , \cos ( t ) \right\rangle + \left\langle 4 , e ^ { t } , t ^ { 2 } \right\rangle$

Find the parametric equations for the vector-valued function $\vec { r } ( t ) = \left\langle \cos ( t ) , t ^ { 2 } - 2 \right\rangle$

Find the parametric equations for the vector-valued function $\vec { r } ( t ) = \langle \sin ( t ) , 14 \rangle$

Give an arc length parameterization of $\vec { r } ( t ) = \left\langle e ^ { t } , 2 e ^ { t } \right\rangle$

Find the curvature of the vector valued function: $\vec { r } ( t ) = \langle \sin ( t ) , \cos ( t ) , t \rangle$

Evaluate and simplify the quantity. $t < \cos ( t ) , e ^ { t } >$

For $\vec { r } ( t ) = \langle \sin ( 3 t ) , \cos ( 3 t ) \rangle$ find the displacement vector as $t$ goes from $t = 0$ to $t = \pi$

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 1 + \cos ( 3 t ) , 1 + \sin ( 3 t ) , 2 \rangle$ over $[ 1,5 ]$

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Exam 11: Vector Functions

Find the arc length of the curve defined by r⃗(t)=⟨1+cos⁡(3t),1+sin⁡(3t)⟩\vec { r } ( t ) = \langle 1 + \cos ( 3 t ) , 1 + \sin ( 3 t ) \rangler(t)=⟨1+cos(3t),1+sin(3t)⟩ over [4,7][ 4,7 ][4,7]

Find an equation for the line tangent to the given curve at the specified point. r⃗(t)=⟨et,cos⁡(t),t⟩,(1,1,0)\vec { r } ( t ) = \left\langle e ^ { t } , \cos ( t ) , t \right\rangle , ( 1,1,0 )r(t)=⟨et,cos(t),t⟩,(1,1,0)

Find the unit tangent for r⃗(t)=⟨cos⁡(3t),sin⁡(3t),et⟩\vec { r } ( t ) = \left\langle \cos ( 3 t ) , \sin ( 3 t ) , e ^ { t } \right\rangler(t)=⟨cos(3t),sin(3t),et⟩

Find the tangential and normal components for the position function r⃗(t)=⟨t,et,t2⟩\vec { r } ( t ) = \left\langle t , e ^ { t } , t ^ { 2 } \right\rangler(t)=⟨t,et,t2⟩

Find the acceleration for the given position vector, r⃗(t)=⟨2t,(1+t2)3,t⟩\vec { r } ( t ) = \left\langle 2 t , \left( 1 + t ^ { 2 } \right) ^ { 3 } , t \right\rangler(t)=⟨2t,(1+t2)3,t⟩

Evaluate and simplify the quantity. ⟨t,7t,cos⁡(t)⟩×⟨1,t,t2⟩\langle t , 7 t , \cos ( t ) \rangle \times \left\langle 1 , t , t ^ { 2 } \right\rangle⟨t,7t,cos(t)⟩×⟨1,t,t2⟩

Find the parametric equations for the vector-valued function r⃗(t)=⟨t,2t2,3t3⟩\vec { r } ( t ) = \left\langle t , 2 t ^ { 2 } , 3 t ^ { 3 } \right\rangler(t)=⟨t,2t2,3t3⟩

For r⃗(t)=⟨sin⁡(π2t),cos⁡(π2t),t⟩\vec { r } ( t ) = \left\langle \sin \left( \frac { \pi } { 2 } t \right) , \cos \left( \frac { \pi } { 2 } t \right) , t \right\rangler(t)=⟨sin(2π​t),cos(2π​t),t⟩ find the displacement vector as ttt goes from t=−1t = - 1t=−1 to t=1t = 1t=1

Find the unit tangent for r⃗(t)=⟨t,et⟩\vec { r } ( t ) = \left\langle t , e ^ { t } \right\rangler(t)=⟨t,et⟩

Given the velocity vector v⃗(t)=⟨1t+1,et,t⟩\vec { v } ( t ) = \left\langle \frac { 1 } { t + 1 } , e ^ { t } , t \right\ranglev(t)=⟨t+11​,et,t⟩ and r⃗(0)=⟨1,0,2⟩\vec { r } ( 0 ) = \langle 1,0,2 \rangler(0)=⟨1,0,2⟩ , find r⃗(t)\vec { r } ( t )r(t)

Find the parametric equations for the vector-valued function r⃗(t)=⟨2t,t2⟩\vec { r } ( t ) = \left\langle 2 t , t ^ { 2 } \right\rangler(t)=⟨2t,t2⟩

Find the principle binormal unit vector for r⃗(t)=⟨sin⁡(t),cos⁡(t),t⟩\vec { r } ( t ) = \langle \sin ( t ) , \cos ( t ) , t \rangler(t)=⟨sin(t),cos(t),t⟩

Evaluate and simplify the quantity. ⟨t,et,cos⁡(t)⟩+⟨4,et,t2⟩\left\langle t , e ^ { t } , \cos ( t ) \right\rangle + \left\langle 4 , e ^ { t } , t ^ { 2 } \right\rangle⟨t,et,cos(t)⟩+⟨4,et,t2⟩

Find the parametric equations for the vector-valued function r⃗(t)=⟨cos⁡(t),t2−2⟩\vec { r } ( t ) = \left\langle \cos ( t ) , t ^ { 2 } - 2 \right\rangler(t)=⟨cos(t),t2−2⟩

Find the parametric equations for the vector-valued function r⃗(t)=⟨sin⁡(t),14⟩\vec { r } ( t ) = \langle \sin ( t ) , 14 \rangler(t)=⟨sin(t),14⟩

Give an arc length parameterization of r⃗(t)=⟨et,2et⟩\vec { r } ( t ) = \left\langle e ^ { t } , 2 e ^ { t } \right\rangler(t)=⟨et,2et⟩

Find the curvature of the vector valued function: r⃗(t)=⟨sin⁡(t),cos⁡(t),t⟩\vec { r } ( t ) = \langle \sin ( t ) , \cos ( t ) , t \rangler(t)=⟨sin(t),cos(t),t⟩

Evaluate and simplify the quantity. t<cos⁡(t),et>t < \cos ( t ) , e ^ { t } >t<cos(t),et>

For r⃗(t)=⟨sin⁡(3t),cos⁡(3t)⟩\vec { r } ( t ) = \langle \sin ( 3 t ) , \cos ( 3 t ) \rangler(t)=⟨sin(3t),cos(3t)⟩ find the displacement vector as ttt goes from t=0t = 0t=0 to t=πt = \pit=π

Find the arc length of the curve defined by r⃗(t)=⟨1+cos⁡(3t),1+sin⁡(3t),2⟩\vec { r } ( t ) = \langle 1 + \cos ( 3 t ) , 1 + \sin ( 3 t ) , 2 \rangler(t)=⟨1+cos(3t),1+sin(3t),2⟩ over [1,5][ 1,5 ][1,5]

Limits

Derivatives

Applications of the Derivative

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vectors

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 1 + \cos ( 3 t ) , 1 + \sin ( 3 t ) \rangle$ over $[ 4,7 ]$

Find an equation for the line tangent to the given curve at the specified point. $\vec { r } ( t ) = \left\langle e ^ { t } , \cos ( t ) , t \right\rangle , ( 1,1,0 )$

Find the unit tangent for $\vec { r } ( t ) = \left\langle \cos ( 3 t ) , \sin ( 3 t ) , e ^ { t } \right\rangle$

Find the tangential and normal components for the position function $\vec { r } ( t ) = \left\langle t , e ^ { t } , t ^ { 2 } \right\rangle$

Find the acceleration for the given position vector, $\vec { r } ( t ) = \left\langle 2 t , \left( 1 + t ^ { 2 } \right) ^ { 3 } , t \right\rangle$

Evaluate and simplify the quantity. $\langle t , 7 t , \cos ( t ) \rangle \times \left\langle 1 , t , t ^ { 2 } \right\rangle$

Find the parametric equations for the vector-valued function $\vec { r } ( t ) = \left\langle t , 2 t ^ { 2 } , 3 t ^ { 3 } \right\rangle$

For $\vec { r } ( t ) = \left\langle \sin \left( \frac { \pi } { 2 } t \right) , \cos \left( \frac { \pi } { 2 } t \right) , t \right\rangle$ find the displacement vector as $t$ goes from $t = - 1$ to $t = 1$

Find the unit tangent for $\vec { r } ( t ) = \left\langle t , e ^ { t } \right\rangle$

Given the velocity vector $\vec { v } ( t ) = \left\langle \frac { 1 } { t + 1 } , e ^ { t } , t \right\rangle$ and $\vec { r } ( 0 ) = \langle 1,0,2 \rangle$ , find $\vec { r } ( t )$

Find the parametric equations for the vector-valued function $\vec { r } ( t ) = \left\langle 2 t , t ^ { 2 } \right\rangle$

Find the principle binormal unit vector for $\vec { r } ( t ) = \langle \sin ( t ) , \cos ( t ) , t \rangle$

Evaluate and simplify the quantity. $\left\langle t , e ^ { t } , \cos ( t ) \right\rangle + \left\langle 4 , e ^ { t } , t ^ { 2 } \right\rangle$

Find the parametric equations for the vector-valued function $\vec { r } ( t ) = \left\langle \cos ( t ) , t ^ { 2 } - 2 \right\rangle$

Find the parametric equations for the vector-valued function $\vec { r } ( t ) = \langle \sin ( t ) , 14 \rangle$

Give an arc length parameterization of $\vec { r } ( t ) = \left\langle e ^ { t } , 2 e ^ { t } \right\rangle$

Find the curvature of the vector valued function: $\vec { r } ( t ) = \langle \sin ( t ) , \cos ( t ) , t \rangle$

Evaluate and simplify the quantity. $t < \cos ( t ) , e ^ { t } >$

For $\vec { r } ( t ) = \langle \sin ( 3 t ) , \cos ( 3 t ) \rangle$ find the displacement vector as $t$ goes from $t = 0$ to $t = \pi$

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 1 + \cos ( 3 t ) , 1 + \sin ( 3 t ) , 2 \rangle$ over $[ 1,5 ]$