Question 1

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 1 + \cos ( 3 t ) , 1 + \sin ( 3 t ) , 5 \rangle$ over $[ 4,7 ]$

Accepted Answer

A)  4 
B)  7 
C)  9 
D)  11 
A)  4 
B)  7 
C)  9 
D)  11

Question 2

Find the arc length of the curve defined by $\vec { r } ( t ) = \left\langle e ^ { t } \cos ( t ) , e ^ { t } , e ^ { t } \sin ( t ) \right\rangle$ over $[ \ln ( 2 ) , \ln ( 5 ) ]$

Accepted Answer

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

Question 3

Evaluate the limit: $$\lim _ { x \rightarrow 1 } \left\langle \frac { t ^ { 2 } - 1 } { t - 1 } , \cos ( \pi t ) , t ^ { 2 } \right\rangle$$

Accepted Answer

The answer of Evaluate the limit: \[\lim _ { x...

Question 4

Find the principle binormal unit vector for $\vec { r } ( t ) = \langle \cos ( 3 t ) , \sin ( 3 t ) , 3 \rangle$

Accepted Answer

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

Question 5

Evaluate the integral. $\int  d t$

Accepted Answer

A)  $\left.  + \vec { c }$ 
C)  $\left\langle \frac { t ^ { 2 } } { 2 } , - \sin ( t ) , e ^ { t } \right\rangle$ 
D)  $ + \vec { c }$ 
A)  $\left.  + \vec { c }$ 
C)  $\left\langle \frac { t ^ { 2 } } { 2 } , - \sin ( t ) , e ^ { t } \right\rangle$ 
D)  $ + \vec { c }$

Question 6

Give an arc length parameterization of $\vec { r } ( t ) = \langle \sin ( 2 t ) , \cos ( 2 t ) \rangle$

Accepted Answer

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

Question 7

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

Accepted Answer

A)  $T ( t ) = \left\langle 1 , e ^ { t } \right\rangle$ 
B)  $\vec { T } ( t ) = \frac { \left\langle 1 , e ^ { t } \right\rangle } { \sqrt { 1 + e ^ { t } } }$ 
C)  $\vec { T } ( t ) = \frac { \left\langle 1 , e ^ { t } \right\rangle } { \sqrt { 1 + e ^ { 2 t } } }$ 
D)  $\vec { T } ( t ) = \frac { \left\langle 1 , e ^ { t } \right\rangle } { \sqrt { t ^ { 2 } + e ^ { 2 t } } }$ 
A)  $T ( t ) = \left\langle 1 , e ^ { t } \right\rangle$ 
B)  $\vec { T } ( t ) = \frac { \left\langle 1 , e ^ { t } \right\rangle } { \sqrt { 1 + e ^ { t } } }$ 
C)  $\vec { T } ( t ) = \frac { \left\langle 1 , e ^ { t } \right\rangle } { \sqrt { 1 + e ^ { 2 t } } }$ 
D)  $\vec { T } ( t ) = \frac { \left\langle 1 , e ^ { t } \right\rangle } { \sqrt { t ^ { 2 } + e ^ { 2 t } } }$

Question 8

Give an arc length parameterization of $\vec { r } ( t ) = \langle 1 + \sin ( 2 t ) , 2 + \cos ( 2 t ) , 3 \rangle$

Accepted Answer

A)  $\vec { r } ( s ) = \langle \cos ( s ) , \sin ( s ) , 3 \rangle$ 
B)  $\vec { r } ( s ) = \langle 1 + \cos ( 2 s ) , 2 + \sin ( 2 s ) , 3 \rangle$ 
C)  $\vec { r } ( s ) = \langle 1 + \cos ( s ) , 2 + \sin ( s ) , 3 \rangle$ 
D)  $\vec { r } ( s ) = \langle \cos ( 2 s ) , \sin ( 2 s ) , 0 \rangle$ 
A)  $\vec { r } ( s ) = \langle \cos ( s ) , \sin ( s ) , 3 \rangle$ 
B)  $\vec { r } ( s ) = \langle 1 + \cos ( 2 s ) , 2 + \sin ( 2 s ) , 3 \rangle$ 
C)  $\vec { r } ( s ) = \langle 1 + \cos ( s ) , 2 + \sin ( s ) , 3 \rangle$ 
D)  $\vec { r } ( s ) = \langle \cos ( 2 s ) , \sin ( 2 s ) , 0 \rangle$

Question 9

Evaluate and simplify the quantity. $$3 $$

Accepted Answer

The answer of Evaluate and simplify the quantity. $$3 $$...

Question 10

Find the distance traveled by a particle moving on the curve given by $\vec { r } ( t ) = \left\langle \sin \left( \frac { \pi } { 2 } t \right) , \cos \left( \frac { \pi } { 2 } t \right) , t \right\rangle$ as $t$ goes from $t = - 1$ to $t = 1$

Accepted Answer

The answer of Find the distance traveled by a particle...

Question 11

Evaluate and simplify the quantity. $$\langle \cos ( t ) , \sin ( t ) , 3 \rangle \cdot \langle \cos ( t ) , \sin ( t ) , t \rangle$$

Accepted Answer

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

Question 12

Find the distance traveled by a particle moving on the curve given by $\vec { r } ( t ) = \langle \sin ( 3 t ) , \cos ( 3 t ) \rangle$ find the displacement vector as $t$ goes from $t = 0$ to $t = \pi$

Accepted Answer

A)  $\pi \sqrt { 3 }$ 
B)  $3 \pi$ 
C)  $2$ 
D)  $2 \pi$ 
A)  $\pi \sqrt { 3 }$ 
B)  $3 \pi$ 
C)  $2$ 
D)  $2 \pi$

Question 13

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

Accepted Answer

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

Question 14

Given the acceleration vector $\vec { a } ( t ) = \left\langle 2 t , \cos ( t ) , 12 t ^ { 2 } \right\rangle$ and initial velocity and position $\vec { v } ( 0 ) = \langle 1,2,1 \rangle$ and $\vec { r } ( 0 ) = \langle 1,1,1 \rangle$ , find $\vec { r } ( t )$

Accepted Answer

The answer of Given the acceleration vector \(\vec { a...

Question 15

Find the tangential and normal components of acceleration for the position function $\vec { r } ( t ) = \langle \cos ( 3 t ) , t , \sin ( 3 t ) \rangle$

Accepted Answer

None

Question 16

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

Accepted Answer

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

Question 17

Evaluate the limit: $$\lim _ { x \rightarrow 1 } \left\langle t , e ^ { t } \right\rangle$$

Accepted Answer

The answer of Evaluate the limit: \[\lim _ { x...

Question 18

Find the principle normal unit vector for $\vec { r } ( t ) = \langle \cos ( 3 t ) , \sin ( 3 t ) , 3 \rangle$

Accepted Answer

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

Question 19

Find the curvature of the vector valued function: $$\vec { r } ( t ) = \left\langle 1 , t ^ { 2 } , t ^ { 2 } \right\rangle$$

Accepted Answer

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

Question 20

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

Accepted Answer

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

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 1 + \cos ( 3 t ) , 1 + \sin ( 3 t ) , 5 \rangle$ over $[ 4,7 ]$

Find the arc length of the curve defined by $\vec { r } ( t ) = \left\langle e ^ { t } \cos ( t ) , e ^ { t } , e ^ { t } \sin ( t ) \right\rangle$ over $[ \ln ( 2 ) , \ln ( 5 ) ]$

Evaluate the limit: $\lim _ { x \rightarrow 1 } \left\langle \frac { t ^ { 2 } - 1 } { t - 1 } , \cos ( \pi t ) , t ^ { 2 } \right\rangle$

Find the principle binormal unit vector for $\vec { r } ( t ) = \langle \cos ( 3 t ) , \sin ( 3 t ) , 3 \rangle$

Evaluate the integral. $\int < t , \cot ( t ) , e ^ { t } > d t$

Give an arc length parameterization of $\vec { r } ( t ) = \langle \sin ( 2 t ) , \cos ( 2 t ) \rangle$

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

Give an arc length parameterization of $\vec { r } ( t ) = \langle 1 + \sin ( 2 t ) , 2 + \cos ( 2 t ) , 3 \rangle$

Evaluate and simplify the quantity. $3 < 2 t , \cos ( t ) >$

Find the distance traveled by a particle moving on the curve given by $\vec { r } ( t ) = \left\langle \sin \left( \frac { \pi } { 2 } t \right) , \cos \left( \frac { \pi } { 2 } t \right) , t \right\rangle$ as $t$ goes from $t = - 1$ to $t = 1$

Evaluate and simplify the quantity. $\langle \cos ( t ) , \sin ( t ) , 3 \rangle \cdot \langle \cos ( t ) , \sin ( t ) , t \rangle$

Find the distance traveled by a particle moving on the curve given by $\vec { r } ( t ) = \langle \sin ( 3 t ) , \cos ( 3 t ) \rangle$ find the displacement vector as $t$ goes from $t = 0$ to $t = \pi$

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

Given the acceleration vector $\vec { a } ( t ) = \left\langle 2 t , \cos ( t ) , 12 t ^ { 2 } \right\rangle$ and initial velocity and position $\vec { v } ( 0 ) = \langle 1,2,1 \rangle$ and $\vec { r } ( 0 ) = \langle 1,1,1 \rangle$ , find $\vec { r } ( t )$

Find the tangential and normal components of acceleration for the position function $\vec { r } ( t ) = \langle \cos ( 3 t ) , t , \sin ( 3 t ) \rangle$

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

Evaluate the limit: $\lim _ { x \rightarrow 1 } \left\langle t , e ^ { t } \right\rangle$

Find the principle normal unit vector for $\vec { r } ( t ) = \langle \cos ( 3 t ) , \sin ( 3 t ) , 3 \rangle$

Find the curvature of the vector valued function: $\vec { r } ( t ) = \left\langle 1 , t ^ { 2 } , t ^ { 2 } \right\rangle$

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

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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),5⟩\vec { r } ( t ) = \langle 1 + \cos ( 3 t ) , 1 + \sin ( 3 t ) , 5 \rangler(t)=⟨1+cos(3t),1+sin(3t),5⟩ over [4,7][ 4,7 ][4,7]

Find the arc length of the curve defined by r⃗(t)=⟨etcos⁡(t),et,etsin⁡(t)⟩\vec { r } ( t ) = \left\langle e ^ { t } \cos ( t ) , e ^ { t } , e ^ { t } \sin ( t ) \right\rangler(t)=⟨etcos(t),et,etsin(t)⟩ over [ln⁡(2),ln⁡(5)][ \ln ( 2 ) , \ln ( 5 ) ][ln(2),ln(5)]

Evaluate the limit: lim⁡x→1⟨t2−1t−1,cos⁡(πt),t2⟩\lim _ { x \rightarrow 1 } \left\langle \frac { t ^ { 2 } - 1 } { t - 1 } , \cos ( \pi t ) , t ^ { 2 } \right\ranglelimx→1​⟨t−1t2−1​,cos(πt),t2⟩

Find the principle binormal unit vector for r⃗(t)=⟨cos⁡(3t),sin⁡(3t),3⟩\vec { r } ( t ) = \langle \cos ( 3 t ) , \sin ( 3 t ) , 3 \rangler(t)=⟨cos(3t),sin(3t),3⟩

Evaluate the integral. ∫<t,cot⁡(t),et>dt\int < t , \cot ( t ) , e ^ { t } > d t∫<t,cot(t),et>dt

Give an arc length parameterization of r⃗(t)=⟨sin⁡(2t),cos⁡(2t)⟩\vec { r } ( t ) = \langle \sin ( 2 t ) , \cos ( 2 t ) \rangler(t)=⟨sin(2t),cos(2t)⟩

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

Give an arc length parameterization of r⃗(t)=⟨1+sin⁡(2t),2+cos⁡(2t),3⟩\vec { r } ( t ) = \langle 1 + \sin ( 2 t ) , 2 + \cos ( 2 t ) , 3 \rangler(t)=⟨1+sin(2t),2+cos(2t),3⟩

Evaluate and simplify the quantity. 3<2t,cos⁡(t)>3 < 2 t , \cos ( t ) >3<2t,cos(t)>

Evaluate and simplify the quantity. ⟨cos⁡(t),sin⁡(t),3⟩⋅⟨cos⁡(t),sin⁡(t),t⟩\langle \cos ( t ) , \sin ( t ) , 3 \rangle \cdot \langle \cos ( t ) , \sin ( t ) , t \rangle⟨cos(t),sin(t),3⟩⋅⟨cos(t),sin(t),t⟩

Find the distance traveled by a particle moving on the curve given by 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

Given the velocity vector v⃗(t)=⟨t,−sin⁡(t),t2⟩\vec { v } ( t ) = \left\langle t , - \sin ( t ) , t ^ { 2 } \right\ranglev(t)=⟨t,−sin(t),t2⟩ and r⃗(0)=⟨1,1,1⟩\vec { r } ( 0 ) = \langle 1,1,1 \rangler(0)=⟨1,1,1⟩ find r⃗(t)\vec { r } ( t )r(t)

Find the tangential and normal components of acceleration for the position function r⃗(t)=⟨cos⁡(3t),t,sin⁡(3t)⟩\vec { r } ( t ) = \langle \cos ( 3 t ) , t , \sin ( 3 t ) \rangler(t)=⟨cos(3t),t,sin(3t)⟩

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

Evaluate the limit: lim⁡x→1⟨t,et⟩\lim _ { x \rightarrow 1 } \left\langle t , e ^ { t } \right\ranglelimx→1​⟨t,et⟩

Find the principle normal unit vector for r⃗(t)=⟨cos⁡(3t),sin⁡(3t),3⟩\vec { r } ( t ) = \langle \cos ( 3 t ) , \sin ( 3 t ) , 3 \rangler(t)=⟨cos(3t),sin(3t),3⟩

Find the curvature of the vector valued function: r⃗(t)=⟨1,t2,t2⟩\vec { r } ( t ) = \left\langle 1 , t ^ { 2 } , t ^ { 2 } \right\rangler(t)=⟨1,t2,t2⟩

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

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 $\vec { r } ( t ) = \langle 1 + \cos ( 3 t ) , 1 + \sin ( 3 t ) , 5 \rangle$ over $[ 4,7 ]$

Find the arc length of the curve defined by $\vec { r } ( t ) = \left\langle e ^ { t } \cos ( t ) , e ^ { t } , e ^ { t } \sin ( t ) \right\rangle$ over $[ \ln ( 2 ) , \ln ( 5 ) ]$

Evaluate the limit: $\lim _ { x \rightarrow 1 } \left\langle \frac { t ^ { 2 } - 1 } { t - 1 } , \cos ( \pi t ) , t ^ { 2 } \right\rangle$

Find the principle binormal unit vector for $\vec { r } ( t ) = \langle \cos ( 3 t ) , \sin ( 3 t ) , 3 \rangle$

Evaluate the integral. $\int < t , \cot ( t ) , e ^ { t } > d t$

Give an arc length parameterization of $\vec { r } ( t ) = \langle \sin ( 2 t ) , \cos ( 2 t ) \rangle$

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

Give an arc length parameterization of $\vec { r } ( t ) = \langle 1 + \sin ( 2 t ) , 2 + \cos ( 2 t ) , 3 \rangle$

Evaluate and simplify the quantity. $3 < 2 t , \cos ( t ) >$

Evaluate and simplify the quantity. $\langle \cos ( t ) , \sin ( t ) , 3 \rangle \cdot \langle \cos ( t ) , \sin ( t ) , t \rangle$

Find the distance traveled by a particle moving on the curve given by $\vec { r } ( t ) = \langle \sin ( 3 t ) , \cos ( 3 t ) \rangle$ find the displacement vector as $t$ goes from $t = 0$ to $t = \pi$

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

Find the tangential and normal components of acceleration for the position function $\vec { r } ( t ) = \langle \cos ( 3 t ) , t , \sin ( 3 t ) \rangle$

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

Evaluate the limit: $\lim _ { x \rightarrow 1 } \left\langle t , e ^ { t } \right\rangle$

Find the principle normal unit vector for $\vec { r } ( t ) = \langle \cos ( 3 t ) , \sin ( 3 t ) , 3 \rangle$

Find the curvature of the vector valued function: $\vec { r } ( t ) = \left\langle 1 , t ^ { 2 } , t ^ { 2 } \right\rangle$

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