Question 1

Evaluate the limit: $$\lim _ { x \rightarrow 0 } \left\langle t , t ^ { 2 } , \cot ( t ) \right\rangle$$

Accepted Answer

DNE

Question 2

Find the derivative $\frac { d \vec { r } } { d s }$ $\vec { r } ( t ) = \left\langle t , t ^ { 3 } , \sqrt { t } \right\rangle , t = \left( s ^ { 2 } + 1 \right) ^ { 3 }$

Accepted Answer

A)  $\vec { r } ^ { \prime } ( s ) = \left\langle 1,3 \left( s ^ { 2 } + 1 \right) ^ { 6 } , \frac { 1 } { 2 } \left( s ^ { 2 } + 1 \right) ^ { - \frac { 3 } { 2 } } \right\rangle$ 
B)  $\vec { r } ^ { \prime } ( s ) = \left\langle 1,3 t ^ { 2 } , \frac { 1 } { 2 } t ^ { - \frac { 1 } { 2 } } \right\rangle$ 
C)  $\vec { r } ^ { \prime } ( s ) = \left\langle 1,3 \left( s ^ { 2 } + 1 \right) ^ { 6 } , \frac { 1 } { 2 } \left( s ^ { 2 } + 1 \right) ^ { - \frac { 3 } { 2 } } > 6 s \left( s ^ { 2 } + 1 \right) ^ { 2 } \right.$ 
D)  $\vec { r } ^ { \prime } ( s ) = \left\langle t , t ^ { 3 } , \sqrt { t } \right\rangle 6 s \left( s ^ { 2 } + 1 \right)$ 
A)  $\vec { r } ^ { \prime } ( s ) = \left\langle 1,3 \left( s ^ { 2 } + 1 \right) ^ { 6 } , \frac { 1 } { 2 } \left( s ^ { 2 } + 1 \right) ^ { - \frac { 3 } { 2 } } \right\rangle$ 
B)  $\vec { r } ^ { \prime } ( s ) = \left\langle 1,3 t ^ { 2 } , \frac { 1 } { 2 } t ^ { - \frac { 1 } { 2 } } \right\rangle$ 
C)  $\vec { r } ^ { \prime } ( s ) = \left\langle 1,3 \left( s ^ { 2 } + 1 \right) ^ { 6 } , \frac { 1 } { 2 } \left( s ^ { 2 } + 1 \right) ^ { - \frac { 3 } { 2 } } > 6 s \left( s ^ { 2 } + 1 \right) ^ { 2 } \right.$ 
D)  $\vec { r } ^ { \prime } ( s ) = \left\langle t , t ^ { 3 } , \sqrt { t } \right\rangle 6 s \left( s ^ { 2 } + 1 \right)$ C

Question 3

Evaluate and simplify the quantity. $$t $$

Accepted Answer

$\left\langle t \cos ( t ) , t \sin ( t ) , t e ^ { t } \right\rangle$

Question 4

Find the curvature of the vector valued function: $$\vec { r } ( t ) = \left\langle e ^ { t } , e ^ { 2 t } , e ^ { 3 t } \right\rangle$$

Accepted Answer

A)  $\kappa ( t ) = \frac { 1 } { 2 }$ 
B)  $\kappa ( t ) = \sqrt { 36 e ^ { 10 t } + 36 e ^ { 8 t } + 4 e ^ { 6 t } }$ 
C)  $\kappa ( t ) = \frac { 2 \sqrt { 9 e ^ { 4 t } + 9 e ^ { 2 t } + 1 } } { \left( 1 + 4 e ^ { 2 t } + 9 e ^ { 4 t } \right) ^ { \frac { 3 } { 2 } } }$ 
D)  $\kappa ( t ) = \sqrt { 9 e ^ { 4 t } + 9 e ^ { 4 t } + 1 }$ 
A)  $\kappa ( t ) = \frac { 1 } { 2 }$ 
B)  $\kappa ( t ) = \sqrt { 36 e ^ { 10 t } + 36 e ^ { 8 t } + 4 e ^ { 6 t } }$ 
C)  $\kappa ( t ) = \frac { 2 \sqrt { 9 e ^ { 4 t } + 9 e ^ { 2 t } + 1 } } { \left( 1 + 4 e ^ { 2 t } + 9 e ^ { 4 t } \right) ^ { \frac { 3 } { 2 } } }$ 
D)  $\kappa ( t ) = \sqrt { 9 e ^ { 4 t } + 9 e ^ { 4 t } + 1 }$

Question 5

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 1 + 7 \cos ( 3 t ) , 1 + 7 \sin ( 3 t ) \rangle$ over $[ 2,3 ]$

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

Find the derivative $\frac { d \vec { r } } { d s }$ $\vec { r } ( t ) = \left\langle \cos ( t ) , e ^ { t } , t \right\rangle , t = e ^ { s }$

Accepted Answer

The answer of Find the derivative \(\frac { d \vec...

Question 8

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

Accepted Answer

A)  $3 \pi$ 
B)  $\pi \sqrt { 10 }$ 
C)  $\pi \sqrt { 3 }$ 
D)  $10 \pi$ 
A)  $3 \pi$ 
B)  $\pi \sqrt { 10 }$ 
C)  $\pi \sqrt { 3 }$ 
D)  $10 \pi$

Question 9

Find the osculating plane for $\vec { r } ( t ) = \langle \cos ( 3 t ) , \sin ( 3 t ) , 3 \rangle$ for $t = 1$

Accepted Answer

The answer of Find the osculating plane for \(\vec {...

Question 10

Given the velocity vector $\vec { v } ( t ) = \left\langle 1 , t , t ^ { 2 } \right\rangle$ and $\vec { r } ( 0 ) = \langle 1,2,3 \rangle$ , find $\vec { r } ( t )$

Accepted Answer

A)  $\vec { r } ( t ) = \left\langle t , t ^ { 2 } , t ^ { 3 } \right\rangle + \langle 1,2,3 \rangle$ 
B)  $\vec { r } ( t ) = \left\langle t , \frac { t ^ { 2 } } { 2 } , \frac { t ^ { 3 } } { 3 } \right\rangle + \left\langle 0 , \frac { 3 } { 2 } , \frac { 8 } { 3 } \right\rangle$ 
C)  $\vec { r } ( t ) = \left\langle 1 , t ^ { 2 } , t ^ { 3 } \right\rangle + \langle 0,2,3 \rangle$ 
D)  $\vec { r } ( t ) = \left\langle t , \frac { t ^ { 2 } } { 2 } , \frac { t ^ { 3 } } { 3 } \right\rangle + \langle 1,2,3 \rangle$ 
A)  $\vec { r } ( t ) = \left\langle t , t ^ { 2 } , t ^ { 3 } \right\rangle + \langle 1,2,3 \rangle$ 
B)  $\vec { r } ( t ) = \left\langle t , \frac { t ^ { 2 } } { 2 } , \frac { t ^ { 3 } } { 3 } \right\rangle + \left\langle 0 , \frac { 3 } { 2 } , \frac { 8 } { 3 } \right\rangle$ 
C)  $\vec { r } ( t ) = \left\langle 1 , t ^ { 2 } , t ^ { 3 } \right\rangle + \langle 0,2,3 \rangle$ 
D)  $\vec { r } ( t ) = \left\langle t , \frac { t ^ { 2 } } { 2 } , \frac { t ^ { 3 } } { 3 } \right\rangle + \langle 1,2,3 \rangle$

Question 11

Evaluate the integral. $\int \left\langle t , e ^ { t } , \sin ( t ) \right\rangle d t$

Accepted Answer

The answer of Evaluate the integral. \(\int \left\langle t ,...

Question 12

Evaluate and simplify the quantity. $$\left\langle t , e ^ { t } \right\rangle \cdot \left\langle - e ^ { t } , - t \right\rangle$$

Accepted Answer

A)  $t ^ { 2 } + e ^ { 2 t }$ 
B)  $2 t e ^ { 2 t }$ 
C)  0 
D)  $- 2 t e ^ { t }$ 
A)  $t ^ { 2 } + e ^ { 2 t }$ 
B)  $2 t e ^ { 2 t }$ 
C)  0 
D)  $- 2 t e ^ { t }$

Question 13

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

Accepted Answer

The answer of Find the acceleration for the given position...

Question 14

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

Accepted Answer

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

Question 15

Find $\vec { r } ( t ) = \langle x ( t ) , y ( t ) \rangle$ determined by $\begin{array} { l } 
x ^ { \prime } ( t ) = x ( t ) , y ^ { \prime } ( t ) = t \
x ( 0 ) = 1 , y ( 0 ) = 1
\end{array}$

Accepted Answer

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

Question 16

Find the velocity for the given position vector, $\vec { r } ( t ) = \left\langle \ln ( t ) , e ^ { t ^ { 2 } } , \ln \left( e ^ { t } \right) \right\rangle$

Accepted Answer

The answer of Find the velocity for the given position...

Question 17

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

Accepted Answer

A)   
B)  $\langle 0,1 \rangle$ 
C)   
D)  $\left. < \frac { \pi ^ { 2 } } { 4 } , 0 \right\rangle$ 
A)   
B)  $\langle 0,1 \rangle$ 
C)   
D)  $\left. < \frac { \pi ^ { 2 } } { 4 } , 0 \right\rangle$

Question 18

Evaluate the limit: $$\lim _ { x \rightarrow 0 } \left\langle t ^ { 2 } , \cos ( t ) \right\rangle$$

Accepted Answer

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

Question 19

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

Accepted Answer

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

Question 20

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 2 \sin ( 5 t ) , 2 \cos ( 5 t ) \rangle$ over $[ 2,4 ]$

Accepted Answer

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

Evaluate the limit: $\lim _ { x \rightarrow 0 } \left\langle t , t ^ { 2 } , \cot ( t ) \right\rangle$

Find the derivative $\frac { d \vec { r } } { d s }$ $\vec { r } ( t ) = \left\langle t , t ^ { 3 } , \sqrt { t } \right\rangle , t = \left( s ^ { 2 } + 1 \right) ^ { 3 }$

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

Find the curvature of the vector valued function: $\vec { r } ( t ) = \left\langle e ^ { t } , e ^ { 2 t } , e ^ { 3 t } \right\rangle$

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 1 + 7 \cos ( 3 t ) , 1 + 7 \sin ( 3 t ) \rangle$ over $[ 2,3 ]$

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

Find the derivative $\frac { d \vec { r } } { d s }$ $\vec { r } ( t ) = \left\langle \cos ( t ) , e ^ { t } , t \right\rangle , t = e ^ { s }$

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

Find the osculating plane for $\vec { r } ( t ) = \langle \cos ( 3 t ) , \sin ( 3 t ) , 3 \rangle$ for $t = 1$

Given the velocity vector $\vec { v } ( t ) = \left\langle 1 , t , t ^ { 2 } \right\rangle$ and $\vec { r } ( 0 ) = \langle 1,2,3 \rangle$ , find $\vec { r } ( t )$

Evaluate the integral. $\int \left\langle t , e ^ { t } , \sin ( t ) \right\rangle d t$

Evaluate and simplify the quantity. $\left\langle t , e ^ { t } \right\rangle \cdot \left\langle - e ^ { t } , - t \right\rangle$

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

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

Find $\vec { r } ( t ) = \langle x ( t ) , y ( t ) \rangle$ determined by (t)=x(t),(t)=t x(0)=1,y(0)=1

Find the velocity for the given position vector, $\vec { r } ( t ) = \left\langle \ln ( t ) , e ^ { t ^ { 2 } } , \ln \left( e ^ { t } \right) \right\rangle$

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

Evaluate the limit: $\lim _ { x \rightarrow 0 } \left\langle t ^ { 2 } , \cos ( t ) \right\rangle$

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

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 2 \sin ( 5 t ) , 2 \cos ( 5 t ) \rangle$ over $[ 2,4 ]$

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Exam 11: Vector Functions

Evaluate the limit: lim⁡x→0⟨t,t2,cot⁡(t)⟩\lim _ { x \rightarrow 0 } \left\langle t , t ^ { 2 } , \cot ( t ) \right\ranglelimx→0​⟨t,t2,cot(t)⟩

Find the derivative dr⃗ds\frac { d \vec { r } } { d s }dsdr​ r⃗(t)=⟨t,t3,t⟩,t=(s2+1)3\vec { r } ( t ) = \left\langle t , t ^ { 3 } , \sqrt { t } \right\rangle , t = \left( s ^ { 2 } + 1 \right) ^ { 3 }r(t)=⟨t,t3,t​⟩,t=(s2+1)3

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

Find the curvature of the vector valued function: r⃗(t)=⟨et,e2t,e3t⟩\vec { r } ( t ) = \left\langle e ^ { t } , e ^ { 2 t } , e ^ { 3 t } \right\rangler(t)=⟨et,e2t,e3t⟩

Find the arc length of the curve defined by r⃗(t)=⟨1+7cos⁡(3t),1+7sin⁡(3t)⟩\vec { r } ( t ) = \langle 1 + 7 \cos ( 3 t ) , 1 + 7 \sin ( 3 t ) \rangler(t)=⟨1+7cos(3t),1+7sin(3t)⟩ over [2,3][ 2,3 ][2,3]

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

Find the derivative dr⃗ds\frac { d \vec { r } } { d s }dsdr​ r⃗(t)=⟨cos⁡(t),et,t⟩,t=es\vec { r } ( t ) = \left\langle \cos ( t ) , e ^ { t } , t \right\rangle , t = e ^ { s }r(t)=⟨cos(t),et,t⟩,t=es

Find the distance traveled by a particle moving on the curve given by r⃗(t)=⟨sin⁡(3t),cos⁡(3t),t⟩\vec { r } ( t ) = \langle \sin ( 3 t ) , \cos ( 3 t ) , t \rangler(t)=⟨sin(3t),cos(3t),t⟩ as ttt goes from t=0t = 0t=0 to t=πt = \pit=π

Find the osculating plane 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⟩ for t=1t = 1t=1

Given the velocity vector v⃗(t)=⟨1,t,t2⟩\vec { v } ( t ) = \left\langle 1 , t , t ^ { 2 } \right\ranglev(t)=⟨1,t,t2⟩ and r⃗(0)=⟨1,2,3⟩\vec { r } ( 0 ) = \langle 1,2,3 \rangler(0)=⟨1,2,3⟩ , find r⃗(t)\vec { r } ( t )r(t)

Evaluate the integral. ∫⟨t,et,sin⁡(t)⟩dt\int \left\langle t , e ^ { t } , \sin ( t ) \right\rangle d t∫⟨t,et,sin(t)⟩dt

Evaluate and simplify the quantity. ⟨t,et⟩⋅⟨−et,−t⟩\left\langle t , e ^ { t } \right\rangle \cdot \left\langle - e ^ { t } , - t \right\rangle⟨t,et⟩⋅⟨−et,−t⟩

Find the acceleration for the given position vector, r⃗(t)=⟨ln⁡(t),et2,ln⁡(et)⟩\vec { r } ( t ) = \left\langle \ln ( t ) , e ^ { t ^ { 2 } } , \ln \left( e ^ { t } \right) \right\rangler(t)=⟨ln(t),et2,ln(et)⟩

Find the principle normal 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⟩

Find r⃗(t)=⟨x(t),y(t)⟩\vec { r } ( t ) = \langle x ( t ) , y ( t ) \rangler(t)=⟨x(t),y(t)⟩ determined by (t)=x(t),(t)=t x(0)=1,y(0)=1

Find the velocity for the given position vector, r⃗(t)=⟨ln⁡(t),et2,ln⁡(et)⟩\vec { r } ( t ) = \left\langle \ln ( t ) , e ^ { t ^ { 2 } } , \ln \left( e ^ { t } \right) \right\rangler(t)=⟨ln(t),et2,ln(et)⟩

Evaluate the limit: lim⁡x→π2⟨t2,cos⁡(t)⟩\lim _ { x \rightarrow \frac { \pi } { 2 } } \left\langle t ^ { 2 } , \cos ( t ) \right\ranglelimx→2π​​⟨t2,cos(t)⟩

Evaluate the limit: lim⁡x→0⟨t2,cos⁡(t)⟩\lim _ { x \rightarrow 0 } \left\langle t ^ { 2 } , \cos ( t ) \right\ranglelimx→0​⟨t2,cos(t)⟩

Find the arc length of the curve defined by r⃗(t)=⟨2sin⁡(5t),2cos⁡(5t)⟩\vec { r } ( t ) = \langle 2 \sin ( 5 t ) , 2 \cos ( 5 t ) \rangler(t)=⟨2sin(5t),2cos(5t)⟩ over [2,4][ 2,4 ][2,4]

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

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Evaluate the limit: $\lim _ { x \rightarrow 0 } \left\langle t , t ^ { 2 } , \cot ( t ) \right\rangle$

Find the derivative $\frac { d \vec { r } } { d s }$ $\vec { r } ( t ) = \left\langle t , t ^ { 3 } , \sqrt { t } \right\rangle , t = \left( s ^ { 2 } + 1 \right) ^ { 3 }$

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

Find the curvature of the vector valued function: $\vec { r } ( t ) = \left\langle e ^ { t } , e ^ { 2 t } , e ^ { 3 t } \right\rangle$

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 1 + 7 \cos ( 3 t ) , 1 + 7 \sin ( 3 t ) \rangle$ over $[ 2,3 ]$

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

Find the derivative $\frac { d \vec { r } } { d s }$ $\vec { r } ( t ) = \left\langle \cos ( t ) , e ^ { t } , t \right\rangle , t = e ^ { s }$

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

Find the osculating plane for $\vec { r } ( t ) = \langle \cos ( 3 t ) , \sin ( 3 t ) , 3 \rangle$ for $t = 1$

Given the velocity vector $\vec { v } ( t ) = \left\langle 1 , t , t ^ { 2 } \right\rangle$ and $\vec { r } ( 0 ) = \langle 1,2,3 \rangle$ , find $\vec { r } ( t )$

Evaluate the integral. $\int \left\langle t , e ^ { t } , \sin ( t ) \right\rangle d t$

Evaluate and simplify the quantity. $\left\langle t , e ^ { t } \right\rangle \cdot \left\langle - e ^ { t } , - t \right\rangle$

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

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

Find $\vec { r } ( t ) = \langle x ( t ) , y ( t ) \rangle$ determined by (t)=x(t),(t)=t x(0)=1,y(0)=1

Find the velocity for the given position vector, $\vec { r } ( t ) = \left\langle \ln ( t ) , e ^ { t ^ { 2 } } , \ln \left( e ^ { t } \right) \right\rangle$

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

Evaluate the limit: $\lim _ { x \rightarrow 0 } \left\langle t ^ { 2 } , \cos ( t ) \right\rangle$

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 2 \sin ( 5 t ) , 2 \cos ( 5 t ) \rangle$ over $[ 2,4 ]$