Exam 10: Vector Functions

A particle is moving along the curve described by the parametric equations $x = 5 t , y = 2 t ^ { 3 } , z = \frac { 3 } { 5 } t ^ { 5 }$ . Determine the velocity and acceleration vectors as well as the speed of the particle when t = 3.

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Question 1

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$\mathbf { v } ( 3 ) = 5 \mathbf { i } + 54 \mathbf { j } + 243 \mathbf { k } ; \mathbf { a } ( 3 ) = 36 \mathbf { j } + 324 \mathbf { k } ; | \mathbf { v } ( 3 ) | = \sqrt { 61,990 }$

Let $\mathbf { r } ( t ) = \cos 2 t \mathbf { i } + 2 t \mathbf { j } + \sin 2 t \mathbf { k }$ . Show that the acceleration vector is parallel to the normal vector N(t).

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Question 2

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$v ( t ) = | \mathbf { v } ( t ) | = \sqrt { 4 \left( \cos ^ { 2 } 2 t + \sin ^ { 2 } 2 t \right) + 2 } = \sqrt { 6 } , \text { so } a _ { T } = v ^ { \prime } ( t ) = 0 \text { and thus } \mathbf { a } ( t ) = a _ { N } \mathrm {~N} \text { is parallel to } \mathbf { N }$

Find a parametric representation for the surface consisting of that part of the hyperboloid $- x ^ { 2 } - y ^ { 2 } + z ^ { 2 } = 1$ that lies below the rectangle $[ - 1,1 ] \times [ - 3,3 ]$ .

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Question 3

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$x = x$ , $y = y$ , $z = - \sqrt { 1 + x ^ { 2 } + y ^ { 2 } }$ where $- 1 \leq x \leq 1$ and $- 3 \leq y \leq 3$

Find the length of the circular helix described by $x = 2 \cos t , y = 2 \sin t , z = \sqrt { 5 } t , 0 \leq t \leq 2 \pi$

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Question 4

Find the equation of the osculating circle of the curve $y = e ^ { x } \text { at } x = 0$

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Question 5

The position of a particle at time t is given parametrically by y = t² and x = $\frac { 1 } { 3 }$ (t³ - 3t). Show that the particle crosses the y-axis three times.

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Question 6

Use the curvature formula to compute the curvature of a straight line y = mx + b.

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Question 7

Let the acceleration of a particle be $\mathbf { a } ( t ) = \mathbf { i } + \mathbf { k }$ , and let its velocity when t = 0 be $\mathbf { v } ( 0 ) = \mathbf { j }$ . Find its velocity when t = 1.

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Question 8

Find the tangent vector $\mathbf { r } ^ { \prime }$ (t) of the function r (t) = sin 2t i - cos 2t j when t = $\frac { \pi } { 6 }$ .

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Question 9

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t ) = \langle 3 \cos t , 3 \sin t , t \rangle$ .

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Question 10

Find the curvature $K$ of the curve $\mathbf { r } ( t ) = \left( t , t , 1 - t ^ { 2 } \right)$ at t = 0.

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Question 11

Find the curvature $K$ of the curve $\mathbf { r } ( t ) = \langle \sin 2 t , 3 t , \cos 2 t \rangle \text { when } t = \frac { \pi } { 2 }$

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Question 12

Find an expression for $\frac { d } { d t } [ ( \mathbf { u } ( t ) \times \mathbf { v } ( t ) ) \cdot \mathbf { w } ( t ) ]$

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Question 13

Consider $y = \sin x , - \pi < x < \pi$ . Determine graphically where the curvature is maximal and minimal.

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Question 14

Suppose a particle moves in the plane according to the vector-valued function $\mathbf { r } ( t ) = 2 e ^ { t } \mathbf { i } + e ^ { - t } \mathbf { j }$ , where t represents time. Find $\mathbf { v } ( t ) , | \mathbf { v } ( t ) | , \text { and } \mathbf { a } ( t )$ , and sketch a graph showing the path taken by the particle indicating the direction of motion. $Suppose a particle moves in the plane according to the vector-valued function \mathbf { r } ( t ) = 2 e ^ { t } \mathbf { i } + e ^ { - t } \mathbf { j } , where t represents time. Find \mathbf { v } ( t ) , | \mathbf { v } ( t ) | , \text { and } \mathbf { a } ( t ) , and sketch a graph showing the path taken by the particle indicating the direction of motion.$

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Question 15

Where does the tangent line to the curve $\mathbf { r } ( t ) = \left\langle e ^ { - 2 t } , \cos t , 3 \sin t \right\rangle$ at (1,1,0) intersect the yz-plane?

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Question 16

Find the derivative of the vector function r (t) = t i + sin t j when t = 0.

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Question 17

Find a parametric representation for the surface consisting of that part of the cylinder $x ^ { 2 } + z ^ { 2 } = 1$ that lies between the planes $y = - 1$ and y = 3.

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Question 18

Find the domain of vector function $\mathbf { r } = \left\langle \sqrt { 1 - t } , \ln t , e ^ { 2 t } \right\rangle$

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Question 19

Find the unit normal vector N(t) to the curve r (t) = $\left\langle t , 2 t , t ^ { 2 } \right\rangle$ when t = 1.

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Question 20

A particle is moving along the curve described by the parametric equations $x = 5 t , y = 2 t ^ { 3 } , z = \frac { 3 } { 5 } t ^ { 5 }$ . Determine the velocity and acceleration vectors as well as the speed of the particle when t = 3.

Let $\mathbf { r } ( t ) = \cos 2 t \mathbf { i } + 2 t \mathbf { j } + \sin 2 t \mathbf { k }$ . Show that the acceleration vector is parallel to the normal vector N(t).

Find a parametric representation for the surface consisting of that part of the hyperboloid $- x ^ { 2 } - y ^ { 2 } + z ^ { 2 } = 1$ that lies below the rectangle $[ - 1,1 ] \times [ - 3,3 ]$ .

Find the length of the circular helix described by $x = 2 \cos t , y = 2 \sin t , z = \sqrt { 5 } t , 0 \leq t \leq 2 \pi$

Find the equation of the osculating circle of the curve $y = e ^ { x } \text { at } x = 0$

The position of a particle at time t is given parametrically by y = t² and x = $\frac { 1 } { 3 }$ (t³ - 3t). Show that the particle crosses the y-axis three times.

Use the curvature formula to compute the curvature of a straight line y = mx + b.

Let the acceleration of a particle be $\mathbf { a } ( t ) = \mathbf { i } + \mathbf { k }$ , and let its velocity when t = 0 be $\mathbf { v } ( 0 ) = \mathbf { j }$ . Find its velocity when t = 1.

Find the tangent vector $\mathbf { r } ^ { \prime }$ (t) of the function r (t) = sin 2t i - cos 2t j when t = $\frac { \pi } { 6 }$ .

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t ) = \langle 3 \cos t , 3 \sin t , t \rangle$ .

Find the curvature $K$ of the curve $\mathbf { r } ( t ) = \left( t , t , 1 - t ^ { 2 } \right)$ at t = 0.

Find the curvature $K$ of the curve $\mathbf { r } ( t ) = \langle \sin 2 t , 3 t , \cos 2 t \rangle \text { when } t = \frac { \pi } { 2 }$

Find an expression for $\frac { d } { d t } [ ( \mathbf { u } ( t ) \times \mathbf { v } ( t ) ) \cdot \mathbf { w } ( t ) ]$

Consider $y = \sin x , - \pi < x < \pi$ . Determine graphically where the curvature is maximal and minimal.

Where does the tangent line to the curve $\mathbf { r } ( t ) = \left\langle e ^ { - 2 t } , \cos t , 3 \sin t \right\rangle$ at (1,1,0) intersect the yz-plane?

Find the derivative of the vector function r (t) = t i + sin t j when t = 0.

Find a parametric representation for the surface consisting of that part of the cylinder $x ^ { 2 } + z ^ { 2 } = 1$ that lies between the planes $y = - 1$ and y = 3.

Find the domain of vector function $\mathbf { r } = \left\langle \sqrt { 1 - t } , \ln t , e ^ { 2 t } \right\rangle$

Find the unit normal vector N(t) to the curve r (t) = $\left\langle t , 2 t , t ^ { 2 } \right\rangle$ when t = 1.

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Exam 10: Vector Functions

A particle is moving along the curve described by the parametric equations x=5t,y=2t3,z=35t5x = 5 t , y = 2 t ^ { 3 } , z = \frac { 3 } { 5 } t ^ { 5 }x=5t,y=2t3,z=53​t5 . Determine the velocity and acceleration vectors as well as the speed of the particle when t = 3.

Let r(t)=cos⁡2ti+2tj+sin⁡2tk\mathbf { r } ( t ) = \cos 2 t \mathbf { i } + 2 t \mathbf { j } + \sin 2 t \mathbf { k }r(t)=cos2ti+2tj+sin2tk . Show that the acceleration vector is parallel to the normal vector N(t).

Find a parametric representation for the surface consisting of that part of the hyperboloid −x2−y2+z2=1- x ^ { 2 } - y ^ { 2 } + z ^ { 2 } = 1−x2−y2+z2=1 that lies below the rectangle [−1,1]×[−3,3][ - 1,1 ] \times [ - 3,3 ][−1,1]×[−3,3] .

Find the length of the circular helix described by x=2cos⁡t,y=2sin⁡t,z=5t,0≤t≤2πx = 2 \cos t , y = 2 \sin t , z = \sqrt { 5 } t , 0 \leq t \leq 2 \pix=2cost,y=2sint,z=5​t,0≤t≤2π

Find the equation of the osculating circle of the curve y=ex at x=0y = e ^ { x } \text { at } x = 0y=ex at x=0

The position of a particle at time t is given parametrically by y = t2 and x = 13\frac { 1 } { 3 }31​ (t3 - 3t). Show that the particle crosses the y-axis three times.

Use the curvature formula to compute the curvature of a straight line y = mx + b.

Let the acceleration of a particle be a(t)=i+k\mathbf { a } ( t ) = \mathbf { i } + \mathbf { k }a(t)=i+k , and let its velocity when t = 0 be v(0)=j\mathbf { v } ( 0 ) = \mathbf { j }v(0)=j . Find its velocity when t = 1.

Find the tangent vector r′\mathbf { r } ^ { \prime }r′ (t) of the function r (t) = sin 2t i - cos 2t j when t = π6\frac { \pi } { 6 }6π​ .

Identify the geometric object that is represented by parametric equations r(t)=⟨3cos⁡t,3sin⁡t,t⟩\mathbf { r } ( t ) = \langle 3 \cos t , 3 \sin t , t \rangler(t)=⟨3cost,3sint,t⟩ .

Find the curvature KKK of the curve r(t)=(t,t,1−t2)\mathbf { r } ( t ) = \left( t , t , 1 - t ^ { 2 } \right)r(t)=(t,t,1−t2) at t = 0.

Find the curvature KKK of the curve r(t)=⟨sin⁡2t,3t,cos⁡2t⟩ when t=π2\mathbf { r } ( t ) = \langle \sin 2 t , 3 t , \cos 2 t \rangle \text { when } t = \frac { \pi } { 2 }r(t)=⟨sin2t,3t,cos2t⟩ when t=2π​

Find an expression for ddt[(u(t)×v(t))⋅w(t)]\frac { d } { d t } [ ( \mathbf { u } ( t ) \times \mathbf { v } ( t ) ) \cdot \mathbf { w } ( t ) ]dtd​[(u(t)×v(t))⋅w(t)]

Consider y=sin⁡x,−π<x<πy = \sin x , - \pi < x < \piy=sinx,−π<x<π . Determine graphically where the curvature is maximal and minimal.

Where does the tangent line to the curve r(t)=⟨e−2t,cos⁡t,3sin⁡t⟩\mathbf { r } ( t ) = \left\langle e ^ { - 2 t } , \cos t , 3 \sin t \right\rangler(t)=⟨e−2t,cost,3sint⟩ at (1,1,0) intersect the yz-plane?

Find the derivative of the vector function r (t) = t i + sin t j when t = 0.

Find a parametric representation for the surface consisting of that part of the cylinder x2+z2=1x ^ { 2 } + z ^ { 2 } = 1x2+z2=1 that lies between the planes y=−1y = - 1y=−1 and y = 3.

Find the domain of vector function r=⟨1−t,ln⁡t,e2t⟩\mathbf { r } = \left\langle \sqrt { 1 - t } , \ln t , e ^ { 2 t } \right\rangler=⟨1−t​,lnt,e2t⟩

Find the unit normal vector N(t) to the curve r (t) = ⟨t,2t,t2⟩\left\langle t , 2 t , t ^ { 2 } \right\rangle⟨t,2t,t2⟩ when t = 1.

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A particle is moving along the curve described by the parametric equations $x = 5 t , y = 2 t ^ { 3 } , z = \frac { 3 } { 5 } t ^ { 5 }$ . Determine the velocity and acceleration vectors as well as the speed of the particle when t = 3.

Let $\mathbf { r } ( t ) = \cos 2 t \mathbf { i } + 2 t \mathbf { j } + \sin 2 t \mathbf { k }$ . Show that the acceleration vector is parallel to the normal vector N(t).

Find a parametric representation for the surface consisting of that part of the hyperboloid $- x ^ { 2 } - y ^ { 2 } + z ^ { 2 } = 1$ that lies below the rectangle $[ - 1,1 ] \times [ - 3,3 ]$ .

Find the length of the circular helix described by $x = 2 \cos t , y = 2 \sin t , z = \sqrt { 5 } t , 0 \leq t \leq 2 \pi$

Find the equation of the osculating circle of the curve $y = e ^ { x } \text { at } x = 0$

The position of a particle at time t is given parametrically by y = t² and x = $\frac { 1 } { 3 }$ (t³ - 3t). Show that the particle crosses the y-axis three times.

Let the acceleration of a particle be $\mathbf { a } ( t ) = \mathbf { i } + \mathbf { k }$ , and let its velocity when t = 0 be $\mathbf { v } ( 0 ) = \mathbf { j }$ . Find its velocity when t = 1.

Find the tangent vector $\mathbf { r } ^ { \prime }$ (t) of the function r (t) = sin 2t i - cos 2t j when t = $\frac { \pi } { 6 }$ .

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t ) = \langle 3 \cos t , 3 \sin t , t \rangle$ .

Find the curvature $K$ of the curve $\mathbf { r } ( t ) = \left( t , t , 1 - t ^ { 2 } \right)$ at t = 0.

Find the curvature $K$ of the curve $\mathbf { r } ( t ) = \langle \sin 2 t , 3 t , \cos 2 t \rangle \text { when } t = \frac { \pi } { 2 }$

Find an expression for $\frac { d } { d t } [ ( \mathbf { u } ( t ) \times \mathbf { v } ( t ) ) \cdot \mathbf { w } ( t ) ]$

Consider $y = \sin x , - \pi < x < \pi$ . Determine graphically where the curvature is maximal and minimal.

Where does the tangent line to the curve $\mathbf { r } ( t ) = \left\langle e ^ { - 2 t } , \cos t , 3 \sin t \right\rangle$ at (1,1,0) intersect the yz-plane?

Find a parametric representation for the surface consisting of that part of the cylinder $x ^ { 2 } + z ^ { 2 } = 1$ that lies between the planes $y = - 1$ and y = 3.

Find the domain of vector function $\mathbf { r } = \left\langle \sqrt { 1 - t } , \ln t , e ^ { 2 t } \right\rangle$

Find the unit normal vector N(t) to the curve r (t) = $\left\langle t , 2 t , t ^ { 2 } \right\rangle$ when t = 1.