Exam 17: Parameterization and Vector Fields

Let S be a circular cylinder of radius 0.2, such that the center of one end is at the origin and the center of the other end is at the point (2, 0, 7). Find two unit vectors $\vec { u }$ and $\vec { v }$ in the plane, P, containing the base of the cylinder (i.e., the plane through the origin perpendicular to the axis of the cylinder)which are perpendicular to each other.

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Question 21

Find the coordinates of the point where the line tangent to the curve $\vec { r } ( t ) = t \vec { i } + t ^ { 2 } \vec { j } + t ^ { 3 } \vec { k }$ at the point (4, 16, 64)crosses the xy-plane.

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Question 22

Use cylindrical coordinates to parameterize the part of the plane x + y - z = 10 inside the cylinder $x ^ { 2 } + y ^ { 2 } = 4$ .

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Question 23

A surveyor wants to measure the height of a building.At point A = ( 733, -369, 0)on the ground, she observes that the vector $\overrightarrow { A C }$ is parallel to $- 3 \vec { i } + 2 \vec { j } + 6 \vec { k }$ , where C is the highest point of the building.At point B = (418, 471, 0)she observes that the vector $\overrightarrow { B C }$ is parallel to $- 3 \vec { i } - 4 \vec { j } + 12 \vec { k }$ .Given that ground level is the plane z = 0 and the units are feet, find the height of the building.

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Question 24

Consider the plane x - 4y + -2z = 5 and the line x = a + bt, y = 2 + -2t, z = 2 - t. Find the value of b such that the line is perpendicular to the plane.

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Question 25

The path of an object moving in xyz-space is given by $( x ( t ) , y ( t ) , z ( t ) ) = \left( 3 t ^ { 2 } , t + 1 , t ^ { 3 } \right)$ . The temperature at a point (x, y, z)in space is given by $f ( x , y , z ) = x ^ { 2 } y - 3 z$ Calculate the directional derivative of f in the direction of $\vec { v }$ at the point (12, 3, 8), where $\vec { v }$ is the velocity vector of the object..

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Question 26

The lines $( 2 t + 1 ) \vec { i } + ( 1 - 3 t ) \vec { j } + ( 2 + 2 t ) \vec { k }$ and $( 2 t - 3 ) \vec { i } + ( 7 - 2 t ) \vec { j } + ( - 2 + a t ) \vec { k }$ are perpendicular.Find a.

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Question 27

The vector field $\vec { F } ( x , y ) = x \vec { i } - y \vec { j }$ represents an ocean current.An iceberg is at the point (1, 2)at t = 0. Determine the position of the iceberg at time t = 2.

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Question 28

Calculate the length of the curve $y = \cosh x$ from x = -3 to x = 3.

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Question 29

How many parameters are needed to parameterize a surface in 3-space?

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Question 30

Let S be a circular cylinder of radius 0.2, such that the center of one end is at the origin and the center of the other end is at the point (5, 0, 4). Let P be the plane containing the base of the cylinder (i.e., the plane through the origin perpendicular to the axis of the cylinder). In each case, give a parameterization $( x ( \theta ) , y ( \theta ) , z ( \theta ) )$ and specify the range of values your parameters must take on. (i)the circle in which the cylinder, S, cuts the plane, P. (ii)the surface of the cylinder S.

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Question 31

Use spherical coordinates to parameterize the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ above the plane z = 1.

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Question 32

Let f(x, y, z)= xy + 6yz + zx.Then f(2, 2, 3)= 46. Give an equation to the tangent plane to xy + 6yz + zx = 46.

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Question 33

The curve $\vec { r } ( t ) = a t \vec { i } + ( t + 4 ) \vec { j } + \left( b t ^ { 2 } + 6 \right) \vec { k }$ passes through the point (-12, 1, 51).Find a and b.

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Question 34

Match the vector field $\vec{F}(x, y)=x \vec{i}+y \vec{j}$ with the descriptions (a)-(d).

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Question 35

What curve, C, is traced out by the parameterization $\vec { r } = 2 \vec { i } + ( \cos t ) \vec { j } + ( \sin t ) \vec { k }$ for 0 $\le$ t $\le$ 2 $\pi$ ? Either give a very complete verbal description or sketch the curve (or both).

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Question 36

The parametric vector form of the position of a roller coaster is $\vec { r } ( t ) = 30 \sin ( t ) \vec { i } + 30 \cos ( t ) \vec { j } + 15 \cos ( t ) \vec { k }$ Answer the following questions about the ride. (a)The scariest point of the ride is when it is traveling fastest.For which value of t > 0 does this occur first? (b)Does the velocity vector of the roller coaster ever point directly downward?

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Question 37

If a particle is moving along a parameterized curve $\vec { r } ( t )$ , then the acceleration vector at any point cannot be parallel to the velocity vector at that point.

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Question 38

Let $\vec { v } _ { 1 } = 2 \vec { i } - 1 \vec { j } + \vec { k }$ and $\vec { v } _ { 2 } = 1 \vec { i } + \vec { j } + \vec { k }$ Find a parametric equation for the plane through the point (1, 2, -1)and containing the vectors $\vec { v } _ { 1 }$ and $\vec { v } _ { 2 }$ Select all that apply.

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Question 39

Sketch the vector fields $\vec { v } = x \vec { i }$

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Question 40

Find the coordinates of the point where the line tangent to the curve $\vec { r } ( t ) = t \vec { i } + t ^ { 2 } \vec { j } + t ^ { 3 } \vec { k }$ at the point (4, 16, 64)crosses the xy-plane.

Use cylindrical coordinates to parameterize the part of the plane x + y - z = 10 inside the cylinder $x ^ { 2 } + y ^ { 2 } = 4$ .

Consider the plane x - 4y + -2z = 5 and the line x = a + bt, y = 2 + -2t, z = 2 - t. Find the value of b such that the line is perpendicular to the plane.

The lines $( 2 t + 1 ) \vec { i } + ( 1 - 3 t ) \vec { j } + ( 2 + 2 t ) \vec { k }$ and $( 2 t - 3 ) \vec { i } + ( 7 - 2 t ) \vec { j } + ( - 2 + a t ) \vec { k }$ are perpendicular.Find a.

The vector field $\vec { F } ( x , y ) = x \vec { i } - y \vec { j }$ represents an ocean current.An iceberg is at the point (1, 2)at t = 0. Determine the position of the iceberg at time t = 2.

Calculate the length of the curve $y = \cosh x$ from x = -3 to x = 3.

How many parameters are needed to parameterize a surface in 3-space?

Use spherical coordinates to parameterize the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ above the plane z = 1.

Let f(x, y, z)= xy + 6yz + zx.Then f(2, 2, 3)= 46. Give an equation to the tangent plane to xy + 6yz + zx = 46.

The curve $\vec { r } ( t ) = a t \vec { i } + ( t + 4 ) \vec { j } + \left( b t ^ { 2 } + 6 \right) \vec { k }$ passes through the point (-12, 1, 51).Find a and b.

Match the vector field $\vec{F}(x, y)=x \vec{i}+y \vec{j}$ with the descriptions (a)-(d).

What curve, C, is traced out by the parameterization $\vec { r } = 2 \vec { i } + ( \cos t ) \vec { j } + ( \sin t ) \vec { k }$ for 0 $\le$ t $\le$ 2 $\pi$ ? Either give a very complete verbal description or sketch the curve (or both).

If a particle is moving along a parameterized curve $\vec { r } ( t )$ , then the acceleration vector at any point cannot be parallel to the velocity vector at that point.

Let $\vec { v } _ { 1 } = 2 \vec { i } - 1 \vec { j } + \vec { k }$ and $\vec { v } _ { 2 } = 1 \vec { i } + \vec { j } + \vec { k }$ Find a parametric equation for the plane through the point (1, 2, -1)and containing the vectors $\vec { v } _ { 1 }$ and $\vec { v } _ { 2 }$ Select all that apply.

Sketch the vector fields $\vec { v } = x \vec { i }$

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Key Concept- the Definite Integral

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Approximating Functions Using Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Differentiating Functions of Several Variables

Optimization- Local and Global Extrema

Integrating Functions of Several Variables

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Exam 17: Parameterization and Vector Fields

Find the coordinates of the point where the line tangent to the curve r⃗(t)=ti⃗+t2j⃗+t3k⃗\vec { r } ( t ) = t \vec { i } + t ^ { 2 } \vec { j } + t ^ { 3 } \vec { k }r(t)=ti+t2j​+t3k at the point (4, 16, 64)crosses the xy-plane.

Use cylindrical coordinates to parameterize the part of the plane x + y - z = 10 inside the cylinder x2+y2=4x ^ { 2 } + y ^ { 2 } = 4x2+y2=4 .

Consider the plane x - 4y + -2z = 5 and the line x = a + bt, y = 2 + -2t, z = 2 - t. Find the value of b such that the line is perpendicular to the plane.

The vector field F⃗(x,y)=xi⃗−yj⃗\vec { F } ( x , y ) = x \vec { i } - y \vec { j }F(x,y)=xi−yj​ represents an ocean current.An iceberg is at the point (1, 2)at t = 0. Determine the position of the iceberg at time t = 2.

Calculate the length of the curve y=cosh⁡xy = \cosh xy=coshx from x = -3 to x = 3.

How many parameters are needed to parameterize a surface in 3-space?

Use spherical coordinates to parameterize the part of the sphere x2+y2+z2=4x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4x2+y2+z2=4 above the plane z = 1.

Let f(x, y, z)= xy + 6yz + zx.Then f(2, 2, 3)= 46. Give an equation to the tangent plane to xy + 6yz + zx = 46.

The curve r⃗(t)=ati⃗+(t+4)j⃗+(bt2+6)k⃗\vec { r } ( t ) = a t \vec { i } + ( t + 4 ) \vec { j } + \left( b t ^ { 2 } + 6 \right) \vec { k }r(t)=ati+(t+4)j​+(bt2+6)k passes through the point (-12, 1, 51).Find a and b.

Match the vector field F⃗(x,y)=xi⃗+yj⃗\vec{F}(x, y)=x \vec{i}+y \vec{j}F(x,y)=xi+yj​ with the descriptions (a)-(d).

If a particle is moving along a parameterized curve r⃗(t)\vec { r } ( t )r(t) , then the acceleration vector at any point cannot be parallel to the velocity vector at that point.

Sketch the vector fields v⃗=xi⃗\vec { v } = x \vec { i }v=xi

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Key Concept- the Definite Integral

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Approximating Functions Using Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Differentiating Functions of Several Variables

Optimization- Local and Global Extrema

Integrating Functions of Several Variables

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Find the coordinates of the point where the line tangent to the curve $\vec { r } ( t ) = t \vec { i } + t ^ { 2 } \vec { j } + t ^ { 3 } \vec { k }$ at the point (4, 16, 64)crosses the xy-plane.

Use cylindrical coordinates to parameterize the part of the plane x + y - z = 10 inside the cylinder $x ^ { 2 } + y ^ { 2 } = 4$ .

The vector field $\vec { F } ( x , y ) = x \vec { i } - y \vec { j }$ represents an ocean current.An iceberg is at the point (1, 2)at t = 0. Determine the position of the iceberg at time t = 2.

Calculate the length of the curve $y = \cosh x$ from x = -3 to x = 3.

Use spherical coordinates to parameterize the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ above the plane z = 1.

The curve $\vec { r } ( t ) = a t \vec { i } + ( t + 4 ) \vec { j } + \left( b t ^ { 2 } + 6 \right) \vec { k }$ passes through the point (-12, 1, 51).Find a and b.

Match the vector field $\vec{F}(x, y)=x \vec{i}+y \vec{j}$ with the descriptions (a)-(d).

If a particle is moving along a parameterized curve $\vec { r } ( t )$ , then the acceleration vector at any point cannot be parallel to the velocity vector at that point.

Sketch the vector fields $\vec { v } = x \vec { i }$