x
y
2.5 3.0 3.5
-1.0 6 7 8
1.0 0 1 2
3.0 -6 -5 -4
MATH 223
FINAL EXAM REVIEW PACKET ANSWERS
(Fall 2012)
1. (a) increasing (b) decreasing
2. (a) 2 2( 3) 25y z− + = This is a cylinder parallel to the x-axis with radius 5.
(b) 3x = , 3x = − . These are vertical planes parallel to the yz-plane.
(c) 2 2 2z x y= + . This is a cone (one opening up and one opening down) centered on the z-axis.
3. There are many possible answers.
(a) 0x = produces the curve 23y z= − .
(b) 1y = produces the curves 23 cosz x= − and 23 cosz x= − − .
(c)
2
x
π
= produces the curves 3z = and 3z = − .
4. (a) (b) (i) 1 (ii) Increase (iii) Decrease
5. (a) Paraboloids centered on the x-axis, opening up in the positive x direction. 2 2x y z c= + +
(b) Spheres centered at the origin with radius 1 ln c− for 0 c e< ≤ . 2 2 2 1 lnx y z c+ + = −
6. (a) 6 am 11:30 am
(b) Temperature as a function of time at a depth of 20 cm.
(c) Temperature as a function of depth at noon.
7. ( , ) 2 3 2z f x y x y= = − −
8. (a) II, III, IV, VI (b) I (c) I, III, VI (d) VI (e) I, V
9. (a)
12
4 12
5
z x y= − + (b) There are many possible answers.
12
4
5
i j k+ −
(c)
3 569
2
10. (a) iii, vii (b) iv (c) viii (d) ii (e) v, vi (f) i, ix
11. There are many possible answers.
(a) ( )5 4 3
26
i j k− +
or ( )5 4 3
26
i j k− − +
(b) 2 3i j− +
(c)
4
cos
442
θ = , 1.38θ ≈ radians (d) ( )4 4 3
26
i j k− +
(e) 4 11 17i j k− − −
12. (a)
3
5
a = − (b)
1
3
a = (c) 2( 1) ( 2) 3( 3) 0x y z− − + + − = (d)
1 2 , 2 , 3 3x t y t z t= + = − − = +
13. 6 39i
or 6 39i−
14. (a)
( )
2
23 2 2
3 2
3 1
z x y x
x x y x y
∂
= −
∂ + + +
(b)
( )4
10 4 3
5
H
H T
f
H
+ +
=
−
(c)
2
2 2
1 1z
x y y x
∂
= − −
∂ ∂
15. (a) 2 2 24 ( 1) 3 ( 2) 2z e x e y e= − + − + (b) 4( 3) 8( 3) 6( 6) 0x y z− + − + − =
16. (a)
2sin(2 ) cos(2 )
5 5
v v
ds dv d
α α
α= +
(b) The distance s decreases if the angle α increases and the initial speed v remains constant.
(c) 0.0886α∆ ≈ − . The angle decreases by about 0.089 radians.
17. (a) The water is getting shallower.
4
( 1, 2)
17
uh − = −
(b) There are many possible answers. 3i j+
(c) 72 ft/min
18. (a)
( )
2 2 2
22 2 22
2 2
1 1 11
yz xyz z yz
grad i j k
x x xx
= − + + + + + +
(b) ( ) ( ) ( )( )2 2 2 2curl x y z i y z j xz k i zj yk+ + − + + = + −
(c) ( ) ( ) ( )( )2 3 3cos sec 2 cos sin sec tan 3z zdiv x i x y j e k x x x y y e+ + = − + +
(d)
37
3
(e) ( , , ) sin zg x y z xy e c= + +
19. ( , ) 4 3vG a b = −
20. (a) positive (b) negative (c) negative (d) negative (e) positive (f) zero
21. (a)
(.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2004. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
Fungi reproduce ___________________________ by fragmentation, buddin.docxericbrooks84875
Fungi reproduce ___________________________ by fragmentation, budding, or producing spores.
10. In ___________________________ , pieces of hyphae grow into new mycelia.
11. The process of a parent cell undergoing mitosis and producing a new individual that pinches off,
matures, and separates from the parent is called ___________________________ .
12. When environmental conditions are right, a ___________________________ may germinate and
produce a threadlike ___________________________ that will grow into a mycelium.
13. Some hyphae grow away from the mycelium to produce a spore-containing structure called a
___________________________ .
14. In most fungi, the structures that support ___________________________ are the only part of the
fungus that can be seen.
15. Fungi may produce spores by ___________________________ or ___________________________ .
16. Many adaptations of fungi for survival involve ___________________________ .
17. ___________________________ protect spores and keep them from from drying out until they
are released.
18. A single puffball may produce a cloud containing as many as ___________________________ spores.
19. Producing a large number of spores increases a species’ chances of ___________________________ .
20. Fungal spores can be dispersed by ___________________________ , ___________________________ ,
and ___________________________ .
.
Full-Circle LearningMyLab™ Learning Full Circle for Mar.docxericbrooks84875
Full-Circle Learning
MyLab™: Learning Full Circle for Marketing,
Management, Business Communication,
and Intro to Business
BEFORE
CLASS
AFTER
CLASS DURING
CLASS
Decision
Sims, Videos,
and Learning
Catalytics
DSMs,
pre-lecture
homework,
eText
Writing
Space, Video
Cases, Quizzes/
Tests
MyLab
Critical Thinking
MyManagementLab®: Improves Student
Engagement Before, During, and After Class
Decision Making
BREAKTHROUGH
Prep and
Engagement
BREAK
THRO
UGH
To better resultsTo better results
• NEW! VIDEO LIBRARY – Robust video library with over 100 new book-specific videos that include
easy-to-assign assessments, the ability for instructors to add YouTube or other sources, the ability for
students to upload video submissions, and the ability for polling and teamwork.
• Decision-making simulations – NEW and improved feedback for students. Place your students
in the role of a key decision-maker! Simulations branch based on the decisions students make, providing
a variation of scenario paths. Upon completion students receive a grade, as well as a detailed report of
the choices and the associated consequences of those decisions.
• Video exercises – UPDATED with new exercises. Engaging videos that bring business concepts to
life and explore business topics related to the theory students are learning in class. Quizzes then assess
students’ comprehension of the concepts covered in each video.
• Learning Catalytics – A “bring your own device”
student engagement, assessment, and classroom
intelligence system helps instructors analyze
students’ critical-thinking skills during lecture.
• Dynamic Study Modules (DSMs) – UPDATED
with additional questions. Through adaptive
learning, students get personalized guidance where
and when they need it most, creating greater
engagement, improving knowledge retention, and
supporting subject-matter mastery. Also available
on mobile devices.
• Writing Space – UPDATED with new commenting tabs, new prompts, and a new tool
for students called Pearson Writer. A single location to develop and assess concept mastery
and critical thinking, the Writing Space offers automatic graded, assisted graded, and create your own
writing assignments, allowing you to exchange personalized feedback with students quickly and easily.
Writing Space can also check students’ work for improper citation or plagiarism by comparing it
against the world’s most accurate text comparison database available from Turnitin.
• Additional Features – Included with the MyLab are a powerful homework and test manager, robust
gradebook tracking, Reporting Dashboard, comprehensive online course content, and easily scalable
and shareable content.
http://www.pearsonmylabandmastering.com
Strategic
ManageMent
concepts and cases
A Competitive AdvAntAge ApproACh
This page intentionally left blank
Fred r. David
Francis Marion University
Florence, South Carolina
Forest r. David
Strategic Planning C.
Functional Requirements Document
Template
Version
Description of Change
Author
Date
CONTENTS
41
INTRODUCTION
1.1
Purpose
4
1.2
Scope
4
1.3
Background
4
1.4
References
4
1.5
Assumptions and Constraints
4
1.6
Document Overview
5
2
METHODOLOGY
5
3
FUNCTIONAL REQUIREMENTS
5
4.1
Context
5
4.2
User Requirements
5
4.3
Data Flow Diagrams
6
4.4
Logical Data Model/Data Dictionary
6
4.5
Functional Requirements
6
5
OTHER REQUIREMENTS
6
5.1
Interface Requirements
6
5.2
Data Conversion Requirements
7
5.3
Hardware/Software Requirements
7
5.4
Operational Requirements
7
APPENDIX A - GLOSSARY
11
1 INTRODUCTION
[Provide an overview of the system and some additional information to place the system in context.]
1.1 Purpose
[Provide an overall description of the FRD, its purpose. Reference the system name and identifying information about the system to be implemented.]
1.2 Scope
[Discuss the scope of the document and how it accomplishes its purpose.]
1.3 Background
[Describe the organization and its overall responsibilities. Describe who is producing the document and why.]
1.4 References
[List references and controlling documents, including: meeting summaries, white papers, other deliverables, etc.]
1.5 Assumptions and Constraints
[Provide a list of contractual or task level assumptions and/or constraints that are preconditions to preparation of the FRD. Assumptions are future situations beyond the control of the project, whose outcomes influence the success of a project.]
1.5.1 Assumptions
Examples of assumptions include: availability of a technical platform, legal changes and policy decisions.
1.5.2 Constraints
Constraints are boundary conditions on how the system must be designed and constructed. Examples include: legal requirements, technical standards, strategic decisions.
· Constraints exist because of real business conditions. For example, a delivery date is a constraint only if there are real business consequences that will happen as a result of not meeting the date. If failing to have the subject application operational by the specified date places the organization in legal default, the date is a constraint.
· Preferences are arbitrary. For example, a date chosen arbitrarily is a preference. Preferences, if included in the FRD, should be noted as such.]
1.6 Document Overview
[Provide a description of the document organization.]
2 METHODOLOGY
[Describe the overall approach used in the determination of the FRD contents. Describe the modeling method(s) so non-technical readers can understand what they are conveying.]
3 FUNCTIONAL REQUIREMENTS
4.1 Context
[Provide a context diagram of the system, with explanations as applicable. The context of a system refers to the connections and relationships between the system and its environment.]Exhibit 2 - Generic Context Diagram
Data 6
Data 1
Data 3
Data 4
Data 7
Data 2
Data 8
System/
Application
Name
Interface
Name 2
Interface
Name 4
Interface
Name 1
(User)
Interface
Name 3
Da.
Fully answer any ONE of the following essay questions1. Is the.docxericbrooks84875
Fully answer any ONE of the following essay questions:
1. Is there an obligation to promote the common good? Contrast the views of Bentham, Kant, and Ross on that question. Whose view do you think is best? Why?
2. What does it mean to claim that ethics is an
a priori
discipline? Who agrees with that view? What is Bentham's view of the nature of ethics (hint: empirical science)? How does Ross combine those views?
3. Both Kant and Ross are nonconsequentialists, yet they disagree about the role of consequences in determining the morally right act. Compare and contrast their views regarding the role of consequences. which view is preferable? Why?
.
Fully answer any ONE of the following essay questions1. Is t.docxericbrooks84875
Fully answer any ONE of the following essay questions:
1. Is there an obligation to promote the common good? Contrast the views of Bentham, Kant, and Ross on that question. Whose view do you think is best? Why?
2. What does it mean to claim that ethics is an
a priori
discipline? Who agrees with that view? What is Bentham's view of the nature of ethics (hint: empirical science)? How does Ross combine those views?
3. Both Kant and Ross are nonconsequentialists, yet they disagree about the role of consequences in determining the morally right act. Compare and contrast their views regarding the role of consequences. which view is preferable? Why?
.
From the weeks chapter reading, we learn from the authors that,.docxericbrooks84875
From the week's chapter reading, we learn from the authors that, the use of mobile devices in our society today has indeed become ubiquitous. In addition, CTIA asserted that over 326 million mobile devices were in use within The United States as of December 2012 – an estimated growth of more than 100 percent penetration rate with users carrying more than one device with notable continues growth. From this research, it’s evident that mobile computing has vastly accelerated in popularity over the last decade due to several factors noted by the authors in our chapter reading.
Q1: In consideration with this revelation, identify and name these factors, and provide a brief discussion about them?
.
FTER watching the videos and reviewing the other materials in this.docxericbrooks84875
FTER
watching the videos and reviewing the other materials in this Learning Unit, answer BOTH of the questions below.
1) Discuss the use of imagery in the poem "kitchenette building" by Gwendolyn Brooks. In your answer, make sure to identify specific images and explain how or why the poet uses them.
2) Discuss the use of diction in the poem "The Secretary Chant" by Marge Piercy. In your answer, make sure to identify specific word choices and explain how or why the poet uses them.
.
More Related Content
Similar to xy2.5 3.0 3.5-1.0 6 7 81.0 0 1 23.0 -6 -5 .docx
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2004. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
Fungi reproduce ___________________________ by fragmentation, buddin.docxericbrooks84875
Fungi reproduce ___________________________ by fragmentation, budding, or producing spores.
10. In ___________________________ , pieces of hyphae grow into new mycelia.
11. The process of a parent cell undergoing mitosis and producing a new individual that pinches off,
matures, and separates from the parent is called ___________________________ .
12. When environmental conditions are right, a ___________________________ may germinate and
produce a threadlike ___________________________ that will grow into a mycelium.
13. Some hyphae grow away from the mycelium to produce a spore-containing structure called a
___________________________ .
14. In most fungi, the structures that support ___________________________ are the only part of the
fungus that can be seen.
15. Fungi may produce spores by ___________________________ or ___________________________ .
16. Many adaptations of fungi for survival involve ___________________________ .
17. ___________________________ protect spores and keep them from from drying out until they
are released.
18. A single puffball may produce a cloud containing as many as ___________________________ spores.
19. Producing a large number of spores increases a species’ chances of ___________________________ .
20. Fungal spores can be dispersed by ___________________________ , ___________________________ ,
and ___________________________ .
.
Full-Circle LearningMyLab™ Learning Full Circle for Mar.docxericbrooks84875
Full-Circle Learning
MyLab™: Learning Full Circle for Marketing,
Management, Business Communication,
and Intro to Business
BEFORE
CLASS
AFTER
CLASS DURING
CLASS
Decision
Sims, Videos,
and Learning
Catalytics
DSMs,
pre-lecture
homework,
eText
Writing
Space, Video
Cases, Quizzes/
Tests
MyLab
Critical Thinking
MyManagementLab®: Improves Student
Engagement Before, During, and After Class
Decision Making
BREAKTHROUGH
Prep and
Engagement
BREAK
THRO
UGH
To better resultsTo better results
• NEW! VIDEO LIBRARY – Robust video library with over 100 new book-specific videos that include
easy-to-assign assessments, the ability for instructors to add YouTube or other sources, the ability for
students to upload video submissions, and the ability for polling and teamwork.
• Decision-making simulations – NEW and improved feedback for students. Place your students
in the role of a key decision-maker! Simulations branch based on the decisions students make, providing
a variation of scenario paths. Upon completion students receive a grade, as well as a detailed report of
the choices and the associated consequences of those decisions.
• Video exercises – UPDATED with new exercises. Engaging videos that bring business concepts to
life and explore business topics related to the theory students are learning in class. Quizzes then assess
students’ comprehension of the concepts covered in each video.
• Learning Catalytics – A “bring your own device”
student engagement, assessment, and classroom
intelligence system helps instructors analyze
students’ critical-thinking skills during lecture.
• Dynamic Study Modules (DSMs) – UPDATED
with additional questions. Through adaptive
learning, students get personalized guidance where
and when they need it most, creating greater
engagement, improving knowledge retention, and
supporting subject-matter mastery. Also available
on mobile devices.
• Writing Space – UPDATED with new commenting tabs, new prompts, and a new tool
for students called Pearson Writer. A single location to develop and assess concept mastery
and critical thinking, the Writing Space offers automatic graded, assisted graded, and create your own
writing assignments, allowing you to exchange personalized feedback with students quickly and easily.
Writing Space can also check students’ work for improper citation or plagiarism by comparing it
against the world’s most accurate text comparison database available from Turnitin.
• Additional Features – Included with the MyLab are a powerful homework and test manager, robust
gradebook tracking, Reporting Dashboard, comprehensive online course content, and easily scalable
and shareable content.
http://www.pearsonmylabandmastering.com
Strategic
ManageMent
concepts and cases
A Competitive AdvAntAge ApproACh
This page intentionally left blank
Fred r. David
Francis Marion University
Florence, South Carolina
Forest r. David
Strategic Planning C.
Functional Requirements Document
Template
Version
Description of Change
Author
Date
CONTENTS
41
INTRODUCTION
1.1
Purpose
4
1.2
Scope
4
1.3
Background
4
1.4
References
4
1.5
Assumptions and Constraints
4
1.6
Document Overview
5
2
METHODOLOGY
5
3
FUNCTIONAL REQUIREMENTS
5
4.1
Context
5
4.2
User Requirements
5
4.3
Data Flow Diagrams
6
4.4
Logical Data Model/Data Dictionary
6
4.5
Functional Requirements
6
5
OTHER REQUIREMENTS
6
5.1
Interface Requirements
6
5.2
Data Conversion Requirements
7
5.3
Hardware/Software Requirements
7
5.4
Operational Requirements
7
APPENDIX A - GLOSSARY
11
1 INTRODUCTION
[Provide an overview of the system and some additional information to place the system in context.]
1.1 Purpose
[Provide an overall description of the FRD, its purpose. Reference the system name and identifying information about the system to be implemented.]
1.2 Scope
[Discuss the scope of the document and how it accomplishes its purpose.]
1.3 Background
[Describe the organization and its overall responsibilities. Describe who is producing the document and why.]
1.4 References
[List references and controlling documents, including: meeting summaries, white papers, other deliverables, etc.]
1.5 Assumptions and Constraints
[Provide a list of contractual or task level assumptions and/or constraints that are preconditions to preparation of the FRD. Assumptions are future situations beyond the control of the project, whose outcomes influence the success of a project.]
1.5.1 Assumptions
Examples of assumptions include: availability of a technical platform, legal changes and policy decisions.
1.5.2 Constraints
Constraints are boundary conditions on how the system must be designed and constructed. Examples include: legal requirements, technical standards, strategic decisions.
· Constraints exist because of real business conditions. For example, a delivery date is a constraint only if there are real business consequences that will happen as a result of not meeting the date. If failing to have the subject application operational by the specified date places the organization in legal default, the date is a constraint.
· Preferences are arbitrary. For example, a date chosen arbitrarily is a preference. Preferences, if included in the FRD, should be noted as such.]
1.6 Document Overview
[Provide a description of the document organization.]
2 METHODOLOGY
[Describe the overall approach used in the determination of the FRD contents. Describe the modeling method(s) so non-technical readers can understand what they are conveying.]
3 FUNCTIONAL REQUIREMENTS
4.1 Context
[Provide a context diagram of the system, with explanations as applicable. The context of a system refers to the connections and relationships between the system and its environment.]Exhibit 2 - Generic Context Diagram
Data 6
Data 1
Data 3
Data 4
Data 7
Data 2
Data 8
System/
Application
Name
Interface
Name 2
Interface
Name 4
Interface
Name 1
(User)
Interface
Name 3
Da.
Fully answer any ONE of the following essay questions1. Is the.docxericbrooks84875
Fully answer any ONE of the following essay questions:
1. Is there an obligation to promote the common good? Contrast the views of Bentham, Kant, and Ross on that question. Whose view do you think is best? Why?
2. What does it mean to claim that ethics is an
a priori
discipline? Who agrees with that view? What is Bentham's view of the nature of ethics (hint: empirical science)? How does Ross combine those views?
3. Both Kant and Ross are nonconsequentialists, yet they disagree about the role of consequences in determining the morally right act. Compare and contrast their views regarding the role of consequences. which view is preferable? Why?
.
Fully answer any ONE of the following essay questions1. Is t.docxericbrooks84875
Fully answer any ONE of the following essay questions:
1. Is there an obligation to promote the common good? Contrast the views of Bentham, Kant, and Ross on that question. Whose view do you think is best? Why?
2. What does it mean to claim that ethics is an
a priori
discipline? Who agrees with that view? What is Bentham's view of the nature of ethics (hint: empirical science)? How does Ross combine those views?
3. Both Kant and Ross are nonconsequentialists, yet they disagree about the role of consequences in determining the morally right act. Compare and contrast their views regarding the role of consequences. which view is preferable? Why?
.
From the weeks chapter reading, we learn from the authors that,.docxericbrooks84875
From the week's chapter reading, we learn from the authors that, the use of mobile devices in our society today has indeed become ubiquitous. In addition, CTIA asserted that over 326 million mobile devices were in use within The United States as of December 2012 – an estimated growth of more than 100 percent penetration rate with users carrying more than one device with notable continues growth. From this research, it’s evident that mobile computing has vastly accelerated in popularity over the last decade due to several factors noted by the authors in our chapter reading.
Q1: In consideration with this revelation, identify and name these factors, and provide a brief discussion about them?
.
FTER watching the videos and reviewing the other materials in this.docxericbrooks84875
FTER
watching the videos and reviewing the other materials in this Learning Unit, answer BOTH of the questions below.
1) Discuss the use of imagery in the poem "kitchenette building" by Gwendolyn Brooks. In your answer, make sure to identify specific images and explain how or why the poet uses them.
2) Discuss the use of diction in the poem "The Secretary Chant" by Marge Piercy. In your answer, make sure to identify specific word choices and explain how or why the poet uses them.
.
fter completing the reading this week, we reflect on a few key conce.docxericbrooks84875
fter completing the reading this week, we reflect on a few key concepts this week and answer question number
How does culture impact leadership? Can culture be seen as a constraint on leadership?
Please be sure to answer all the questions above in the initial post.
Please ensure the initial post and two response posts are substantive. Substantive posts will do at least TWO of the following:
Ask an interesting, thoughtful question pertaining to the topic
Expand on the topic, by adding additional thoughtful information
Answer a question posted by another student in detail
Share an applicable personal experience
Provide an outside source
Make an argument
At least one scholarly (peer-reviewed) resource should be used in the initial discussion thread. Please ensure to use information from your readings and other sources from the UC Library. Use APA references and in-text citations.
.
FS-3FORD MOTOR COMPANY AND SUBSIDIARIESCONSOLIDATED INCO.docxericbrooks84875
FS-3
FORD MOTOR COMPANY AND SUBSIDIARIES
CONSOLIDATED INCOME STATEMENT
(in millions, except per share amounts)
For the years ended December 31,
2016 2017 2018
Revenues
Automotive $ 141,546 $ 145,653 $ 148,294
Ford Credit 10,253 11,113 12,018
Mobility 1 10 26
Total revenues (Note 4) 151,800 156,776 160,338
Costs and expenses
Cost of sales 126,195 131,321 136,269
Selling, administrative, and other expenses 10,972 11,527 11,403
Ford Credit interest, operating, and other expenses 8,847 9,047 9,463
Total costs and expenses 146,014 151,895 157,135
Interest expense on Automotive debt 894 1,133 1,171
Interest expense on Other debt 57 57 57
Other income/(loss), net (Note 5) 169 3,267 2,247
Equity in net income of affiliated companies 1,780 1,201 123
Income before income taxes 6,784 8,159 4,345
Provision for/(Benefit from) income taxes (Note 7) 2,184 402 650
Net income 4,600 7,757 3,695
Less: Income/(Loss) attributable to noncontrolling interests 11 26 18
Net income attributable to Ford Motor Company $ 4,589 $ 7,731 $ 3,677
EARNINGS PER SHARE ATTRIBUTABLE TO FORD MOTOR COMPANY COMMON AND CLASS B STOCK (Note 8)
Basic income $ 1.16 $ 1.94 $ 0.93
Diluted income 1.15 1.93 0.92
CONSOLIDATED STATEMENT OF COMPREHENSIVE INCOME
(in millions)
For the years ended December 31,
2016 2017 2018
Net income $ 4,600 $ 7,757 $ 3,695
Other comprehensive income/(loss), net of tax (Note 21)
Foreign currency translation (1,024) 314 (523)
Marketable securities (8) (34) (11)
Derivative instruments 219 (265) 183
Pension and other postretirement benefits 56 37 (56)
Total other comprehensive income/(loss), net of tax (757) 52 (407)
Comprehensive income 3,843 7,809 3,288
Less: Comprehensive income/(loss) attributable to noncontrolling interests 10 24 18
Comprehensive income attributable to Ford Motor Company $ 3,833 $ 7,785 $ 3,270
The accompanying notes are part of the consolidated financial statements.
FS-4
FORD MOTOR COMPANY AND SUBSIDIARIES
CONSOLIDATED BALANCE SHEET
(in millions)
December 31,
2017
December 31,
2018
ASSETS
Cash and cash equivalents (Note 9) $ 18,492 $ 16,718
Marketable securities (Note 9) 20,435 17,233
Ford Credit finance receivables, net (Note 10) 52,210 54,353
Trade and other receivables, less allowances of $412 and $94 10,599 11,195
Inventories (Note 12) 11,176 11,220
Other assets 3,889 3,930
Total current assets 116,801 114,649
Ford Credit finance receivables, net (Note 10) 56,182 55,544
Net investment in operating leases (Note 13) 28,235 29,119
Net property (Note 14) 35,327 36,178
Equity in net assets of affiliated companies (Note 15) 3,085 2,709
Deferred income taxes (Note 7) 10,762 10,412
Other assets 8,104 7,929
Total assets $ 258,496 $ 256,540
LIABILITIES
Payables $ 23,282 $ 21,520
Other liabilities and deferred revenue (Note 16) 19,697 20,556
Automotive debt payable within one year (Note 18) 3,356 2,314
Ford Credit debt payable within one year (Note 18) 48,265 51,179
Total current liabilities 94,600 95,56.
Fromm’s concept of the syndrome of decay included three personality .docxericbrooks84875
Fromm’s concept of the syndrome of decay included three personality disorders: (1) necrophilia, or love of death; (2) malignant narcissism, or extreme self-interest; and (3) incestuous symbiosis, or a passionate devotion to one’s mother or a mother substitute. Fromm identified Adolf Hitler as the most conspicuous example of a person with the syndrome of decay. If Fromm’s conception is valid, one would see the syndrome of decay traits in contemporary personalities. Identify recent examples of well-known people with the syndrome of decay, including serial killers or heads of state who manifest the love of death. Pay special attention to malignant narcissism and incestuous symbiosis.
.
From your readings in Chapter 4, choose one of the organizational sy.docxericbrooks84875
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From your daily briefs, Kaiser Health News Morning Briefing or P.docxericbrooks84875
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List the interest groups that are mentioned in the brief and what their interest is in the piece.
Categorize the issues in the brief according to the following—politics and politicians, access to health care, health care insurance, health care legislation, money, drugs, or other.
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From the perspective of the public safety field youre in, aspire to.docxericbrooks84875
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From the following terms Orthodox Judaism, Hassidic Judaism. Brief.docxericbrooks84875
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Required Resources
Text
Bozarth, C. C., & Handfield, R. B. (2016).
Introduction to operations and supply chain management
(4th ed.). Upper Saddle River, NJ: Pearson.
Chapter 3: Process choice and layout decisions in manufacturing and services
Chapter 6: Managing capacity
Chapter 14: Managing projects
Chapter 15: Developing products and services
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From Chapter Seven How does horizontal growth differ from v.docxericbrooks84875
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From the e-Activity, determine the fundamental differences between t.docxericbrooks84875
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The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
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Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
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The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
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Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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1. x
y
2.5 3.0 3.5
-1.0 6 7 8
1.0 0 1 2
3.0 -6 -5 -4
MATH 223
FINAL EXAM REVIEW PACKET ANSWERS
(Fall 2012)
1. (a) increasing (b) decreasing
2. (a) 2 2( 3) 25y z− + = This is a cylinder parallel to the x-axis
with radius 5.
(b) 3x = , 3x = − . These are vertical planes parallel to the yz-
plane.
(c) 2 2 2z x y= + . This is a cone (one opening up and one
opening down) centered on the z-axis.
3. There are many possible answers.
(a) 0x = produces the curve 23y z= − .
2. (b) 1y = produces the curves 23 cosz x= − and 23 cosz x= − −
.
(c)
2
x
π
= produces the curves 3z = and 3z = − .
4. (a) (b) (i) 1 (ii) Increase (iii) Decrease
5. (a) Paraboloids centered on the x-axis, opening up in the
positive x direction. 2 2x y z c= + +
(b) Spheres centered at the origin with radius 1 ln c− for 0 c e<
≤ . 2 2 2 1 lnx y z c+ + = −
6. (a) 6 am 11:30 am
(b) Temperature as a function of time at a depth of 20 cm.
(c) Temperature as a function of depth at noon.
3. 7. ( , ) 2 3 2z f x y x y= = − −
8. (a) II, III, IV, VI (b) I (c) I, III, VI (d) VI
(e) I, V
9. (a)
12
4 12
5
z x y= − + (b) There are many possible answers.
12
4
5
i j k+ −
(c)
3 569
2
4. 10. (a) iii, vii (b) iv (c) viii (d) ii (e) v, vi (f)
i, ix
11. There are many possible answers.
(a) ( )5 4 3
26
i j k− +
or ( )5 4 3
26
i j k− − +
(b) 2 3i j− +
(c)
4
cos
442
θ = , 1.38θ ≈ radians (d) ( )4 4 3
5. 26
i j k− +
(e) 4 11 17i j k− − −
12. (a)
3
5
a = − (b)
1
3
a = (c) 2( 1) ( 2) 3( 3) 0x y z− − + + − = (d)
1 2 , 2 , 3 3x t y t z t= + = − − = +
13. 6 39i
or 6 39i−
14. (a)
6. ( )
2
23 2 2
3 2
3 1
z x y x
x x y x y
∂
= −
∂ + + +
(b)
( )4
10 4 3
5
H
H T
f
H
+ +
=
−
(c)
2
7. 2 2
1 1z
x y y x
∂
= − −
∂ ∂
15. (a) 2 2 24 ( 1) 3 ( 2) 2z e x e y e= − + − + (b) 4( 3)
8( 3) 6( 6) 0x y z− + − + − =
16. (a)
2sin(2 ) cos(2 )
5 5
v v
ds dv d
α α
α= +
(b) The distance s decreases if the angle α increases and the
initial speed v remains constant.
(c) 0.0886α∆ ≈ − . The angle decreases by about 0.089 radians.
17. (a) The water is getting shallower.
4
8. ( 1, 2)
17
(b) There are many possible answers. 3i j+
(c) 72 ft/min
18. (a)
( )
2 2 2
22 2 22
2 2
1 1 11
yz xyz z yz
grad i j k
x x xx
9. (b) ( ) ( ) ( )( )2 2 2 2curl x y z i y z j xz k i zj yk+ + − + + = +
−
(c) ( ) ( ) ( )( )2 3 3cos sec 2 cos sin sec tan 3z zdiv x i x y j e k
x x x y y e+ + = − + +
(d)
37
3
(e) ( , , ) sin zg x y z xy e c= + +
20. (a) positive (b) negative (c) negative (d) negative
(e) positive (f) zero
21. (a)
(1, 1 2)
11. dt e=
= −
22. ( )1, 2 37rz π =
23. ( 1, 3)− − local maximum, (2,1) local minimum, ( 1,1)−
and (2, 3)− saddle points
24. (b) 4K < , saddle point, 4K > local minimum, no values of
K for local maximum.
25. The minimum distance from the surface to the origin is 6 .
This occurs at the points (2, 1,1)− and
(2, 1, 1)− − .
26. (a) 8r = (b)
4
π
θ = (c)
5
4
π
φ = (d)
10
cos
ρ
12. φ
=
27. (a) positive (b) positive (c) negative (d) negative
28. (a) ( )1 sin18 cos 5 cos11
6
− (b) 3
2
(5)
3
π volume of a half sphere
(c) ( )3 22 28 1
9
− change the order (d) 8
1 1
2 3 3
e
convert to spherical
13. (e)
7
3
− (f)
25
2
area of a triangle
29. (a)
2 2 2
2 2 2
2 2 4
2 2
x x y
x x y
dzdydx
− − −
− − − +∫ ∫ ∫
2 2 2 4
0 0
r
r
14. rdzdrd
π
θ
−
∫ ∫ ∫
2 4 2 2
0 0 0
sin d d d
π π
ρ φ ρ φ θ∫ ∫ ∫
(b)
2 2 2
2 2 2
3 9 20
3 9 2
z y z
z y z
dxdydz
− − −
16. 0 0 0
rdydrd
π
θ∫ ∫ ∫
(d)
0 1 3 2 2 6 12
6 0 0
x x y
dzdydx
+ − +
−∫ ∫ ∫
30.
2 4 4
0 0 0
(4 )
r
k z rdzdrd
π
θ
−
17. −∫ ∫ ∫
31. There are many possible answers.
(a) 3 cos 2, 1, 3sin 0 2x t y z t t π= + = = − ≤ ≤
(b) 1 2 , 2 3 , 3 x t y t z t t= + = − − = − −∞ < < ∞
(c) ( )3, 2 2 0x t y t t= − = − + − ≤ ≤
(d) 2 cos , 2 sin , 2 0 2x t y t z t π= = = ≤ ≤
32. (a) 4t = seconds (b) 10 feet per second
(c) There are many possible answers. 3, 3( 2 ), 10 2 ( 2 )
2x y t z t tπ π π π= = − = − − − ≥
33. (a) There are many possible answers. 1 2 , 1 6 , 7+ 0x
t y t z t t= + = + = ≥ (b) No
34. (a) iii (b) v (c) vi (d) i (e) ii (f) iv
35.
1 2 3C C C
F dr F dr F dr⋅ < ⋅ < ⋅ ∫ ∫ ∫
19. 39. 0p = , 500flux π=
40. 22, 000, 000 40, 000, 000 18, 000, 000π π π− = −
41. (a) (i) 0 (ii) 0 (iii) zero k
component (iv) could be a gradient field
(b) (i) positive (ii) 0 (iii) positive k
component (iv) could not be a gradient field
(c) (i) 0 (ii) positive (iii) zero k
component (iv) could be a gradient field
42. (a) On a sphere of radius 5. (b)
6 5 1S S S
F dA F dA F dA⋅ < ⋅ < ⋅ ∫ ∫ ∫
43. 75 12 63π π π− =
20. 44. (a) 10− (b) 29
45. (a) V (b) S (c) S (d) V (e) S (f) V
(g) S (h) ND (i) ND
46. (a) false (b) true (c) true (d) true (e) true
47. There are many possible answers.
(a)
10 10
3 2
i j−
(b)
8
2
− (c) (18.5, 74.5) (d) 10 (e) 6(60 80 50 70)
1560+ + + =
MATH 223
FINAL EXAM REVIEW PACKET
21. (Fall 2012)
The following questions can be used as a review for Math 223.
These questions are not actual samples
of questions that will appear on the final exam, but they will
provide additional practice for the
material that will be covered on the final exam. When solving
these problems keep the following in
mind: Full credit for correct answers will only be awarded if all
work is shown. Exact values must be
given unless an approximation is required. Credit will not be
given for an approximation when an
exact value can be found by techniques covered in the course.
The answers, along with comments, are
posted as a separate file on http://math.arizona.edu/~calc.
1. A sonic boom carpet is a region on the ground where the
sonic boom is heard directly from the
airplane and not as a reflection. The width of the carpet, W, can
be expressed as a function of the air
temperature on the ground directly below the airplane, t, and the
vertical temperature gradient at the
airplane’s altitude, d. Suppose ( , )
t
W t d k
d
= for some positive constant k.
(a) If d is fixed, is the width of the carpet an increasing or
decreasing function of t.
22. (b) If t is fixed, is the width of the carpet an increasing or
decreasing function of d.
2. Describe the following sets of points in words, write an
equation, and sketch a graph:
(a) The set of points whose distance from the line L is five. The
line L is the intersection of the
plane 3y = and the xy-plane.
(b) The set of points whose distance from the yz-plane is three.
(c) The set of points whose distance from the z-axis and the xy-
plane are equal.
3. By setting one variable constant, find a plane that intersects
2 2cos 3y x z+ = in a:
(a) parabola (b) waves (related to cosine curves) (c) line(s)
4. Consider the function 2( , )f x y y x= − .
(a) Plot the level curves of the function for 2, 1, 0,1, 2z = − − .
(b) Imagine the surface whose height above any point ( , )x y is
given by ( , )f x y . Suppose you are
standing on the surface at the point where 1, 2x y= = .
(i) What is your height?
(ii) If you start to move on the surface parallel to the y-axis in
the direction of increasing y, does
your height increase or decrease?
(iii) Does your height increase or decrease if you start to move
on the surface parallel to the x-
axis in the direction of increasing x?
5. Describe the level surfaces of each:
(a) 2 2( , , )f x y z x y z= − − (b)
23. 2 2 21( , , ) x y zg x y z e − − −=
http://math.arizona.edu/~calc�
2.5 3.0 3.5
-1.0 6 8
1.0 1 2
3.0 -6
6. The figure at the right shows the level curves of the
temperature T in degrees Celsius as a function
of t hours and depth h in centimeters beneath the surface of the
ground from midnight ( 0t = ) one day
to midnight ( 24t = ) the next.
(a) Approximately what time did the sunrise?
When do you think the sun is directly overhead?
(b) Sketch a graph of the temperature as a
function of time at 20 centimeters.
(c) Sketch a graph of the temperature as a
function of the depth at noon.
from S. J. Williamson, Fundamentals of Air Pollution,
(Reading: Addison-Wesley, 1973)
7. Given the table of some values of a linear function, complete
24. the table and find a formula for the
function.
8. Consider the planes:
I. 3 5 2x y z− − = II. 5 3x y= + III. 5 3 2x y+ =
IV. 3 5 2x y+ = V. 3 5 2x y z+ + = VI. 1 0y + =
List all of the planes which:
(a) Are parallel to the z-axis.
(b) Are parallel to 3 5 7x y z= + + .
(c) Contain the point (1, 1, 6)− .
(d) Are normal to ( ) ( )2 3 3i k i k+ × −
.
(e) Could be the tangent plane to a surface ( , )z f x y= , where f
is some function which has finite
partial derivatives everywhere.
9. A portion of the graph of a linear function is shown.
(a) Find an equation for the linear function.
(b) Find a vector perpendicular to the plane.
(c) Find the area of the shaded triangular region.
25. x
-2 -1 0 1 2
2 0.111 0.167 0.200 0.167 0.111
1 0.167 0.333 0.500 0.333 0.167
y 0 0.200 0.500 1.000 0.500 0.200
-1 0.167 0.333 0.500 0.333 0.167
-2 0.111 0.167 0.200 0.167 0.111
x
-2 -1 0 1 2
2 0.00 -3.00 -4.00 -3.00 0.00
1 3.00 0.00 -1.00 0.00 3.00
y 0 4.00 1.00 0.00 1.00 4.00
-1 3.00 0.00 -1.00 0.00 3.00
-2 0.00 -3.00 -4.00 -3.00 0.00
x
y
x
y
x
y
x
26. -2 -1 0 1 2
2 2.828 2.236 2.000 2.236 2.828
1 2.236 1.414 1.000 1.414 2.236
y 0 2.000 1.000 0.000 1.000 2.000
-1 2.236 1.414 1.000 1.414 2.236
-2 2.828 2.236 2.000 2.236 2.828
10. Match each of the following functions (a) – (f), given by a
formula, to the corresponding tables,
graphs, and/or contour diagrams (i) – (ix). There may be more
than one representation or no
representations for a formula.
(a) 2 2( , )f x y x y= − (b) ( , ) 6 2 3f x y x y= − + (c) 2 2( , )
1f x y x y= − −
(d) 2 2
1
( , )
1
f x y
x y
=
+ +
(e) ( , ) 6 2 3f x y x y= − − (f) 2 2( , )f x y x y= +
(i) Table 1 (ii) Table 2 (iii) Table 3
28. 12. Consider the vectors 2 3u i j k= − +
(a) For what value(s) o
the point (1, 2, 3)− .
containing the point (1, 2, 3)− .
13. Let 3 2u j k= −
-plane such that the
3
π
29. 14. Find the following:
(a) ( ) ( )( )3 2 2ln 3 arctanx y x y
x
∂
+ − +
∂
(b) Hf if
( )3
2
( , )
5
H T
f H T
H
+
=
−
(c)
2 x y
x y y x
15. Find an equation for the tangent plane to:
30. (a) ( , ) xyf x y ye= at ( , ) (1, 2)x y = (b) 2 2 2( 1) 4( 2) ( 3)
17x y z− + − + − = at (3, 3, 6)
16. A ball is thrown from ground level with initial speed v
(m/sec) and at an angle of α with the
horizontal. It hits the ground at a distance
2 sin(2 )
( , )
v
s v
g
α
α = where 10 g ≈ m/sec2.
(a) Find the differential ds .
(b) What does the sign of (20, 3)sα π tell you?
(c) Use the linearization of s about (20, 3)π to estimate the
change in α that is needed to get
approximately the same distance if the initial speed changes to
19 m/sec.
17. The depth of a lake at the point ( , )x y is given by 2 2( , ) 2
3h x y x y= + feet. A boat is at (-1,2).
(a) If the boat sails in the direction of the point (3, 3) , is the
water getting deeper or shallower?
(b) In which direction should the boat sail for the depth to
remain constant? Give your answer as a
vector.
(c) If the boat moves on the curve ( ) ( )2( ) 1 2r t t i t j= − + +
31. changing when 2t = ?
18. Calculate the following:
(a)
2
21
yz
grad
x
(b) ( ) ( ) ( )( )2 2 2curl x y z i y z j xz k+ + − + +
(c) ( ) ( ) ( )( )2 3cos sec zdiv x i x y j e k+ +
(d) The greatest rate of change of 3( , , ) tanf x y z x z= + at (
4 , 3,1)π .
32. (e) The potential function for cos( )z zG yi xj e e k= + +
.
19. Sup
direction u
5
6
π
.
20. The contour plot for ( , )f x y is shown at the right.
Determine if each quantity is positive, negative, or zero.
(a) (1,1)xf (b) ( 1,1)xf −
(c) ( 2, 2)yf − − (d) (1,1) (1, 2)x xf f−
(e) (1,1)yyf (f) ( 2, 2)xyf − −
21. Let ( , ) 3 cos( )w x y x yπ= .
33. (a) Find
(1, 1 2)
w
u
∂
∂
and
(1, 1 2)
w
v
∂
∂
if 2 2x u v= + and
v
y
u
= .
(b) Find
1t
dw
dt =
if tx e−= and lny t= .
34. 22. Suppose ( , )z f x y= , ( , )x g r θ= , and ( , )y h r θ= . Find (
)1, 2rz π if given the following:
( )1, 2 0g π = , ( )1, 2 1h π = , (0,1) 2xf = , (0,1) 3yf = ,
( )1, 2 5rg π = , ( )1, 2 7gθ π = , ( )1, 2 9rh π = , ( )1, 2 11hθ
π =
23. Find and classify all of the critical points for 3 2 3 2( , ) 2 3
12 3 9f x y x x x y y y= − − + + − .
24. Let 2 2( , ) 4f x y Kx y xy= + − .
(a) Verify that the point (0, 0) is a critical point.
(b) Determine the values of K, if any, for which (0, 0) can be
classified as the following.
(i) a saddle point (ii) a local minimum (iii) a local maximum
25. Find the minimum distance from the surface 2 3 9z x xy+ −
= to the origin.
26. Find an equation for each surface:
(a) 2 2 8x y+ = in cylindrical coordinates (b) y x= in
cylindrical coordinates
(c) 2 2z x y= − + in spherical coordinates (d) 10z = in
spherical coordinates
x
2
27. Determine (without calculation) whether the integrals are
35. positive, negative, or zero. Let D be the
region inside the unit circle centered at the origin, T be the top
half of the region, B be the bottom half
of the region, L be the left half of the region, and R be the right
half of the region.
(a) x
T
e dA−∫ (b) cosB ydA∫ (c) ( )L x y dA+∫ (d)
y
R
ye dA−∫
28. Evaluate each of the integrals:
(a)
3 6
0 0
cos(3 ) sin(2 5)y x dydx+∫ ∫ (b)
5 2
0 0 0
sin d d d
π π
ρ φ ρ φ θ∫ ∫ ∫
(c)
9 3 3
36. 0
1
y
x dxdy+∫ ∫ (d)
( )3 22 2 2 2 2 2 2 4 4
0 0 0
x x y x y z
e dzdydx
− − − − + +
∫ ∫ ∫
(e)
R
xdA∫ where R is shown below. (f)
2 5 sin
4 0
rdrd
π θ
π
θ∫ ∫ Hint: sketch R first.
37. 29. Set up integrals needed to find the following:
(a) The volume between the sphere 2ρ = and the cone z r= .
(Cartesian, cylindrical, and spherical)
(b) The volume between 2 220x y z= − − and 2 2 2x y z= + + .
(Cartesian and cylindrical)
(c) The volume of the solid in the first octant bounded from
above by 2 2 16x z+ = and 12y = .
(Cartesian in the order dxdydz and Cylindrical in the order
rdydrdθ )
(d) The volume of the tetrahedron under the portion of the plane
shown at the right, bounded by the planes 0y = , 0x = , and
0z = . (Cartesian)
30. A pile of dirt is approximately in the shape of 2 24z x y= −
+ , where x, y, and z are in meters. The
density (kg/m3) of the dirt is proportional to the distance from
the top of the pile of dirt. Set up an
integral for the mass of the pile of dirt.
31. Give parametric equations for the following curves:
(a) A circle of radius 3 on the plane 1y = centered at (2,1, 0)
oriented clockwise when viewed from
the origin.
(b) A line perpendicular to 2 3 7z x y= − + and through the
point (1, 2, 3)− .
(c) The curve ( )32y x= + oriented from (2, 64) to (0, 8) .
(d) The intersection of the surfaces 2 2 2z x y= + and 2 26z x y=
− − .
38. y
2
6−
12
1
32. A child is sliding down a helical slide. Her position at time t
seconds after the start is given in feet
by ( ) ( )3 cos 3sin (10 )r t i t j t k= + + −
-plane.
(a) When is the child 6 feet from the ground?
(b) How fast is the child traveling at 2 seconds?
(c) At time 2t π= seconds, the child leaves the slide tangent to
the slide at that point. What is the
equation of the tangent line?
33. The surface of a hill is represented by 2 212 3z x y= − − ,
where x and y are measured horizontally.
A projectile is launched from the point (1,1, 7) and travels in a
line perpendicular to the surface at that
point.
(a) Find parametric equations for the path.
(b) Does the projectile pass through the point (6,16,10) ?
39. 34. Match the vector field to its sketch.
(a) xi yj+
(b) xi yj−
(c) yi xj+
(d) yi
(e) i xj+
(f) 2x i xyj+
(i) (ii) (iii)
(iv) (v) (vi)
2C
35. Given the plot of the vector field, F
40. , list the following quantities in increasing order. Also give a
possible formula for F
.
(i)
1C
F dr⋅ ∫
(ii)
2C
F dr⋅ ∫
(iii)
3C
F dr⋅ ∫
36. Evaluate
41. C
F dr⋅ ∫
(a) ( )F x z i zj yk= + + +
. C is the line from (2, 4, 4) to (1, 5, 2) .
(b) 2 sin( ) sin( )F x i z yz j y yz k= + +
. C is the curve from (0, 0,1)A to (3,1, 2)B as shown below.
(c) F yi xj zk= − +
. C is the circle of radius 3 centered on the z-axis in the plane
42. 4z = oriented
clockwise when viewed from above.
(d) 34 ( )F x i x y j= + +
. C is the curve sin(2 )y x= from (0, 0) to ( 2 , 0)π .
(e) ( ) ( )3 2 3 2sin( ) ln( 1)F y x i x y j= − + + − +
. C is the circle of radius 5 centered at (0, 0) in the xy-
plane oriented counterclockwise.
37. Evaluate
S
F dA⋅ ∫
:
(a) 3 4 ( )F i j z x k= + + −
. S is a square of side 2 on the plane z x= oriented upward.
(b) 5F i zj yk= − + −
43. . S is 2 2x y z= + for 0 8x≤ ≤ , oriented in the negative x-
direction.
(c) 3 52 ( ) ( 7 )F xi z y j x z k= − − + +
. S is the closed cylinder centered on the y-axis with radius 3,
length 5, oriented outward.
(d) F xi yj zk= + +
. S is the part of the surface 2 225 ( )z x y= − + above the disk
of radius 5 centered
at the origin, oriented upward.
38. (a) Evaluate 2 3( )
C
grad x yz dr⋅ ∫
(1,1) in the xy-
plane,
44. oriented counterclockwise.
(b) Evaluate 2( ( ) )
S
curl x i y z j xzk dA− + + ⋅ ∫
where S is the cube of side 4 centered at (2,1, 3) , oriented
outward.
1C
3C
P• P•
39. Consider the flux of the vector field p
r
H
r
=
for 0p ≥ out of the sphere of radius 5 centered at
45. the origin. For what value of p is the flux a maximum? What is
that maximum value?
40. An oceanographic vessel suspends a paraboloid-shaped net
below the ocean at a depth of 1000 feet,
held open at the top by a circular metal ring of radius 20 feet,
with bottom 100 feet below the ring and
just touching the ocean floor. Set up coordinates with the origin
at the point where the net touches the
ocean floor and with z measured upward. Water is flowing with
velocity
2
2 (1100 ) (1100 )xv xzi xe j z z k−= − + + −
top of the paraboloid shape to find the flux of water through the
net (oriented from inside to outside).
41. The vector fields below have the form 1 2F F i F j= +
. Assume 1F and 2F depend only on x and y.
For each vector field, circle the best answers.
(a) (b) (c)
46. (i)
C
F dr⋅ ∫
zero
(ii) ( )divF P
is positive negative zero
(iii) curlF
at P has positive k
component negative k
component zero k
component
(iv) F
47. could be a gradient field could not be a gradient field
42. Let 3 3 3(75 )F x x i y j z k= − − −
and let 1S , 5S , and 6S be spheres of radius 1, 5, and 6
respectively,
centered at the origin.
(a) Where is 0divF =
?
(b) Without computing the flux, order the flux out of the
spheres from smallest to largest.
43. In the region between the circles 2 21 : 4C x y+ = and
2 2
2 : 25C x y+ = in the xy-plane, the vector
field F
has 3curlF k=
48. . If 1C and 2C are both oriented counterclockwise when viewed
from above,
find the value of
2 1C C
F dr F dr⋅ − ⋅ ∫ ∫
P•
y
44. Let ( )27 7F xyi x y j= + +
.
(a) Find the circulation density of F
around k
at (2,1, 3)
(b) Find the flux density of F
49. at ( 1, 4, 2)−
45. Determine if each of the following quantities is a vector
(V), a scalar (S), or is not defined (ND).
-D vectors, r xi yj zk= + +
is
a differentiable 3-D vector field, and f is a differentiable scalar
function of x, y, and z.
(a) ( )curlG r×
u v
r
⋅
50. (f) ( )curl fG
(g) ( )
C
curlG dr⋅ ∫
S
divG dA⋅ ∫
(i) gradG
46. True or False?
(a) If all of the contours of a function ( , )g x y are parallel
lines, then the function must be linear.
(b) If curlF
is parallel to the x-axis for all x, y, and z and if C is a circle in
the xy-plane, then the
circulation of F
51. around C must be zero.
(c) If f is a differentiable function, then ( , ) ( , )uf a b f a b≥ −
∇
(d) If F
is a divergence free vector field defined everywhere and S is a
closed surface oriented inward,
then 0
S
F dA⋅ =∫
.
(e) If G
is a curl free vector field defined everywhere and C is a simple
closed path, then 0
C
G dr⋅ =∫
47. Use the portion of the contour diagram of ( , )f x y shown
below to estimate the following:
52. he direction i j− +
(c) A critical point of ( , )f x y . (d)
C
f dr∇ ⋅ ∫
(e) ( , )
R
f x y dA∫ where R is the rectangle 9 15x≤ ≤ , 76 80y≤ ≤ .
x