KING’S OWN INSTITUTE Success in Higher Education IC.docx

KING’S OWN INSTITUTE* Success in Higher Education ICT 200 DATABASE DESIGN AND DEVELOPMENT T219 All information contained within this Subject Outline applies to all students enrolled in the trimester as indicated. 1. General Information 1.1 Administrative Details Associated HE Award(s) Duration Level Subject Coordinator Bachelor of Information Technology (BIT) 1 trimester Level 2 Mr Utpal Nanavati [email protected] Consultation: via Moodle or by appointment. 1.2 Core / Elective Core subject in the BIT 1.3. Subject Weighting Indicated below is the weighting of this subject and the total course points. Subject Credit Points Total Course Credit Points 4 BIT (96 Credit Points) 1.4 Student Workload Indicated below is the expected student workload per week for this subject No. timetabled hours/week* No. personal study hours/week** Total workloa d hours/w 4 hours/week containing 1 x 2 hour Lecture 1 x 2 hour Tutorial 6 hours/week 10 hours/week * Total time spent per week at lectures and tutorials ** Total time students are expected to spend per week in studying, completing assignments, etc. *** Combination of timetable hours and personal study. 1.5 Mode of Delivery On-campus 1.6 Pre-requisites ICT 103 Systems Analysis and Design 1.7 General Study and Resource Requirements o Dedicated computer laboratories are available for student use. Normally, tutorial classes are conducted in the computer laboratories. o Students are expected to attend classes with the requisite textbook and must read specific chapters prior to each tutorial. This will allow them to actively take part in discussions. Students should have elementary skills in both word processing and electronic spreadsheet software, such as OFFICE 365 or MS Word and MS Excel. o Computers and WIFI facilities are extensively available for student use throughout KOI. Students are encouraged to make use of the campus Library for reference materials. o Students will require access to the internet and email. Where students use their own computers, they should have internet access. KOI will provide access to required software. ICT 200 DATABASE DESIGN AND DEVELOPMENT T219 19/06/2019 16:20 PAGE 1 OF 15 *AUSTRALIAN INSTITUTE OF BUSINESS AND MANAGEMENT PTY LTD © ABN: 72 132 629 979 CRICOS 03171A Approved by KOI Academic Board for T2 2019 mailto:[email protected] ICT200 Software Resource requirements specific to this subject: Office 365, MS Imagine, SQL Server 2017. 2. Academic Details 2.1 Overview of the Subject This subject will provide the student with an overall understanding of database development, concepts and theory. Students will learn to design and build a database from data analysis, normalisation, mapping a specific database model. The relational model is emphasised and introduced using structured queried language (SQL) for creating and manipulating databases in both MS Access and ...

KING’S OWN INSTITUTE*
Success in Higher Education
ICT 200 DATABASE DESIGN AND DEVELOPMENT T219
All information contained within this Subject Outline applies to
all students enrolled in the trimester as indicated.
1. General Information
1.1 Administrative Details
Associated HE Award(s) Duration Level Subject Coordinator
Bachelor of Information Technology (BIT) 1 trimester Level 2
Mr Utpal Nanavati
[email protected]
Consultation: via Moodle or by
appointment.
1.2 Core / Elective
Core subject in the BIT
1.3. Subject Weighting
Indicated below is the weighting of this subject and the total
course points.
Subject Credit Points Total Course Credit Points
4 BIT (96 Credit Points)
1.4 Student Workload
Indicated below is the expected student workload per week for
this subject
No. timetabled hours/week* No. personal
study
hours/week**
Total
workloa
d
hours/w
4 hours/week containing
1 x 2 hour Lecture
1 x 2 hour Tutorial
6 hours/week 10 hours/week
* Total time spent per week at lectures and tutorials
** Total time students are expected to spend per week in
studying, completing assignments, etc.
*** Combination of timetable hours and personal study.
1.5 Mode of Delivery On-campus
1.6 Pre-requisites ICT 103 Systems Analysis and Design
1.7 General Study and Resource Requirements
o Dedicated computer laboratories are available for student use.
Normally, tutorial classes are
conducted in the computer laboratories.
o Students are expected to attend classes with the requisite
textbook and must read specific chapters
prior to each tutorial. This will allow them to actively take part
in discussions. Students should have
elementary skills in both word processing and electronic
spreadsheet software, such as OFFICE 365
or MS Word and MS Excel.
o Computers and WIFI facilities are extensively available for
student use throughout KOI. Students are
encouraged to make use of the campus Library for reference
materials.
o Students will require access to the internet and email. Where
students use their own computers, they
should have internet access. KOI will provide access to required
software.
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19/06/2019 16:20 PAGE 1 OF 15
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Approved by KOI Academic Board for T2 2019
mailto:[email protected]
ICT200
Software Resource requirements specific to this subject: Office
365, MS Imagine, SQL Server 2017.
2. Academic Details
2.1 Overview of the Subject
This subject will provide the student with an overall
understanding of database development, concepts
and theory. Students will learn to design and build a database
from data analysis, normalisation,
mapping a specific database model. The relational model is
emphasised and introduced using
structured queried language (SQL) for creating and
manipulating databases in both MS Access and
SQL Server environments. Assignment work includes the
analysis, design, and implementation of a
database using SQL queries in SQL Server environment.
2.2 Graduate Attributes for Undergraduate Courses
Graduates of Bachelor courses from King’s Own Institute (KOI)
will be able to demonstrate the
attributes of a successful Bachelor degree graduate as outlined
in the Australian Qualifications
Framework (2nd edition, January 2013). Graduates at this level
will be able to apply an advanced
body of knowledge across a range of contexts for the purposes
of professional practice or academic
scholarship, and as a pathway for further learning.
King’s Own Institute’s key generic graduate attributes for a
Bachelor’s level degree are summarised
below:
Across the course, these skills are developed progressively at
three levels:
o Level 1 Foundation – Students learn the basic skills, theories
and techniques of the subject and
apply them in basic, standalone contexts
o Level 2 Intermediate – Students further develop the skills,
theories and techniques of the subject
and apply them in more complex contexts, and begin to
integrate this application with other subjects.
o Level 3 Advanced – Students demonstrate an ability to plan,
research and apply the skills, theories
and techniques of the subject in complex situations, integrating
the subject content with a range of
other subject disciplines within the context of the course.
KOI Bachelor Degree
Graduate Attributes Detailed Description
Knowledge Current, comprehensive, and coherent and
connected knowledge
Critical Thinking Critical thinking and creative skills to analyse
and synthesise information and evaluate new problems
Communication
Communication skills for effective reading, writing, listening
and presenting in varied modes and contexts and for the
transferring of knowledge and skills to others
Information Literacy Information and technological skills for
accessing, evaluating, managing and using information
professionally
Problem Solving Skills
Skills to apply logical and creative thinking to solve problems
and evaluate solutions
Ethical and Cultural Sensitivity Appreciation of ethical
principles, cultural sensitivity and social responsibility, both
personally and professionally
Teamwork Leadership and teamwork skills to collaborate,
inspire colleagues and manage responsibly with positive results
Professional Skills
Professional skills to exercise judgement in planning,
problem solving and decision making
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2.3 Subject Learning Outcomes
This is a Level 2 subject.
On successful completion of this subject, students should be
able to:
2.4 Subject Content and Structure
Below are details of the subject content and how it is structured,
including specific topics covered in
lectures and tutorials. Reading refers to the text unless
otherwise indicated.
Weekly Planner:
Week
(beginning)
Topic covered in each week’s
lecture Reading(s)
Expected work as
listed in Moodle
1
08 Jul
Introduction to DBMS:
o history of database processing
o emergence of relational model
o post-relational developments
o DBMS concepts
o MS Access and SQL
Chapter 1 and
Database History
Article
(See section 2.6)
Chapter 1 Discussion.
Formative Not Graded
Introduction to MS Access
and SQL environments.
Tutorial exercises
2
15 Jul
Introduction to Structured Query
Language:
o creating SQL statements
o using SQL in MS Access
o using SQL in SQL Server
o Querying tables
Chapter 2 Activities, Database
exercises - execute simple
SQL queries in MS Access
and SQL Server. Formative
Not Graded. Tutorial
exercises
3
22 Jul
Data modelling with the Entity-
Relationship model:
o purpose of a data model
o the E-R model and diagrams
o variations of the E-R model
o entities and data modelling
process
Chapter 5 Activities, Data Modelling
ERD exercises. VISIO and
UML (Appendix C and D).
Formative Not Graded.
Tutorial exercises
Subject Learning Outcomes Contribution to Course Graduate
Attributes
a) Explain the history and development of database technologies
and the
emergence of the relational database model and SQL.
b) Model business information requirements and produce a
logical
database design using entity relationship diagrams (ERD) and
extended
relationship diagrams (EERD).
c) Design, develop, test and prove the functionality of a
database using MS
Access and SQL.
d) Describe and carry out the necessary steps to develop an
effective
physical database design
e) Formulate, write and execute SQL queries in an SQL Server
environment.
f) Explain the main functions of database administration and
data
warehousing.
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4
29 Jul
The relational model and
normalisation:
o terminology
o characteristics of relationships
o normal forms normalisation
categories
Chapter 3 Activities, Normalisation
exercises on 1NF,2NF, 3NF
and BCNF.
Formative Not Graded
Tutorial exercises
Assessment 2 Quiz A
5
05 Aug
Database design using
normalisation:
o advantages and
disadvantages
o normalising with SQL
o common design problems
Chapter 4 Simple SQL Activities,
normalisation and de-
normalisation exercises.
Formative Not Graded
Tutorial exercises
6
12 Aug
Transforming data models into
database designs:
o purpose of database design
o tables, entities,
primary/alternate keys
o verify normalisation
o create relationships
o design for minimum cardinality
Chapter 6 Activities creating tables and
relationship in SQL,
Database Design exercises.
Formative Not Graded
Database Exercises from
textbook
18 Aug 2019
–
25 Aug 2019 Mid trimester break
7
26 Aug
SQL for database construction
and application processing:
o using SQL scripts
o Advanced SQL statements
Chapter 7 Chapter Activities complex
SQL queries using join,
union, sub-queries. Tutorial
exercises
Formative Not Graded
8
02 Sep
Database redesign:
o the need for database
redesign
o analysing existing databases
o database backup and test
databases
o making changes to tables,
columns, constraints,
cardinalities, relationships
Chapter 8 Chapter Activities
Testing database and query
optimisation for database
efficiency. Formative Not
Graded Tutorial exercises
Assessment 3 Quiz B.
9
09 Sep
Managing multiuser databases:
o database administration
o DBMS and application
security
o database backup and
recovery
o managing the DBMS and data
repository
Chapter 9 Tutorial exercises
Chapter Activities Formative
Not Graded
10
16 Sep
Managing databases with
Microsoft SQL server:
o installing the DBMS
o using the DBMS utilities
o creating a database
o creating and running SQL
scripts
o DBMS security
Chapter 10 Tutorial exercises
Assessment 4 due - Report
11
23 Sep
The web server environment:
o the web database processing
environment
o database server access
standards
o OBDC standard
o web database processing with
PHP and NetBeans IDE
o challenges and SQL injection
Chapter 11 Tutorial exercises SQL
Server Project. Formative
Not Graded
Assessment 4 due -
Presentation
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attacks
o the importance of XML
12
30 Sep
Big data, data warehousing and
business intelligence systems:
o business intelligence systems
o reporting and data mining
systems
o data warehousing and data
marts
o components of a data
warehouses
o OLAP and data mining
o distributed databases
o cloud computing
o big data and the Not Only
SQL movement
Chapter 12 Discussion. Formative Not
Graded.
Tutorial exercises
13
06 Oct Study Review Week
14
14 Oct Final Exam Week
Please see Exam Timetable for exam date, time
and location
15
20 Oct
Student Vacation begins
Enrolments for T319 open
16
28 Oct
Results Released 29 Oct 2019
Certification of Grades 01 Nov 2019
T319 begins 04 Nov 2019
1
04 Nov
Week 1 of classes for T319
Friday 01 Nov 2019 – Review of Grade Day for T219 – see
Sections 2.6 and 3.6
below for more information.
2.5 Public Holiday Amendments
Please note: KOI is closed on all scheduled NSW Public
Holidays.
T219 has one (1) public holiday (Labour Day) that occurs
during this trimester. Classes scheduled for this
public holiday (Calendar Class Dates) will be rescheduled as
per the table below.
This applies to ALL subjects taught in T219.
Please see the table below and adjust your class timing as
required. Please make sure you have
arrangements in place to attend the rescheduled classes if
applicable to your T219 enrolment.
Classes will be conducted at the same time and in the same
location as your normally scheduled class
except these classes will be held on the date shown below.
Calendar Class Date Rescheduled Class Date
Monday 07 October 2019 (Week 13) Study
Review Week
Not required
2.6 Review of Grade, Deferred Exams & Supplementary
Exams/Assessments
Review of Grade:
There may be instances when you believe that your final grade
in a subject does not accurately reflect
your performance against the subject criteria. Section 8 of the
Assessment and Assessment Appeals
Policy (www.koi.edu.au) describes the grounds on which you
may apply for a Review of Grade.
If this happens and you are unable to resolve it with the
Academic staff concerned then you can apply for
a formal Review of Grade within the timeframes indicated in the
following sections of this subject outline -
Supplementary Assessments, 3.6 Appeals Process as well as the
Assessment and Assessment Appeals
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ICT200
Policy. Please ensure you read the Review of Grade information
before submitting an application.
Review of Grade Day:
KOI will hold the Review of Grade Day for all subjects studied
in T219 on
Friday 01 November 2019
Only final exams will be discussed as all other assessments
should have been reviewed during the
trimester.
If you fail one or more subjects and you wish to consider
applying for a Review of Grade you MUST
attend the Review of Grade Day. You will have the chance to
discuss your final exam with your lecturer,
and will be advised if you have valid reasons for applying for a
Review of Grade (see Section 3.6 below
and Assessment and Assessment Appeals Policy).
If you do not attend the Review of Grade Day you are
considered to have accepted your results for T219.
Deferred Exams:
If you wish to apply for a deferred exam, you should submit an
Application for Assignment Extension or
Deferred Exam Form before the prescribed deadline.
If you miss your mid-trimester or final exam there is no
guarantee you will be offered a deferred exam.
You must apply within the stated timeframe and satisfy the
conditions for approval to be offered a
deferred exam (see Section 8.1 of the Assessment and
Assessment Appeals Policy and the Application
for Assignment Extension or Deferred Exam Forms). In
assessing your request for a deferred exam, KOI
will take into account the information you provide, the severity
of the event or circumstance, your
performance on other items of assessment in the subject, class
attendance and your history of previous
applications for special consideration.
Deferred mid-trimester exams will be held before the end of
week 9. Deferred final exams will be held on
two days during week 1 or 2 in the next trimester. You will not
normally be granted a deferred exam on
the grounds that you mistook the time, date or place of an
examination, or that you have made
arrangements to be elsewhere at that time; for example, have
booked plane tickets.
If you are offered a deferred exam, but do not attend you will be
awarded 0 marks for the exam. This may
mean it becomes difficult for you to pass the subject. If you
apply for a deferred exam within the required
time frame and satisfy the conditions you will be advised by
email (to your KOI student email address) of
the time and date for the deferred exam. Please ensure that you
are available to take the exam at this
time.
Marks awarded for the deferred exam will be the marks awarded
for that item of assessment towards
your final mark in the subject.
Supplementary Assessments (Exams and Assessments):
A supplementary assessment may be offered to students to
provide a final opportunity to demonstrate
successful achievement of the learning outcomes of a subject.
Supplementary assessments are only
offered at the discretion of the Board of Examiners. In
considering whether or not to offer a
supplementary assessment, KOI will take into account your
performance on all the major assessment
items in the subject, your attendance, participation and your
history of any previous special
considerations.
Students are eligible for a supplementary assessment for their
final subject in a course where they fail the
subject but have successfully completed all other subjects in the
course. You must have completed all
major assessment tasks for the subject and obtained a passing
mark on at least one of the major
assessment tasks to be eligible for a supplementary assessment.
If you believe you meet the criteria for a supplementary
assessment for the final subject in your course,
but have not received an offer, complete the “Complaint,
Grievance, Appeal Form” and send your form to
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[email protected] The deadline for applying for supplementary
assessment is the Friday of the first
week of classes in the next trimester.
If you are offered a supplementary assessment, you will be
advised by email to your KOI student email
address of the time and due date for the supplementary
assessment – supplementary exams will
normally be held at the same time as deferred final exams
during week 1 or week 2 of the next trimester.
You must pass the supplementary assessment to pass the
subject. The maximum grade you can achieve
in a subject based on a supplementary assessment is a PASS
grade.
If you:
o are offered a supplementary assessment, but fail it;
o are offered a supplementary exam, but do not attend; or
o are offered a supplementary assessment but do not submit by
the due date;
you will receive a FAIL grade for the subject.
2.7 Teaching Methods/Strategies
Briefly described below are the teaching methods/strategies
used in this subject:
o On-campus lectures (2 hours/week) are conducted in seminar
style and address the subject content,
provide motivation and context and draw on the students’
experience and preparatory reading.
o Tutorials (2 hours/week) include class discussion of case
studies and research papers, practice sets
and problem-solving and syndicate work on group projects.
Tutorial participation is an essential
component of the subject and contributes to the development of
graduate attributes (see section 2.2
above). It is intended that specific tutorial material such as case
studies, recommended readings,
review questions etc. will be made available each week in
Moodle.
o Online teaching resources include class materials, readings,
model answers to assignments and
exercises and discussion boards. All online materials for this
subject as provided by KOI will be
found in the Moodle page for this subject. Students should
access Moodle regularly as material may
be updated at any time during the trimester
o Other contact - academic staff may also contact students
either via Moodle messaging, or via email
to the email address provided to KOI on enrolment.
Grading
Final grades are awarded by the Faculty Assessment Committee
in accordance with KOI's
Assessment Regulations. The definitions and guidelines for the
awarding of final grades within the
BIT degree are:
o HD High distinction (85-100%) an outstanding level of
achievement in relation to the assessment
process.
o DI Distinction (75-84%) a high level of achievement in
relation to the assessment process.
o CR Credit (65-74%) a better than satisfactory level of
achievement in relation to the assessment
process.
o PS Pass (50-64%) a satisfactory level of achievement in
relation to the assessment process.
o FL Fail (0-49%) an unsatisfactory level of achievement in
relation to the assessment process.
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mailto:[email protected]
ICT200
2.8 Student Assessment
Provided below is a schedule of formal assessment tasks and
major examinations for the subject.
Assessment Type When assessed Weighting Learning Outcomes
Assessed
Assessment 1
Weekly Tutorial Week 11 10% a, b, c, d, e, f
Assessment 2
Quiz A (opens week 2) Week 4 5% a
Assessment 3
Quiz B (opens week 8) Week 8 10% d
Assessment 4
Group Project (Report and
Presentation) - Problem
Based Scenario to Design
(ERD), Implement and Query
a SQL Server Database
Week 10
Week 11
Report 20%
Presentation 5% d, e
Assessment 5
Final Exam (3 hours) Final exam period 50% b, c, d, e, f
Requirements to Pass the Subject:
To gain a pass or better in this subject, students must gain a
minimum of 50% of the total available
subject marks.
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2.9 Prescribed and Recommended Readings
Prescribed Text:
Kroenke, D.M., Auer, D., 2016. Database Processing. 14th ed.,
Pearson
Also available as eText at Pearson Online:
http://www.mypearsonstore.com/bookstore/database-processing-
fundamentals-design-and-
implementation-9780133878998?xid=PSED
Recommended Readings:
Elmasri, R. and Navathe, S. B., 2016. Fundamentals of Database
Systems. 7th ed. Pearson
Publishing.
Hoffer, J., Venkataraman, R. and Topi, H., 2016. Modern
Database Management. 12th ed. Prentice
Hall.
Useful Websites:
The following websites are useful sources covering a range of
information
useful for this subject. As we are using MS Access and SQL
Server there are numerous resources
available online. A number are listed below.
For core concepts and platform-independent (irrelevant to the
DBMS being used) Data Modelling and
SQL useful links are also provided. The prescribed textbook
provides all the information required for
students to do well in this unit. Students are encouraged to
research additional information where
required.
During lectures and tutorials, students will be informed where
example databases and related files
may be found. Students are also expected to use the library and
the internet for additional learning.
The following links are primarily for additional learning and
future reference (be sure to bookmark
them). Remember to use your textbook first – all of the
information you should need is in the text.
Database resources and articles
Journals:
o ACM Transactions on Computer Systems. Available from
EBSCOhost research databases
o ACM Transactions On Database Systems. Available from
EBSCOhost research databases
o Journal of Electronic Commerce Research www.jecr.org
o Journal of International Technology and Information
Management
https://www.sciencedirect.com/journal/international-journal-of-
information-management
A larger resource of database information, tutorials, and
exercises
o Access 2013 Microsoft Official site for Access 2013
o Basic design from MS Office Database Design Basics
o SQL Server Central http://www.sqlservercentral.com/Articles
o Allen Browne Tips. Allen Browne is an Australian based
Access developer with many years of
experience. More for VBA and advanced coding problems.
o MSDN Community Forum. Microsoft Developer Network
(MSDN)
o TechOnTheNet. More advanced on VBA forms
3. Assessment Details
3.1 Details of Each Assessment Item
The assessments for this subject are described below. The
description includes the type of assessment,
its purpose, weighting, due date and submission requirements,
the topic of the assessment, details of the
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http://www.mypearsonstore.com/bookstore/database-processing-
fundamentals-design-and-implementation-
9780133878998?xid=PSED
http://www.mypearsonstore.com/bookstore/database-processing-
fundamentals-design-and-implementation-
9780133878998?xid=PSED
http://www.jecr.org/
https://www.sciencedirect.com/journal/international-journal-of-
information-management
http://www.sqlservercentral.com/Articles
ICT200
task and detailed marking criteria, including a marking rubric
for essays, reports and presentations.
Supplementary assessment information and assistance can be
found in Moodle.
KOI expects students to submit their own original work in both
assignments and exams, or the original
work of their group in the case of group assignments.
Marking guides for assessments follow the assessment
descriptions. Students should compare final
drafts of their assessment against the marking guide before
submission.
Please note that the final exam is not open book. Students are
not permitted to bring any reference
materials into the final exam. Students are not permitted to use
mobile phones or other communication
devices during tests and exams.
Assessment 1
Assessment type: Tutorial Exercises - individual assessment
Assessment purpose: To answer weekly tutorial exercises on the
topics covered in lectures. This
assessment contributes to learning outcomes a, b, c, d, e and f.
Value 10%
Submission requirements details: The tutorial exercises for
weeks 2-11
Assessment topic: Tutorial Exercises
Task details: Students need to complete the weekly tutorial
exercises and upload the answers on Moodle
Assessment 2
Assessment type: Quiz A (opens week 4) - individual
assignment.
Assessment purpose: This assessment will allow students to
demonstrate that they have understood the
concepts covered in weeks 1, 2 and 3. This assessment
contributes to learning outcome a.
Value: 5% Due Date: End of Week 4
Assessment topic: Quiz A
Task details: The quiz will consist of questions and problems
relating to subject content from weeks 1 – 3
inclusive.
Assessment 2 Marking Rubric: Quiz A – Worth 5 Marks
Criteria Fail
(0 – 49%)
Pass
(50 – 64%)
Credit
(65 – 74%)
Distinction
(75 – 84%)
High
Distinction
(85 – 100%)
Score (out of 10) 0-4 5-6 7 8 9-10
Assessment 3
Assessment type: Quiz B (opens week 8) - individual
assignment.
Purpose: This assessment will allow students to demonstrate
that they have understood the concepts
covered in weeks 4 to 7. This assessment contributes to learning
outcome d.
Value: 10% Due Date: Week 8
Task details: The quiz will consist of questions and problems
relating to subject content from weeks 4 – 7
inclusive.
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Assessment 3 Marking Rubric: Short Answer Quiz B – Worth 10
Marks
Criteria Fail
(0 – 49%)
Pass
(50 – 64%)
Credit
(65 – 74%)
Distinction
(75 – 84%)
High
Distinction
(85 – 100%)
Score (out of 10) 0-4 5-6 7 8 9-10
Assessment 4
Assessment type: Group Project Report and Presentation
Purpose: Demonstrate students can effectively work in a group
to design, normalise, implement and
query an SQL Server database according to a project scenario
specification which they may select from a
case study from the textbook. This assessment contributes to
learning outcomes d, e.
Value: Report 20%
Presentation 5%
Total: 25%
Due Date: Report Week 10; Presentation in class Week 11
Assessment topic: Topics covered in week 1 - 10
Submission: Report, Group Charter and presentation file via
Moodle; Presentation in class.
Task details: Group Project (Report and Presentation) - Problem
Based Scenario to Design (ERD),
Implement and Query a SQL Server Database.
Students will choose a case study from the textbook for this
assignment. An ERD, database, scripts for
creating tables, views, triggers, functions, stored procedures,
insert, update, alter, delete and successful
query results as requested in the case study will be assessed
according to the marking criteria below.
Students must normalize the data from 1 NF to BCNF and show
each and every normalization with
diagrams/tables and appropriate sample data to explain their
understanding.
Students will submit one group written report of 1000 words
detailing the tasks they carried out in the
assignment. 2 to 3 students can be in one group. This report will
be due in the end of week 10. Students
will also deliver a 15 mins presentation in class and explain
their understanding and the approach they
took to work on the assignment. This will be arranged in Week
11. The report requires a title page with
student group members’ names and email, the assessment title
and date. A brief introduction to the
assignment can then be followed by the initial ERD, the
Normalised version and the SQL statements
used as requested by the case study chosen. Report must be
well-written, have in-text and detailed
Harvard style referencing and presented professionally,
containing:
o Title page
o Table of Contents
o Introduction
o Appropriate use of headings within report
o Overall structure, presentation and formatting.
All group members must remain present for the demonstration
in tutorial and must be able to explain the
scripts written for the tasks outlined in case study.
Marking Guide: A detailed marking guide for the Database
Project (Assessment 4) is provided. See the
following page.
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Assessment 4 Marking Rubric: SQL Server Database Project –
Worth 20 Marks
Criteria Fail
(0-49%)
Pass
(50-64%)
Credit
(65-74%)
Distinction
(75-84%)
High
Distinction
(85-100%)
ER Diagram
5 marks
ER diagram not
correct or not
submitted
Identified all entities
and built
relationships using
crow’s foot notation
Data is in 3NF
Identified entities and
relationships,
representation using
crow’s foot notation
correct
Data is in BCNF
ERD accompanied
with explanation of
relationships
Data is in BCNF
Demonstrated all the
normalization from 1 NF
to BCNF along with
diagram and table data
DDL, DML and PSM
scripts
10 marks
Scripts are incorrect
or not submitted
DDL and DML
Scripts are correct
Tables are created,
data is inserted,
updated, deleted
DDL and DML Scripts
are correct
Tables are created,
data is inserted,
updated, deleted
Tables have default
values and
constraints are
enabled on tables
DDL and DML
Scripts are correct
Tables are created,
data is inserted,
updated, deleted
Tables have default
values and
constraints are
enabled on tables
Functions and
Triggers are written
to perform tasks
mentioned in case
study
DDL and DML Scripts
are correct
Tables are created, data
is inserted, updated,
deleted
Tables have default
values and constraints
are enabled on tables
Functions, Triggers and
Stored Procedures are
written to perform tasks
mentioned in case study
Report and Presentation
10 marks
Report is not
adhering to
guidelines provided,
tasks are not done
satisfactorily
Team member(s)
have not shown
convincing evidence
of having performed
the tasks outlined
the case study
Report is submitted,
meeting word count
and presentation is
delivered.
There is sufficient
evidence of
participation of team
member(s)
Database, tables
and other objects
are created on SQL
Server
Report is submitted,
meeting word count,
well referenced and
presentation is
delivered.
There is sufficient
evidence of
participation of team
member(s)
Database, tables and
other objects are
created on SQL
Server
Students must be
able to provide an
explanation of the
scripts they’ve written.
Report is submitted,
meeting word count,
well referenced and
presentation is
delivered.
There is sufficient
evidence of
participation of team
member(s)
Students must be
able to provide an
explanation of the
scripts they’ve
written.
Students are able to
demonstrate the
tasks outlined in the
case study.
Report is well-written
and presented
professionally,
containing:
o Title page
o Table of Contents
o Introduction
o Appropriate use of
headings within
report
o Overall structure,
presentation and
formatting.
All tasks outlined in
case study are carried
out and the group has
shown evidence of
having done the task on
their own.
Assessment 5
Assessment type: Final Exam (three hours) individual
assessment – closed book exam
Purpose: The final exam is intended to test that students have
understood key concepts for relational
database design, development and administration. This
assessment contributes specifically to learning
outcomes b, c, d, e, and f.
Value: 50% Due Date: The final exam will be held in the
official KOI exam period in Week 14
of the trimester. The specific date and time will be posted
towards the end of the
trimester.
Topic: The examination may cover content from any part of the
entire subject related to learning
outcomes b, c, d, e and f.
Task Details: Students will be expected to answer written
response questions derived from all lectures
and tutorials in the trimester.
3.2 Late Penalties and Extensions
An important part of business life and key to achieving KOI’s
graduate outcome of Professional Skills is
the ability to manage workloads and meet deadlines.
Consequently, any assessment items such as in-
class quizzes and assignments missed or submitted after the due
date/time will attract a penalty (see
below).
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Students who miss mid-trimester tests and final exams without a
valid and accepted reason (see below)
may not be granted a deferred exam and will be awarded 0
marks for assessment item. These penalties
are designed to encourage students to develop good time
management practices, and create equity for
all students.
Any penalties applied will only be up to the maximum marks
available for the specific piece of
assessment attracting the penalty.
Late penalties, granting of extensions and deferred exams are
based on the following:
In Class Tests (excluding Mid-Trimester Tests)
o No extensions permitted or granted – a make-up test may only
be permitted under very special
circumstances where acceptable supporting evidence is
provided. The procedures and timing to
apply for a make-up test (only if available) are as shown in
Section 3.3 Applying for an Extension
(below).
o Missing a class test will result in 0 marks for that assessment
element unless the above applies.
Written Assessments
o 5% of the total available marks per calendar day unless an
extension is approved (see Section 3.3
below)
Presentations
o No extensions permitted or granted – no presentation = 0
marks. The rules for make-up presentations
are the same as for missing in-class tests (described above).
Mid-Trimester Tests and Final Exams
o If students are unable to attend mid-trimester tests or final
exams due to illness or some other event
(acceptable to KOI), they must:
− Advise KOI in writing (email: [email protected]) as soon as
possible, but no later than three
(3) working days after the exam date, that they will be / were
absent and the reasons. They will
be advised in writing (return email) as to whether the
circumstances are acceptable.
− Complete the appropriate Application for Extension or
Deferred Exam Form available from the
Student Information Centre in Moodle, on the KOI Website
(Policies and Forms) and the
Reception Desk (Market St and Kent St), as soon as possible
and email with attachments to
[email protected]
− Provide acceptable documentary evidence in the form of a
satisfactorily detailed medical
certificate, police report or some other evidence that will be
accepted by KOI.
− Agree to attend the deferred exam as set by KOI.
Deferred exam
o There will only be one deferred exam offered.
o Marks awarded for the deferred exam will be the marks
awarded for that assessment.
o If you miss the deferred exam you will be awarded 0 marks
for the assessment. This may mean you
are unable to complete (pass) the subject.
3.3 Applying for an Extension
If students are unable to submit or attend an assessment when
due, and extensions are possible, they
must apply by completing the appropriate Application for
Extension form available from the Student
Information Centre in Moodle, the KOI Website (Policies and
Forms) and the Reception Desk (Market St
and Kent St), as soon as possible but no later than three (3)
working days of the assessment due date.
The completed form must be emailed with supporting
documentation to [email protected]
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mailto:[email protected]
mailto:[email protected]
mailto:[email protected]
ICT200
Students and lecturers / tutors will be advised of the outcome of
the extension request as soon as
practicable.
Appropriate documentary evidence to support the request for an
extension must be supplied. Please
remember there is no guarantee of an extension being granted,
and poor organisation is not a
satisfactory reason to be granted an extension.
3.4 Referencing and Plagiarism
Please remember that all sources used in assessment tasks must
be suitably referenced.
Failure to acknowledge sources is plagiarism, and as such is a
very serious academic issue. Students
plagiarising run the risk of severe penalties ranging from a
reduction through to 0 marks for a first offence
for a single assessment task, to exclusion from KOI in the most
serious repeat cases. Exclusion has
serious visa implications. The easiest way to avoid plagiarising
is to reference all sources.
Harvard referencing is the required method – in-text referencing
using Author’s Surname (family name)
and year of publication. A Referencing Guide, “Harvard
Referencing”, and a Referencing Tutorial can be
found on the right hand menu strip in Moodle on all subject
pages.
An effective way to reference correctly is to use Microsoft
Word’s referencing function (please note that
other versions and programs are likely to be different). To use
the referencing function, click on the
References Tab in the menu ribbon – students should choose
Harvard.
Authorship is also an issue under plagiarism – KOI expects
students to submit their own original work in
both assessment and exams, or the original work of their group
in the case of a group project. All
students agree to a statement of authorship when submitting
assessments online via Moodle, stating that
the work submitted is their own original work.
The following are examples of academic misconduct and can
attract severe penalties:
o Handing in work created by someone else (without
acknowledgement), whether copied from another
student, written by someone else, or from any published or
electronic source, is fraud, and falls under
the general Plagiarism guidelines.
o Copying / cheating in tests and exams is academic
misconduct. Such incidents will be treated just as
seriously as other forms of plagiarism.
o Students who willingly allow another student to copy their
work in any assessment may be considered
to assisting in copying/cheating, and similar penalties may be
applied.
Where a subject coordinator considers that a student might have
engaged in academic misconduct, KOI
may require the student to undertake an additional oral exam as
a part of the assessment for the subject,
as a way of testing the student’s understanding of their work.
Further information can be found on the KOI website.
3.5 Reasonable Adjustment
The Commonwealth Disability Discrimination Act (1992) makes
it unlawful to treat people with a disability
less fairly than people without a disability. In the context of
this subject, the principle of Reasonable
Adjustment is applied to ensure that participants with a
disability have equitable access to all aspects of
the learning situation. For assessment, this means that artificial
barriers to their demonstrating
competence are removed.
Examples of reasonable adjustment in assessment may include:
o provision of an oral assessment, rather than a written
assessment
o provision of extra time
o use of adaptive technology.
The focus of the adjusted assessment should be on enabling the
participants to demonstrate that they
have achieved the subject purpose, rather than on the method
used.
3.6 Appeals Process
Full details of the KOI Assessment and Assessment Appeals
Policy may be obtained in hard copy from
the Library, and on the KOI website www.koi.edu.au under
Policies and Forms.
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http://www.koi.edu.au/
ICT200
Assessments and Mid-Trimester Exams:
Where students are not satisfied with the results of an
assessment, including mid-trimester exams, they
have the right to appeal. The process is as follows:
o Discuss the assessment with their tutor or lecturer – students
should identify where they feel more
marks should have been awarded – students should provide
valid reasons based on the marking
guide provided for the assessment. Reasons such as “I worked
really hard” are not considered valid.
o If still not satisfied, students should complete an Application
for Review of Assessment Marks form,
detailing the reason for review. This form can be found on the
KOI website and is also available at
KOI Reception (Market St and Kent St).
o Application for Review of Assessment Marks forms must be
submitted as explained on the form
within ten (10) working days of the return of the marked
assessment, or within five (5) working days
after the return of the assessment if the assessment is returned
after the end of the trimester.
Review of Grade – whole of subject and final exams:
Where students are not satisfied with the results of the whole
subject or with their final exam results, they
have the right to request a Review of Grade – see the
Assessment and Assessment Appeals Policy for
more information.
An Application for Review of Grade/Assessment Form
(available from the KOI Website under Policies
and Forms and from KOI Reception, Market St and Kent St)
should be completed clearly explaining the
grounds for the application. The completed application should
be submitted as explained on the form,
with supporting evidence attached, to the Academic Manager.
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2. Academic Details2.1 Overview of the Subject2.2 Graduate
Attributes for Undergraduate CoursesKing’s Own Institute’s key
generic graduate attributes for a Bachelor’s level degree are
summarised below:2.3 Subject Learning OutcomesThis is a
Level 2 subject.
PowerPoint Slides prepared by:
V. Andreea CHIRITESCU
Eastern Illinois University
Walter Nicholson | Christopher Snyder 12th edition
CHAPTER Mathematics
2 for Microeconomics
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
1
Maximization of a Function of One Variable
• Economic theories assume that
– An economic agent is seeking to find the
optimal value of somefunction
• Consumers seek to maximize utility
• Firms seek to maximize profit
• Simple example: � = �(�)
– Manager of a firm wants to maximize profits
• The profits (π) received depend only on the
quantity (q) of the good sold
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
2
2.1 Hypothetical Relationship between Quantity
Produced and Profits
If a manager wishes to produce the level of
output that maximizes profits, then q*
should be produced. Notice that at q*, dπ/dq = 0.
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
3
p = f(q)
p
Quantity
p*
q*
p2
q2
p1
q1
p3
q3
Maximization of a Function of One Variable
• Vary q to see where maximum profit occurs
– An increase from q1 toq2 leads to a rise in p
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
4
2 1
2 1
0 or 0
q q q
p p p- D
> >
- D
Maximization of a Function of One Variable
• If output is increased beyond q*, profit will
decline
– An increase from q* to q3 leads to a drop in p
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
5
0
q
pD
<
D
Maximization of a Function of One Variable
• Derivatives
– The derivative of p = f(q) is the limit of Dp/Dq
for very small changes in q
– Is the slope of the curve
– The value depends on the value of q1
– The derivative of p = f(q) at the pointq1 is:
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessiblewebsite, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
6
1 1
0
( ) ( )
lim
h
f q h f qd df
dq dq h
p
®
+ -
= =
Maximization of a Function of One Variable
• Value of a derivative at a point(the slope)
– The evaluation of the derivative at the point
q = q1 can be denoted
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
7
– In our previous example,
1q q
d
dq
p
=
1
0
q q
d
dq
p
=
>
3
0
q q
d
dq
p
=
<
*
0
q q
d
dq
p
=
=
Maximization of a Function of One Variable
• First-order condition for a maximum
– For a function of one variable to attain its
maximum value at somepoint, the derivative
at that pointmust be zero
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
8
*
0
q q
df
dq
=
=
Maximization of a Function of One Variable
• The first order condition (dp/dq=0)
– Is a necessary condition for a maximum
– But it is not a sufficient condition
• The second order condition
– In order for q* to be the optimum,
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
9
– That is, at q*, d�/dq must be decreasing
• The derivative of dπ/dq must be negative at q*
and0 for * 0 for *
d d
q q q q
dq dq
p p
> < < >
2.2 Two Profit Functions That Give Misleading
Results If the First Derivative Rule Is
Applied Uncritically
In (a), the application of the first derivative rule
would result in point��∗ being chosen.
This
pointis in fact a pointof minimum profits.
Similarly, in (b), output level ��∗ would be
recommended by the first derivative rule, but this
pointis inferior to all outputs greater than ��∗ .
This demonstrates graphically that finding a pointat
which the derivative is equal to 0 is a
necessary, but not a sufficient,condition for a
function to attain its maximum value.
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
10
p
(a) Quantity
pa*
qa*
p
(b) Quantity
pb*
qb*
Maximization of a Function of One Variable
• Second derivative
– The derivative of a derivative
– Denotedby:
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
11
• The second order condition
– q* (solution to f.o.c.) is a local maximum if:
2 2
2 2 or ) o "(r
d d f
f q
dq dq
p
2
2 *
*
"( ) 0
q q
q q
d
f q
dq
p
=
=
= <
Rules for Finding Derivatives
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
12
13. If is a constant, then
1. If i
[ ( )]
2. If is a constant, then '( )
5. ln for any constant
- special case
s a constant, then 0
ln
4.
1
:
x
x
x
x
a
adxa ax
d
d af x
a af x
dx
da
a a a
dx
da
a
dx
d x
d
de
e
dx
x
x x
-=
=
=
=
=
=
Rules for Finding Derivatives
• Suppose that f(x) and g(x) are two functions
of x and f’(x) and g’(x)exist, then:
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
13
[ ]2
( )
( ) ' ( ) ( ) ( ) ' ( )
8. pro
[ ( ) ( )]
6. '(
[ ( ) ( )]
7. ( ) '( ) '
vided t
( ) ( )
hat ( )
'
)
)
0
g
( )
(
f x
d
g x f x g x f x g x
d f x g x
f x g x f x g x
d f x g x
f x g x
dx
g x
dx x
dx
×
= +
æ ö
ç ÷
-è ø =
+
+
¹
=
Rules for Finding Derivatives
• If y = f(x) and x = g(z) and if both f’(x) and
g’(x)exist, then:
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
14
• This is called the chain rule
– Allows us to study how one variable (z) affects
another variable (y) through its influence on
someintermediate variable (x)
9.
dy dy dx df dg
dz dx dz dx dz
= × = ×
Rules for Finding Derivatives
• Some examples of the chain rule include:
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
15
[ ] [ ]
2 2 2
2 2
ln ( ) ln ( ) ( ) 1 1
11.
( )
( )
10.
[ln( )] [ln( )] ( ) 1 2
12. 2
(
(
)
)
ax ax
ax ax
d ax d ax d ax
a
dx d ax dx ax
d x d x d x
x
dx dx xd x x
de de d ax
e a ae
dx x x
x
d a d
= ×
= × = ×
= ×
= × =
=
=
× =
2.1 Profit Maximization
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
16
• Suppose that the relationship between profit
and output is
p = 1,000q - 5q2
• The first order condition for a maximum is
dp/dq = 1,000 - 10q = 0
q* = 100
• Since the second derivative is always -10, then
q = 100 is a global maximum
Functions of Several Variables
• Most goals of economic agents depend on
several variables
– Trade-offs must be made
• The dependence of one variable (y) on a
series of othervariables (x1,x2,…,xn) is
denoted by
y = f(x1, x2,…,xn)
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
17
Functions of Several Variables
• Partial derivatives
– Partial derivative of y with respect to x1:
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copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
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18
– Allof the otherx’s are held constant
• A more formal definition is
1 1
1 1
or or or x
y f
f f
x x
¶ ¶
¶ ¶
2
2 21 1
0
1 ..., ,
( , ,..., ) ( , ,..., )
lim
n
n n
h
x x
f x h x x f x x xf
x h®
+ -¶
=
¶
Calculating Partial Derivatives
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
19
1 2
1 2 1 2
1 2
1 2
1 2 1 2
1
1
2 2
1 2 1 1 2 2
1 1 2 2 1
1 2
2
1 2
3
2. If ( , ) , then
and
1. If (
.
, ) , then
2 and
If ( , ) ln ln , then
2
ax bx
ax bx ax bx
y f x x ax bx x cx
f f
f ax bx f bx cx
x x
y f x x e
f f
f ae f be
x
y f x x a x b x
f a
x
x
f
+
+ +
= = + +
¶ ¶
= = + = = +
¶ ¶
= =
¶ ¶
=
= = +
¶
=
¶
= = =
¶
=
¶
2
1 2 2
and
f b
f
x x x
¶
= =
¶
Functions of Several Variables
• Partial derivatives
– Are the mathematical expression of the ceteris
paribus assumption
• Law of demand: ��/��<0, ceteris paribus
– Show how changes in one variable affect some
outcome when othervariables are held
constant
• We must be concerned with units of
measurement
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20
Functions of Several Variables
• Elasticity
– Measures the proportional effect of a change
in one variable on another
– Unit free
– Of y with respect to x is
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21
,
( )
y x
y
y x dy x xy
e
x x y dx y
x
D
D
= = × = ×
D D
2.2 Elasticity and Functional Form
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22
• For: � = � + �� + ��ℎ�� �����
• The elasticity is:
• ey,x is not constant
– It is important to pick the pointat which the
elasticity is to be computed
,y x
dy x x x
e b b
dx y y a bx
= × = × = ×
+ + ×××
2.2 Elasticity and Functional Form
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23
• For � = ���
– The elasticity is a constant:
• For ln � = ln � + � ln �
– The elasticity is:
• Elasticities can be calculated through
logarithmic differentiation
1
,
b
y x b
dy x x
e abx b
dx y ax
-= × = × =
,
ln
lny x
d y
e b
d x
= =
<= log of above equation
Functions of Several Variables
• Second-order partial derivatives
– The partial derivative of a partial derivative
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24
2( / )i
ij
j j i
f x f
f
x x x
¶ ¶ ¶ ¶
= =
¶ ¶
Second-order partial derivatives
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25
1 2
1 2 1 2
1 2 1 2
2 2
1 2
1 2 1 2
2
11 1
2
1
1
1 12
2
21 2
2 1 1 2 2
11 12 21 22
2
1. ( , ) ,
2 ; ;
2. ( , )
3. (
, ) ln ln ,
;
;
,
; ;
;
2
ax bx
ax bx ax bx
ax bx ax bx
y f x x ax bx x cx then
f a f
If y f x x a x b x the
y f x x e then
f a e f abe
f ab
b
e f
n
f ax
f b f
b
c
e
+
+ +
+ +
-
= =
= =
=
= = +
= -
= = + +
= = = =
=
2
12 21 22 2 0; 0; f f f bx
-= = = -
Functions of Several Variables
• Young’s theorem
– Under general conditions
– The order in which partial differentiation is
conducted to evaluate second-order partial
derivatives does not matter
���= ���
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26
Functions of Several Variables
• Uses of second-order partial derivatives
– Play an important role in many economic
theories
– A variable’s own second-order partial, fii
• Shows how ¶y/¶xi changes as the value of xi
increases
• fii < 0 indicates diminishing marginal effectiveness
– Cross-partial fij indicates how the marginal
effectiveness of xi changes as xj increases
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27
Functions of Several Variables
• The chain rule with many variables
– � = �(�1,�2,�3)
• Each of thesex’s is itselfa function of a
single
parameter, a
– � = �[�1(�),�2(�),�3(�)]
– How a change in a affects the value of y:
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28
31 2
1 2 3
dxdx dxdy f f f
da x da x da x da
¶ ¶ ¶
= × + × + ×
¶ ¶ ¶
Functions of Several Variables
• Special case:if �3 =�, then:
� = �[�1(�),�2(�),�]
– The effect of a on y:
• A direct effect (which is given by fa)
• An indirect effect that operates only through the
ways in which a affects the x’s
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29
1 2
1 2
dx dxdy f f f
da x da x da a
¶ ¶ ¶
= × + × +
¶ ¶ ¶
Functions of Several Variables
• Implicit functions
– If the value of a function is held constant
• An implicit relationship is created among the
independent variables that enterinto the
function
• The independent variables can no longer take on
any values
–But must instead take on only that set of
values that result in the function’s retaining
the required value
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30
Functions of Several Variables
• Implicit functions
– Use to quantify the trade-offs inherent in
economic models
• y = f(x1,x2); Implicit function: x2=g(x1)
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31
1
1 2
1
1 2
1 2 1
1 1
1 1
1
2
( )
0
0 ( , ) ( , ( ))
Differentiate with respect to :
Rearranging
( )
terms :
y f x x f
dg x
f f
dx
dg x
x g
dx f
x
x
dx dx f
= = =
= + ×
= = -
2.3 Using the Chain Rule
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32
• A pizza fanatic
– Each week, he consumes threekinds of pizza,
denoted by x1, x2, and x3
• Cost of type 1 pizza is p perpie
• Cost of type 2 pizza is 2p
• Cost of type 3 pizza is 3p
– Allocates $30 each weekto each type of pizza
– How the total number of pizzas purchased is
affected by the underlying pricep
2.3 Using the Chain Rule
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33
• Quantitypurchased of each type:
x1=30/p; x2=30/2p; x3=30/3p
• Total pizza purchases:
y = f[x1(p), x2(p), x3(p)] = x1(p) +
x2(p) + x3(p)
• Applying the chain rule:
31 2
1 2 3
2 2 2 230 15 10 55
dxdx dxdy
f f f
dp dp dp dp
p p p p- - - -
= × + × + × =
= - - - = -
2.4 A Production Possibility
Frontier—Again
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34
• A production possibility frontier for two goods
of the form
x2+0.25y2=200
• The implicit function:
2 4
0.5
x
y
fdy x x
dx f y y
- - -
= = =
Functions of Several Variables
• Comparative statics analysis
– From �=0=�(�1,�2)=�(�1,�(�1 ))
– Exogenous variable, a , the implicit form is
�(�(�), �)=0
– Applying the implicit function theorem yields:
��(�)/��=−��/��
• This shows directly how changes in the
exogenous variable a affect the endogenous
variable x.
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35
2.5 Comparative Statics of a Price-Taking Firm
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36
• The first order condition for a profit firm
that
takesmarket priceas given:
�(�(�),�)=�−�ʹ (�(�))=0
• Applying the Implicit Function Theorem to this
expression yields:
��(�)
��
= −
�;
�<
= −
1
�(−�>(�)) ��⁄
=
1
�@(�)
> 0
Maximization of Functions of Several Variables
• Suppose an agent wishes to maximize
y = f
(x1,x2,…,xn)
– The change in y from a change in x1
(holding
all otherx’s constant) is
• Equal to the change in x1 times the slope
(measuredin the x1 direction)
�� =
��
��C
��C = �C��C
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37
Maximization of Functions of Several Variables
• First-order conditions for a maximum
– Necessary condition for a maximum of the
function�(�1,�2,…,��)is that �� = 0 for any
combination of small changes in the x’s:
�1=�2=…=��=0
• Critical pointof the function
– Not sufficient to ensure a maximum
• Second-order conditions, ��� < 0
– Second partial derivatives must be negative
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38
2.6 Finding a Maximum
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39
• Suppose that y is a function of x1 and x2
� = − (�1 − 1)2 − (�2 − 2)2 + 10
� = − �12 + 2�1 − �22 + 4�2 + 5
• First-order conditions imply that
1
1
2
2
2 2 0
2 4 0
y
x
x
y
x
x
¶
= - + =
¶
¶
= - + =
¶
OR
*
1
*
2
1
2
x
x
=
=
The Envelope Theorem
• The envelope theorem
– How an optimized function changes when a
parameter of the function changes
• A specific example: � = −�2 + ��
– Represents a family of inverted parabolas
• For different values of a
– Is a function of x only
• If a is assigned a specific value
• Can calculate the value of x that maximizes
y
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40
2.1 Optimal values of y and x for alternative
values of a in y=-x2+ax
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41
2.3 Illustration of the Envelope Theorem
The envelope theorem states
that the slope of the
relationship between y (the
maximum value of y) and the
parameter a can be found by
calculating the slope of the
auxiliary relationship found
by substituting the respective
optimal values for x into the
objective function and
calculating ∂y/∂a.
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42
The Envelope Theorem
• If we are interested in how y* changes as a
changes
– Calculate the slope of y directly (time-
consuming approach)
– Hold x constantat its optimal value and
calculate ¶y/¶a directly (the envelope
shortcut)
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43
The Envelope Theorem
• Calculate the slope of y directly
– Must solve for the optimal value of x for
any
value of a: ��/�� = − 2� +� = 0; �∗ =�/2
– Substituting, we get:
�∗ = −(�∗ )2 + �(�∗ ) = −(�/2)2 +
�(�/2);
�∗ = −�2/4 + �2/2 = �2/4
• Therefore, ��∗ /�� = 2�/4= �/2
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44
The Envelope Theorem
• The envelope shortcut
– For small changes in a, dy*/da can be
computed by holding x at its optimal value
(x*) and calculating ¶y/¶a directly from y
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45
– Holding x = x*: LM
LN
= �∗ (�) = �/2
* *
* 2
*
( ) ( )
( )
( )
x x a x x a
dy y x ax
x a
da a a= =
¶ ¶ - +
= = =
¶ ¶
The Envelope Theorem
• The envelope theorem
– The change in the value of an optimized
function with respect to a parameter of that
function
– Can be found by partially differentiating the
objective function while holding x (or several
x’s) at its optimal value
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46
*
{ *( )}
dy y
x x a
da a
¶
= =
¶
The Envelope Theorem
• Many-variable case
– y is a function of several variables
y = f(x1,…,xn,a)
– Finding an optimal value for y: solve n first-
order equations:
¶y/¶xi = 0 (i = 1,…,n)
– Optimal values for thesex’s would be a
function of a
x1* = x1*(a); x2* = x2*(a); …; xn* = xn*(a)
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47
The Envelope Theorem
• Many-variable case
– Substituting into the original objective
function gives us the optimal value of y (say,
y*), we get the value function:
y* = f [x1*(a), x2*(a),…, xn*(a),a]
– Differentiating yields
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48
*
1 2
1 2
( )
*
...
*
for all
i i
n
n
i
x x a
dxdx dxdy f f f f
da x da x da x da a
dy f
x
da a =
¶ ¶ ¶ ¶
= × + × + + × +
¶ ¶ ¶ ¶
¶
=
¶
2.7 A Price-Taking Firm’s Supply Function
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49
• Cost function �(�)=5�2
– First order condition: �=�ʹ (�)=10�, we get
�∗ =0.1�
• Alternatively:
– Profits are given by: �(�,�)=��−�(�)
– Calculate the optimal value of the firm’s
profits:
�∗ (�)=��∗ −�(�∗ )=�(0.1�)−5(0.1�)2=.05�2
• The envelope theorem states that
*
*
*
*( ) ( , )0.1 |
q q
q q
d p p q
p q q
dp p
p p
=
=
¶
= = = =
¶
Constrained Maximization
• What if all values for the x’s are not
feasible?
– The values of x may all have to be > 0
– A consumer’s choices are limited by the
amount of purchasing power available
• Lagrange multiplier method
– One method used to solve constrained
maximization problems
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50
Lagrange Multiplier Method
• Lagrange multiplier method
– Supposethat we wish to find the values of x1,
x2,…, xn that maximize: � = �(�1, �2,…,
��)
– Subject to a constraint: �(�1, �2,…, ��) =
0
• The Lagrangian expression
ℒ = �(�1, �2,…, �� ) + l �(�1, �2,…, ��)
– l iscalled the Lagrange multiplier
– ℒ = �, because �(�1, �2,…, ��) = 0
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51
Lagrange Multiplier Method
• First-order conditions
– Conditions for a critical pointfor function ℒ
¶ℒ /¶�1 = �1 + l�1 = 0
¶ℒ /¶�2 = �2 + l�2 = 0
…
¶ℒ /¶�� = �� + l�� = 0
¶ℒ /¶l = �(�1,�2,…,��) = 0
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52
Lagrange Multiplier Method
• First-order conditions
– Can generally be solved for x1, x2,…, xn and
l
– The solution will have two properties:
• The x’s will obey the constraint
• These x’s will make the value of ℒ (and
therefore
f) as largeas possible
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53
Lagrange Multiplier Method
• The Lagrange multiplier (l)
– Important economic interpretation
– The first-orderconditions imply that
�1/−�1 = �2/−�2 = ⋯ = ��/−�� = l
• The numerators measure the marginal benefit of
one more unit of xi
• The denominators reflect the added burden on
the constraint of using more xi
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54
Lagrange Multiplier Method
• The Lagrange multiplier (l)
– At the optimal xi’s, the ratio of the marginal
benefit to the marginal cost of xi should be
the same for every xi
– l isthe common cost-benefit ratio for all xi
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55
marginal benefit of
marginal cost of
i
i
x
x
l =
Lagrange Multiplier Method
• The Lagrange multiplier (l)
– A high value of l indicates that each xi has a
high cost-benefit ratio
– A low value of l indicates that each xi has a
low cost-benefit ratio
– l = 0 implies that the constraint is not binding
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56
Constrained Maximization
• Duality
– Any constrained maximization problem has a
dual problem in constrained minimization
• Focuses attention on the constraints in the
original (primal) problem
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57
Constrained Maximization
• Individuals maximize utility subject to a
budget constraint
– Dual problem: individuals minimize the
expenditure needed to achieve a given level
of utility
• Firms minimize the cost of inputs to produce
a given level of output
– Dual problem: firms maximize output for a
given cost of inputs purchased
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
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part, except for use
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58
2.8 Optimal Fences and Constrained
Maximization
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59
• A farmer had a certain length of fence
(P)
– Wishes to enclose the largest possible rectangular
area, with x and y the lengths of the sides
– Choose x and y to maximize the area (A = x·y)
– Subject to the constraint that the perimeter is fixed
at P = 2x + 2y
2.8 Optimal Fences and Constrained
Maximization
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
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part, except for use
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60
• The Lagrangian expression:
ℒ = � · � + l(� − 2� − 2�)
• First-order conditions
¶ℒ /¶� = � − 2l = 0
¶ℒ /¶� = � − 2l = 0
¶ℒ /¶l = � − 2� − 2� = 0
�/2 = �/2 = λ, so � = �, and the field should be
square
� = � and � = 2l, then
� = � = �/4 ��� l = �/8
2.8 Optimal Fences and Constrained
Maximization
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
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61
• Interpretation of the Lagrange multiplier
– l suggests that an extrayard of fencing would
add
P/8 to the area
– Provides information about the implicit value
of the
constraint
• Duality
– Choose x and y to minimize the amount of fence
required to surround the field
minimize � = 2� + 2� subject to � = �
· �
– Setting up the Lagrangian:
ℒ � = 2� + 2� + l�(� − � × �)
2.8 Optimal Fences and Constrained
Maximization
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
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part, except for use
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62
• Duality
– First-order conditions:
¶ℒ �/¶� = 2 − l� · � = 0
¶ℒ �/¶� = 2 − l� · � = 0
¶ℒ �/¶l� = � − � · � = 0
– Solving,we get: � = � = ��
– The Lagrangian multiplier l� = X
Y
= X
M
= X
Z�
Envelope Theorem in Constrained
Maximization Problems
• Suppose that we want to maximize
� = �(�1,…,��;�)
Subject to the constraint: �(�1,…,��;�) = 0
• One way to solve
– Setup the Lagrangian expression
– Solve the first-orderconditions
• The envelope theorem:
��∗
��
=
¶ℒ
¶�
(�C∗ ,…,�[∗ ;�)
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
63
2.9 Optimal Fences and the Envelope
Theorem
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
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management system for classroom use.
64
• The fencing problem (example 2.8), the value
function: �∗ = �∗ ⋅ �∗ = ]
^
⋅ ]
^
= ]
_
C`
– The Lagrangian: L = �� + �(� − 2� −
2�)
– Applying envelope theorem: ��∗ /��=�/8=∂�/∂�=�
• The Lagrange multiplier in a constrained
maximization problem
– Shows the marginal gain in the objective function
that can be obtained from a slight relaxation of
the
constraint
Inequality Constraints
• Maximize � = �(�1,�2) subject to
�(�1,�2)≥ 0, �1 ≥0, ��� �2 ≥ 0
• Slack variables
– Introduce threenew variables (a, b, and c)
that convert the inequalities into equalities
– Square thesenew variables
�(�1,�2)− �2 =0; �1 −�2 =0; ��� �2 − �2 =0
– Any solution that obeys thesethreeequality
constraints
will also obey the inequality constraints
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
65
Inequality Constraints
• Maximize � = �(�1,�2)subject to
�(�1,�2)≥ 0, �1 ≥ 0, ��� �2 ≥ 0
• Lagrange multipliers
ℒ = f(x1,x2)+ l1[g(x1,x2) - a2]+l2[x1 - b2]+ l3[x2
- c2]
– There will be 8 first-orderconditions
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
66
¶ℒ /¶x1 = f1 + l1g1 + l2 = 0 ¶ℒ /¶c = -2cl3 = 0
¶ℒ /¶x2 = f1 + l1g2 + l3 = 0 ¶ℒ /¶l1 = g(x1,x2) - a2 = 0
¶ℒ /¶a = -2al1 = 0 ¶ℒ /¶l2 = x1 - b2 = 0
¶ℒ /¶b = -2bl2 = 0 ¶ℒ /¶l3 = x2 - c2 = 0
Inequality Constraints
• Complementary slackness
– According to the third condition, either a = 0
or l1 = 0
• If a = 0, the constraint �(�1,�2)holds exactly
• If l1 = 0, the availability of someslackness of
the
constraint implies that its value to the objective
function is 0
– Similar complementary slackness
relationships also hold for x1 and x2
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
67
Inequality Constraints
• Complementary slackness
– These results are sometimes called Kuhn-
Tucker conditions
• Show that solutions to problems involving
inequality constraints will differ from those
involving equality constraints in rather simple
ways
• Allows us to work primarily with constraints
involving equalities
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
68
Second-Order Conditions and Curvature
• Functions of one variable, � = �(�)
– A necessary condition for a maximum:
��/�� = � ′(�) = 0
• y must be decreasing for movements awayfrom it
– Change in y: �� = � ′(�) ��
• To be at a maximum, dy must be decreasing
for
small increases in x
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
69
Second-Order Conditions and Curvature
• Functions of one variable, � = �(�)
– Second derivative of y
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
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management system for classroom use.
70
• Since �2 � < 0 ,� ’’(�)��2 < 0
• Since ��2 mustbe > 0, � ’’(�) < 0
• This means that the function f must have a
concave shape at the critical point
2 2[ '( ) ]( ) "( ) "( )
d f x dx
d dy d y dx f x dx dx f x dx
dx
= = × = × =
Second-Order Conditions and Curvature
• Functions of two variables, � = �(�1,�2)
– First order conditions for a maximum:
¶�/¶�1 = �1 = 0
¶�/¶�2 = �2 = 0
– f1 and f2 must be diminishing at the critical
point
– Conditions must also be placed on the cross-
partial derivative (�12 = �21)
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
71
Second-Order Conditions and Curvature
• The total differential of y: �� = �1 ��1 +
�2 ��2
�2� = �11��1 + �12��2 ��1 + �21��1 +
�22��2 ��2
�2� = �11��12 + �12��2��1 + �21��1
��2 + �22��2
2
• By Young’s theorem, �12 = �21 and
�2� = �11��12 + 2�12��1��2 + �22��22
–d2y < 0 for any dx1 and dx2, if f11<0 and
f22<0
–If neither dx1 nor dx2 is zero, then d2y < 0
for any
dx1 and dx2, if f11 f22 - f122 > 0.
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
72
Multivariable Optimization (New)
• First Order Condition
• �h Li
LY
= 0,for all � > 0,where � is a vector of
length n, and
Li
LY
is a vector of partial derivatives
of length n evaluated at �∗ , and T denotes
transpose.
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
73
• Second Order Condition
• �h�� < 0,for all � > 0, where � is a vector
of
length n, and � is a matrix of second
partial
derivatives of size �×�evaluated at �∗ .
Multivariable Optimization (New)
• First Order Condition
• �r =
Li
LYs
= 0,for all �,when evaluated at �∗
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
74
• Second Order Condition
• �, evaluated at �∗ , is a negative definite matrix.
Negative definite means that the leading principal
minors have
alternative signs, starting with a minus.
� =
�X�
��C��C
⋯
�X�
��C��[
⋮ ⋱ ⋮
�X�
��[��C
⋯
�X�
��[��[
=
�CC ⋯ �C[
⋮ ⋱ ⋮
�[C ⋯ �[[
2.10 Second-Order Conditions: Health status
for the Last Time
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
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management system for classroom use.
75
� = �(�1,�2) = − �12 + 2�1 − �22 + 4�2 +
5
– First-order conditions
• �1 = −2�1 + 2 = 0 and �2 = −2�2 + 4 = 0
• Or: �1
∗ = 1,�1
∗ = 2
– Second-order partial derivatives
• �11 = −2
• �22 = −2
• �12 = 0
Second-Order Conditions and Curvature
• Concave functions
• The own second partial derivatives ( f11 and
f22)
be sufficiently negative
• So that their product will outweigh any possible
perverse effects from the cross-partial derivatives
( f12 = f21)
– Have the property that they always lie below
any plane that is tangent to them
• The plane defined by the maximum value of
the
function is simply a special case of this
property
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
76
Second-Order Conditions and Curvature
• Constrained maximization
– Choose x1 and x2 tomaximize: � = �(�1,�2)
– Linear constraint: c − b1x1 − b2x2 = 0
– Lagrangian: ℒ = �(�1,�2) + l(� − �1�1 − �2�2)
– The first-orderconditions:
�1 − l�1 = 0,
�2 − l�2 = 0,
� − �1�1 − �2�2 = 0
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
77
Second-Order Conditions and Curvature
• Constrained maximization
– Use the “second” total differential:
�2� = �11��12 + 2�12��1��2 + �22��22
• Only values of x1 and x2 that satisfy the
constraint
can be considered validalternatives to the critical
point
– Total differential of the constraint
−�1 ��1 − �2 ��2 = 0, ��2 = −(�1/�2)��1
• Allowable relative changes in x1 and x2
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
learning
management system for classroom use.
78
Second-Order Conditions and Curvature
• Constrained maximization
– First-order conditions imply that f1/f2 = b1/b2,
we get: ��2 = −(�1/�2) ��1
– Since: �2� = �11��12 + 2�12��1��2 +
�22��22
– Substitute for dx2 and get �2� = �11��12 −
2�12(�1/�2)��12 + �22(�12/�22)��12
– Combining terms and rearranging, we get
�2� = �11 �2
2
− 2�12�1�2 + �22�12
��12
�22
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
certain product or service or otherwise on a
password-protected website or school-approved
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management system for classroom use.
79
Second-Order Conditions and Curvature
• Constrained maximization
– Therefore, for �2� < 0, it must be true that
�11 �2
2
− 2�12�1�2 + �22�12 < 0
– This equation characterizes a set of functions
termed quasi-concave functions
• Quasi-concave functions
– Any two points within the upper contour
set
can be joined by a straight line contained
completely in the upper contour set
© 2017 Cengage Learning®. May not be scanned,
copied or duplicated, or posted to a
publicly accessible website, in whole or in
part, except for use
as permitted in a license distributed with a
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80
Second Order Condition for Constrained Optimization
with One Constraint (new)
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81
• �}, evaluated at �∗ , is a bordered Hessian
which is a matrix of
second derivatives of the Lagrangian, including �.
If Λ is the
Lagrangian, then the bordered Hessian is:
• (−1)�} must be negative definite.
2.11 Concave and Quasi-Concave Functions
© 2017 Cengage Learning®. May not be scanned,
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82
• � = �(�1,�2)= (�1×�2)�
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  • 1. KING’S OWN INSTITUTE* Success in Higher Education ICT 200 DATABASE DESIGN AND DEVELOPMENT T219 All information contained within this Subject Outline applies to all students enrolled in the trimester as indicated. 1. General Information 1.1 Administrative Details Associated HE Award(s) Duration Level Subject Coordinator Bachelor of Information Technology (BIT) 1 trimester Level 2 Mr Utpal Nanavati [email protected] Consultation: via Moodle or by appointment. 1.2 Core / Elective Core subject in the BIT 1.3. Subject Weighting Indicated below is the weighting of this subject and the total course points.
  • 2. Subject Credit Points Total Course Credit Points 4 BIT (96 Credit Points) 1.4 Student Workload Indicated below is the expected student workload per week for this subject No. timetabled hours/week* No. personal study hours/week** Total workloa d hours/w 4 hours/week containing 1 x 2 hour Lecture 1 x 2 hour Tutorial 6 hours/week 10 hours/week * Total time spent per week at lectures and tutorials ** Total time students are expected to spend per week in studying, completing assignments, etc. *** Combination of timetable hours and personal study. 1.5 Mode of Delivery On-campus 1.6 Pre-requisites ICT 103 Systems Analysis and Design
  • 3. 1.7 General Study and Resource Requirements o Dedicated computer laboratories are available for student use. Normally, tutorial classes are conducted in the computer laboratories. o Students are expected to attend classes with the requisite textbook and must read specific chapters prior to each tutorial. This will allow them to actively take part in discussions. Students should have elementary skills in both word processing and electronic spreadsheet software, such as OFFICE 365 or MS Word and MS Excel. o Computers and WIFI facilities are extensively available for student use throughout KOI. Students are encouraged to make use of the campus Library for reference materials. o Students will require access to the internet and email. Where students use their own computers, they should have internet access. KOI will provide access to required software. ICT 200 DATABASE DESIGN AND DEVELOPMENT T219 19/06/2019 16:20 PAGE 1 OF 15 *AUSTRALIAN INSTITUTE OF BUSINESS AND MANAGEMENT PTY LTD © ABN: 72 132 629 979 CRICOS 03171A Approved by KOI Academic Board for T2 2019 mailto:[email protected] ICT200
  • 4. Software Resource requirements specific to this subject: Office 365, MS Imagine, SQL Server 2017. 2. Academic Details 2.1 Overview of the Subject This subject will provide the student with an overall understanding of database development, concepts and theory. Students will learn to design and build a database from data analysis, normalisation, mapping a specific database model. The relational model is emphasised and introduced using structured queried language (SQL) for creating and manipulating databases in both MS Access and SQL Server environments. Assignment work includes the analysis, design, and implementation of a database using SQL queries in SQL Server environment. 2.2 Graduate Attributes for Undergraduate Courses Graduates of Bachelor courses from King’s Own Institute (KOI) will be able to demonstrate the attributes of a successful Bachelor degree graduate as outlined in the Australian Qualifications Framework (2nd edition, January 2013). Graduates at this level will be able to apply an advanced body of knowledge across a range of contexts for the purposes of professional practice or academic scholarship, and as a pathway for further learning. King’s Own Institute’s key generic graduate attributes for a Bachelor’s level degree are summarised below:
  • 5. Across the course, these skills are developed progressively at three levels: o Level 1 Foundation – Students learn the basic skills, theories and techniques of the subject and apply them in basic, standalone contexts o Level 2 Intermediate – Students further develop the skills, theories and techniques of the subject and apply them in more complex contexts, and begin to integrate this application with other subjects. o Level 3 Advanced – Students demonstrate an ability to plan, research and apply the skills, theories and techniques of the subject in complex situations, integrating the subject content with a range of other subject disciplines within the context of the course. KOI Bachelor Degree Graduate Attributes Detailed Description Knowledge Current, comprehensive, and coherent and connected knowledge Critical Thinking Critical thinking and creative skills to analyse and synthesise information and evaluate new problems Communication Communication skills for effective reading, writing, listening and presenting in varied modes and contexts and for the
  • 6. transferring of knowledge and skills to others Information Literacy Information and technological skills for accessing, evaluating, managing and using information professionally Problem Solving Skills Skills to apply logical and creative thinking to solve problems and evaluate solutions Ethical and Cultural Sensitivity Appreciation of ethical principles, cultural sensitivity and social responsibility, both personally and professionally Teamwork Leadership and teamwork skills to collaborate, inspire colleagues and manage responsibly with positive results Professional Skills Professional skills to exercise judgement in planning, problem solving and decision making ICT 200 DATABASE DESIGN AND DEVELOPMENT T219 19/06/2019 16:20 PAGE 2 OF 15 *AUSTRALIAN INSTITUTE OF BUSINESS AND MANAGEMENT PTY LTD © ABN: 72 132 629 979 CRICOS 03171A Approved by KOI Academic Board for T2 2019 ICT200
  • 7. 2.3 Subject Learning Outcomes This is a Level 2 subject. On successful completion of this subject, students should be able to: 2.4 Subject Content and Structure Below are details of the subject content and how it is structured, including specific topics covered in lectures and tutorials. Reading refers to the text unless otherwise indicated. Weekly Planner: Week (beginning) Topic covered in each week’s lecture Reading(s) Expected work as listed in Moodle 1 08 Jul Introduction to DBMS: o history of database processing o emergence of relational model o post-relational developments o DBMS concepts o MS Access and SQL
  • 8. Chapter 1 and Database History Article (See section 2.6) Chapter 1 Discussion. Formative Not Graded Introduction to MS Access and SQL environments. Tutorial exercises 2 15 Jul Introduction to Structured Query Language: o creating SQL statements o using SQL in MS Access o using SQL in SQL Server o Querying tables Chapter 2 Activities, Database exercises - execute simple SQL queries in MS Access and SQL Server. Formative Not Graded. Tutorial exercises 3 22 Jul Data modelling with the Entity- Relationship model: o purpose of a data model o the E-R model and diagrams
  • 9. o variations of the E-R model o entities and data modelling process Chapter 5 Activities, Data Modelling ERD exercises. VISIO and UML (Appendix C and D). Formative Not Graded. Tutorial exercises Subject Learning Outcomes Contribution to Course Graduate Attributes a) Explain the history and development of database technologies and the emergence of the relational database model and SQL. b) Model business information requirements and produce a logical database design using entity relationship diagrams (ERD) and extended relationship diagrams (EERD). c) Design, develop, test and prove the functionality of a database using MS Access and SQL. d) Describe and carry out the necessary steps to develop an effective
  • 10. physical database design e) Formulate, write and execute SQL queries in an SQL Server environment. f) Explain the main functions of database administration and data warehousing. ICT 200 DATABASE DESIGN AND DEVELOPMENT T219 19/06/2019 16:20 PAGE 3 OF 15 *AUSTRALIAN INSTITUTE OF BUSINESS AND MANAGEMENT PTY LTD © ABN: 72 132 629 979 CRICOS 03171A Approved by KOI Academic Board for T2 2019 ICT200 4 29 Jul The relational model and normalisation: o terminology o characteristics of relationships
  • 11. o normal forms normalisation categories Chapter 3 Activities, Normalisation exercises on 1NF,2NF, 3NF and BCNF. Formative Not Graded Tutorial exercises Assessment 2 Quiz A 5 05 Aug Database design using normalisation: o advantages and disadvantages o normalising with SQL o common design problems Chapter 4 Simple SQL Activities, normalisation and de- normalisation exercises. Formative Not Graded Tutorial exercises 6 12 Aug Transforming data models into database designs: o purpose of database design o tables, entities,
  • 12. primary/alternate keys o verify normalisation o create relationships o design for minimum cardinality Chapter 6 Activities creating tables and relationship in SQL, Database Design exercises. Formative Not Graded Database Exercises from textbook 18 Aug 2019 – 25 Aug 2019 Mid trimester break 7 26 Aug SQL for database construction and application processing: o using SQL scripts o Advanced SQL statements Chapter 7 Chapter Activities complex SQL queries using join, union, sub-queries. Tutorial exercises Formative Not Graded 8 02 Sep Database redesign: o the need for database
  • 13. redesign o analysing existing databases o database backup and test databases o making changes to tables, columns, constraints, cardinalities, relationships Chapter 8 Chapter Activities Testing database and query optimisation for database efficiency. Formative Not Graded Tutorial exercises Assessment 3 Quiz B. 9 09 Sep Managing multiuser databases: o database administration o DBMS and application security o database backup and recovery o managing the DBMS and data repository Chapter 9 Tutorial exercises Chapter Activities Formative Not Graded
  • 14. 10 16 Sep Managing databases with Microsoft SQL server: o installing the DBMS o using the DBMS utilities o creating a database o creating and running SQL scripts o DBMS security Chapter 10 Tutorial exercises Assessment 4 due - Report 11 23 Sep The web server environment: o the web database processing environment o database server access standards o OBDC standard o web database processing with PHP and NetBeans IDE o challenges and SQL injection Chapter 11 Tutorial exercises SQL Server Project. Formative Not Graded
  • 15. Assessment 4 due - Presentation ICT 200 DATABASE DESIGN AND DEVELOPMENT T219 19/06/2019 16:20 PAGE 4 OF 15 *AUSTRALIAN INSTITUTE OF BUSINESS AND MANAGEMENT PTY LTD © ABN: 72 132 629 979 CRICOS 03171A Approved by KOI Academic Board for T2 2019 ICT200 attacks o the importance of XML 12 30 Sep Big data, data warehousing and business intelligence systems: o business intelligence systems o reporting and data mining systems o data warehousing and data marts o components of a data warehouses o OLAP and data mining o distributed databases o cloud computing
  • 16. o big data and the Not Only SQL movement Chapter 12 Discussion. Formative Not Graded. Tutorial exercises 13 06 Oct Study Review Week 14 14 Oct Final Exam Week Please see Exam Timetable for exam date, time and location 15 20 Oct Student Vacation begins Enrolments for T319 open 16 28 Oct Results Released 29 Oct 2019 Certification of Grades 01 Nov 2019 T319 begins 04 Nov 2019 1 04 Nov Week 1 of classes for T319 Friday 01 Nov 2019 – Review of Grade Day for T219 – see
  • 17. Sections 2.6 and 3.6 below for more information. 2.5 Public Holiday Amendments Please note: KOI is closed on all scheduled NSW Public Holidays. T219 has one (1) public holiday (Labour Day) that occurs during this trimester. Classes scheduled for this public holiday (Calendar Class Dates) will be rescheduled as per the table below. This applies to ALL subjects taught in T219. Please see the table below and adjust your class timing as required. Please make sure you have arrangements in place to attend the rescheduled classes if applicable to your T219 enrolment. Classes will be conducted at the same time and in the same location as your normally scheduled class except these classes will be held on the date shown below. Calendar Class Date Rescheduled Class Date Monday 07 October 2019 (Week 13) Study Review Week Not required 2.6 Review of Grade, Deferred Exams & Supplementary
  • 18. Exams/Assessments Review of Grade: There may be instances when you believe that your final grade in a subject does not accurately reflect your performance against the subject criteria. Section 8 of the Assessment and Assessment Appeals Policy (www.koi.edu.au) describes the grounds on which you may apply for a Review of Grade. If this happens and you are unable to resolve it with the Academic staff concerned then you can apply for a formal Review of Grade within the timeframes indicated in the following sections of this subject outline - Supplementary Assessments, 3.6 Appeals Process as well as the Assessment and Assessment Appeals ICT 200 DATABASE DESIGN AND DEVELOPMENT T219 19/06/2019 16:20 PAGE 5 OF 15 *AUSTRALIAN INSTITUTE OF BUSINESS AND MANAGEMENT PTY LTD © ABN: 72 132 629 979 CRICOS 03171A Approved by KOI Academic Board for T2 2019 http://www.koi.edu.au/ ICT200 Policy. Please ensure you read the Review of Grade information before submitting an application. Review of Grade Day:
  • 19. KOI will hold the Review of Grade Day for all subjects studied in T219 on Friday 01 November 2019 Only final exams will be discussed as all other assessments should have been reviewed during the trimester. If you fail one or more subjects and you wish to consider applying for a Review of Grade you MUST attend the Review of Grade Day. You will have the chance to discuss your final exam with your lecturer, and will be advised if you have valid reasons for applying for a Review of Grade (see Section 3.6 below and Assessment and Assessment Appeals Policy). If you do not attend the Review of Grade Day you are considered to have accepted your results for T219. Deferred Exams: If you wish to apply for a deferred exam, you should submit an Application for Assignment Extension or Deferred Exam Form before the prescribed deadline. If you miss your mid-trimester or final exam there is no guarantee you will be offered a deferred exam. You must apply within the stated timeframe and satisfy the conditions for approval to be offered a deferred exam (see Section 8.1 of the Assessment and Assessment Appeals Policy and the Application
  • 20. for Assignment Extension or Deferred Exam Forms). In assessing your request for a deferred exam, KOI will take into account the information you provide, the severity of the event or circumstance, your performance on other items of assessment in the subject, class attendance and your history of previous applications for special consideration. Deferred mid-trimester exams will be held before the end of week 9. Deferred final exams will be held on two days during week 1 or 2 in the next trimester. You will not normally be granted a deferred exam on the grounds that you mistook the time, date or place of an examination, or that you have made arrangements to be elsewhere at that time; for example, have booked plane tickets. If you are offered a deferred exam, but do not attend you will be awarded 0 marks for the exam. This may mean it becomes difficult for you to pass the subject. If you apply for a deferred exam within the required time frame and satisfy the conditions you will be advised by email (to your KOI student email address) of the time and date for the deferred exam. Please ensure that you are available to take the exam at this time. Marks awarded for the deferred exam will be the marks awarded for that item of assessment towards your final mark in the subject. Supplementary Assessments (Exams and Assessments): A supplementary assessment may be offered to students to provide a final opportunity to demonstrate successful achievement of the learning outcomes of a subject.
  • 21. Supplementary assessments are only offered at the discretion of the Board of Examiners. In considering whether or not to offer a supplementary assessment, KOI will take into account your performance on all the major assessment items in the subject, your attendance, participation and your history of any previous special considerations. Students are eligible for a supplementary assessment for their final subject in a course where they fail the subject but have successfully completed all other subjects in the course. You must have completed all major assessment tasks for the subject and obtained a passing mark on at least one of the major assessment tasks to be eligible for a supplementary assessment. If you believe you meet the criteria for a supplementary assessment for the final subject in your course, but have not received an offer, complete the “Complaint, Grievance, Appeal Form” and send your form to ICT 200 DATABASE DESIGN AND DEVELOPMENT T219 19/06/2019 16:20 PAGE 6 OF 15 *AUSTRALIAN INSTITUTE OF BUSINESS AND MANAGEMENT PTY LTD © ABN: 72 132 629 979 CRICOS 03171A Approved by KOI Academic Board for T2 2019 ICT200 [email protected] The deadline for applying for supplementary assessment is the Friday of the first
  • 22. week of classes in the next trimester. If you are offered a supplementary assessment, you will be advised by email to your KOI student email address of the time and due date for the supplementary assessment – supplementary exams will normally be held at the same time as deferred final exams during week 1 or week 2 of the next trimester. You must pass the supplementary assessment to pass the subject. The maximum grade you can achieve in a subject based on a supplementary assessment is a PASS grade. If you: o are offered a supplementary assessment, but fail it; o are offered a supplementary exam, but do not attend; or o are offered a supplementary assessment but do not submit by the due date; you will receive a FAIL grade for the subject. 2.7 Teaching Methods/Strategies Briefly described below are the teaching methods/strategies used in this subject: o On-campus lectures (2 hours/week) are conducted in seminar style and address the subject content, provide motivation and context and draw on the students’ experience and preparatory reading. o Tutorials (2 hours/week) include class discussion of case studies and research papers, practice sets and problem-solving and syndicate work on group projects. Tutorial participation is an essential
  • 23. component of the subject and contributes to the development of graduate attributes (see section 2.2 above). It is intended that specific tutorial material such as case studies, recommended readings, review questions etc. will be made available each week in Moodle. o Online teaching resources include class materials, readings, model answers to assignments and exercises and discussion boards. All online materials for this subject as provided by KOI will be found in the Moodle page for this subject. Students should access Moodle regularly as material may be updated at any time during the trimester o Other contact - academic staff may also contact students either via Moodle messaging, or via email to the email address provided to KOI on enrolment. Grading Final grades are awarded by the Faculty Assessment Committee in accordance with KOI's Assessment Regulations. The definitions and guidelines for the awarding of final grades within the BIT degree are: o HD High distinction (85-100%) an outstanding level of achievement in relation to the assessment process. o DI Distinction (75-84%) a high level of achievement in relation to the assessment process.
  • 24. o CR Credit (65-74%) a better than satisfactory level of achievement in relation to the assessment process. o PS Pass (50-64%) a satisfactory level of achievement in relation to the assessment process. o FL Fail (0-49%) an unsatisfactory level of achievement in relation to the assessment process. ICT 200 DATABASE DESIGN AND DEVELOPMENT T219 19/06/2019 16:20 PAGE 7 OF 15 *AUSTRALIAN INSTITUTE OF BUSINESS AND MANAGEMENT PTY LTD © ABN: 72 132 629 979 CRICOS 03171A Approved by KOI Academic Board for T2 2019 mailto:[email protected] ICT200 2.8 Student Assessment Provided below is a schedule of formal assessment tasks and major examinations for the subject.
  • 25. Assessment Type When assessed Weighting Learning Outcomes Assessed Assessment 1 Weekly Tutorial Week 11 10% a, b, c, d, e, f Assessment 2 Quiz A (opens week 2) Week 4 5% a Assessment 3 Quiz B (opens week 8) Week 8 10% d Assessment 4 Group Project (Report and Presentation) - Problem Based Scenario to Design (ERD), Implement and Query a SQL Server Database Week 10 Week 11 Report 20% Presentation 5% d, e Assessment 5 Final Exam (3 hours) Final exam period 50% b, c, d, e, f Requirements to Pass the Subject: To gain a pass or better in this subject, students must gain a minimum of 50% of the total available
  • 26. subject marks. ICT 200 DATABASE DESIGN AND DEVELOPMENT T219 19/06/2019 16:20 PAGE 8 OF 15 *AUSTRALIAN INSTITUTE OF BUSINESS AND MANAGEMENT PTY LTD © ABN: 72 132 629 979 CRICOS 03171A Approved by KOI Academic Board for T2 2019 ICT200 2.9 Prescribed and Recommended Readings Prescribed Text: Kroenke, D.M., Auer, D., 2016. Database Processing. 14th ed., Pearson Also available as eText at Pearson Online: http://www.mypearsonstore.com/bookstore/database-processing- fundamentals-design-and- implementation-9780133878998?xid=PSED Recommended Readings: Elmasri, R. and Navathe, S. B., 2016. Fundamentals of Database Systems. 7th ed. Pearson
  • 27. Publishing. Hoffer, J., Venkataraman, R. and Topi, H., 2016. Modern Database Management. 12th ed. Prentice Hall. Useful Websites: The following websites are useful sources covering a range of information useful for this subject. As we are using MS Access and SQL Server there are numerous resources available online. A number are listed below. For core concepts and platform-independent (irrelevant to the DBMS being used) Data Modelling and SQL useful links are also provided. The prescribed textbook provides all the information required for students to do well in this unit. Students are encouraged to research additional information where required. During lectures and tutorials, students will be informed where example databases and related files may be found. Students are also expected to use the library and the internet for additional learning. The following links are primarily for additional learning and future reference (be sure to bookmark them). Remember to use your textbook first – all of the information you should need is in the text. Database resources and articles Journals: o ACM Transactions on Computer Systems. Available from
  • 28. EBSCOhost research databases o ACM Transactions On Database Systems. Available from EBSCOhost research databases o Journal of Electronic Commerce Research www.jecr.org o Journal of International Technology and Information Management https://www.sciencedirect.com/journal/international-journal-of- information-management A larger resource of database information, tutorials, and exercises o Access 2013 Microsoft Official site for Access 2013 o Basic design from MS Office Database Design Basics o SQL Server Central http://www.sqlservercentral.com/Articles o Allen Browne Tips. Allen Browne is an Australian based Access developer with many years of experience. More for VBA and advanced coding problems. o MSDN Community Forum. Microsoft Developer Network (MSDN) o TechOnTheNet. More advanced on VBA forms 3. Assessment Details 3.1 Details of Each Assessment Item The assessments for this subject are described below. The description includes the type of assessment, its purpose, weighting, due date and submission requirements, the topic of the assessment, details of the ICT 200 DATABASE DESIGN AND DEVELOPMENT T219 19/06/2019 16:20 PAGE 9 OF 15
  • 29. *AUSTRALIAN INSTITUTE OF BUSINESS AND MANAGEMENT PTY LTD © ABN: 72 132 629 979 CRICOS 03171A Approved by KOI Academic Board for T2 2019 http://www.mypearsonstore.com/bookstore/database-processing- fundamentals-design-and-implementation- 9780133878998?xid=PSED http://www.mypearsonstore.com/bookstore/database-processing- fundamentals-design-and-implementation- 9780133878998?xid=PSED http://www.jecr.org/ https://www.sciencedirect.com/journal/international-journal-of- information-management http://www.sqlservercentral.com/Articles ICT200 task and detailed marking criteria, including a marking rubric for essays, reports and presentations. Supplementary assessment information and assistance can be found in Moodle. KOI expects students to submit their own original work in both assignments and exams, or the original work of their group in the case of group assignments. Marking guides for assessments follow the assessment descriptions. Students should compare final drafts of their assessment against the marking guide before submission. Please note that the final exam is not open book. Students are not permitted to bring any reference
  • 30. materials into the final exam. Students are not permitted to use mobile phones or other communication devices during tests and exams. Assessment 1 Assessment type: Tutorial Exercises - individual assessment Assessment purpose: To answer weekly tutorial exercises on the topics covered in lectures. This assessment contributes to learning outcomes a, b, c, d, e and f. Value 10% Submission requirements details: The tutorial exercises for weeks 2-11 Assessment topic: Tutorial Exercises Task details: Students need to complete the weekly tutorial exercises and upload the answers on Moodle Assessment 2 Assessment type: Quiz A (opens week 4) - individual assignment. Assessment purpose: This assessment will allow students to demonstrate that they have understood the concepts covered in weeks 1, 2 and 3. This assessment contributes to learning outcome a. Value: 5% Due Date: End of Week 4 Assessment topic: Quiz A
  • 31. Task details: The quiz will consist of questions and problems relating to subject content from weeks 1 – 3 inclusive. Assessment 2 Marking Rubric: Quiz A – Worth 5 Marks Criteria Fail (0 – 49%) Pass (50 – 64%) Credit (65 – 74%) Distinction (75 – 84%) High Distinction (85 – 100%) Score (out of 10) 0-4 5-6 7 8 9-10 Assessment 3 Assessment type: Quiz B (opens week 8) - individual assignment. Purpose: This assessment will allow students to demonstrate that they have understood the concepts covered in weeks 4 to 7. This assessment contributes to learning outcome d.
  • 32. Value: 10% Due Date: Week 8 Task details: The quiz will consist of questions and problems relating to subject content from weeks 4 – 7 inclusive. ICT 200 DATABASE DESIGN AND DEVELOPMENT T219 19/06/2019 16:20 PAGE 10 OF 15 *AUSTRALIAN INSTITUTE OF BUSINESS AND MANAGEMENT PTY LTD © ABN: 72 132 629 979 CRICOS 03171A Approved by KOI Academic Board for T2 2019 ICT200 Assessment 3 Marking Rubric: Short Answer Quiz B – Worth 10 Marks Criteria Fail (0 – 49%) Pass (50 – 64%) Credit (65 – 74%) Distinction (75 – 84%)
  • 33. High Distinction (85 – 100%) Score (out of 10) 0-4 5-6 7 8 9-10 Assessment 4 Assessment type: Group Project Report and Presentation Purpose: Demonstrate students can effectively work in a group to design, normalise, implement and query an SQL Server database according to a project scenario specification which they may select from a case study from the textbook. This assessment contributes to learning outcomes d, e. Value: Report 20% Presentation 5% Total: 25% Due Date: Report Week 10; Presentation in class Week 11 Assessment topic: Topics covered in week 1 - 10 Submission: Report, Group Charter and presentation file via Moodle; Presentation in class. Task details: Group Project (Report and Presentation) - Problem Based Scenario to Design (ERD), Implement and Query a SQL Server Database.
  • 34. Students will choose a case study from the textbook for this assignment. An ERD, database, scripts for creating tables, views, triggers, functions, stored procedures, insert, update, alter, delete and successful query results as requested in the case study will be assessed according to the marking criteria below. Students must normalize the data from 1 NF to BCNF and show each and every normalization with diagrams/tables and appropriate sample data to explain their understanding. Students will submit one group written report of 1000 words detailing the tasks they carried out in the assignment. 2 to 3 students can be in one group. This report will be due in the end of week 10. Students will also deliver a 15 mins presentation in class and explain their understanding and the approach they took to work on the assignment. This will be arranged in Week 11. The report requires a title page with student group members’ names and email, the assessment title and date. A brief introduction to the assignment can then be followed by the initial ERD, the Normalised version and the SQL statements used as requested by the case study chosen. Report must be well-written, have in-text and detailed Harvard style referencing and presented professionally, containing: o Title page o Table of Contents o Introduction o Appropriate use of headings within report o Overall structure, presentation and formatting. All group members must remain present for the demonstration in tutorial and must be able to explain the scripts written for the tasks outlined in case study.
  • 35. Marking Guide: A detailed marking guide for the Database Project (Assessment 4) is provided. See the following page. ICT 200 DATABASE DESIGN AND DEVELOPMENT T219 19/06/2019 16:20 PAGE 11 OF 15 *AUSTRALIAN INSTITUTE OF BUSINESS AND MANAGEMENT PTY LTD © ABN: 72 132 629 979 CRICOS 03171A Approved by KOI Academic Board for T2 2019 ICT200 Assessment 4 Marking Rubric: SQL Server Database Project – Worth 20 Marks Criteria Fail (0-49%) Pass (50-64%) Credit
  • 36. (65-74%) Distinction (75-84%) High Distinction (85-100%) ER Diagram 5 marks ER diagram not correct or not submitted Identified all entities and built relationships using crow’s foot notation Data is in 3NF Identified entities and relationships, representation using crow’s foot notation correct Data is in BCNF ERD accompanied with explanation of relationships Data is in BCNF Demonstrated all the
  • 37. normalization from 1 NF to BCNF along with diagram and table data DDL, DML and PSM scripts 10 marks Scripts are incorrect or not submitted DDL and DML Scripts are correct Tables are created, data is inserted, updated, deleted DDL and DML Scripts are correct Tables are created, data is inserted, updated, deleted Tables have default values and constraints are enabled on tables DDL and DML Scripts are correct Tables are created, data is inserted, updated, deleted Tables have default values and constraints are enabled on tables
  • 38. Functions and Triggers are written to perform tasks mentioned in case study DDL and DML Scripts are correct Tables are created, data is inserted, updated, deleted Tables have default values and constraints are enabled on tables Functions, Triggers and Stored Procedures are written to perform tasks mentioned in case study Report and Presentation 10 marks Report is not adhering to guidelines provided, tasks are not done satisfactorily Team member(s) have not shown convincing evidence of having performed the tasks outlined the case study Report is submitted,
  • 39. meeting word count and presentation is delivered. There is sufficient evidence of participation of team member(s) Database, tables and other objects are created on SQL Server Report is submitted, meeting word count, well referenced and presentation is delivered. There is sufficient evidence of participation of team member(s) Database, tables and other objects are created on SQL Server Students must be able to provide an explanation of the scripts they’ve written. Report is submitted, meeting word count, well referenced and presentation is delivered. There is sufficient
  • 40. evidence of participation of team member(s) Students must be able to provide an explanation of the scripts they’ve written. Students are able to demonstrate the tasks outlined in the case study. Report is well-written and presented professionally, containing: o Title page o Table of Contents o Introduction o Appropriate use of headings within report o Overall structure, presentation and formatting. All tasks outlined in case study are carried out and the group has shown evidence of having done the task on their own.
  • 41. Assessment 5 Assessment type: Final Exam (three hours) individual assessment – closed book exam Purpose: The final exam is intended to test that students have understood key concepts for relational database design, development and administration. This assessment contributes specifically to learning outcomes b, c, d, e, and f. Value: 50% Due Date: The final exam will be held in the official KOI exam period in Week 14 of the trimester. The specific date and time will be posted towards the end of the trimester. Topic: The examination may cover content from any part of the entire subject related to learning outcomes b, c, d, e and f. Task Details: Students will be expected to answer written response questions derived from all lectures and tutorials in the trimester. 3.2 Late Penalties and Extensions An important part of business life and key to achieving KOI’s graduate outcome of Professional Skills is the ability to manage workloads and meet deadlines. Consequently, any assessment items such as in- class quizzes and assignments missed or submitted after the due date/time will attract a penalty (see
  • 42. below). ICT 200 DATABASE DESIGN AND DEVELOPMENT T219 19/06/2019 16:20 PAGE 12 OF 15 *AUSTRALIAN INSTITUTE OF BUSINESS AND MANAGEMENT PTY LTD © ABN: 72 132 629 979 CRICOS 03171A Approved by KOI Academic Board for T2 2019 ICT200 Students who miss mid-trimester tests and final exams without a valid and accepted reason (see below) may not be granted a deferred exam and will be awarded 0 marks for assessment item. These penalties are designed to encourage students to develop good time management practices, and create equity for all students. Any penalties applied will only be up to the maximum marks available for the specific piece of assessment attracting the penalty. Late penalties, granting of extensions and deferred exams are based on the following: In Class Tests (excluding Mid-Trimester Tests) o No extensions permitted or granted – a make-up test may only be permitted under very special circumstances where acceptable supporting evidence is
  • 43. provided. The procedures and timing to apply for a make-up test (only if available) are as shown in Section 3.3 Applying for an Extension (below). o Missing a class test will result in 0 marks for that assessment element unless the above applies. Written Assessments o 5% of the total available marks per calendar day unless an extension is approved (see Section 3.3 below) Presentations o No extensions permitted or granted – no presentation = 0 marks. The rules for make-up presentations are the same as for missing in-class tests (described above). Mid-Trimester Tests and Final Exams o If students are unable to attend mid-trimester tests or final exams due to illness or some other event (acceptable to KOI), they must: − Advise KOI in writing (email: [email protected]) as soon as possible, but no later than three (3) working days after the exam date, that they will be / were absent and the reasons. They will be advised in writing (return email) as to whether the circumstances are acceptable.
  • 44. − Complete the appropriate Application for Extension or Deferred Exam Form available from the Student Information Centre in Moodle, on the KOI Website (Policies and Forms) and the Reception Desk (Market St and Kent St), as soon as possible and email with attachments to [email protected] − Provide acceptable documentary evidence in the form of a satisfactorily detailed medical certificate, police report or some other evidence that will be accepted by KOI. − Agree to attend the deferred exam as set by KOI. Deferred exam o There will only be one deferred exam offered. o Marks awarded for the deferred exam will be the marks awarded for that assessment. o If you miss the deferred exam you will be awarded 0 marks for the assessment. This may mean you are unable to complete (pass) the subject. 3.3 Applying for an Extension If students are unable to submit or attend an assessment when due, and extensions are possible, they must apply by completing the appropriate Application for Extension form available from the Student Information Centre in Moodle, the KOI Website (Policies and Forms) and the Reception Desk (Market St and Kent St), as soon as possible but no later than three (3) working days of the assessment due date. The completed form must be emailed with supporting
  • 45. documentation to [email protected] ICT 200 DATABASE DESIGN AND DEVELOPMENT T219 19/06/2019 16:20 PAGE 13 OF 15 *AUSTRALIAN INSTITUTE OF BUSINESS AND MANAGEMENT PTY LTD © ABN: 72 132 629 979 CRICOS 03171A Approved by KOI Academic Board for T2 2019 mailto:[email protected] mailto:[email protected] mailto:[email protected] ICT200 Students and lecturers / tutors will be advised of the outcome of the extension request as soon as practicable. Appropriate documentary evidence to support the request for an extension must be supplied. Please remember there is no guarantee of an extension being granted, and poor organisation is not a satisfactory reason to be granted an extension. 3.4 Referencing and Plagiarism Please remember that all sources used in assessment tasks must be suitably referenced. Failure to acknowledge sources is plagiarism, and as such is a very serious academic issue. Students plagiarising run the risk of severe penalties ranging from a reduction through to 0 marks for a first offence for a single assessment task, to exclusion from KOI in the most serious repeat cases. Exclusion has
  • 46. serious visa implications. The easiest way to avoid plagiarising is to reference all sources. Harvard referencing is the required method – in-text referencing using Author’s Surname (family name) and year of publication. A Referencing Guide, “Harvard Referencing”, and a Referencing Tutorial can be found on the right hand menu strip in Moodle on all subject pages. An effective way to reference correctly is to use Microsoft Word’s referencing function (please note that other versions and programs are likely to be different). To use the referencing function, click on the References Tab in the menu ribbon – students should choose Harvard. Authorship is also an issue under plagiarism – KOI expects students to submit their own original work in both assessment and exams, or the original work of their group in the case of a group project. All students agree to a statement of authorship when submitting assessments online via Moodle, stating that the work submitted is their own original work. The following are examples of academic misconduct and can attract severe penalties: o Handing in work created by someone else (without acknowledgement), whether copied from another student, written by someone else, or from any published or electronic source, is fraud, and falls under the general Plagiarism guidelines. o Copying / cheating in tests and exams is academic
  • 47. misconduct. Such incidents will be treated just as seriously as other forms of plagiarism. o Students who willingly allow another student to copy their work in any assessment may be considered to assisting in copying/cheating, and similar penalties may be applied. Where a subject coordinator considers that a student might have engaged in academic misconduct, KOI may require the student to undertake an additional oral exam as a part of the assessment for the subject, as a way of testing the student’s understanding of their work. Further information can be found on the KOI website. 3.5 Reasonable Adjustment The Commonwealth Disability Discrimination Act (1992) makes it unlawful to treat people with a disability less fairly than people without a disability. In the context of this subject, the principle of Reasonable Adjustment is applied to ensure that participants with a disability have equitable access to all aspects of the learning situation. For assessment, this means that artificial barriers to their demonstrating competence are removed. Examples of reasonable adjustment in assessment may include: o provision of an oral assessment, rather than a written assessment o provision of extra time o use of adaptive technology. The focus of the adjusted assessment should be on enabling the
  • 48. participants to demonstrate that they have achieved the subject purpose, rather than on the method used. 3.6 Appeals Process Full details of the KOI Assessment and Assessment Appeals Policy may be obtained in hard copy from the Library, and on the KOI website www.koi.edu.au under Policies and Forms. ICT 200 DATABASE DESIGN AND DEVELOPMENT T219 19/06/2019 16:20 PAGE 14 OF 15 *AUSTRALIAN INSTITUTE OF BUSINESS AND MANAGEMENT PTY LTD © ABN: 72 132 629 979 CRICOS 03171A Approved by KOI Academic Board for T2 2019 http://www.koi.edu.au/ ICT200 Assessments and Mid-Trimester Exams: Where students are not satisfied with the results of an assessment, including mid-trimester exams, they have the right to appeal. The process is as follows: o Discuss the assessment with their tutor or lecturer – students should identify where they feel more marks should have been awarded – students should provide valid reasons based on the marking
  • 49. guide provided for the assessment. Reasons such as “I worked really hard” are not considered valid. o If still not satisfied, students should complete an Application for Review of Assessment Marks form, detailing the reason for review. This form can be found on the KOI website and is also available at KOI Reception (Market St and Kent St). o Application for Review of Assessment Marks forms must be submitted as explained on the form within ten (10) working days of the return of the marked assessment, or within five (5) working days after the return of the assessment if the assessment is returned after the end of the trimester. Review of Grade – whole of subject and final exams: Where students are not satisfied with the results of the whole subject or with their final exam results, they have the right to request a Review of Grade – see the Assessment and Assessment Appeals Policy for more information. An Application for Review of Grade/Assessment Form (available from the KOI Website under Policies and Forms and from KOI Reception, Market St and Kent St) should be completed clearly explaining the grounds for the application. The completed application should be submitted as explained on the form, with supporting evidence attached, to the Academic Manager.
  • 50. ICT 200 DATABASE DESIGN AND DEVELOPMENT T219 19/06/2019 16:20 PAGE 15 OF 15 *AUSTRALIAN INSTITUTE OF BUSINESS AND MANAGEMENT PTY LTD © ABN: 72 132 629 979 CRICOS 03171A Approved by KOI Academic Board for T2 2019 2. Academic Details2.1 Overview of the Subject2.2 Graduate Attributes for Undergraduate CoursesKing’s Own Institute’s key generic graduate attributes for a Bachelor’s level degree are summarised below:2.3 Subject Learning OutcomesThis is a Level 2 subject. PowerPoint Slides prepared by: V. Andreea CHIRITESCU Eastern Illinois University Walter Nicholson | Christopher Snyder 12th edition CHAPTER Mathematics 2 for Microeconomics © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 1
  • 51. Maximization of a Function of One Variable • Economic theories assume that – An economic agent is seeking to find the optimal value of somefunction • Consumers seek to maximize utility • Firms seek to maximize profit • Simple example: � = �(�) – Manager of a firm wants to maximize profits • The profits (π) received depend only on the quantity (q) of the good sold © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 2 2.1 Hypothetical Relationship between Quantity Produced and Profits If a manager wishes to produce the level of output that maximizes profits, then q*
  • 52. should be produced. Notice that at q*, dπ/dq = 0. © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 3 p = f(q) p Quantity p* q* p2 q2 p1 q1 p3 q3
  • 53. Maximization of a Function of One Variable • Vary q to see where maximum profit occurs – An increase from q1 toq2 leads to a rise in p © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 4 2 1 2 1 0 or 0 q q q p p p- D > > - D Maximization of a Function of One Variable • If output is increased beyond q*, profit will decline
  • 54. – An increase from q* to q3 leads to a drop in p © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 5 0 q pD < D Maximization of a Function of One Variable • Derivatives – The derivative of p = f(q) is the limit of Dp/Dq for very small changes in q – Is the slope of the curve – The value depends on the value of q1 – The derivative of p = f(q) at the pointq1 is: © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a
  • 55. publicly accessiblewebsite, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 6 1 1 0 ( ) ( ) lim h f q h f qd df dq dq h p ® + - = = Maximization of a Function of One Variable • Value of a derivative at a point(the slope) – The evaluation of the derivative at the point q = q1 can be denoted © 2017 Cengage Learning®. May not be scanned,
  • 56. copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 7 – In our previous example, 1q q d dq p = 1 0 q q d dq p = > 3 0 q q
  • 57. d dq p = < * 0 q q d dq p = = Maximization of a Function of One Variable • First-order condition for a maximum – For a function of one variable to attain its maximum value at somepoint, the derivative at that pointmust be zero © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a
  • 58. certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 8 * 0 q q df dq = = Maximization of a Function of One Variable • The first order condition (dp/dq=0) – Is a necessary condition for a maximum – But it is not a sufficient condition • The second order condition – In order for q* to be the optimum, © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a
  • 59. password-protected website or school-approved learning management system for classroom use. 9 – That is, at q*, d�/dq must be decreasing • The derivative of dπ/dq must be negative at q* and0 for * 0 for * d d q q q q dq dq p p > < < > 2.2 Two Profit Functions That Give Misleading Results If the First Derivative Rule Is Applied Uncritically In (a), the application of the first derivative rule would result in point��∗ being chosen. This pointis in fact a pointof minimum profits. Similarly, in (b), output level ��∗ would be recommended by the first derivative rule, but this pointis inferior to all outputs greater than ��∗ . This demonstrates graphically that finding a pointat which the derivative is equal to 0 is a necessary, but not a sufficient,condition for a function to attain its maximum value.
  • 60. © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 10 p (a) Quantity pa* qa* p (b) Quantity pb* qb* Maximization of a Function of One Variable • Second derivative – The derivative of a derivative – Denotedby:
  • 61. © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 11 • The second order condition – q* (solution to f.o.c.) is a local maximum if: 2 2 2 2 or ) o "(r d d f f q dq dq p 2 2 * * "( ) 0 q q q q d
  • 62. f q dq p = = = < Rules for Finding Derivatives © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 12 13. If is a constant, then 1. If i [ ( )] 2. If is a constant, then '( ) 5. ln for any constant - special case
  • 63. s a constant, then 0 ln 4. 1 : x x x x a adxa ax d d af x a af x dx da a a a dx da a
  • 64. dx d x d de e dx x x x -= = = = = = Rules for Finding Derivatives • Suppose that f(x) and g(x) are two functions of x and f’(x) and g’(x)exist, then: © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a
  • 65. publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 13 [ ]2 ( ) ( ) ' ( ) ( ) ( ) ' ( ) 8. pro [ ( ) ( )] 6. '( [ ( ) ( )] 7. ( ) '( ) ' vided t ( ) ( ) hat ( ) ' ) ) 0
  • 66. g ( ) ( f x d g x f x g x f x g x d f x g x f x g x f x g x d f x g x f x g x dx g x dx x dx × = + æ ö ç ÷ -è ø = + + ¹
  • 67. = Rules for Finding Derivatives • If y = f(x) and x = g(z) and if both f’(x) and g’(x)exist, then: © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 14 • This is called the chain rule – Allows us to study how one variable (z) affects another variable (y) through its influence on someintermediate variable (x) 9. dy dy dx df dg dz dx dz dx dz = × = × Rules for Finding Derivatives
  • 68. • Some examples of the chain rule include: © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 15 [ ] [ ] 2 2 2 2 2 ln ( ) ln ( ) ( ) 1 1 11. ( ) ( ) 10. [ln( )] [ln( )] ( ) 1 2 12. 2 ( (
  • 69. ) ) ax ax ax ax d ax d ax d ax a dx d ax dx ax d x d x d x x dx dx xd x x de de d ax e a ae dx x x x d a d = × = × = × = × = × = = =
  • 70. × = 2.1 Profit Maximization © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 16 • Suppose that the relationship between profit and output is p = 1,000q - 5q2 • The first order condition for a maximum is dp/dq = 1,000 - 10q = 0 q* = 100 • Since the second derivative is always -10, then q = 100 is a global maximum Functions of Several Variables • Most goals of economic agents depend on
  • 71. several variables – Trade-offs must be made • The dependence of one variable (y) on a series of othervariables (x1,x2,…,xn) is denoted by y = f(x1, x2,…,xn) © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 17 Functions of Several Variables • Partial derivatives – Partial derivative of y with respect to x1: © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning
  • 72. management system for classroom use. 18 – Allof the otherx’s are held constant • A more formal definition is 1 1 1 1 or or or x y f f f x x ¶ ¶ ¶ ¶ 2 2 21 1 0 1 ..., , ( , ,..., ) ( , ,..., ) lim n n n h x x f x h x x f x x xf
  • 73. x h® + -¶ = ¶ Calculating Partial Derivatives © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 19 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1
  • 74. 2 2 1 2 1 1 2 2 1 1 2 2 1 1 2 2 1 2 3 2. If ( , ) , then and 1. If ( . , ) , then 2 and If ( , ) ln ln , then 2 ax bx ax bx ax bx y f x x ax bx x cx f f
  • 75. f ax bx f bx cx x x y f x x e f f f ae f be x y f x x a x b x f a x x f + + + = = + + ¶ ¶ = = + = = + ¶ ¶ = = ¶ ¶ = = = + ¶
  • 76. = ¶ = = = ¶ = ¶ 2 1 2 2 and f b f x x x ¶ = = ¶ Functions of Several Variables • Partial derivatives – Are the mathematical expression of the ceteris paribus assumption • Law of demand: ��/��<0, ceteris paribus – Show how changes in one variable affect some outcome when othervariables are held
  • 77. constant • We must be concerned with units of measurement © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 20 Functions of Several Variables • Elasticity – Measures the proportional effect of a change in one variable on another – Unit free – Of y with respect to x is © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved
  • 78. learning management system for classroom use. 21 , ( ) y x y y x dy x xy e x x y dx y x D D = = × = × D D 2.2 Elasticity and Functional Form © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.
  • 79. 22 • For: � = � + �� + ��ℎ�� ����� • The elasticity is: • ey,x is not constant – It is important to pick the pointat which the elasticity is to be computed ,y x dy x x x e b b dx y y a bx = × = × = × + + ××× 2.2 Elasticity and Functional Form © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 23 • For � = ���
  • 80. – The elasticity is a constant: • For ln � = ln � + � ln � – The elasticity is: • Elasticities can be calculated through logarithmic differentiation 1 , b y x b dy x x e abx b dx y ax -= × = × = , ln lny x d y e b d x = = <= log of above equation Functions of Several Variables
  • 81. • Second-order partial derivatives – The partial derivative of a partial derivative © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 24 2( / )i ij j j i f x f f x x x ¶ ¶ ¶ ¶ = = ¶ ¶ Second-order partial derivatives © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in
  • 82. part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 25 1 2 1 2 1 2 1 2 1 2 2 2 1 2 1 2 1 2 2 11 1 2 1 1 1 12 2 21 2 2 1 1 2 2
  • 83. 11 12 21 22 2 1. ( , ) , 2 ; ; 2. ( , ) 3. ( , ) ln ln , ; ; , ; ; ; 2 ax bx ax bx ax bx ax bx ax bx y f x x ax bx x cx then f a f If y f x x a x b x the
  • 84. y f x x e then f a e f abe f ab b e f n f ax f b f b c e + + + + + - = = = = = = = + = -
  • 85. = = + + = = = = = 2 12 21 22 2 0; 0; f f f bx -= = = - Functions of Several Variables • Young’s theorem – Under general conditions – The order in which partial differentiation is conducted to evaluate second-order partial derivatives does not matter ���= ��� © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 26
  • 86. Functions of Several Variables • Uses of second-order partial derivatives – Play an important role in many economic theories – A variable’s own second-order partial, fii • Shows how ¶y/¶xi changes as the value of xi increases • fii < 0 indicates diminishing marginal effectiveness – Cross-partial fij indicates how the marginal effectiveness of xi changes as xj increases © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 27 Functions of Several Variables • The chain rule with many variables – � = �(�1,�2,�3) • Each of thesex’s is itselfa function of a single
  • 87. parameter, a – � = �[�1(�),�2(�),�3(�)] – How a change in a affects the value of y: © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 28 31 2 1 2 3 dxdx dxdy f f f da x da x da x da ¶ ¶ ¶ = × + × + × ¶ ¶ ¶ Functions of Several Variables • Special case:if �3 =�, then: � = �[�1(�),�2(�),�] – The effect of a on y:
  • 88. • A direct effect (which is given by fa) • An indirect effect that operates only through the ways in which a affects the x’s © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 29 1 2 1 2 dx dxdy f f f da x da x da a ¶ ¶ ¶ = × + × + ¶ ¶ ¶ Functions of Several Variables • Implicit functions – If the value of a function is held constant • An implicit relationship is created among the independent variables that enterinto the
  • 89. function • The independent variables can no longer take on any values –But must instead take on only that set of values that result in the function’s retaining the required value © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 30 Functions of Several Variables • Implicit functions – Use to quantify the trade-offs inherent in economic models • y = f(x1,x2); Implicit function: x2=g(x1) © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use
  • 90. as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 31 1 1 2 1 1 2 1 2 1 1 1 1 1 1 2 ( ) 0 0 ( , ) ( , ( )) Differentiate with respect to : Rearranging ( ) terms :
  • 91. y f x x f dg x f f dx dg x x g dx f x x dx dx f = = = = + × = = - 2.3 Using the Chain Rule © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved
  • 92. learning management system for classroom use. 32 • A pizza fanatic – Each week, he consumes threekinds of pizza, denoted by x1, x2, and x3 • Cost of type 1 pizza is p perpie • Cost of type 2 pizza is 2p • Cost of type 3 pizza is 3p – Allocates $30 each weekto each type of pizza – How the total number of pizzas purchased is affected by the underlying pricep 2.3 Using the Chain Rule © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 33 • Quantitypurchased of each type: x1=30/p; x2=30/2p; x3=30/3p
  • 93. • Total pizza purchases: y = f[x1(p), x2(p), x3(p)] = x1(p) + x2(p) + x3(p) • Applying the chain rule: 31 2 1 2 3 2 2 2 230 15 10 55 dxdx dxdy f f f dp dp dp dp p p p p- - - - = × + × + × = = - - - = - 2.4 A Production Possibility Frontier—Again © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.
  • 94. 34 • A production possibility frontier for two goods of the form x2+0.25y2=200 • The implicit function: 2 4 0.5 x y fdy x x dx f y y - - - = = = Functions of Several Variables • Comparative statics analysis – From �=0=�(�1,�2)=�(�1,�(�1 )) – Exogenous variable, a , the implicit form is �(�(�), �)=0 – Applying the implicit function theorem yields: ��(�)/��=−��/�� • This shows directly how changes in the exogenous variable a affect the endogenous variable x.
  • 95. © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 35 2.5 Comparative Statics of a Price-Taking Firm © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 36 • The first order condition for a profit firm that takesmarket priceas given: �(�(�),�)=�−�ʹ (�(�))=0 • Applying the Implicit Function Theorem to this expression yields:
  • 96. ��(�) �� = − �; �< = − 1 �(−�>(�)) ��⁄ = 1 �@(�) > 0 Maximization of Functions of Several Variables • Suppose an agent wishes to maximize y = f (x1,x2,…,xn) – The change in y from a change in x1 (holding all otherx’s constant) is • Equal to the change in x1 times the slope (measuredin the x1 direction) �� = �� ��C
  • 97. ��C = �C��C © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 37 Maximization of Functions of Several Variables • First-order conditions for a maximum – Necessary condition for a maximum of the function�(�1,�2,…,��)is that �� = 0 for any combination of small changes in the x’s: �1=�2=…=��=0 • Critical pointof the function – Not sufficient to ensure a maximum • Second-order conditions, ��� < 0 – Second partial derivatives must be negative © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a
  • 98. password-protected website or school-approved learning management system for classroom use. 38 2.6 Finding a Maximum © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 39 • Suppose that y is a function of x1 and x2 � = − (�1 − 1)2 − (�2 − 2)2 + 10 � = − �12 + 2�1 − �22 + 4�2 + 5 • First-order conditions imply that 1 1 2 2 2 2 0
  • 99. 2 4 0 y x x y x x ¶ = - + = ¶ ¶ = - + = ¶ OR * 1 * 2 1 2 x x = =
  • 100. The Envelope Theorem • The envelope theorem – How an optimized function changes when a parameter of the function changes • A specific example: � = −�2 + �� – Represents a family of inverted parabolas • For different values of a – Is a function of x only • If a is assigned a specific value • Can calculate the value of x that maximizes y © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 40 2.1 Optimal values of y and x for alternative values of a in y=-x2+ax © 2017 Cengage Learning®. May not be scanned,
  • 101. copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 41 2.3 Illustration of the Envelope Theorem The envelope theorem states that the slope of the relationship between y (the maximum value of y) and the parameter a can be found by calculating the slope of the auxiliary relationship found by substituting the respective optimal values for x into the objective function and calculating ∂y/∂a. © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.
  • 102. 42 The Envelope Theorem • If we are interested in how y* changes as a changes – Calculate the slope of y directly (time- consuming approach) – Hold x constantat its optimal value and calculate ¶y/¶a directly (the envelope shortcut) © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 43 The Envelope Theorem • Calculate the slope of y directly – Must solve for the optimal value of x for any value of a: ��/�� = − 2� +� = 0; �∗ =�/2
  • 103. – Substituting, we get: �∗ = −(�∗ )2 + �(�∗ ) = −(�/2)2 + �(�/2); �∗ = −�2/4 + �2/2 = �2/4 • Therefore, ��∗ /�� = 2�/4= �/2 © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 44 The Envelope Theorem • The envelope shortcut – For small changes in a, dy*/da can be computed by holding x at its optimal value (x*) and calculating ¶y/¶a directly from y © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a
  • 104. password-protected website or school-approved learning management system for classroom use. 45 – Holding x = x*: LM LN = �∗ (�) = �/2 * * * 2 * ( ) ( ) ( ) ( ) x x a x x a dy y x ax x a da a a= = ¶ ¶ - + = = = ¶ ¶ The Envelope Theorem • The envelope theorem
  • 105. – The change in the value of an optimized function with respect to a parameter of that function – Can be found by partially differentiating the objective function while holding x (or several x’s) at its optimal value © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 46 * { *( )} dy y x x a da a ¶ = = ¶ The Envelope Theorem
  • 106. • Many-variable case – y is a function of several variables y = f(x1,…,xn,a) – Finding an optimal value for y: solve n first- order equations: ¶y/¶xi = 0 (i = 1,…,n) – Optimal values for thesex’s would be a function of a x1* = x1*(a); x2* = x2*(a); …; xn* = xn*(a) © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 47 The Envelope Theorem • Many-variable case – Substituting into the original objective function gives us the optimal value of y (say, y*), we get the value function: y* = f [x1*(a), x2*(a),…, xn*(a),a] – Differentiating yields
  • 107. © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 48 * 1 2 1 2 ( ) * ... * for all i i n n i x x a
  • 108. dxdx dxdy f f f f da x da x da x da a dy f x da a = ¶ ¶ ¶ ¶ = × + × + + × + ¶ ¶ ¶ ¶ ¶ = ¶ 2.7 A Price-Taking Firm’s Supply Function © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 49 • Cost function �(�)=5�2 – First order condition: �=�ʹ (�)=10�, we get �∗ =0.1� • Alternatively:
  • 109. – Profits are given by: �(�,�)=��−�(�) – Calculate the optimal value of the firm’s profits: �∗ (�)=��∗ −�(�∗ )=�(0.1�)−5(0.1�)2=.05�2 • The envelope theorem states that * * * *( ) ( , )0.1 | q q q q d p p q p q q dp p p p = = ¶ = = = = ¶ Constrained Maximization
  • 110. • What if all values for the x’s are not feasible? – The values of x may all have to be > 0 – A consumer’s choices are limited by the amount of purchasing power available • Lagrange multiplier method – One method used to solve constrained maximization problems © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 50 Lagrange Multiplier Method • Lagrange multiplier method – Supposethat we wish to find the values of x1, x2,…, xn that maximize: � = �(�1, �2,…, ��) – Subject to a constraint: �(�1, �2,…, ��) = 0 • The Lagrangian expression ℒ = �(�1, �2,…, �� ) + l �(�1, �2,…, ��)
  • 111. – l iscalled the Lagrange multiplier – ℒ = �, because �(�1, �2,…, ��) = 0 © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 51 Lagrange Multiplier Method • First-order conditions – Conditions for a critical pointfor function ℒ ¶ℒ /¶�1 = �1 + l�1 = 0 ¶ℒ /¶�2 = �2 + l�2 = 0 … ¶ℒ /¶�� = �� + l�� = 0 ¶ℒ /¶l = �(�1,�2,…,��) = 0 © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved
  • 112. learning management system for classroom use. 52 Lagrange Multiplier Method • First-order conditions – Can generally be solved for x1, x2,…, xn and l – The solution will have two properties: • The x’s will obey the constraint • These x’s will make the value of ℒ (and therefore f) as largeas possible © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 53 Lagrange Multiplier Method • The Lagrange multiplier (l)
  • 113. – Important economic interpretation – The first-orderconditions imply that �1/−�1 = �2/−�2 = ⋯ = ��/−�� = l • The numerators measure the marginal benefit of one more unit of xi • The denominators reflect the added burden on the constraint of using more xi © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 54 Lagrange Multiplier Method • The Lagrange multiplier (l) – At the optimal xi’s, the ratio of the marginal benefit to the marginal cost of xi should be the same for every xi – l isthe common cost-benefit ratio for all xi © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a
  • 114. publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 55 marginal benefit of marginal cost of i i x x l = Lagrange Multiplier Method • The Lagrange multiplier (l) – A high value of l indicates that each xi has a high cost-benefit ratio – A low value of l indicates that each xi has a low cost-benefit ratio – l = 0 implies that the constraint is not binding
  • 115. © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 56 Constrained Maximization • Duality – Any constrained maximization problem has a dual problem in constrained minimization • Focuses attention on the constraints in the original (primal) problem © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 57
  • 116. Constrained Maximization • Individuals maximize utility subject to a budget constraint – Dual problem: individuals minimize the expenditure needed to achieve a given level of utility • Firms minimize the cost of inputs to produce a given level of output – Dual problem: firms maximize output for a given cost of inputs purchased © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 58 2.8 Optimal Fences and Constrained Maximization © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a
  • 117. certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 59 • A farmer had a certain length of fence (P) – Wishes to enclose the largest possible rectangular area, with x and y the lengths of the sides – Choose x and y to maximize the area (A = x·y) – Subject to the constraint that the perimeter is fixed at P = 2x + 2y 2.8 Optimal Fences and Constrained Maximization © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 60 • The Lagrangian expression: ℒ = � · � + l(� − 2� − 2�)
  • 118. • First-order conditions ¶ℒ /¶� = � − 2l = 0 ¶ℒ /¶� = � − 2l = 0 ¶ℒ /¶l = � − 2� − 2� = 0 �/2 = �/2 = λ, so � = �, and the field should be square � = � and � = 2l, then � = � = �/4 ��� l = �/8 2.8 Optimal Fences and Constrained Maximization © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 61 • Interpretation of the Lagrange multiplier – l suggests that an extrayard of fencing would add P/8 to the area – Provides information about the implicit value
  • 119. of the constraint • Duality – Choose x and y to minimize the amount of fence required to surround the field minimize � = 2� + 2� subject to � = � · � – Setting up the Lagrangian: ℒ � = 2� + 2� + l�(� − � × �) 2.8 Optimal Fences and Constrained Maximization © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 62 • Duality – First-order conditions: ¶ℒ �/¶� = 2 − l� · � = 0 ¶ℒ �/¶� = 2 − l� · � = 0 ¶ℒ �/¶l� = � − � · � = 0
  • 120. – Solving,we get: � = � = �� – The Lagrangian multiplier l� = X Y = X M = X Z� Envelope Theorem in Constrained Maximization Problems • Suppose that we want to maximize � = �(�1,…,��;�) Subject to the constraint: �(�1,…,��;�) = 0 • One way to solve – Setup the Lagrangian expression – Solve the first-orderconditions • The envelope theorem: ��∗ �� = ¶ℒ ¶� (�C∗ ,…,�[∗ ;�)
  • 121. © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 63 2.9 Optimal Fences and the Envelope Theorem © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 64 • The fencing problem (example 2.8), the value function: �∗ = �∗ ⋅ �∗ = ] ^ ⋅ ] ^
  • 122. = ] _ C` – The Lagrangian: L = �� + �(� − 2� − 2�) – Applying envelope theorem: ��∗ /��=�/8=∂�/∂�=� • The Lagrange multiplier in a constrained maximization problem – Shows the marginal gain in the objective function that can be obtained from a slight relaxation of the constraint Inequality Constraints • Maximize � = �(�1,�2) subject to �(�1,�2)≥ 0, �1 ≥0, ��� �2 ≥ 0 • Slack variables – Introduce threenew variables (a, b, and c) that convert the inequalities into equalities – Square thesenew variables �(�1,�2)− �2 =0; �1 −�2 =0; ��� �2 − �2 =0 – Any solution that obeys thesethreeequality constraints will also obey the inequality constraints © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in
  • 123. part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 65 Inequality Constraints • Maximize � = �(�1,�2)subject to �(�1,�2)≥ 0, �1 ≥ 0, ��� �2 ≥ 0 • Lagrange multipliers ℒ = f(x1,x2)+ l1[g(x1,x2) - a2]+l2[x1 - b2]+ l3[x2 - c2] – There will be 8 first-orderconditions © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 66 ¶ℒ /¶x1 = f1 + l1g1 + l2 = 0 ¶ℒ /¶c = -2cl3 = 0 ¶ℒ /¶x2 = f1 + l1g2 + l3 = 0 ¶ℒ /¶l1 = g(x1,x2) - a2 = 0
  • 124. ¶ℒ /¶a = -2al1 = 0 ¶ℒ /¶l2 = x1 - b2 = 0 ¶ℒ /¶b = -2bl2 = 0 ¶ℒ /¶l3 = x2 - c2 = 0 Inequality Constraints • Complementary slackness – According to the third condition, either a = 0 or l1 = 0 • If a = 0, the constraint �(�1,�2)holds exactly • If l1 = 0, the availability of someslackness of the constraint implies that its value to the objective function is 0 – Similar complementary slackness relationships also hold for x1 and x2 © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 67 Inequality Constraints • Complementary slackness
  • 125. – These results are sometimes called Kuhn- Tucker conditions • Show that solutions to problems involving inequality constraints will differ from those involving equality constraints in rather simple ways • Allows us to work primarily with constraints involving equalities © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 68 Second-Order Conditions and Curvature • Functions of one variable, � = �(�) – A necessary condition for a maximum: ��/�� = � ′(�) = 0 • y must be decreasing for movements awayfrom it – Change in y: �� = � ′(�) �� • To be at a maximum, dy must be decreasing for small increases in x
  • 126. © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 69 Second-Order Conditions and Curvature • Functions of one variable, � = �(�) – Second derivative of y © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 70 • Since �2 � < 0 ,� ’’(�)��2 < 0 • Since ��2 mustbe > 0, � ’’(�) < 0 • This means that the function f must have a concave shape at the critical point
  • 127. 2 2[ '( ) ]( ) "( ) "( ) d f x dx d dy d y dx f x dx dx f x dx dx = = × = × = Second-Order Conditions and Curvature • Functions of two variables, � = �(�1,�2) – First order conditions for a maximum: ¶�/¶�1 = �1 = 0 ¶�/¶�2 = �2 = 0 – f1 and f2 must be diminishing at the critical point – Conditions must also be placed on the cross- partial derivative (�12 = �21) © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 71
  • 128. Second-Order Conditions and Curvature • The total differential of y: �� = �1 ��1 + �2 ��2 �2� = �11��1 + �12��2 ��1 + �21��1 + �22��2 ��2 �2� = �11��12 + �12��2��1 + �21��1 ��2 + �22��2 2 • By Young’s theorem, �12 = �21 and �2� = �11��12 + 2�12��1��2 + �22��22 –d2y < 0 for any dx1 and dx2, if f11<0 and f22<0 –If neither dx1 nor dx2 is zero, then d2y < 0 for any dx1 and dx2, if f11 f22 - f122 > 0. © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 72
  • 129. Multivariable Optimization (New) • First Order Condition • �h Li LY = 0,for all � > 0,where � is a vector of length n, and Li LY is a vector of partial derivatives of length n evaluated at �∗ , and T denotes transpose. © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 73 • Second Order Condition • �h�� < 0,for all � > 0, where � is a vector of length n, and � is a matrix of second partial derivatives of size �×�evaluated at �∗ .
  • 130. Multivariable Optimization (New) • First Order Condition • �r = Li LYs = 0,for all �,when evaluated at �∗ © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 74 • Second Order Condition • �, evaluated at �∗ , is a negative definite matrix. Negative definite means that the leading principal minors have alternative signs, starting with a minus. � = �X� ��C��C ⋯ �X�
  • 131. ��C��[ ⋮ ⋱ ⋮ �X� ��[��C ⋯ �X� ��[��[ = �CC ⋯ �C[ ⋮ ⋱ ⋮ �[C ⋯ �[[ 2.10 Second-Order Conditions: Health status for the Last Time © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 75 � = �(�1,�2) = − �12 + 2�1 − �22 + 4�2 + 5
  • 132. – First-order conditions • �1 = −2�1 + 2 = 0 and �2 = −2�2 + 4 = 0 • Or: �1 ∗ = 1,�1 ∗ = 2 – Second-order partial derivatives • �11 = −2 • �22 = −2 • �12 = 0 Second-Order Conditions and Curvature • Concave functions • The own second partial derivatives ( f11 and f22) be sufficiently negative • So that their product will outweigh any possible perverse effects from the cross-partial derivatives ( f12 = f21) – Have the property that they always lie below any plane that is tangent to them • The plane defined by the maximum value of the function is simply a special case of this property © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in
  • 133. part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 76 Second-Order Conditions and Curvature • Constrained maximization – Choose x1 and x2 tomaximize: � = �(�1,�2) – Linear constraint: c − b1x1 − b2x2 = 0 – Lagrangian: ℒ = �(�1,�2) + l(� − �1�1 − �2�2) – The first-orderconditions: �1 − l�1 = 0, �2 − l�2 = 0, � − �1�1 − �2�2 = 0 © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 77
  • 134. Second-Order Conditions and Curvature • Constrained maximization – Use the “second” total differential: �2� = �11��12 + 2�12��1��2 + �22��22 • Only values of x1 and x2 that satisfy the constraint can be considered validalternatives to the critical point – Total differential of the constraint −�1 ��1 − �2 ��2 = 0, ��2 = −(�1/�2)��1 • Allowable relative changes in x1 and x2 © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 78 Second-Order Conditions and Curvature • Constrained maximization – First-order conditions imply that f1/f2 = b1/b2,
  • 135. we get: ��2 = −(�1/�2) ��1 – Since: �2� = �11��12 + 2�12��1��2 + �22��22 – Substitute for dx2 and get �2� = �11��12 − 2�12(�1/�2)��12 + �22(�12/�22)��12 – Combining terms and rearranging, we get �2� = �11 �2 2 − 2�12�1�2 + �22�12 ��12 �22 © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 79 Second-Order Conditions and Curvature
  • 136. • Constrained maximization – Therefore, for �2� < 0, it must be true that �11 �2 2 − 2�12�1�2 + �22�12 < 0 – This equation characterizes a set of functions termed quasi-concave functions • Quasi-concave functions – Any two points within the upper contour set can be joined by a straight line contained completely in the upper contour set © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 80 Second Order Condition for Constrained Optimization with One Constraint (new) © 2017 Cengage Learning®. May not be scanned,
  • 137. copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 81 • �}, evaluated at �∗ , is a bordered Hessian which is a matrix of second derivatives of the Lagrangian, including �. If Λ is the Lagrangian, then the bordered Hessian is: • (−1)�} must be negative definite. 2.11 Concave and Quasi-Concave Functions © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 82 • � = �(�1,�2)= (�1×�2)�