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University of New England
Faculty of Arts and Sciences
School of Science and Technology
MATH120
BASIC MATHEMATICAL METHODS IN SCIENCE AND ECONOMICS
Unit Guide
Trimester 1, 2013
c�University of New England
CRICOS Provider No: 00003G
Contents
1 Welcome and Contact Details . . . . . . . . . . . . . . . . . . . . . 4
2 Unit Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Is MATH120 the appropriate Mathematics Unit for You? . . . . . . 5
4 Studying Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . 5
5 Lecture Notes and Recommended Text . . . . . . . . . . . . . . . . 6
6 Intensive School for External Students . . . . . . . . . . . . . . . . 7
7 Lectures, Vodcasts, Tutorials and Practicals . . . . . . . . . . . . . 8
8 Assignments and Assessment Tasks . . . . . . . . . . . . . . . . . . 8
8.1 Assignment 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 9
8.2 Submission of Assessment Tasks . . . . . . . . . . . . . . . . 9
8.3 Due Dates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
9 Assessment and Examination . . . . . . . . . . . . . . . . . . . . . 10
10 Moodle Site and Turing Home Page . . . . . . . . . . . . . . . . . . 11
11 Announcements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
12 Assistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
13 Schedule of Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
14 Study Timetable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
15 Tutorial Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
16 Assignment Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3
4
1 Welcome and Contact Details
Welcome to MATH120 Basic Mathematical Methods in Science and Economics.
In Trimester 1, 2013, the members of sta↵ responsible for MATH120 are Associate
Professor Gerd Schmalz, Dr Bea Bleile and Dr Imi Bokor. Please call us Gerd, Bea
and Imi, respectively, and do not hesitate to contact us regarding matters relating
to any aspect of this unit.
Internal students, that is, students enrolled in on–campus mode, please contact
Gerd, the On–Campus Unit Co-ordinator, with any administrative questions. His
contact details are
[email protected] (preferred)
02 6773 3182
External students, that is, students enrolled in o↵–campus mode please, contact
Bea, the O↵–Campus Unit Co-ordinator, with any administrative questions. Her
contact details are
02 6773 3572
beableile on Skype
All students are welcome to contact any one of us with mathematical questions.
Imi’s contact details are
02 6773 2213
You can also contact us by fax on
02 6773 3312.
For problems relating to your enrolment, please contact the Student Centre:
Web: http://askune.custhelp.com/
2 Unit Overview
MATH120 introduces students of economics, social and life sciences to some of
the mathematical concepts used in their respective areas of study. These concepts
include di↵erentiation and integration of functions of one real variable, di↵erentia-
tion of functions of two real variables, maxima and minima of such functions and
matrices with emphasis on applications.
A schedule of topics of study for MATH120 is provided in Section 13 and a
timetable in Section 14.
Note that after topic seven, that is, in week nine of the trimester, the unit
will divide into two streams, namely the Science Stream for students of rural,
5
environmental and biological sciences, and the Economics Stream for students of
economics and related areas. In the Economics Stream students will also receive
practical instruction in the use of spreadsheets such as Excel.
3 Is MATH120 the appropriate Mathematics Unit
for You?
UNE o↵ers two parallel programs in first year mathematics and statistics. Students
either take MATH101/MATH102 or MATH120/STAT100. MATH101/MATH102
provides the basis for further study in mathematics and is a prerequisite for second
and third year mathematics and statistics. MATH120 is a terminating unit and is
not su�cient background for further study in mathematics.
Depending on your background in mathematics you might find MATH120 easy
or you might find it hard. MATH120 presumes familiarity with the content of
advanced (or 2 unit) mathematics in NSW or its equivalent. It is possible to
succeed without this background, but requires a lot of hard work and close contact
with us.
If you don’t have this background or are struggling, we recommend that you
first complete MATH123 Foundation Mathematics. Assignment 0 provides a warm
up and helps you to gauge your background. Please contact us if you aren’t sure
whether you should enrol in MATH123 first, the sooner, the better.
If you have successfully completed (the equivalent of) Extension 1 (or 3 Unit)
Mathematics in NSW, you should consider enrolling in MATH102/MATH102 rather
than MATH120/STAT100. MATH101/MATH102 are designed for students with
Extension 1 background. They are more advanced and leave open the possibility
of further study of mathematics. You will find them more rewarding.
Please feel free to contact us regarding the choice of the appropriate mathemat-
ics unit for you, the sooner, the better.
4 Studying Mathematics
To learn mathematics one has to do mathematics. Thus the tutorial and assignment
problems are the most important part of this unit, with the tutorial problems in
Section 7 preparing you for the assignment problems in Section 16.
Mathematics is a language designed to deal precisely with general concepts
which can be applied to solving a broad range of concrete problems. Mathematics
is powerful and applicable because it is abstract. The language of mathematics
6
takes time to learn. MATH120 provides an introduction to it and shows you how
to use it.
You cannot expect to just read mathematics, such as in your lecture notes. You
need to work through the lecture notes. It usually takes several readings of the
same material before new concepts start to make sense.
While it can be useful to read more than one source, it can also be confusing,
because di↵erent authors take can take incompatible approaches. It is hard to find
the right balance. Our advice is that you do not spend much time searching for
material, especially not on the internet. The prime focus of your study should be
the lecture notes we provide. If you feel the need for alternative material, turn to
the recommended text for this unit (see Section 5) as it is what we consider to be
the most suitable.
It is more important to read the lecture notes and/or the recommended text,
think about the material and engage with us and your fellow students. Inter-
nal (on–campus) students, please ask your mathematical questions during lectures
and tutorials or come and see us during our o�ce hours. External (o↵–campus)
students, please ask your mathematical questions during the intensive school (see
Section 6) via the Moodle discussion boards (see Section 10) or contact us by e-mail.
There are situations where a tutor might be helpful, for example to help fill
gaps in your background. However, we recommend that you first turn to us for
support. In particular, relying too much on a tutor for assignment problems will
not stand you in good stead for the examination.
There is no such thing as a “stupid question”. When in doubt, ask.
We are here to help you, but we cannot help unless you tell us what the
problem is.
5 Lecture Notes and Recommended Text
You will receive a complete set of printed lecture notes containing all material
covered in this unit. Electronic versions of these notes will be available on the
MATH120 Moodle site and the MATH120 home page on turing (see Section 10).
The information contained in these notes is core material for completion of all
assessment tasks and the examination in MATH120. Make the study of these
notes and mastering the material they contain your prime focus in MATH120.
7
The recommended text for this unit is
Mathematical Methods for Science and Economics
(compiled for MATH120)
Goldstein, L.J. and Haeussler, E., Pearson 2nd ed. 2011
ISBN: 9781442555631
There is also a cheaper electronic version available at
http://www.pearson.com.au/9781486000029
This is a blended edition of two textbooks previously recommended for this unit,
namely Introductory Mathematical Analysis for Business, Economics, and the Life
and Social Sciences, by Haeussler, Paul and Wood (12th Edition), and Calculus
and its Applications, by Goldstein, Lay, Schneider (11th Edition).
Mathematical Methods for Science and Economics has been compiled especially
for MATH120, and covers all the topics of this unit in a single book. The publishers
have agreed to do this in order to save students the cost and inconvenience of
obtaining both books above.
Information about the recommended text, including options for purchase, is
available at
http://www.une.edu.au/studentcentre/textbook-info.php
6 Intensive School for External Students
The Intensive School for external (o↵–campus) students will be held at UNE from
Saturday, 16th, to Monday, 18th February, 2012. The timetable is available via
UNE’s timetabling web site, see
http://www.une.edu.au/timetables/2013/unit/MATH120/I1
For further information regarding intensive schools go to
http://www.une.edu.au/timetable/residentials.php
The Intensive School provides an introduction and warm-up for the unit. Key
concepts are highlighted and explained, providing an overview of MATH120.
We strongly recommend that you attend the Intensive School. It is not com-
pulsory, but for the communication of mathematical concepts there is nothing like
being in the same room. And getting to know each other will make it easier to
8
communicate throughout the trimester. Students attending the Intensive School
usually find it invaluable.
You need to organise travel and accommodation yourself, but you don’t need
to register for the Intensive School. Just turn up with pencil and paper and an
open mind. Questions will be very welcome. Please contact Bea (see Section 1) if
you have any questions regarding the Intensive School.
7 Lectures, Vodcasts, Tutorials and Practicals
There are four lectures and one tutorial per week for internal (on–campus) students.
It is crucial for your progress in this unit that you attend lectures and tutorials
and ask questions if there is anything you don’t understand.
Vodcasts will be available on the MATH120 Moodle site for external (o↵-
campus) students discussing key concepts and providing some worked examples. It
is crucial that you ask questions via e-mail or the relevant discussion forum on the
MATH120 Moodle site if there is anything you don’t understand.
Section 15 of this Unit Information booklet contains tutorial problems, namely
Tutorials 1 to 6 for both Streams and Tutorials 7s to 10s for the Science Stream.
These tutorial problems prepare you for the assignments and sample solutions will
be available on the MATH120 Moodle site.
All students are expected to work through the tutorial problems. Internal
(on–campus) students will discuss these problems during the tutorials. External
(o↵–campus) students are encouraged to discuss the tutorial problems in the cor-
responding discussion forum on the MATH120 Moodle site.
Students in the Economics Stream will receive a separate booklet for practical
instruction in the use of spreadsheets. You will start working with spreadsheets
after topic seven, that is, in week six.
8 Assignments and Assessment Tasks
The assignments are the most important part of this unit. The assignment problems
are contained in Section 16 and will be posted on the MATH120 Moodle site and
the MATH120 home page on turing (see Section 16).
Do not hesitate to contact us for assistance with background information or
guidance with the assignment problems. By all means work together with fellow
students. But make sure that the work you submit is your own. Anyone who
provides you with more or less complete solutions before you have submitted your
own attempts is doing you a gross disservice indeed.
9
The manual containing the short assessment tasks for the computer labs will
also be available on the MATH120 Moodle site and the MATH120 home page on
turing (see Section 16).
8.1 Assignment 0
Assignment 0 is a warm up which will help you decide whether you need to complete
MATH123 Foundation Mathematics before enrolling in MATH120. Assignment 0
will not be marked and is not to be submitted. Complete sample solutions will be
posted on the MATH120 Moodle site.
8.2 Submission of Assessment Tasks
Make sure you keep a copy of everything you submit. Mathematics assignments
should be neatly hand-written, and carefully set out, with ample space left for
corrections and comments.
Internal (on–campus) students will receive instructions regarding submission of
assignments during the first lecture.
External (o↵-campus) students submit their assignments in hard copy by mail
with an assignment cover sheet attached. The cover sheets are available via the
MATH120 Moodle site – first click on “Submit MATH120 Assignment X” and
then on “Download Coversheet”. If you have received an exemption from the
requirement to have computer access, you will receive your assignment cover sheets
in the mail.
Assignments should be mailed to:
Assignment Section
Teaching & Learning Centre
University of New England
Armidale NSW 2351
DO NOT FAX: Faxed assignments will not be accepted.
External (o↵–campus) students who are able to scan their handwritten assign-
ments may submit them electronically via the MATH120 Moodle site by clicking
on “Submit MATH120 Assignment X”.
Students in the Economics Stream may also submit the lab tasks electronically.
Please do not type your mathematics assignments unless you are
using some version of TeX.
10
8.3 Due Dates
Please note that the dates below are post-by dates for external (o↵–campus) stu-
dents, that is, your assessment task will be on time as long as you put it into the
mail by the specified date.
Assignment Due Date Computer Lab Due Date
0 4th March 1 30th March
1 11th March 2 3rd April
2 18th March 3 7th April
3 25th March 4 10th April
4 1st April 5 14th April
5 8th April 6 17th April
6 29th April 7 20th April
7s, 7e 6th May
8s 13th May
9s, 8e 20th May
10s 27th May
9 Assessment and Examination
There are ten weekly assignments for the Science Stream and eight for the Eco-
nomics Stream, with the first seven identical for both Streams.
The assignments contribute 30% to the overall assessment for students in the
Science Stream and 20% for students in the Economics Steam. There are an ad-
ditional 7 small assessment tasks from the computer labs for students in the Eco-
nomics Stream, contributing 10% to the final assessment. Students in the Science
Stream are not required to complete these lab tasks. Detailed worked solutions
to all assignment problems will be provided on return of the marked assignment
scripts.
There will be one 2–hour exam paper for both streams, but the paper will be
divided into Part A (Common questions), Part B (Science–Stream questions), and
Part C (Economics–Stream questions). The exam will contribute 70% to the final
assessment for both streams. All students must achieve a mark of at least
50% on the final exam in order to pass the unit.
You will be allowed to take three (3) sheets or six (6) single sided pages of
hand–written notes into the examination, no photocopies, no printed or scanned
notes will be allowed. You will also be allowed to take a silent type calculator into
the exam which is not programmable and does not have an alpha key pad.
11
Preparing the six-page summary for the examination is an important tool for
learning the material. You should start writing your summary as you learn the
material, revising it week by week. You will probably find it starts o↵ being too
long, but repeatedly revising it will make it easy to condense it. Experience shows
that students who follow this advice benefit greatly — those who draw up their
own summary rarely need to use it.
10 Moodle Site and Turing Home Page
The lecture notes, this unit guide and the computer lab manual will be available
on the MATH120 Moodle site and the MATH120 turing home page
http://mcs.une.edu.au/~math120/
The MATH120 Moodle site also provides access to announcements concerning this
unit, to the MATH120 discussion boards, the sample solutions for tutorial problems
and to vodcasts.
11 Announcements
All students must check their UNE e–mail accounts at least once a week for im-
portant information and announcements regarding their studies at UNE. Further
information about MATH120 will be provided on the News and Announcements
board on the MATH120 Moodle site for external (o↵–campus) students and during
lectures for internal (on–campus) students.
12 Assistance
Do not hesitate to contact us with any questions, problems, complaints or sugges-
tions. While it is important that you read through the material a few times and
struggle with the tutorial and assignment problems, it is equally important to seek
advice when you are stuck (see Section 1) for our contact details.
We look forward to meeting you.
Gerd, Bea and Imi
12
13 Schedule of Topics
Below is an outline and approximate schedule of topics covered during the course
of the semester.
Part 1. Common topics for Science and Economics Streams
1. Functions (week 1)
1.1. The equation of a straight line
1.2. Translations
1.3. Quadratic functions
1.4. Power functions
1.5. Exponential functions
1.6. Logarithmic functions
1.7. Trigonometric functions
1.8. Combining functions
2. Solving Equations (week 2)
2.1. Equations involving elementary functions
2.2. Simultaneous equations
3. Di↵erential Calculus (weeks 3 and 4)
3.1. Definition of the derivative
3.2. Rules for di↵erentiation
3.3. Higher derivatives
3.4. Derivatives of exponential, logarithmic and trigonometric functions
4. Maxima and minima (week 5)
4.1. Stationary points
4.2. Derivative of sine and cosine revisited
4.3. Second derivative test for classifying stationary points
5. Integration (week 6)
5.1. Indefinite integrals
13
5.2. Definite integrals
5.3. More on areas
5.4. Where does the natural logarithm really come from?
5.5. The average value of a function
6. Functions of several variables (week 7)
6.1. Partial derivatives
6.2. Maxima and minima of two-variable functions
7. Matrices and linear systems of equations (weeks 8 and 9)
7.1. Matrices
7.2. Operations of matrices
7.3. Multiplication of matrices
7.4. The inverse of a matrix and determinant
7.5. Solutions of linear systems
Part II. Separate Streams
A.Science Stream:
8. Applications of Matrices in Biology (weeks 9 and 10)
8.1. The Leslie Matrix Model of Population Growth
8.2. Powers of Matrices
9. Di↵erential Equations (weeks 11 and 12)
9.1. Introduction to di↵erential equations
9.2. Exponential growth and decay
9.3. Restricted exponential growth and decay
9.4. The logistic equation
B.Economics Stream (including Excel labs):
8. Applications in Economics (weeks 9 to 12)
8.1. The input-output model
14
8.2. Linear inequalities
8.3. Linear optimization
8.4. Compound interest
8.5. Geometric progressions and savings plans
8.6. Annuities and Amortization
8.7. Other applications
Excel Labs. (commencing week 7 - economics stream only)
1. Getting started, entering data and formulae
2. Drawing graphs
3. Using Solver to find zeros, minima and maxima
4. Using matrix commands in Excel
5. Solving systems of linear equations by using Excel
6. Using Solver for linear programming problems
7. Using financial functions in Excel
14 Study Timetable
Week Topics
1 Functions 1.1-1.8
2 Solving Equations 2.1-2.2
3 Di↵erential Calculus 3.1-3.2
4 Di↵erential Calculus 3.3-3.4
5 Maxima and Minima 4.1-4.3
6 Integration 5.1-5.5, Excel Labs commence (Economics Stream)
7 Functions of Several Variables 6.1-6.2
Mid semester break (two weeks)
8 Matrices and Linear Systems 7.1-7.5
9 Parallel Lectures Commence (on campus students)
10 Science 8.1-9.1, Economics 8.1-8.3
11 Science 9.2-9.4, Economics 8.4-8.7
12 Revision
Examination period commences
15 Tutorial Problems
MATH120 (2013)
TUTORIAL 1
Question 1. Sketch the graphs of the functions given by
(a) y = 4x� 1;
(b) y = �1
2x;
(c) y = x
2 + 3.
Question 2. Find an equation for the straight line through the points with
coordinates (�1, 2) and (2, 8).
Question 3. Sketch the graphs of the functions given by
(a) y = �2x2 + 6x+ 3;
(b) y = x
2 + x+ 1.
In each case find the maximum or minimum value of the function and its zeros,
that is, all x for which y = 0.
Question 3. Sketch the graphs of the functions given by
(a) f1(x) = 2 sin x;
(b) f2(x) = �2 sin x;
(c) f3(x) = 2 sin 4x;
(d) f4(x) = 2 sin(4x+ ⇡);
(e) f5(x) = 2 sin x+ 1.
between �⇡ and 2⇡.
Question 4. Sketch on the one diagram the graphs of the functions given by
(a) g1(x) = ln x, x > 0;
(b) g2(x) = � ln x, x > 0 ;
(c) g3(x) = ln(x� 1), x > 1.
MATH120 (2013)
TUTORIAL 2
Question 1. Solve the following equations for x.
(a) 7x+ 15 = 6;
(b) x� 23 = 3;
(c) 3e2x = 9;
(d)1
2ln x� 3 = 0;
(e) |x| = 3;
(f) |x� 2| = 3;
(f) (x+ 1)x(x� 4) = 0.
Question 2. Consider the two curves given by
y = x+ 2 and y = x
2 � 2x+ 1.
Decide whether the two curves intersect. If they do intersect, find their points of
intersection.
Question 3. Assuming only that ln 3 is approximately 1.0999 and ln 4 approx-
imately 1.386, calculate an approximation for
(a) ln 18 and
(b) ln4
9.
Question 4. Find the inverses of the functions given by
(a) y = 4x+ 3;
(b) y = 2ex � 1.
For each function sketch the graph of the function and that of its inverse in one
diagram.
MATH120 (2013)
TUTORIAL 3
Question 1. Find the derivatives of the functions given by
(a) y = 3x4;
(b) y = e
101x + 0.1;
(c) y = x sin x;
(d) y = 5sin x
x
, x 6= 0;
(e) y = ln(2� x), x < 2;
(f) y = cos(x4);
(e) y = (cosx)4.
Question 2. Find the third derivative of the function f given by
f(x) = 2x6 � 4x3 + 3x� 10x�2.
Question 3. Find an equation of the tangent to the curve given by
y = x
2 + 4x+ 4
at the point where x = 1.
MATH120 (2013)
TUTORIAL 4
Question 1. We want to build a closed rectangular box with square base. There
are 6 square meters of material available. What dimensions will result in a box
with the largest possible volume?
Question 2. Find and classify the stationary points of the functions f given by
(a) f(x) = x
4 + 2x3 + 4;
(b) f(x) = xe
x;
(c) f(x) =1
5(cos x)2.
Question 3. Find the maximum and the minimum of the function
f(x) = �(2x� 1)2 + 100
on the interval 0 x 5.
MATH120 (2013)
TUTORIAL 5
Question 1. Find the following indefinite integrals
(a)
Z(6x2 � 8x+
3
x
) dx, x 6= 0;
(b)
Z3px dx, x � 0.
Question 2. Find the following definite integrals
(a)
Z ⇡2
0
sin(2x) dx;
(b)
Z 1
0
e
�2xdx.
Question 3. The vertical speed of a rocket t seconds after lift–o↵ is t + 6t2
meters per second. How high is the rocket 2 minutes after lift–o↵?
Question 4. Find the area between the graphs of the functions given by y = x
2
and y = x+ 2.
Question 5. Find the area enclosed by the curve given by y = x
3 � x
2 and the
x-axis.
Question 6. Find the average of the function f(t) = sin t for ⇡ < t < 2⇡.
MATH120 (2013)
TUTORIAL 6
Question 1. Find the partial derivatives@z
@x
,@z
@y
,@
2z
@x
2,@
2z
@y
2and
@
2z
@x@y
for
(a) z(x, y) = 5x2 � 4y5 + 75,
(b) z(x, y) = xy
2 � xy + x
2,
(c) z(x, y) = y cos x.
Question 2. Find and classify the stationary points of the function given by
(a) z(x, y) = xy
2 � xy + x
2,
(b) z(x, y) = y cos x.
MATH120 (2013)
TUTORIAL 7s
Question 1. Suppose
A =
2
64�1 0
2 3
1 2
3
75 , B =
2
642 0 �1
3 1 2
0 �1 1
3
75 , C =
2
644 1 3
0 5 11
1 1 13
3
75 and X =
"5
2
#.
(a) Write down the size of each of these matrices.
(b) Find the entries a13, b31 and c22.
(c) Using your answer to (a), determine which of the expressions A+B, 2B�C,
AB, BA and AX are defined. Calculate the ones that are defined.
(d) Calculate BC and CB. Are they equal to each other?
Question 2. A certain style of shirt comes in small, medium and large. A
company has two stores and in 2010 the sales figures for these shirts at the two
stores were
Small Medium Large
Store 1 201 402 236
Store 2 310 632 471
.
The corresponding sales figures for 2011 are given by
Small Medium Large
Store 1 256 410 198
Store 2 241 574 332
.
The profit per shirt sold is $8 for the small ones, $7 for the medium ones and $6
for the large ones.
Let A be the matrix representing the sales data for 2010 and B the matrix
representing the sales data for 2011.
(a) Write down a matrix expression for the total sales over 2010 and 2011.
(b) Write down and evaluate a matrix expression for the average sales over the
two years.
(c) Write down and evaluate a matrix expression for the profit made at each
store in 2010.
Question 3. Suppose
A =
"1 1
2 0
#.
(a) Find the determinant, detA, of the matrix A.
(b) Decide whether A is invertible. If it is invertible, find A
�1.
MATH120 (2013)
TUTORIAL 8s
Question 1. Suppose
B =
2
643 1 0
0 4 1
1 0 0
3
75 .
(a) Show that B is invertible with B
�1 =
2
640 0 1
1 0 �3
�4 1 12
3
75.
(b) Suppose X is a three dimensional vector with BX =
2
641
1
4
3
75. Find X.
(c) Calculate B
2.
Question 2. A species lives for 4 months and the birth and survival rates are
given by the Leslie matrix L and the initial population at the beginning of August
2012 is given by the vector P0, where
L =
2
6664
0 30 30 40
0.9 0 0 0
0 0.5 0 0
0 0 0.5 0
3
7775and P0 =
2
6664
1000
9
5
5
3
7775.
(a) What are the birth rates for the animals aged 0 � 1 months, 1 � 2 months
and 2� 3 months?
(b) What are the survival rates for the animals aged 0� 1 months, 1� 2 months
and 2� 3 months?
(c) Find L
2 and use your answer to calculate the population vector P2 at the
beginning of October 2012.
(d) Show that the inverse of L is given by
M =
2
6664
0 109 0 0
0 0 2 0
0 0 0 2140 0 �3
2 �32
3
7775
and use M to find the population vector P�1 at the beginning of July 2012.
MATH120 (2013)
TUTORIAL 9s
Question 1. Show that f(t) = 2e3t3 � t
2 solves the di↵erential equation
y
0 � 9t2y = 9t4 � 2t.
Question 2. Show that the function f(t) = (e2t + 1)�1 satisfies
y
0 � 2y2 = �2y and y(0) =1
2.
Question 3. Use separation of variables to solve the following di↵erential equa-
tions:
(a)dy
dx
=3x2 � 1
y
2,
(b)
y
0 = y
2 � e
2ty
2,
(c)
3y0 + y = 2, y(0) = 5.
MATH120 (2013)
TUTORIAL 10s
Question 1. The probability of an atom of carbon-14 decaying within one year
is 0.00012. What is the half–life of carbon-14? What proportion of a sample will
be left after 10, 000 years?
Question 2. The size of a population satisfies the di↵erential equation
dy
dt
= 0.1(1000� y),
where y(t) is the number of organisms at time t measured in minutes.
If the initial population size is 100, when will the populations size reach 300, and
what is the population size after 10 minutes?
Question 3. A rabbit population satisfies the logistic equation
dy
dt
= 3⇥ 10�7y (105 � y),
where t is time measured in months and y is the number of individuals at time t.
Myxamatosis suddenly reduces the population to 30% of its steady state size.
If the myxamatosis then has no further e↵ect, how long will it take for the popu-
lation to build up to 80% of its steady state size?
16 Assignment Problems
MATH120 (2013) (Date Due: 4th March 2013)
ASSIGNMENT 0
Question 1. Write the following as one fraction and cancel common factors in
the numerator and denominator
(a)3
8+
1
10; (b)
3
8� 1
10; (c)
3
8⇥ 1
10; (d)
3
8÷ 1
10.
Question 2. Expand the following expressions
(a) x(x2 + 2); (b) (x� 2)(x+ 3); (c) (2x+ 1)2.
Question 3. Factorise the following expressions
(a) 6x+ 8x3; (b) mn+ 4n; (c) 4a2 � 9.
Question 4. Cancel common factors in the numerator and denominator
(a)mn
2
3m3n
; (b)mn+ 4n
4n.
Question 5. Find all real numbers x such that
(a) x+ 2 = 0; (b) (x+ 2)(x� 3) = 0; (c)px = 3; (d) x2 = 16.
Question 6. Evaluate the following expressions
(a)
r1
16; (b) (27)
13 ; (c) a
32 ⇥ a
12 ; (d) b
32 ÷ b
12 .
Question 7. For a cube with side length 4 cm find
(a) the volume; (b) the surface area.
MATH120 (2013) (Date Due: 11th March)
ASSIGNMENT 1
Question 1. Sketch the graphs of the functions given by [4 marks]
(a) y = 2x� 3; (b) y = �1
2x+ 4; (c) y = 2x2; (d) y = 2x2 � 8.
For each graph find the points of intersection with the x-axis and the y-axis.
Question 2. Find an equation for the straight line with slope �2 through the
point with coordinates (1, 3). [4 marks]
Question 2. Sketch the graphs of the function given by
(a) y = �4x2 + 7x+ 2; [3 marks]
(b) y = x
2 � x� 1. [3 marks]
In each case find the maximum or minimum value of the function and its zeros,
that is, all x for which y = 0.
Question 3. Sketch the graphs of the functions given by
(a) f1(x) = �3 cosx; [2 marks]
(b) f2(x) = �3 cos(2x); [2 marks]
(c) f3(x) = �3 cos(2x� ⇡
2 ). [2 marks]
between �2⇡ and 2⇡.
MATH120 (2013) (Date Due: 11th March)
ASSIGNMENT 2
Question 1. Solve the following equations for x
(a) 15x+ 8 = 53; [1 mark]
(b) x
� 14 = 2; [1 mark]
(c) 3e5x�2 = 3; [1 mark]
(d)1
2ln(x� 3) = 4; [1 mark]
(e) |x| = 100; [1 mark]
(f) |x� 1| = 100. [1 mark]
Question 2. Consider the two curves given by
y = x
2 � 2x and y = 2x� 16.
Decide whether the two curves intersect. If they do intersect, find their points of
intersection. [4 marks]
Question 3. Assuming only that ln 3 is approximately 1.0999 and ln 4 approx-
imately 1.386, calculate an approximation for
(a) ln 6; [2marks]
(b) ln9
16. [2 marks]
Question 4. Find the inverses of the functions given by
(a) y = 2x� 10; [3marks]
(b) y = 3ex + 5. [3 marks]
For each function sketch the graph of the function and that of its inverse on one
diagram.
MATH120 (2013) (Date Due: 25th March)
ASSIGNMENT 3
Question 1. Find the derivatives of the functions given by
(a) y(x) = x
19 + 2px+ 3, x > 0; [2 marks]
(b) h(x) =7
6x4� 1
x
, x 6= 0; [2 marks]
(c) f(x) =1
5cos(x2); [2 marks]
(d) g(x) = e
3x cos x; [2 marks]
(e) y(t) =x+ 1
x� 1, x 6= 1; [2 marks]
(f) y(t) = ln(5t5 � t+ 6). [2 marks]
Question 2. Find the third derivative of the function f given by [3 marks]
f(x) = 3x4 + 179 + 2 ln x� 2x�1, x > 0.
Question 3. Find the slope of the tangent to the curve given by y = sin x at
x = ⇡
2 . [2 marks]
Question 4. Find the equation of the tangent to the curve given by
y = x
3 � 3
at the point where x = �1. [3 marks].
MATH120 (2013) (Date Due: 1st April)
ASSIGNMENT 4
Question 1. We want to build a 600 square metre rectangular enclosure, three
sides of which will be built of wooden fencing at a cost of $14 per metre. The
remaining side will be made of cement blocks at $28 per metre. How will we set
the dimensions of the enclosure in order to minimize the cost of materials?
[5 marks]
Question 2. Find and classify the stationary points of the functions given by
(a) g(x) = x
2 � 3x4; [5 marks]
(b) f(x) = (x2 � 1)ex. [5 marks]
Question 3. Find the absolute maximum and the absolute minimum of the
function
f(x) = x
3 � 3x
on the interval �3 x 1. [5 marks]
MATH120 (2013) (Date Due: 8th April)
ASSIGNMENT 5
Question 1. Find the following indefinite integrals
(a)
Z(x200 + 199�
px
9 +1
x
3)dx; [2 marks]
(b)
Z1
4cos⇣x
12
⌘dx. [2 marks]
Question 2. [4 marks]
Find the following definite integrals
(a)
Z �1
�e
dx
2x; [3 marks]
(b)
Z⇡
�⇡
2 sin(x
2)dx. [3 marks]
Question 3. [3 marks]
Find the area between the graphs of the functions given by y = x
3 + x
2 � 1 and
y = x
3 + 3.
Question 5. [3 marks]
The size of a population grows according to the formula P (t) = 3000e0.2t, where
t is measured in months. What is the average of the population in the first ten
months?
MATH120 (2013) (Date Due: 29th April)
ASSIGNMENT 6
Question 1. Consider the function of two variables given by
z(x, y) = 2x2 + y
3 � x� 12y + 7.
(a) Find the partial derivatives@z
@x
,@z
@y
,@
2z
@x
2,@
2z
@y
2and
@
2z
@x@y
. [4 marks]
(b) Find the stationary points of this function. [3 marks]
(c) Use the second-derivative test to classify the stationary points. [3 marks]
Question 2. Consider the function of two variables given by
z(x, y) = x sin y.
(a) Find the partial derivatives@z
@x
,@z
@y
,@
2z
@x
2,@
2z
@y
2and
@
2z
@x@y
. [4 marks]
(b) Find the stationary points of this function. [3 marks]
(c) Use the second-derivative test to classify the stationary points. [3 marks]
MATH120 (2013) (Date Due: 6th May)
ASSIGNMENT 7s
Question 1. Suppose
A =
"1 �2 3
0 3 �4
#, B =
"2 0
1 �3
#, C =
"1 3
0 11
#and X =
2
642
5
�1
3
75 .
(a) Write down the size of each of these matrices. [1mark]
(b) Find the entries a13, b21 and c22. [1 mark]
(c) Using your answer to (a), determine which of the expressions A+B, 2B�C,
AB, BA and AX are defined. Calculate the ones that are defined. [3 marks]
(d) Calculate BC and CB. Are they equal to each other? [3 marks]
Question 2. The sales figures for two pet stores in July were
Kittens Puppies Parrots
Store 1 6 4 12
Store 2 3 5 2
.
The corresponding sales figures for August are given by
Kittens Puppies Parrots
Store 1 8 7 10
Store 2 3 0 3
.
The value of each kitten is $55, each puppy is $ 150, and each parrot is $35.
Let A be the matrix representing the sales data for July and B the matrix
representing the sales data for August.
(a) Write down and evaluate a matrix expression for the total sale quantities over
July and August. [2 marks]
(b) Write down and evaluate a matrix expression for the average sale quantities
over the two months. [2 marks]
(c) Write down and evaluate a matrix expression for the total value of the pets
sold at each store in August. [2 marks]
Question 3. Suppose
A =
"1001 1
2001 2
#, B =
"�1 3
�2 5
#.
(a) Find the determinants, detA, detB of the matricis A and B.
(b) Decide whether A and B are invertible. If they are invertible, find A
�1 and
B
�1.
MATH120 (2013) (Date Due: 6th May)
ASSIGNMENT 7e
Question 1. [5 marks]
A firm produces three sizes of recording tapes in two di↵erent qualities. The
production (in thousands) at its Melbourne plant is given by the following matrix.
Size 1 Size 2 Size 3
Quality 1 x1 x2 x3
Quality 2 y1 y2 y3
!
The production (in thousands) at its Sydney plant is given by the matrix:
Size 1 Size 2 Size 3
Quality 1 u1 u2 u3
Quality 2 v1 v2 v3
!
(a) Write a matrix that represents the total production of recording tapes at
both plants.
(b) The firm’s management is planning to buy an existing plant at Armidale
as its third plant. It is known that the Armidale plant has one third of
the capacity of the Melbourne plant, but has fifth of the capacity of the
Sydney plant. Find the relationship between the matrices representing the
productions in Melbourne and Sydney respectively.
(c) Find the total production of all the three plants if the Armidale plant has
the production level given by
Size 1 Size 2 Size 3
Quality 1 9 8 11
Quality 2 10 11 8
!
Question 2. [5 marks]
Determine whether the following products of matrices are defined and find the sizes
of the resulting matrices where the product is defined.
(a) "2 1 4 2
5 3 6 3
#2
641 0 2 4
3 �1 0 1
0 2 1 3
3
75
(b)2
641 0 2
0 2 �1
3 1 0
3
75
2
642 �1
1 0
0 3
3
75
"0 1 �2
3 0 1
#
Question 3. [5 marks]
Express the following systems of linear equations in matrix form.
(a) 3x+ 8y + z = 9
x� 5y � 6z = �7
(b) 2x� y = 1
3y + 4z = 3
5z + x = 5
Question 4. [5 marks]
The interaction between two industries P and Q that form a hypothetical economy
is given in the following diagram:
Industry P Industry Q Consumer Demands Total Output
Industry P 46 342 72 460
Industry Q 322 114 134 570
Labour Inputs 92 114
(a) Determine the input-output matrix A.
(b) Find the matrix equation satisfied by the output matrix if the consumer
demands change to 129 for P and 213 for Q. (You are not required to solve the
equation.)
Question 5. [5 marks]
A mining Company has two mines, P and Q. Each tonne of ore from the mine P
yields 26 kilograms of copper, 2 kilograms of zinc, and 0.5 kilogram of molybdenum.
Each tonne of ore from Q yields 12.5 kilograms of copper, 5 kilograms of zinc, and
1.5 kilograms of molybdenum. The company must produce at least 88,500, 18,000,
and 5000 kilograms per week of these three metals, respectively. If it costs $550
per tonne to obtain ore from P and $670 per tonne from Q, express this as a
Linear Programming problem for the manager of the company who wants to know
how much ore should be obtained from each mine in order to meet the production
requirements at minimum cost.
MATH120 (2013) (Date Due: 13th May)
ASSIGNMENT 8s
Question 1. Suppose
B =
2
641 0 �2
4 �2 1
1 2 �10
3
75 .
(a) Show that B is invertible with B
�1 =
2
64�9 2 2
�412 4 9
2
�5 1 1
3
75. [3 marks]
(b) Suppose X is a three dimensional vector with BX =
2
641
2
�1
3
75. Find X.
[4 marks]
(c) Calculate B
2. [3 marks]
Question 2. A species lives for 4 months and the birth and survival rates are
given by the Leslie matrix L. The initial population at the beginning of August
2012 is given by the vector P0, where
L =
2
6664
0 40 40 30
0.5 0 0 0
0 0.9 0 0
0 0 0.5 0
3
7775and P0 =
2
6664
1500
15
9
5
3
7775.
[10 marks]
(a) What are the birth rates for the animals aged 1�2 months and 3�4 months?
(b) What are the survival rates for the animals aged 0 � 1 months and 1 � 2
months?
(c) What is the number of animals aged 1�2 months at the beginning of Septem-
ber?
(d) Show that the inverse of L is given by M =
2
6664
0 2 0 0
0 0 109 0
0 0 0 2130 0 �40
27 �83
3
7775. Find the
population vector P�1 at the beginning of July 2012. How many animals
were born in July?
MATH120 (2013) (Date Due: 20th May)
ASSIGNMENT 8e
Question 1. A sum of $1,000 is invested at a nominal interest rate of 12%.
Calculate the capital
(a) after 1 year if compounding is monthly; [1 mark]
(b) after 1 year if compounding is quarterly; [1 mark]
(c) after 2 years if compounding occurs every 6 months and [1 mark]
(d) after 2 years with quarterly compounding. [1 mark]
Question 2. Which option is better for an investor, that is, which e↵ective
interest rate is bihigher, six-monthly compounding with a nominal rate of 12.5%
or a monthly compounding with a nominal rate of 12.2%? [4 marks]
Question 3. A machine is purchased for $25,000. The depreciation is calculated
on the diminishing value at 10% for the first 5 years and at 8% for the next 5 years.
Find the value of the machine after a period of 10 years. [4 marks]
Question 4. A loan of $16,000 is to be repaid by regular monthly installments
over 20 months. If the interest rate is 1% per month, what are the monthly pay-
ments? [4 marks]
Question 5.
(a) The demand for a certain product is given by the equation p
2 + x
2 = 2500,
where x units can be sold at a price of p dollars each. Determine the marginal
demand at a price level of 40 dollars. [2 marks]
(b) The demand equation of a certain product is p = 3000e�x/20, where x units
are sold at a price of p dollars each. If the manufacturer has a fixed cost
of $500 and the variable cost of $20 per unit, find the marginal revenue and
marginal profit functions. [2 marks]
MATH120 (2013) (Date Due: 20th May)
ASSIGNMENT 9s
Question 1. Show that the function given by f(t) = 32e
t
2 � 12 is a solution of
the di↵erential equation
y
0 � 2ty = t.
[4 marks]
Question 2. Show that the function given by f(t) = (e�t + 1)�1 satisfies
y
0 + y
2 = y and y(0) =1
2.
[4 marks]
Question 3. Use separation of variables to solve the following di↵erential equa-
tions.
(a)dy
dx
=5� x
y
2; [4 marks]
(b) y0 = y
2 � e
3ty
2, y(0) = 1 and [4 marks]
(c) y0 + y = 1, y(0) = 2. [4 marks]
MATH120 (2013) (Date Due: 27th May)
ASSIGNMENT 10s
Question 1. Some species of bacteria follows an exponential growth pattern
with rate k = 0.02 (per hour).
(a) How long will it take for the population to double? [3 marks]
(b) If at the start of an experiment there are 100 bacteria, what will the popula-
tion be after 5 hours? [3 marks]
Question 2. A rabbit population satisfies the logistic equation
dy
dt
= 2⇥ 10�7y(106 � y),
where t is the time measured in months. The population is suddenly reduced to
40% of its steady state size by myxamatosis.
(a) If the myxamatosis then has no further e↵ect, how large is the population 8
months later? [3 marks]
(b) How long will it take for the population to build up again to 90% of its steady
state size? [3 marks]
Question 3. In ecology, the logistic equation is often written in the form
dN
dt
= rN(1� N
K
),
where N = N(t) stands for the size of the population at time t, the constants r
and K stand for the intrinsic growth rate and the carrying capacity of the species,
respectively.
A pond on a fish farm has carrying capacity of 1000 fish, intrinsic growth rate
0.3 (when time is measured in months) and is originally stocked with 120 fish.
(a) Set up a logistic equation for the fish population N(t) in the pond, with t
measured in months. [2 marks]
(b) Find the size of the population when t = 10. [3 marks]
(c) Can the population reach 1000 at any future time? [3 marks]
