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Lecture 1 Introducing QABE School of Economics, UNSW

Contents 1 Introduction 1 1.1 Tuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Assessment

4

3 Miscellany 4 3.1 On the lecture materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 This is NOT the Course Outline . . . . . . . . . . . . . . . . . . . . . . . 5 3.3 Studying at university . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4 Functions of One Variable

6

5 What is a Function? 7 5.1 Some Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.2 Functions are not functions! . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.3 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 6 Functions 8 6.1 Common Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 6.2 Combining Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 6.3 Composite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 7 Special Functions 11 7.1 Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 7.2 Exponential & Logarithmic . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1

Introduction

Welcome to Quantitative Analysis for Business and Economics (QABE). QABE deals with the fundamentals of mathematics for business and economics. It replaces the old course Quantitative Methods A, and reflects our goal of continual course improvement. For those of you who have studied QMA in the past, there are similarities but also differences between the two courses. You may be wondering if all the material taught in QABE will be applicable to you. The short answer is that all of it will be applicable, but not always immediately. That is, the course material has been chosen to reflect the core mathematical skills that you will need for further study in the quantitative courses, particularly those taught by the

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ECON 1202/ECON 2291: QABE

c School of Economics, UNSW

School of Economics. We have students studying marketing and hospitality, through to economics and econometrics. The spectrum of disciplines requires a range of mathematical tools. For this reason, we will try to point to applications of our mathematics for your further studies wherever possible, drawing on business and economic scenarios and problems to bring out the relevance of the techniques we’ll be learning. Finally, so that we can keep this course improving all the time, we’d appreciate your feedback. So if you have an idea for an improvement, please send an email to the lecturer-in-charge to let us know. All the best with your studies. We hope that you enjoy the course! Agenda 1. Introductions 2. How do I learn in QABE? 3. Assessment 4. Futher help 5. A few notes on studying at university 6. Functions of one variable Introductions 1. Who is your lecturer? 2. Who is the lecturer-in-charge? 3. Who is your tutor?

1.1

Tuition

lectures (2hr per week) Introduce and emphasise key points from the course, see worked examples, ask one or two questions; prepare by reading the lecture notes, reading over reference chapters; tutorials (1hr per week from week 2) Core place of learning, developing understanding, making mistakes, asking many questions; consultation (3hrs per week) Clarifying lecture material, discussing course-program related issues (LIC), focussed tuition.

1.2

Materials

online Go to https://moodle.telt.unsw.edu.au

Lecture 1

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ECON 1202/ECON 2291: QABE

textbooks

c School of Economics, UNSW

1. (required) Haeussler, Paul & Wood, ‘Introductory Mathematical Anal-

2. (strongly recommended) Knox, Zima & Brown, ‘Mathematics of Finance’, McGraw-Hill Book Company, 2nd Edition, 1999. (KZB); 3. (other) see course outline.

More help? pass (many hrs per week from week 3) “Peer Assistance Support Scheme”: Peer assisted study groups, run by second and third year students; education development unit (edu) (UNSW Business School) Learning and language

the learning centre (UNSW) Free and confidential learning support for students;

Is this for you?

Assumed knowledge A level of knowledge equivalent to achieving a mark or at least 60 in HSC Mathematics. Students who have taken General Mathematics will not have achieved the level of knowledge which is assumed for this course. From the (intro) Calculus lectures... d p ax x e dx d p 2x = e +x dx Z b = k(1 − ex ) dx a Z   3 x −2 5e − x + = dx x

ya = yb yc yd

(x 6= 0).

—————————————————————Refresher Resources See revision text in the Reserve section of the library (also available at the UNSW bookshop): • Managing Mathematics: A Refresher Course for Economics and Commerce Students by Judith Watson, 2nd Edition, 2002.

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2

Assessment

Marks Break-down • Passing the course requires 50% or higher! • See Course-outline for further information. Item Online quizzes (4 × 2%) In-tutorial tests (3 × 10%) Tutorial participation Final exam Total

3

% 8 30 5 57 100

Miscellany

3.1

On the lecture materials

Using lecture resources • In the notes – look for chapter references in the margins; • At the end look for key words of interest that you should revise; • Note special text like, Definition | The fundamental theorem of first-year The amount of work w undertaken by a student is inversely related to the difference between the total session time T and time elapsed in the session t, w(t) ∝

1 T −t

(1)

—————————————————————• Examples appear in the notes with a box for working, Example: The world is experiencing exponential growth in population, but declining economic stocks of energy, fresh water and food. Solve.

• Or words of caution to make sure you don’t fall into common traps, Lecture 1

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The Fundamental Theorem of First-year is fundamental for a reason. here

3.2

This is NOT the Course Outline

—————————————————————Read the Course Outline!

Check (and re-check) the course-outline for information provided and more (see list below). • Special consideration (e.g. illness); • Student misconduct and plagiarism policy; • Contact details of key people; • Syllabus – what we’ll be studying, with chapter references.

3.3

Studying at university

—————————————————————Some advice 1. Attend classes – the ‘turning up’ philosophy to education; 2. Use a diary/palm-pilot/organiser/calendar/scraps-of-paper, write in assessments, put down reminders; 3. Make a habit of opening the textbook and reading it weekly; 4. Ask questions – lots of them; 5. Introduce yourself to someone else in a tutorial or lab. —————————————————————And finally... you

Lecture 1

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ECON 1202/ECON 2291: QABE

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• Proportion of world’s population living in a developed country = 18.7%; • Proportion of Australians who have attained a tertiary degree (or higher) = 17.8%; • Proportion of world’s population living in a developed country AND in tertiary education, 0.187 × 0.178 = 3.3% some useful tags

¡chap ref¿

Example: ¡title¿ ¡problem¿

Definition | ¡title¿ ¡content¿

¡title¿ ¡content¿

4

Functions of One Variable

To begin with, we will go back over some fundamental concepts and terminology of the vast world of functions. Following which, we will meet some particular kinds of functions that will keep cropping up in this course, and most likely, in the rest of your studies. Note: Notation We will use a particular notational convention for functions, such in the following example: f (x) = x2 + 4, It should be noted that there is nothing special about ‘f ’ (or ‘x’ for that matter), they are just labels. As we note below, we could just have correctly chosen to name all of our ‘representative’ functions as ‘blah’ with input ‘words’, (giving say, blah(words), which would be read, ‘the function blah of words’). However, this might get confusing, and so we will follow the very established convention of using the function title f (x), or perhaps y(x).

—————————————————————Agenda 1. Function review; 2. Special functions; 3. Exponential and Logarithms; 4. Limits. Lecture 1

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5

What is a Function?

—————————————————————-

HPW 2.1

Definition | Function A function is a rule that assigns to each input number exactly one output number. —————————————————————Example: A linear function Consider what is meant by the simple linear function f(x) = 1 + 0.5x.

5.1

Some Definitions

============================================================ 1. The name of the function is irrelevant. Consider, f ish(shrimp) =

shrimp − 1 ; 2

... still a valid function! 2. Often we talk in terms of dependent and independent variables, or alternatively, in terms of the value and argument respectively: Example: Dependent, Independent Identify the dependent and independent terms, and the value and argument of the function H(a, b) = a2 + 2b + 3.

5.2

Functions are not functions!

============================================================

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1. Functions are part of a broader class called relations. Functions are the special case – they give one output value for a given input value. 2. For this reason, they are also called a mapping, or a transformation.

One of these is a function, one isn’t! f (x) = x2

f (x) =

x

5.3

√

x

x

Domain and Range

============================================================

Definition | Domain and Range The domain of a function is the set of all x values over which the function ‘makes sense’ (works!). The range of a function, is the set of all possible f(x) values, given the domain. Example: Find the domain of the function, y(x) =

6 6.1

2 x2 +3x−4 .

Functions Common Functions

============================================================ HPW 2.2

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Definition | The Constant Function A constant function is of the form: f(x) = c where c is a constant. f (x) = 2

x

Definition | The Polynomial Function A polynomial function is of the form: f(x) = cn xn + cn−1 xn−1 + · · · + c1 x1 + c0 where cn . . . c0 are constants. f (x) = x2

x

========================================================== Definition | The Rational Function A rational function is of the form: f(x) =

p1 (x) p2 (x)

where p1 and p2 are polynomial functions. f (x) =

x2 −6 x+6

x

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Definition | The Absolute Function An absolute value function is of the form: f(x) = |g(x)| where g(x) is some function and | · | indicates ‘positive value’. √ f (x) = | x|

x

6.2

Combining Functions

============================================================ HPW 2.3 Suppose we have two functions, f(x) = 3x + 2, and p(x) = x3 − 3, then we will be interf (x) ested to solve: f(x) + p(x), or f(x) − p(x), or f(x) × p(x), or even p(x) . Definition | Function Combination In general, we have, sum (f + g)(x) = f (x) + g (x) , difference (f − g)(x) = f (x) − g (x) , product (fg)(x) = f (x) · g (x) , f(x) quotient ( fg )(x) for g(x) 6= 0 . = g(x)

========================================================== Example: Combining functions Suppose f(x) = 2x2 − 3x − 2 and g(x) = x − 2, and let h(x) = show that (h − g)(x) = x + 3.

6.3

f g (x),

then

Composite Functions

============================================================ Now suppose we don’t want a combination, but we want to construct a process of more than one function, Lecture 1

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ECON 1202/ECON 2291: QABE

x

f(x)

y

g(y)

x

h(x)

z

z

that is,

Definition | Composite Function If f and g are functions, the composite function of g and f is the function g ◦ f, (g ◦ f)(x) = g (f (x)) , and the domain of g ◦ f is the values of x in the domain of f such that f(x) is in the domain of g . ========================================================== Example: Composite functions√ Let p(x) = x2 − 2, and h(x) = 5x + 1 (for x ≥ 0). Find (p ◦ h)(2).

7 7.1

Special Functions Inverse

============================================================ HPW 2.4 Now suppose that instead of, x

f(x)

y

we want to go back the other way, that is, x

?

y

or in other words, if f(x) = y , then what we are after is the function, f −1 (y) = x . where f −1 is the inverse function of f . ==========================================================

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Example: Suppose f(x) =

x 2

+ 1, find f −1 .

==========================================================

A function may not an inverse function Example: Suppose f(x) =

x2 +1 5

, find f −1 (x).

========================================================== Let’s try that out, suppose x = 2: f (x) = f (2) =

(2)2 + 1 =1 5

... and the other way around, f −1 (x) = f −1 (1) =

p

(5)(1) − 1 = ±2

???!!! we received two answers back: +2, or −2

A function has an inverse if and only if it is a one-to-one function.

Definition | One-to-one Function A function is one-to-one if for all a and b, if a 6= b, then f(a) 6= f(b). Note: Inverse or reciprocal? You’ll have noted that the way that we represent the inverse of a function, f −1 (x) looks a lot like how we might represent the reciprocal of a number, x−1 . So the question is, ‘how do I know what is being talked about?’ The context will be Lecture 1

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most helpful, and the way that the inverse is presented should give some indication. For instance, to represent the reciprocal of the function f (x) we would normally write, [f (x)]−1 =

1 , f (x)

rather than when we just want the inverse we would write, f −1 (x) , see the difference? However, suppose you were confronted with, √ z(x) = x2 + 2 x + 4f −1 x , what would you understand this to mean? It is clearly a bit ambiguous, with ambiguity due to the lack of any indication of whether f is a function (which has been written in the equation without its input value), or if it is just another variable that is taken reciprocally ( f1 ). To avoid such ambiguity, good practice is always to make the inputs to functions very clear (write them in), unless there are many inputs, in which case, make it clear that you are just going to write the function name (‘for convenience, we shall write f (a, b, c, d, e, f ) as just f ’) and be sure not to confuse things in the expression.

7.2

Exponential & Logarithmic

We will have much more to say about exponential functions in coming weeks, since they HPW 4.1, provide an easy way to talk about various kinds of time-dependent processes. Be sure to 4.2 do a number of exercises in exponential and logarithmic functions since it is quite likely that either you have forgotten the rules associated with them, or are meeting them for the first time. It will be of great benefit if the rules and manipulation of these types of functions comes quickly to hand. ========================================================== Definition | Exponential f(x) = ax (A selection of ) Important rules: am an = am+n am = am−n an (am )n = amn f (x) = 2x

x

==========================================================

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Definition | Logarithmic Where b is the base, f(x) = logb x (A selection of ) Important rules: logb (mn) = log b m + logb n m logb ( ) = logb m − log b n n logb (mr ) = r log b m logb 1 = 0 f (x) = log10 x

x

Note: log b x is like saying, ‘what power must I raise b to, to obtain x?’ ========================================================== The connection between logarithms and exponentials... Definition | A very nice rule logb x = y

by = x

corresponds to

log

x

=

y

b

==========================================================

Revise!

1. Go over the lecture notes, chapter refs, tutorial problems, be sure you can do them(not just read them). 2. Go over the lecture notes, chapter refs, tutorial problems, be sure you can do them(not just read them). 3. Go over the lecture notes, chapter refs, tutorial problems, be sure you can do them(not just read them).

Lecture 1

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