Mathematics 101 Transitional Mathematics

Study Guide

MATH 101: Stepping Stones to Higher Mathematics

In this course, we begin with a fundamental observation that certain processes in our collective everyday world consist of intricate relationships between measurable but varying quantities. For example, in our physical world, the area ${\displaystyle A}$ inside a circular disk will change depending on its radius ${\displaystyle r}$. The question is: “How?” In the world of finance, the total value ${\displaystyle I}$ of an investment will vary according to the amount initially invested ${\displaystyle P}$, the compounding period of the investment ${\displaystyle n}$, and the interest rate ${\displaystyle r}$ promised by the financial institution. By using letters to talk about the quantities in question, we are already in the realm of algebra. Once there, we can model these relationships by expressing the primary entity of interest (dependent variable) as an algebraic expression involving the remaining measurable quantities (independent variables). For example, ${\displaystyle A=\pi r^2}$ where ${\displaystyle A}$ is the dependent variable and ${\displaystyle r}$ is the independent variable in the relationship between area of a circle and its radius (note that ${\displaystyle \pi }$ is a fixed real number and does not vary in value). In the second case, ${\displaystyle I=P{(1+r)}^n}$ where ${\displaystyle I}$ is the dependent variable, so-called because it is dependent on the values of ${\displaystyle P}$, ${\displaystyle n}$, and ${\displaystyle r}$, which are considered to be the independent variables. The values of the independent variables which make sense in the context are often called the domain values. The values of the dependent variable constitute the range values of the relationship. For example, none of ${\displaystyle P}$, ${\displaystyle n}$ or ${\displaystyle r}$ can be negative in value. The interest rate ${\displaystyle r}$ is generally not 0 (unless you don’t want to make any money); nor is it over 1 (unless you are the luckiest investor imaginable). Therefore, to be able to model mathematically the many and varied phenomena which arise in our complex world, we have to start at a very basic mathematical foundational level. Units 1 and 2 cover the necessary preparatory background for this course found in classical number theory, Euclidean geometry, and elementary algebra.

The notion of a function as a special kind of relation(ship) between two ‘real’ variables—one dependent, one independent (often labeled ${\displaystyle y}$ and ${\displaystyle x}$)—grew out of many such observations as being, more or less, the simplest of all such relationships. The qualification of a variable being ‘real’ means that the measurements are all to be real numbers, that is, expressible in decimal format. Unit 3, therefore, is an overview of functions of a single independent real-valued variable ${\displaystyle x}$ with a corresponding dependent real variable ${\displaystyle y}$, often denoted as ${\displaystyle y=f(x)}$. In this case, ${\displaystyle f(x)}$ is an abstracted way of denoting an expression in only one independent variable ${\displaystyle x}$. Using functional, instead of formula, notation, the area of a circle would then be expressed as ${\displaystyle A(r)=\pi r^2}$ and the total value of an investment would become ${\displaystyle I(P,n,r)=P{(1+r)}^n}$. In Unit 3, we also study certain important properties of functions, discover how they can be transformed, and consider different ways of working with them numerically (in practical terms), algebraically (in abstract theoretical terms), and geometrically (by visual representations).

Once we have become accustomed to the idea of a function in quite general terms, we then continue our examination of different classifications or families of functions, their distinguishing properties, how to view them graphically, and finally how they may be used to model certain relationships we encounter in our every-day lives, particularly as scientists or other career professionals. Units 4 through 8 discuss some of the types of functions which have been used extensively in modeling the real world around us. Unit 9 covers real-world problems which, in order to be described and solved, require a whole system of equations or inequalities. Unit 10 is an overview of a class of geometric relationships which are not necessarily expressible as functions per se but have been used to model a wide array of our real-world applications from the design of the lenses of microscopes and telescopes, to medical procedures for treating kidney stones (lithotripsy), to predicting the paths of satellites in space, to the construction of roller coasters, billiard tables, and awe-inspiring architectural forms throughout the world.

In short, we may think of MATH 101 as an introductory course in mathematical modeling.

Approaches to this Course: Customized to Suit Your Preferred Learning Style

There are several different ways to study using both the text readings and the interactive online tutorials. Pick the one which suits your learning style best. To find out your preferred learning style, if you don’t already know it, check out the AU Math Centre Learning Styles page. In any case, do what is natural for you to do.

For Active/Sensing/Verbal/Sequential learners:

Read the primary textbook readings and work through the given problem exercises. Then review the concepts by reading/watching the associated online tutorials, finishing by trying some of the online Maple TA exercise sets just to see how well you have learned the topic.

For Reflective/Sensing/Visual /Sequential learners:

Get an overview of the topic by watching the online tutorials. Then go section by section through the textbook readings, doing the assigned exercises after each section. Review by working through some of Maple TA online practice exercises.

For Reflective/Intuitive/Verbal/Global learners:

Read the primary text book readings. Reinforce the concepts by watching the online tutorials. Work through the textbook exercises. Review by working through some of Maple TA online practice exercises.

For Reflective/Intuitive/Visual/Global learners:

Get an overview of the topic by watching the online tutorials. Reinforce the concepts by reading the textbook sections, correlating them with the online tutorials. Work through the practice problems in the text. Review by working through some of Maple TA online practice exercises.

In what follows, the practice exercises are just that. Do as many as you think are necessary in order to understand the concept. Because answers are given only to the odd-numbered exercises in the textbook, only odd-numbered exercises are generally assigned with some exceptions. If you wish to know the answers to any of the others, please email your tutor.