MATH 1LS3 – TEXTBOOK NOTES
INTRODUCTION TO MODELS AND FUNCTIOS:
This chapter introduces the main tools needed to study biology using mathematics:
models and functions. A model is a collection of mathematical objects (such as functions
and equations) that allows us to interpret biological problems in math. Biological
phenomena are often described by measurements: a set of numeric values with units
(such as kilograms or meters). Many relations between measurements are described by
functions, which assign to each input value a unique output value.
After talking about what constitutes a math model and presenting a few examples, we
briefly review functions and their properties.
In the two sections of this chapter, we discuss the domain and range and the graph of a
function, algebraic operations with functions, composition of functions, and the
We build new functions from old using shifting, scaling, and reflections. We catalogue
important elementary functions and note their properties. We introduce the four
approaches algebraic, numeric, geometric, and verbal that we us throughout the book
to discuss functions, their properties, and their applications
0.1: MODELS IN LIFE SCIENCES
In this book, we use the language of mathematics to describe quantitatively how living
systems work and to develop the mathematical tools needed to compute how they change.
From measurements describing the initial state of a system and a set of rules describing
how change occurs, we will attempt to predict what will happen to the system.
For example, by knowing the initial amount of drug taken (caffeine, Tylenol, alcohol etc.)
and how it is processed by the liver (dynamical rules), we can predict how long the drug
will stay in the body and its effects.
To study a life sciences phenomenon using mathematics, we build a model. How does a
First, we identify a problem we need to study, or question we need to answer. Assume
that a virus (say H1N1) appears within a population. Will the virus spread?
How many people will get infected? How many will die? Will our hospitals have
adequate resources to treat increasing numbers of patients? These are just a handful of the
questions that we would like to know the answers to. Underlying all them is the basic
questions: we know (approximately) how many people are infected today. How many
will be infected tomorrow? In three days? I