After studying this unit you should be able to:

• create simple models, given a clear statement of the problem;

• write down the simplifying assumptions that underpin a model;

• identify the key variables and the parameters of a model;

• apply the input–output principle to obtain a mathematical model, where appropiate;

• obtain mathematical relationships between variables, based on or linking back to the simplifying assumptions
Author(s): The Open University

The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook.

Click 'View document' to open the workbook (PDF, 0.2 MB).

This unit provides an overview of the processes involved in developing models. It starts by explaining how to specify the purpose of the model and moves on to look at aspects involved in creating models, such as simplifying problems, choosing variables and parameters, formulating relationships and finding solutions. You will also look at interpreting results and evaluating models.

This unit, the third in a series of five, builds on the ideas introduced and developed in Modelling poll
Author(s): The Open University

Acknowledgements

The content acknowledged below is Proprietary (see terms and conditions) and is used under licence.

## Unit Image

All other materials included in thi
Author(s): The Open University

Introduction

In unit MSXR209_1 you saw how some of the stages of a mathematical modelling process can be applied in the context of modelling pollution in the Great Lakes. In this unit you are asked to relate the stages of the mathematical modelling process to another practical example, this time modelling the skid marks caused by vehicle tyres. By considering the example you should be able to draw out and clarify your ideas of mathematical modelling.

This unit, the second in a series of five, builds
Author(s): The Open University

Acknowledgements

The content acknowledged below is Proprietary (see terms and conditions) and is used under licence.

All materials included in this unit are derived from content originated at the Open University.

Author(s): The Open University

Introduction

This unit explores a real-world system – the Great Lakes – where mathematical modelling has been used to understand what is happening and to predict what will happen if changes are made. The system concerned is extremely complex but, by keeping things as simple as possible, sufficient information will be extracted to allow a mathematical model of the system to be obtained.

This unit is an adapted extract from the course Author(s): The Open University

5.3 Powers

Having discussed nth roots, we are now in a position to define the expression ax, where a is positive and x is a rational power (or exponent).

## Definition

If a > 0, m Author(s): The Open University

5.2 Existence of roots

Just as we usually take for granted the basic arithmetical operations with real numbers, so we usually assume that, given any positive real number a, there is a unique positive real number b =  such that b2 = a. We now discuss the justification
Author(s): The Open University

5.1 Arithmetic with real numbers

At the end of Section 1, we discussed the decimals and asked whether it is possible to add and multiply these numbers to obtain another real number. We now explain how this can be done using the Least Upper Bound Property of Author(s): The Open University

4.4 Further exercises

## Exercise 29

In this exercise, take

Author(s): The Open University

In the examples just given, it was straightforward to guess the values of sup E and inf E. Sometimes, however, this is not the case. For example, if then it can be shown that E is bounded above by 3, but it is not so easy to guess the least upper bound of E.

In such cases, it i
Author(s): The Open University

We have seen that the set [0, 2) has no maximum element. However, [0, 2) has many upper bounds, for example, 2, 3, 3.5 and 157.1. Among all these upper bounds, the number 2 is the least upper bound because any number less than 2 is not an upper bound of [0, 2).

Author(s): The Open University

In this section we show you how to prove inequalities of various types. We use the rules for rearranging inequalities given in Section 2, and also other rules which enable us to deduce ‘new inequalities from old’. We met the first such rule in Author(s): The Open University

Now we consider inequalities involving the modulus of a real number. Recall that if a , then its modulus, or abso
Author(s): The Open University

1.1 Rational numbers

The set of natural numbers is the set of integers is and the set of rational numbers is Author(s): The Open University

10 Conclusion

This unit has introduced you to some aspects of using a scientific or graphics calculator. However, in many ways, it has only scratched the surface. Hopefully your calculator will be your friend throughout your study of mathematics and beyond. Like any friend, you will get to know it better and appreciate its advantages as you become more familiar with it. Don't expect to know everything at the beginning. You may find the instruction booklet, or other help facility, a bit hard going to begin
Author(s): The Open University

9 When to use the calculator

Despite the list of advantages given, here is a word of warning: a calculator is not a substitute for a brain! Even when you are using your calculator, you will still need to sort out what calculation to do to get the answer to a particular problem. However skilled you are at using your calculator, if you do the wrong sum, you will get the wrong answer. The phrase ‘garbage in, garbage out’ applies just as much to calculators as to computers. Your calculator is just that – a calculator!<
Author(s): The Open University

Ease of use

Most aspects of the calculator are straightforward to use. Calculations are entered on the screen in the same order as you would write them down. More complicated mathematical functions and features are also reasonably intuitive, and there are ‘escape’ mechanisms, so that you can explore without worrying about how you will get back to where you were.

Author(s): The Open University

8.2.6 Information

The calculator will give you information about any number that you have entered: for example, its square or cube, its square root or cube root. It will also give you information about a whole list of numbers: for example, the mean (average) or the highest value in the list.

Author(s): The Open University