Learning outcomes 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
1 Developing modelling skills 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). Introduction 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 Acknowledgements The content acknowledged below is Proprietary (see terms and conditions) and is used under licence. Wade_In_Tulsa, photos All other materials included in thi 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 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. 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): 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). If a > 0, m 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 = 5.1 Arithmetic with real numbers At the end of Section 1, we discussed the decimals
4.4 Further exercises In this exercise, take
4.3 Least Upper Bound Property 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
In such cases, it i 4.2 Least upper and greatest lower bounds 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).
3 Proving inequalities 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): 2.3 Inequalities involving modulus signs Now we consider inequalities involving the modulus of a real number. Recall that if a 1.1 Rational numbers The set of natural numbers is
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 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!< 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. 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.
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Definition
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such that b2 = a. We now discuss the justification
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):
Exercise 29
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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.
, then its modulus, or abso
the set of integers is
and the set of rational numbers is
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