1 Functions and modelling

Engineers use mathematics to a considerable extent. Mathematical techniques offer ways of handling mathematical models of an engineering problem and coming up with a solution. Of course it is possible to model a problem in ways that are not mathematical e.g. by physical or scale modelling, but this Workbook is concerned exclusively with mathematical modelling, so we will drop the word ‘mathematical’ and refer just to modelling. This Section is intended to introduce some modelling ideas as well as to show applications of the functions and techniques introduced in HELM booklet  2, HELM booklet  3 and HELM booklet  6. By modelling we mean the process by which we set up a mathematical model of a situation or of an assumed situation, use the model to make some predictions and then interpret the results in the original context. The mathematical techniques themselves contribute only to part of the modelling procedure. The modelling procedure can be regarded as a cycle. If we do not like the outcome for some reason we can try again. Five steps of a modelling cycle can be identified as follows:

Step 1 Specify the purpose of the model.

Step 2 Create the mathematical model after making and stating relevant assumptions.

Step 3 Do the resulting mathematics .

Step 4 Interpret the results .

Step 5 Evaluate the outcome, usually by comparing with reality and/or purpose and, if

necessary, try again!.

Much of this first Section is concerned with steps 2 and 3 of the cycle: creating a mathematical model and doing the maths . Engineering case studies found in many Workbooks will aim to demonstrate the complete cycle. An important part of step 2 may include choosing an appropriate function based on the assumptions made also as part of this step. This choice will influence the kind of mathematical activity that is involved in step 3.

So far in your engineering mathematical studies you might have had little opportunity to think about what is ‘appropriate’, since the type of function to be studied and used has been chosen for you. Sometimes, however, you may be faced with making appropriate choices of function for yourself so it is important to have some understanding of what might be appropriate in any given circumstance. A well chosen function will be appropriate in two different ways. Firstly the function should be consistent with the purpose of the model, with known data or theory or facts , and with known or assumed behaviours . For example, the purpose might be to predict the future behaviour of a quantity which is expected to increase with time. In this case time can be identified as the independent variable since the quantity depends on time. The function chosen for mathematical activity should be one in which the value of the dependent variable increases with time. Secondly, bearing in mind that the modelling process is a cycle and so it is possible, and usual, to go round it more than once , the first choice of function should be as simple as allowed by the modelling context. The main reason for doing this is to avoid complication unless it is really necessary. Philosophically, an initial choice of a simple function is consistent with the fundamental belief that most phenomena may be modelled adequately by simple laws and theories. It is common engineering practice always to use the simplest model possible in a given situation. So, for the first trip around the cycle, the appropriate function should be the simplest that is consistent with known facts, behaviours, theory or data. If the quantity of interest is known not to be constant, this might be a linear function. If the first choice turns out to be inadequate at the stage of the cycle where the result is interpreted or the outcome is evaluated (step 5) then it is reasonable to try something more complicated; a quadratic function might be the second choice if the first choice was linear.

It is important to realise that sophistication is not necessarily a virtue in itself. The merits of complication depend upon the purpose for which the model is being formulated. A model of the weather that enables a decision on whether or not to take an umbrella to work on any particular day will be rather less sophisticated than that required to give an accurate prediction of the amount of rainfall in the vicinity of the workplace on that day.

In the next subsection we will look at various types of functions that have been introduced so far but in a different way, concentrating more on their graphical behaviour and their parameters. As mentioned earlier, appropriateness is determined by the extent to which the behaviour of the chosen function reflects the behaviour to be modelled as the independent variable varies. The behaviour of a function is determined by whether it is linear, non-linear, or periodic and its range of validity. An important task of this Workbook is to get you to think more and more in modelling terms about the forms and associated behaviours of functions. We shall also take the opportunity of deriving some generalities from specific examples.