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Section Ⅲ will be of no help unless the reader______.
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To learn lessons from Nature, the author advocates that humans should______.
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Over the past few decades, there has been a considerable increase in the use of mathematical analysis, both for solving everyday problems and for theoretical developments of many disciplines. For example, economics, biology, geography and medicine have all seen a considerable increase in the use of quantitative techniques. Twenty years ago applied mathematics meant the application of mathematics to problems in mechanics and little else—now, applied mathematics, or as many people prefer to call it, applicable mathematics, could refer to the use of mathematics in many varied areas. The one unifying theme that these applications have is that of mathematical modeling, by which we mean the construction of a mathematical model to describe the situation under study. This process of changing a real life problem into a mathematical one is not at all easy, we hasten to add, although one of the overall aims of this book is to improve your ability as a mathematical modeler.
There have been many books written during the past decade on the topic of mathematical modeling; all these books have been devoted to explaining and developing mathematical models, but very little space has been given to how to construct mathematical models, that is, how to take a real problem and convert it into a mathematical one. Although we appreciate that we might not yet have the best methods for teaching how to tackle real problems, we do at least regard this mastery of model formulation as a crucial step, and much of this book is devoted to attempting to make you more proficient in this process.
Our basic concept is that applied mathematicians become better modelers through more and more experience of tackling real problems. So in order to get the most out of this book, we stress that you must make a positive effort to tackle the many problems posed before looking at the solutions we have given. To help you to gain confidence in the art of modeling we have divided the book into four distinct sections.
In the first section we describe three different examples of how mathematical analysis has been used to solve practical problems. These are all true accounts of how mathematical analysis has helped to provide solutions. We are not expecting you to do much at this stage, except to read through the case studies carefully, paying particular attention to the way in which the problems have been tackled—the process of translating the problem into a mathematical one.
The second section consists of a series of real problems, together with possible solutions and related problems. Each problem has a clear statement, and we very much encourage you to try to solve these problems in the first place without looking at the solutions we have given. The problems require for solution different levels of mathematics, and you might find you have not yet covered some of the mathematical topics required. In general we have tried to order them, so that the level of mathematics required in the solutions increase as you move through the problems. Remember that we are only giving our solutions and, particularly if you don't look at our solution, you might well have a completely different approach which might provide a better solution.
Here, in the third section, we try to give you some advice as to how to approach the tackling of real problem solving, and we give some general concepts involved in mathematical modeling. It must, though, again be stressed that we are all convinced that experience is the all-important ingredient needed for confidence in model formulation. If you have just read Sections I and 11 without making at least attempts at your own solutions to some of the problems set, you will not have gained any real experience in tackling real problems, and this section will not really be of much help. On the other hand, if you have taken the problem solving seriously in Section Ⅱ, you might find the general advice give
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