Learning outcomes By the end of this unit you should be able to: explain the meanings of the emboldened terms and use them appropriately; describe the behaviour of wave packets when they encounter potential energy steps, barriers and wells; describe how stationary-state solutions of the Schrödinger equation can be used to analyse scattering and tunnelling; for a range of simple potential energy functions, obtain the solution of the time-independent Sc
6.1 A difficult evolutionary transition Before going any further, click on 'View document' below and read pages 78–79 and 82–83 from Douglas Palmer's Atlas of the Prehistoric World. 4.2 Crinoids Figure 7 shows the fossilised remains of a type of echinoderm called a crinoid (‘cry-noyed’). Although crinoids occur today, they were far more common in the Palaeozoic and Mesozoic Eras. Most crinoids feed by bending their umbrella-like arrangement of flexible appendages (called ‘arms’) downstream so as to catch a current, rather as in an umbrella being caught in the wind. Tube feet (multipurpose tentacles) on the arms gather food particles suspended in the water, which are th Numbers Introduction In the OpenLearn unit on Developing modelling skills (MSXR209_3), the idea of revising a model was introduced. In this unit you will be taken through the whole modelling process in detail, from creating a first simple model, through evaluating it, to the subsequent revision of the model by changing one of the assumptions. The new aspect here is the emphasis on a revised model, which comes in Section 2. The problem that will be examined is one based on heat transfer. This unit, the four Acknowledgements The content acknowledged below is Proprietary (see terms and conditions) and is available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence All materials included in this unit are derived from content originated at the Open University. 1 Analysing skid marks 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). Acknowledgements All written material contained within this unit originated at the Open University. The audio extract is taken from M208 © Copyright 2006 The Open University. http://www.flickr.com/photos/re_birf/69485963/ [Details correct as of 9th June 2008] 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 = 3.4 Further exercises Use the Triangle Inequality to prove that
3.3.1 Post-audio exercises To practise using the techniques described in the audio, we suggest that you now try the following exercises. Use the Binomial Theorem to prove that
3.3 Worked examples The audio provided below illustrates various methods for proving inequalities. In addition to the techniques already described for proving inequalities, we use mathematical induction and the Binomial Theorem, restated below. If x 3.2 Inequalities involving integers In analysis we often need to prove inequalities involving an integer n. It is a common convention in mathematics that the symbol n is used to denote an integer (frequently a natural number). It is often possible to deal with inequalities involving n using the rules of rearrangement given in Section 2. Here 3.1 Triangle Inequality Our next inequality is also used to deduce ‘new inequalities from old’. It involves the absolute values of three real numbers a, b and a + b, and is called the Triangle Inequality. As you will see, the Triangle Inequality has many applications in the analysis units. 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): 2.1 Rearranging inequalities Much of analysis is concerned with inequalities of various kinds; the aim of this section and the next is to provide practice in their manipulation. The fundamental rule, on which much manipulation of inequalities is based, is that the statement a < b means exactly the same as the statement b −a > 0. 1.6 Further exercises Arrange the following numbers in increasing order: (a) 7/36, 3/20, 1/6, 7/45, 11/60; (b) 1.5 Arithmetic with real numbers We can do arithmetic with recurring decimals by first converting the decimals to fractions. However, it is not obvious how to do arithmetic with non-recurring decimals. For example, assuming that we can represent 1.4 Real numbers and their properties Together, the rational numbers (recurring decimals) and irrational numbers (non-recurring decimals) form the set of real numbers, denoted by As with rational numbers, we can determine which of two real numbers is greater by comparing their decimals and noticing the first pair of corresponding digits 1.3 Irrational numbers There is no rational number which satisfies the equation x2 = 2. A number which is not rational is called irrational. There are many other mathematical quantities which cannot be described exactly by rational numbers; for example,
This unit will help you understand more about real numbers and their properties. It will explain the relationship between real numbers and recurring decimals, explain irrational numbers and discuss inequalities. The unit will help you to use the Triangle Inequality, the Binomial Theorem and the Least Upper Bound Property. First published on Wed, 2
Audio Materials
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such that b2Â =Â a. We now discuss the justification
Exercise 21
Exercise 18
Theorem 3.1 Binomial Theorem
Triangle Inequality
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Exercise 7
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and Author(s):
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where m and n are natural numbers and
