7.2 Square roots Earlier you met the square function and on most calculators the square root is the second function on the same key. Look to see if this is the case for your calculator and check the calculator handbook on how to use this function. In many cases you will need to press the square root key before the number, instead of afterwards, as for the square key. This is the case on the TI-84. Check that you can find the square root of 25 and of 0.49 (you should get 5 and .7 respectively). Now find
Introduction You may have met complex numbers before, but not had experience in manipulating them. This unit gives an accessible introduction to complex numbers, which are very important in science and technology, as well as mathematics. The unit includes definitions, concepts and techniques which will be very helpful and interesting to a wide variety of people with a reasonable background in algebra and trigonometry. This unit is from our archive and is an adapted extract from Complex analysis (M33
3 Graph sketching In Section 2 we describe how the graphs of polynomial and rational functions may be sketched by analysing their behaviour – for example, by using techniques of calculus. We assume that you are familiar with basic calculus and that its use is valid. In particular, we assume that the graphs of the functions under consideration consist of smooth curves. Click 'View document' below to open Section 2 (16 pages, 200KB). 1 Overview A fundamental concept in mathematics is that of a function. Consider, for example, the function f defined by
This is an example of a real function, because it associates with a given real number x the real number 2x2 − 1: it maps real numbers to real n 3.1 Expressing numbers in scientific notation Earlier you looked at place values for numbers, and why they were called powers of ten. 1.3.7 Internet resources There are many websites where you will find useful information on maths and statistics. With all information on the internet you need to make a judgement on its reliability. 1.3.2 The fate of incoming solar radiation Look back at Figure 7. In this schematic representation, what is the fate of incoming solar radiation? It is either reflected back to space (31 units) or absorbed by the su 7.3 Running the models forward What happens when the models are run forward? It depends upon the models used and the scenarios they are asked to run. It seems almost certain, however, that there will be increases in the global mean surface temperature, to the order of +1.5 to +4.5 °C (– possibly more, according to some models and scenarios. These changes are predicted to be associated with increases in sea level, changes to weather conditions (e.g. more regular and violent winter storms in the UK) and changes to t 5.3.4 Plan testing and validation It is one thing to have a plan; it is another thing to have a plan that you can rely on to work. There is an old military maxim that ‘A plan only gets you into first contact with the enemy. After that, you fly by the seat of your pants’ (Anon). A 1993 IBM report on business continuity planning confirmed this when it revealed that ‘half of the plans failed completely or substantially when they were first tested’ (IBM, 1993, p. 5). The IBM report identified three categories of pla 5.7.1 Plan preparation Perhaps the first question to ask is ‘What is an emergency plan?’ Dodswell, in his guide to business continuity management, defined an ‘emergency management plan’ as simply: A plan which supports the emergency management team by providing them with information and guidelines. (Dodswell, 2000, p. 56) Another definition, of an ‘emergency preparedness plan’ prepared in the co 5.6 Business continuity planning An organisation's strategy regarding insurance for its business risks is no substitute for high-quality risk management and emergency preparedness to address all contingencies. Some incidents we have mentioned above. Others may involve IT security for example. While the day-to-day activity of an organisation may not be particularly hazardous, it can still be affected by a hazard not of its own making. Examples might be a natural disaster such as flooding, or a hazardous activity on an adjacen 4.1 Human predation and extinctions There are a number of ways in which humans have altered ecosystems, that have led to the decline of particular species. We will leave to one side any major interference such as felling forests to provide land for agricultural and urban development, and instead begin by looking at examples where we have eroded or eradicated stocks of particular species. This has notably been a consequence of the over-exploitation of food species (prey items). Predators do not normally eliminate their prey (see 3 3 Present energy sources and sustainability What are the principal energy sources at present, and how sustainable are they? About 80 per cent of the world's energy is currently supplied by fossil fuels such as coal, oil and gas. Present estimates suggest that, at current consumption rates, there are over 200-years' worth of coal left, 60-years' of gas, and 40-years' of oil. Fossil fuels are hydrocarbons, and their combustion releases carbon dioxide into the atmosphere, one of the main causes of the human-induced Introduction The search for sustainable energy will dominate the twenty-first century. This unit provides an introductory overview of the present energy systems and takes a brief look at where the world may find energy in the future – cleaner use of fossil fuels or renewable energy sources? This unit is from our archive and is an adapted extract from Energy for a sustainable future (T206) which is no longer taught by The Open University. If you want to study formally with us, you may wish to explo 5.2 New Zealand's changing environment In this study I want to explore some possible effects of this new trade on the environment of one of the countries involved. I've chosen New Zealand, partly because the developments we have just been discussing happened only a few decades after the first large-scale settlements of Europeans, and had a strong influence on the direction of its economy. Some background information will help to set the scene. New Zealand consists of two mountainous islands with a total area similar to that 7 Conclusion One might think of the different interpretations of internationally recognised notions of rights and justice as running along a spectrum, from which we shall now identify four different positions. The first interpretation would argue that, overall, the extension of rights to the international sphere has been benign and effective. It has led and will lead to further successful claims for justice. Evidence for the development of a globa 4.3 Asymmetry between labour and capital Finally, stepping back to get a broader picture, I would like to point to the asymmetry built into the emerging institutional framework governing international economic relations, of which the WTO is one important pillar. The various WTO agreements encourage free movement of goods and certain kinds of services. Possible agreements on cross-border investment and competition policy may allow for freer international movement of capital, already encouraged by the IMF. Yet there is no move towards 5.3 Order and chaos How can we explain a sudden switch of behaviour at a particular temperature? There must be two competing influences (say X and Y) that depend differently on temperature. Figure 23 indicates how a unique temperature (a so-called critical temperature, Tc) arises, 5.2 What's in a phase? In an engineering and scientific context, a phase is an arrangement of atoms that is identifiable through its recurrence – the same pattern is found time and again. For instance, the compound of hydrogen and oxygen that we call water turns up all over the place in the same form as a runny, colourless liquid; this is a specific phase of the compound H2O. In water, the atoms apparently organise themselves according to what they are and the ambient conditions of temperature a 5.8 Plagiarism Referencing is not only useful as a way of sharing information, but also as a means of ensuring that due credit is given to other people’s work. In the electronic information age, it is easy to copy and paste from journal articles and web pages into your own work. But if you do use someone else’s work, you should acknowledge the source by giving a correct reference. Taking someone's work and not indicating where you took it from is termed plagiarism and is regarded as an infringemen
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