3.2 Natural dives The physiology of the diving response can be studied in the laboratory, but investigating the behaviour of a diving mammal in its natural environment can be more of a problem. However, modern physiological techniques have made it possible to record continuously physiological variables (such as heart rate) and information on depth and position during the spontaneous dives in the wild that are part of the animal's normal behaviour. Most such findings show that the majority of an animal's dives
4.5 More about covalent bonding So far, the valencies in Table 1 have just been numbers that we use to predict the formulae of compounds. But in the case of covalent substances they can tell us more. In particular, they can tell us how the atoms are linked together in the molecule. This information is obtained from a two-dimensional drawing of the structural form
4.3 Metallic bonding Two familiar properties of metals point to a simple model of metallic bonding. Firstly, metals have a strong tendency to form positive ions. Thus, when sodium reacts with water, and when magnesium and aluminium react with acids, hydrogen gas is evolved and the ions Na+(aq), Mg2+(aq) and Al3+(aq), respectively, are formed. Secondly, metals are good conductors of electricity: when a voltage difference is applied
3.5 Electron states and box diagrams So far, we have represented the electronic state of an atom as a collection of sub-shells. Now we turn to the states of the electrons within those sub-shells. Just as shells can be broken down into sub-shells, so sub-shells can be broken down into atomic orbitals. Each atomic orbital describes an allowed spatial distribution about the nucleus for an electron in the sub-shell. Here we shall only be concerned with their number. Consider the formula for the sub-shell electron capaci
1.2 Chemical elements Atoms of the same atomic number behave virtually identically in chemical reactions. They are therefore given the same chemical name and chemical symbol. For example, the atom of atomic number 6, which is shown in Figure 1, is a carbon atom, whose symbol is C. All materials are made of atoms, but there is a special class of substan
11 Additional resources Bandolier (2005) Statins: when should you take the tablet? British Red Cross (2007) First aid guidelines in the UK Cardiac Risk in the Young (2003) When a young person dies suddenly Clay, R. A. (2001) Research to the heart of the matter Department of Health (2000) National Service Framework for coronary heart disease, Chapter 4 Department of Health (2007) The coronary heart disease National Service Framework: shaping the future: progress report 2006 The Nat
9 Summary Now you will be very familiar with cardiovascular diseases, their development and their diagnosis. You will also know their treatment and many of the cardiovascular disease risk factors – what they are and how they can be influenced positively to minimise cardiovascular diseases. You will understand the overall importance of a balanced diet, regular exercise and weight management (guided by adiposity measurements) throughout life, to maintain cardiac and vascular health. You will also be a
3.2.1 Fats Fats, also known as lipids, are important components of living tissues, and are used by the body for making cell membranes and for storing energy. Fats come in a variety of different biochemical types, which may be obtained from the diet or can be synthesised within the body. Many cells of the body can convert certain types of fat into others, but by preference, fats will be obtained from the diet, if available. The fatty acids that cannot be synthesised by the body and therefore must
2.10.1 Mean and standard deviation for repeated measurements In everyday terms, everybody is familiar with the word ‘average’, but in science and statistics there are actually several different kinds of average used for different purposes. In the kind of situation exemplified by Table 2, the sort to use is the mean
(or more strictly the ‘arithmetic mean’) For a set of measurements, this is de
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). 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. 5.1 Arithmetic with real numbers At the end of Section 1, we discussed the decimals
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): 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. 8.2.2 The screen You can see the calculations that you have entered as well as the answers. This means you can easily check whether you have made any mistakes. 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 3 Aims The aim of this section is to help you to think about how you study mathematics and consider ways in which you can make your study more effective.
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):
