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.
Learning outcomes After studying this unit you should be able to: create simple models, given a clear statement of the problem; write down the simplifying assumptions that underpin a model; identify the key variables and the parameters of a model; apply the input–output principle to obtain a mathematical model, where appropiate; obtain mathematical relationships between variables, based on or linking back to the simplifying assumptions
5.3 Powers Having discussed nth roots, we are now in a position to define the expression ax, where a is positive and x is a rational power (or exponent). If a > 0, m 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 = Introduction This unit is devoted to the real numbers and their properties. In particular, we discuss inequalities, which play a crucial role in analysis. Acknowledgements Except for third party materials and otherwise stated (see terms and conditions), this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence Grateful acknowledgement is made to the following sources for permission to reproduce material in this unit: The content ackno 8.2.4 Accuracy The calculator does not make mistakes in the way that human brains tend to. Human fingers do, however, make mistakes sometimes; and the calculator may not be doing what you think you have told it to do. So correcting errors and estimating the approximate size of answers are important skills in double-checking your calculator calculations. (Just as they are for checking calculations done in your head or on paper!) 6.2 Getting the feel of big and small numbers Very small and very large numbers can be difficult to comprehend. Nothing in our everyday experience helps us to get a good feel for them. For example numbers such as 1099 are so big that if Figure 1 was drawn to scale, you would be dealing with enormous distances. How big is big? First express 1 000 000 000 in scientific notation as 109. Next, to find out how many times bigger 1099 is, use your calculator to divide 1099 by 109 5.5 Comparing price rises While reading a newspaper article, I noticed some examples of how prices changed in the 10 years from 1984 to 1994. The table below shows the typical prices that you would have expected to pay in 1984 and in 1994 for a pint of milk and a Ford Fiesta motor car. Using your calculator Many people see calculators only as a way of producing answers—indeed some people see them almost as a means of cheating, of short-cutting procedures that can and should be carried out in one's head or on paper. However, the calculator can also be a means of learning mathematics more effectively, something you will come to appreciate more. Many previous mathematics students have found that their graphics calculator, used with understanding and intelligence, has become a most effective aid t Learning outcomes After studying this unit you should: be able to perform basic algebraic manipulation with complex numbers; understand the geometric interpretation of complex numbers; know methods of finding the nth roots of complex numbers and the solutions of simple polynomial equations. 6 Curves from parameters In Section 5 we show how functions may be used to sketch curves in the plane, even when these curves are not necessarily the graphs of functions. Click 'View document' below to open Section 5 (8 pages, 151KB). 5 Hyperbolic functions In Section 4 we introduce the hyperbolic functions sinh, cosh and tanh, which are constructed from exponential functions. These hyperbolic functions share some of the properties of the trigonometric functions but, as you will see, their graphs are very different. Click 'View document' below to open Section 4 (5 pages, 104KB). 2 Real functions In Section 1 we formally define real functions and describe how they may arise when we try to solve equations. We remind you of some basic real functions and their graphs, and describe how some of the properties of these functions are featured in their graphs. Click 'View document' below to open Section 1 (12 pages, 1.8MB). 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 Learning outcomes By the end of this section you should be able to: Section 1: Real functions understand the definition of a real function; use the notation for intervals of the real line; recognise and use the graphs of the basic functions described in the audio section; understand the effect on a graph of translations, scalings, rotations and reflections; understand how the shape o Introduction Many problems are best studied by working with real functions, and the properties of real functions are often revealed most clearly by their graphs. Learning to sketch such graphs is therefore a useful skill, even though computer packages can now perform the task. Computers can plot many more points than can be plotted by hand, but simply ‘joining up the dots’ can sometimes give a misleading picture, so an understanding of how such graphs may be obtained remains important. The object of t 4 Open Mark quiz Now try the quiz and see if there are any areas you need to work on. 3.2.1 Try some yourself 1 Use the method outlined in Example 9 to estimate each of the following, and then use yo 3.2 Using scientific notation Scientific notation can be very useful when estimating the answers to calculations involving very large and/or small decimal numbers. A lottery winner won £7851 000. He put the money straight into a deposit account which earns 7.5% interest per annum (i.e. each year). If he wanted to
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Price in 1984
Price in 1994
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Example 9
