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Written to check the contents of the Cambridge syllabus. natural arithmetic 2 corresponds to devices P2 and P3. It covers algebra, logarithmic and exponential features, trigonometry, differentiation, integration, numerical resolution of equations, vectors, differential equations and intricate numbers.

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Extra resources for Advanced Level Mathematics: Pure Mathematics 2 and 3

Example text

1. f(x) f(x) f(x)= abx, a a x x Fig. 1 Fig. 2 CHAPTER 3: 31 EXPONENTIAL AND LOGARITHMIC FuNCTIONS For exponential growth b has to be greater than 1. If 0 < b < 1 the graph takes the form shown in Fig. 2; for large values of x the graph gets closer to the x-axis but never reaches it. This then represents exponential decay. Examples of this are the level of radioactivity in a lump of uranium ore, and the concentration of an antibiotic in the blood stream. 1 The population of the USA grew exponentially from the end of the War of Independence until the Civil War.

8 Fig. 7 If the graph is drawn with the price shown on a logarithmic scale, you get Fig. 8. If the claim were true, this graph would be a straight line. This seems approximately true for the first three years, but more recently the graph has begun to bend downwards, suggesting that the early promise of exponential growth has not been sustained. The ideas of the last two examples can be combined, not just to investigate whether there is an exponential relationship, but also to find the numerical constants in the equation.

5 =log-= log3 3 -log2 = 3log3- log2 = 3s- r. 2 Find the connection between logb c and loge b . I ! I 1 logbc=x <=> c=bx <=> c' =(bx)x =b 1 <=> b=c' <=> logeb=-. x 1 - c. Therefore loge b- logb Historically logarithms were important because for many years,_ before calculators and computers were available, they provided the most useful form of calculating aid. With a table of logarithms students would, for example, find the cube root of 100 by looking up the value of log 100 and dividing it by 3.