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Exponential changes

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In this section we will learn: Some common examples of exponential changes.Some common examples of exponential changes. What an exponential change signifies.What an exponential change signifies. Two useful ways of testing for exponential curves.Two useful ways of testing for exponential curves.

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Let’s start with a bit of British history………. The Black Death of 1666

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Week ending Plague deaths Week ending Plague deaths Week ending Plague deaths Week ending Plague deaths May 20July 4470Sept 56988Nov 71414 May 90July 11725Sept 126544Nov 141050 May 163July 181089Sept 197165Nov 21652 May 2314July 251843Sept 26 5533Nov 28333 May 3017August 1 2010Oct 34929Dec 5210 June 643August 8 2817Oct 104327Dec 12243 June 13112August 15 3880Oct 172665Dec 19281 June 20 168August 22 4237Oct 241421 June 27 267August 29 6102Oct 311031

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Plot a graph of Plague deaths (y axis) against Week number (or time) on the x axis. Now plot a graph of log of plague deaths against time. You’ll notice that parts of the graph are a straight line. These are the periods when the plague deaths grew EXPONENTIALLY with time. There are lots of examples of exponential growth (and decay) all around us.

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Radioactive decay. Bacterial growth which turns milk into yoghurt. Epidemics. Many chemical reactions. Oscillations of a bridge.

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What is an EXPONENTIAL CHANGE? MULTIPLIED by the same FACTOR in successive time intervals. An exponential curve grows (or decays) so that the amount present is MULTIPLIED by the same FACTOR in successive time intervals. We’ve met this idea already, in RADIOACTIVE DECAY. When a radioactive material decays, the amount of material goes down by the same factor (a half) in successive time intervals.

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Count rate reduces by ½ for every time interval t (about 15 hours in this case) tt We give t a special name – “half-life” – to show that the quantity is halved after this time interval.

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TESTS for exponentiality. How can you tell if your growth or decay is exponential? 1.The RATIO test. Pick several equal time intervals and read off the quantities from the data or graph. Work out the ratio of each quantity to the next. If it’s the same, your change is exponential. Ratios: 720/1094 = 0.66 1094/1663 = 0.66 1663/2528 = 0.66 TimeQuantity 0720 101094 201663 302528

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2. The log graph test. Re-plot your graph but this time use the logarithms (log 10 or log e – called ln on your calculator) of your quantities against time. If your graph is a straight line, you have an exponential relationship. It takes a little more time to do this test. It’s very useful to plot a log test graph for any coursework that you do. Not only will you find an exponential relationship, but you can use the straight line y = mx + c to come up with a maths relationship from the graph.

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