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Whether does the number e come from!?!?

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Suppose the number of bacteria, n 0, in a dish doubles in unit time. If a very simple growth model is adopted, you can think of the number of bacteria remaining constant during the period and then, at t = 1, the number doubles instantaneously. … but this is not very realistic

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To make it more realistic, suppose that the number of bacteria remains constant during the periods 0≤t<0.1, 0.1 ≤t<0.2, 0.2 ≤t<0.3, …, 0.9 ≤t<1 and the number of bacteria grows by 10% (one tenth) at times t=0.1, t=0.2, t=0.3, …, t=0.9, t=1

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Then, when t=1, the number of bacteria is *Note that the number has more than doubled. In fact, it has increased by a factor of almost 2.6 because of the compounding effect.

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Ultimately, the smaller the length of the time interval, the more accurate the growth model is. For example, if the number of bacteria remains constant during the periods 0≤t<0.01, 0.01 ≤t<0.02, 0.02 ≤t<0.03, …, 0.99 ≤t<1 and the number of bacteria grows by 1% (one hundredth) at times t=0.01, t=0.02, t=0.03, …, t=0.99, t=1

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Then what is the formula for the number of bacteria?

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