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Atomos
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I understand why base e can be used when the percent growth per annum is 100%, but I don't understand how it can be justified that the growth for numbers other than 1 can be put into the exponent.
Atomos said:I understand why base e can be used when the percent growth per annum is 100%, but I don't understand how it can be justified that the growth for numbers other than 1 can be put into the exponent.
Continuous exponential growth is a type of growth pattern in which a quantity increases at a rate proportional to its current value. This means that the larger the quantity becomes, the faster it will continue to increase. This type of growth is often observed in biological populations and financial markets.
Regular exponential growth occurs in discrete time intervals, meaning that the quantity increases by a fixed amount at each interval. Continuous exponential growth occurs in continuous time, meaning that the quantity increases constantly without any discrete intervals. In other words, regular exponential growth is like climbing a set of stairs, while continuous exponential growth is like ascending a ramp.
Some examples of continuous exponential growth include the growth of bacteria in a petri dish, the spread of a virus in a population, the increase in size of a tumor, and the growth of a city's population. These examples all involve a quantity increasing at a rate proportional to its current value.
Continuous exponential growth is not sustainable in the long term, as it requires infinite resources and space. In real life, populations and economies cannot grow infinitely. Eventually, there will be a limiting factor or resource that will slow or stop the growth. Therefore, it is important for societies and economies to find ways to achieve sustainable growth rates.
No, continuous exponential growth will always result in a positive quantity. The rate of growth may slow down or even reach a plateau, but it will never become negative. If a quantity decreases over time, it is not exhibiting continuous exponential growth.