What causes exponential growth of a population?
The rate of growth of a population in people per year is the births minus the deaths for that year. But what determines those numbers?
Births are caused by women having babies, and the number of pregnant women is proportional to the size of the population. The deaths are also proportional to the population P(t) at time t, so the rate of change of P(t) is about P'(t) = (births - deaths) in year t.
You can prove that P'(t) = k P(t) by taking the derivative of
To prove these are the only solutions, let g(t) =
g(t) = constant, say A, so
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