# 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.

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Exponential growth of a population occurs when the birth rate exceeds the death rate consistently over time, leading to an increasing population size. This phenomenon is often observed in environments where resources are abundant and there are minimal limiting factors such as disease, predation, or competition for resources. Exponential growth can also occur in situations where individuals have high reproductive rates and the offspring themselves begin reproducing early, contributing to a rapid increase in population size.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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- What is the general solution of the differential equation #xy'' = y'+x(y')^2#?

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