Exponential growth and doubling

When growth is proportional to the current amount, equal time steps multiply the total rather than add a fixed amount.

For continuous exponential growth, N(t) = N(0)e^(rt), and a positive constant rate r gives doubling time ln(2)/r. The constant-rate assumption matters: resource limits or changing conditions can make a different model more appropriate.

For example: A population doubling each hour grows from 1 to 2 to 4 to 8. The next increase is bigger even though the doubling interval is unchanged.

Keep in mind: Few physical populations grow exponentially forever. A visually steep curve alone does not establish an exponential law.

Think about it

Where are you assuming constant additions when the process actually compounds, and what would eventually limit it?

Connected ideas

Feedback loops and stability · Evolution has no foresight · Infinity comes in sizes

Sources and further reading

• OpenStax Precalculus 2e: Exponential Functions

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