Functions
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Exponential Growth & Decay

From bacteria doubling to radioactive decay — explore the power of e^(kx)

Population growth, compound interest, radioactive decay, viral spread — the most dramatic changes in nature and finance all follow exponential patterns. Exponential functions describe quantities that grow or shrink by a constant percentage in each time step, rather than a constant amount. This makes them fundamentally different from linear or polynomial growth — and far more powerful (or dangerous) over time.

The base of the natural exponential function is e ≈ 2.718, a special number that arises naturally in calculus, finance, and physics. The function y = ekx models growth when k > 0 and decay when k < 0.

In this lesson, you'll manipulate a slider to see how the growth constant k transforms the exponential curve, compare different exponential bases, and connect the math to real-world phenomena like compound interest, population growth, and radioactive decay — with an AI tutor guiding you step by step.

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FAQ

What is exponential growth?
Exponential growth occurs when a quantity increases by a constant percentage over equal time intervals. The function y = e^{kx} with k > 0 models this behavior. Unlike linear growth (which adds the same amount each step), exponential growth multiplies by the same factor — so it starts slow but accelerates dramatically. A classic example: bacteria that double every hour start with 1 and reach over a million in just 20 hours.
What is the difference between exponential and polynomial growth?
Polynomial functions like x^2 or x^3 grow by increasing powers of x, but exponential functions like e^x grow by putting x in the exponent. Eventually, any exponential function with base > 1 will overtake any polynomial — no matter how high the polynomial's degree. For example, e^x eventually surpasses x^{100}.
What is the number e?
The number e ≈ 2.71828... is a mathematical constant that appears naturally when you study continuous growth. It can be defined as the limit of (1 + 1/n)^n as n approaches infinity. It is the unique base where the exponential function equals its own derivative: if f(x) = e^x, then f'(x) = e^x as well.
What are real-world examples of exponential functions?
Exponential functions model many real phenomena: compound interest (money grows exponentially when interest is reinvested), population growth (bacteria, viruses, or human populations under ideal conditions), radioactive decay (atoms decay at a rate proportional to how many remain), cooling/heating (Newton's law of cooling), and drug metabolism (medications leave the bloodstream exponentially).

Related Topics

Logarithmic FunctionsQuadratic EquationsPower Functions