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Exponential decay is a process that reduces the quantity of an item by a consistent percentage over time.

This often describes the decrease in amount of some real substance or resource, like radioactive elements.
To characterize exponential decay mathematically, we use the formula \(y(t) = y_0 e^{-kt}\), where:

• \(y(t)\) is the amount at time \(t\).
• \(y_0\) is the initial quantity at \(t = 0\).
• \(k\) is the decay constant.

Here, you're seeing that the amount decreases rapidly at first and then slowly over time. This non-linear but continuous decrease forms the basis for many natural decay processes.

Radioactive carbon-14 is a naturally occurring isotope of carbon that is used extensively in radiocarbon dating.

When living organisms are alive, they constantly exchange carbon with their environment, maintaining a stable amount of carbon-14.
However, once they die, they stop absorbing carbon, and the carbon-14 starts to decay.
The decay of carbon-14 is a process of exponential decay, and it is useful for dating archaeological and geological samples up to about 50,000 years old because of its relatively short half-life.
The half-life of carbon-14 is about 5,730 years, which means after this period, only half of the original carbon-14 atoms remain.
This characteristic makes carbon-14 a valuable tool for scientists in trace dating.

A Taylor series expansion allows us to approximate complex functions with a series of simpler polynomials.

Think of it as breaking down difficult calculations into more manageable chunks.
When we talk specifically about representing functions around zero, we use the Maclaurin series, a special case of the Taylor series.
The general form of a Maclaurin series for a function \(f(x)\) is:

• \(f(0)\) –- the function's value at 0.
• \(f'(0)x\) -– first derivative times \(x\).
• \(\frac{f''(0)}{2!}x^2\) –- second derivative over factorial of 2.
• \(\frac{f'''(0)}{3!}x^3\) –- third derivative over factorial of 3.
• And so on.

In the context of exponential decay, using a Taylor or Maclaurin series helps us understand and predict changes over time with precision through a series approximation, starting with simpler expressions.