AP Calculus AB and BC glossary

Half-life

A half-life is the time an exponentially decaying quantity takes to fall to half of whatever it currently is. Setting the amount equal to half the starting amount makes the starting value cancel, leaving the half-life equal to the natural log of 2 divided by the size of the decay constant.

Ask when the amount reaches half the start: $y_{0}e^{kt} = \frac{1}{2}y_{0}$ . The $y_{0}$ cancels, so $e^{kt} = \frac{1}{2}$ and $kt = -\ln 2$ . In a decay model $k$ is negative, so the time comes out positive.

t_{1/2} = \frac{\ln 2}{\lvert k \rvert} \qquad y = y_{0}\left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}

Because $y_{0}$ cancels, the half-life belongs to the process and not to the sample, and the same interval halves the amount again starting from any moment. It also runs backwards: a stated half-life hands you $\lvert k \rvert = \frac{\ln 2}{t_{1/2}}$ , which is how most problems supply the model without ever printing $k$ .

The mistake

Treating two half-lives as emptying the sample. Each one halves what is left, not what you started with, so after $n$ of them the fraction remaining is $\left(\frac{1}{2}\right)^{n}$ : a quarter after two, an eighth after three, never zero.

Appears in: Unit 7: Differential Equations