Exponential growth and decay

Parametric equations

The exponential $y=C{e}^{kt}$ grows without bound when $k>0$, and decays toward $0$ when $k<0$. The doubling time $T_{+}=\ln 2/k$ and the half-life $T_{1/2}=\ln 2/|k|$ turn the rate $k$ into a time you can see on the axis. The same relation is the slope.

Growth ($k>0$): $y(t+T_{+})=2y(t)$. Decay ($k<0$): $y(t+T_{1/2})=\frac{1}{2}y(t)$.

Interaction

• Growth and Decay: switch the sign of the exponential and watch one formula do both
• Initial value C and Rate |k|: move the starting height and the speed, and compare how steep the curve gets
• Tangent slope: show the tangent at Tangent position t/tmax. The slope is $\frac{dy}{dt}=ky$, proportional to the height there
• ln y scale: switch the vertical axis to $\ln y$ and watch the exponential become a straight line

What to notice

• The vertical marks are doubling times while it grows, and half-lives while it decays. Each mark is one step of $\ln 2/|k|$.
• The tangent slope is the current height times $k$. A taller point is steeper in that same proportion.
• On the $\ln y$ scale the same curve is the straight line $\ln y=\ln C+kt$.

Related

• Logarithmic scale
• Geometric definition of the natural logarithm
• Logistic curve

Further reading

• Exponential growth (Wikipedia)
• Exponential decay (Wikipedia)