Growth and Decay Characteristics Tool

Math Algebra • Exponential and Logarithmic Functions

Written by STEM Calculators Team Published December 17, 2025 Updated February 24, 2026

Model: Preset scenario:

Initial amount P: Rate r (per time unit): Compounds n (per time unit): Discrete base b: Half-life T½: Doubling time Td: Time range:

Percent change interval:

Comparison curve: Show a second curve using continuous rate \(r_2\) Comparison rate r₂: Markers: Show A(t₁), A(t₂) markers

Drag to pan • wheel / pinch to zoom • Use “Auto fit” if the curve goes off-screen.

Ready

Model curve

Effective rate • doubling/half-life • interval percent change

t: 0, A: 0

Pick a model and parameters, then click “Calculate”.

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Theory

Many real-world processes grow or decay proportionally to their current amount, leading to exponential models. This tool focuses on characteristics: effective growth rate, doubling/half-life, and percent change over time intervals.

1) Continuous growth/decay

\[ \frac{dA}{dt}=rA \quad\Longrightarrow\quad \boxed{A(t)=Pe^{rt}} \]

Sign meaning: \(r>0\) growth, \(r<0\) decay, \(r=0\) constant.

2) Other common exponential forms

\(\boxed{A=P(1+\frac{r}{n})^{nt}}\) (compound interest)
\(\boxed{A=P\cdot b^t}\) (discrete base)
\(\boxed{A=P(\frac12)^{t/T_{1/2}}}\) (half-life)
\(\boxed{A=P\cdot 2^{t/T_d}}\) (doubling-time)

All can be compared using an effective continuous rate \(r_{\text{eff}}\).

3) Effective continuous rate and doubling/half-life

\[ \boxed{r_{\text{eff}}= \begin{cases} r & \text{(continuous)}\\ n\ln(1+r/n) & \text{(compound)}\\ \ln(b) & \text{(discrete)}\\ -\ln(2)/T_{1/2} & \text{(half-life)}\\ \ln(2)/T_d & \text{(doubling-time)} \end{cases}} \]

\[ \boxed{T_d=\frac{\ln 2}{r_{\text{eff}}}\ (r_{\text{eff}}>0)} \qquad \boxed{T_{1/2}=\frac{\ln 2}{-r_{\text{eff}}}\ (r_{\text{eff}}<0)} \]

4) Percent change over an interval

\[ \boxed{\%\Delta=\left(\frac{A(t_2)}{A(t_1)}-1\right)\cdot 100\%} \]

The tool also reports a geometric-average “% per time unit” over \([t_1,t_2]\).