eˣExponential Growth & Decay
Exponential growth and decay describe processes where the rate of change is proportional to the current value. Growth occurs when the rate is positive (population growth, compound interest, viral spread). Decay occurs when negative (radioactive decay, drug elimination, cooling). The continuous model uses the natural exponential function P(t) = P₀ × e^(rt).
- 1Growth: P(t) = P₀ × e^(rt), r > 0
- 2Decay: P(t) = P₀ × e^(−rt), r > 0
- 3e is Euler's number ≈ 2.71828
- 4Doubling time (growth): t₂ = ln(2)/r ≈ 0.693/r
- 5Half-life (decay): t₁/₂ = ln(2)/r ≈ 0.693/r
P₀=1000, r=5%, t=10 (growth)=P(10) = 1000 × e^(0.5) ≈ 1649Doubling time = ln(2)/0.05 ≈ 13.9 years
C-14 half-life 5730 yr, find % remaining after 10,000 yr=≈ 29.8% remaining
| Process | r (approx) | Doubling/Half-life |
|---|---|---|
| World population growth | 1.1% per year | ~63 years |
| COVID-19 early spread (R=3) | ~25%/day | ~3 days |
| Carbon-14 decay | 0.012%/year | 5,730 years |
| Uranium-238 decay | 0.000000016%/year | 4.47 billion years |
| Bank interest 5% pa | 5% per year | ~14.4 years |
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