Exponential Growth & DecayFoundation

Doubling times and the rule of 72 — interactive Mathematics simulation for IIT-JEE.

Concept

When a quantity grows in proportion to its current size ( $dP/dt = kP$ ), it grows exponentially: $P = P_0e^{kt}$ . The signature: a constant doubling time $\ln 2/k$ — the same wait doubles it again, whether it's ₹100 or ₹1 crore. Decay is the same law with k < 0 and a constant half-life.

Key formula

P(t) = P_0e^{kt}, \qquad T_{double} = \frac{\ln 2}{k} \approx \frac{72}{\text{rate\%}}

Derivation

Separate the ODE: $\int dP/P = \int k\,dt$ → $\ln P = kt + C$ → $P = P_0e^{kt}$ .

Doubling: $2P_0 = P_0e^{kT}$ → $T = \ln 2/k ≈ 0.693/k$ . With k as a percentage, $T ≈ 69.3/\text{rate}$ — bankers round to 72 for its many divisors.

Scenarios to explore

Exponential Growth & Decay — Doubling times, half-lives and the rule of 72.

Real-world applications

Compound interest & inflation.
Population growth, epidemics (early phase).
Radioactive decay & carbon dating (the k < 0 twin).

JEE exam tips

ln 2 ≈ 0.693 — memorise; doubling time drops out instantly.
After n half-lives: fraction left = (1/2)ⁿ — no exponentials needed.
e^x beats ANY polynomial eventually — limit comparisons rely on it.

Common mistakes

Confusing 8% growth per year with ×8 (it's ×1.083 continuous).
Linear extrapolation of exponential data — the classic underestimate.
Mixing discrete (1+r)ᵗ and continuous e^{rt} conventions carelessly.

Exam traps to avoid

Rule of 72 is an approximation for the DISCRETE compounding world.
Same data on a log scale plots as a straight line — slope = k.