Exponential growth and decay models — half-life, doubling time

3.2.5 · Maths › Exponentials & Logarithms

Exponential behaviour har jagah kyun dikhai deta hai?

KYA model kar rahe hain: ek quantity $N$ (population, paisa, radioactive atoms, blood mein drug) jiska growth/decay speed current amount par depend karta hai.

KYUN yeh exponentials ki taraf le jaata hai: agar aapke paas do-gune bacteria hain, to har second do-gune split hote hain. Agar do-gune radioactive atoms hain, to har second do-gune decay karte hain. To:

$\frac{d N}{d t} \propto N ⟹ \frac{d N}{d t} = k N$

Constant $k$ rate constant hai. Agar $k > 0$ → growth; agar $k < 0$ → decay.

$N (t) = N_{0} e^{k t}$ ko scratch se derive kaise karein

Hum sirf defining property $\frac{d N}{d t} = k N$ se shuru karte hain aur sab kuch derive karte hain.

Step 1 — Variables alag karo. $\frac{1}{N} d N = k d t$ Yeh step kyun? Humne saare $N$ ek side rakh diye, saare $t$ doosri side, taaki dono ko alag-alag integrate kar sakein.

Step 2 — Dono sides integrate karo. $\int \frac{1}{N} d N = \int k d t ⟹ ln ∣ N ∣ = k t + C$ Yeh step kyun? $\int \frac{1}{N} d N = ln ∣ N ∣$ ek standard antiderivative hai; constant $C$ initial info carry karta hai.

Step 3 — $N$ ko free karne ke liye exponentiate karo. $∣ N ∣ = e^{k t + C} = e^{C} e^{k t}$ Yeh step kyun? $e^{x}$ , $ln$ ko undo karta hai. Kyunki physically $N > 0$ hai, modulus hata do aur $A = e^{C}$ likho: $N = A e^{k t}$

Step 4 — $t = 0$ use karke $A$ fix karo. $N (0) = A e^{0} = A ⟹ A = N_{0}$ $N (t) = N_{0} e^{k t}$ Yeh step kyun? Integration ke constant ko starting amount encode karna padta hai — yahi solution ko specific banata hai, general nahi.

Doubling time aur half-life derive karna

Yeh bas special "kitne waqt mein amount ek fixed factor se change hogi?" wale sawaal hain.

Key insight (WHY yeh constant hain): doubling/half-life $N_{0}$ par ya aap kis waqt se shuru karte hain us par depend nahi karta. Har fixed ratio mein utna hi waqt lagta hai — yahi exponential ki pehchaan hai.

Figure — Exponential growth and decay models — half-life, doubling time

Model ko half-life/doubling time se rewrite karna

Kabhi kabhi $k$ nahi, balki half-life di hoti hai. $k = \frac{ln 2}{T _{d}}$ substitute karo:

$N (t) = N_{0} e^{(l n 2) t / T_{d}} = N_{0} 2^{t / T_{d}} (growth)$ $N (t) = N_{0} 2^{- t / T_{1/2}} = N_{0} (\frac{1}{2})^{t / T_{1/2}} (decay)$

Yeh form kyun acchi hai: ek half-life ke baad $t = T_{1/2}$ par, $N_{0} \cdot \frac{1}{2}$ milta hai; do ke baad $N_{0} \cdot \frac{1}{4}$ ; round numbers ke liye turat intuition, koi logs nahi chahiye.

Worked Examples

Common Mistakes (Steel-manned)

Recall Feynman: 12-saal ke bachche ko samjhao

Ek magic pile of coins imagine karo jahan har coin har roz ek nayi coin banati hai. Pehle din thodi hain, lekin kyunki zyada coins aur bhi zyada coins banati hain, pile bahut tezi se badhti hai — yahi exponential growth hai, aur "doubling time" woh waqt hai jitna pile ko do-guna hone mein lagta hai (hamesha utna hi waqt!). Ab ulta karo: glowing pebbles imagine karo jahan aadhe har roz dark ho jaate hain chahe kitne bhi ho. 100 se shuru karo → 50 → 25 → 12 → ... yeh kabhi bilkul zero nahi pahunchta, aur "half-life" woh fixed ek-din ka step hai. Dono baar trick yahi hai: change depend karta hai us waqt aapke paas kitna hai.

Active Recall

Exponential growth/decay ko kaun sa differential equation define karta hai?

\frac{d N}{d t} = k N

, rate current amount ke proportional.

\frac{d N}{d t} = k N

N (0) = N_{0}

ke saath solve karo.

N (t) = N_{0} e^{k t}

(separate karo,

ln N = k t + C

tak integrate karo, exponentiate karo, IC use karo).

Doubling time ka formula aur derivation ki shuruat.

T_{d} = \frac{l n 2}{k}

2 N_{0} = N_{0} e^{k T_{d}}

set karne se.

Half-life ka formula.

T_{1/2} = \frac{l n 2}{∣ k ∣}

\frac{1}{2} N_{0} = N_{0} e^{k T_{1/2}}

se.

Kya half-life initial amount par depend karti hai?

Nahi — rate amount ke saath scale hoti hai isliye

N_{0}

cancel ho jaata hai.

Decay model ko half-life

T_{1/2}

use karke rewrite karo.

N (t) = N_{0} 2^{- t / T_{1/2}}

3 half-lives ke baad kitna fraction bachta hai?

(1/2)^{3} = 1/8

Data se exponential aur linear mein fark kaise karo?

Exponential: equal steps mein constant ratio; linear: constant difference.

Decay ke liye

k

ka sign, aur

∣ k ∣

kahan use kar sakte ho?

Decay ke liye

k < 0

;

∣ k ∣

sirf half-life formula mein use karo, kabhi

N (t)

ke andar nahi.

Agar

C

time

τ

mein half ho jaaye, to

k

kya hai?

k = - \frac{l n 2}{τ}

Connections

Natural logarithm and e — kyun $e$ continuous rates ke liye natural base hai.
Solving first-order separable ODEs — derivation ke peeche ki machinery.
Laws of logarithms — exponent ke andar $t$ ke liye solve karne ke liye zaroori.
Graphs of exponential functions — shape, asymptotes, transformations.
Compound interest — discrete cousin; continuous limit $e^{k t}$ deta hai.
Radioactive decay (Physics) — real-world half-life application.