1. Linear vs. Exponential Growth

• Linear: grows by the same amount each period. y = y₀ + r·t
• Exponential: grows by the same percentage each period. y = y₀(1 + r)^t
• Exponential decay: shrinks by the same percentage each period. y = y₀(1 − r)^t

Signal words: “increases by 642 per year” is linear; “grows 30% per year” or “doubles every…” is exponential.

Examples: 1600 + 642(4) = 4168; 150(1.3)³ ≈ 330; 2000(1.1)³ = 2662; $500 at 8% for 3 years: 500(1.08)³ ≈ $629.86.

2. Doubling Time

The doubling time T is the time for a growing quantity to double.

• Example: 50 spores doubling every hour: after 4 hours, 50 × 2⁴ = 800.
• Rule of 70: T ≈ 70 ÷ (percent growth rate). At 5% per year, T ≈ 70/5 = 14 years. Works best for rates under about 15%.
• Exact: T = ln 2 ÷ ln(1 + r), with r as a decimal.

3. Half-Life

The half-life T is the time for a decaying quantity to fall to half its value.

• After 1 half-life, 50% remains; after 2, 25%; after 3, 12.5%.
• Approximate half-life: T ≈ 70 ÷ (percent decay rate).

4. Population Growth

• Growth rate = birth rate − death rate. Rates per 1000 people: (25 − 10)/1000 = 0.015 = 1.5% per year.
• At 1.5% per year the Rule of 70 gives a doubling time of about 70/1.5 ≈ 47 years.

5. Logistic Growth

Real populations cannot double forever. Logistic growth starts out nearly exponential and levels off at the carrying capacity K, the largest population the environment can sustain.

1. Slow start: few members, so few new ones.
2. Rapid growth: close to exponential.
3. Slowing: limited resources reduce the growth rate.
4. Leveling off: the population approaches K and the growth rate approaches 0.

Resources

Lesson: andrewhvolk.github.io/courses/math114-module11.html  ·  Slides: andrewhvolk.github.io/pdfs/math114-module11-slides-2026-11-02.pdf  ·  Desmos Scientific and Excel Online