Lesson 5.5
Continuous Growth
When compounding happens infinitely often, the compound interest formula simplifies to . Fewer variables, cleaner math — and it models natural phenomena perfectly.
Introduction
From 5.3, as in , the expression converges to . This is continuous compounding — growth that never pauses.
Past Knowledge
The natural base (5.3), compound interest (5.4).
Today's Goal
Apply to growth and decay scenarios.
Future Success
Solving for requires logarithms (Chapter 17).
Key Concepts
The Formula
A = final amount
P = initial amount
r = rate (positive for growth, negative for decay)
t = time
No needed! Continuous means , which simplifies everything.
Growth vs. Decay
→ Continuous Growth
Population, investments, bacteria
→ Continuous Decay
Radioactive decay, cooling, depreciation
Worked Examples
Example 1: Investment
Basic$2,000 invested at 5% compounded continuously for 8 years.
Identify:
Substitute
Calculate
Interest earned: ≈ $983.65
Example 2: Radioactive Decay
Intermediate50 grams of a substance decays at 3% per year. How much remains after 20 years?
Decay → negative rate
Calculate
About 55% remains after 20 years
Example 3: Continuous vs. Monthly
Comparison$10,000 at 6% for 15 years. Compare monthly compounding to continuous.
Monthly (n = 12)
Continuous
Continuous earns only $54.94 more — the gap shrinks as n increases
Common Pitfalls
Forgetting the Negative for Decay
Decay means . Writing instead of gives growth instead of decay.
Mixing Up Formulas
Use only when the problem says "continuously." Otherwise use .
Real-Life Applications
Newton's Law of Cooling, carbon dating, population growth models, and pharmacokinetics (how drugs leave your body) all use . It's the single most used growth/decay model in science.
Practice Quiz
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