Sizu Riatto
Calculus · Chapter 22

Growth and Doubling

When growth feeds itself, compounding more and more often doesn’t run away to infinity: it settles on a ceiling, e. And eˣ is its own slope, which is why continuous growth is written with e.

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Opens on Chapter 22’s example: $1,000 at 5% for one year; $5,000 at 4% for 18 years; a sourdough starter that doubles every 4 hours.

1. Compounding
2. Doubles every …
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How it works

Compounding n times a year multiplies by (1 + r/n) n times a year. As n grows, (1 + r/n)ⁿ settles on e^r, so continuous growth gives P × e^(rt). Because eˣ is its own slope, continuous growth at rate r grows at r × the amount at every moment. The doubling time solves e^(rt) = 2, so t = ln 2 ÷ r ≈ 0.693 ÷ r: the rule of 70 (70 ÷ the percent) is that with a round number. Something that doubles every D hours grows by 2^(t/D), which is the same curve as e^(kt) with k = ln 2 ÷ D.

From the book, Chapter 22. $1,000 at 5% for a year: $1,050.00 yearly, $1,051.16 monthly, $1,051.27 daily, $1,051.27 continuously. $5,000 at 4% for 18 years grows to $10,272.17 continuously ($10,129.08 yearly), growing at $410.89 a year by then. At 7% the doubling time is 9.9 years (rule of 70: 10); at 6% tripling takes 18.3. Doubling every 4 hours, 6 hours gives × 2.83. Every chapter and its calculator →

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