Exponential Growth Models

Business Calculus · Axiom Academy

LESSON Exponential Growth Models Modeling percentage-based growth and decay in business Linear growth adds a fixed amount each period. But many business quantities grow by a fixed percentage —and that leads to exponential growth. Two Forms of Exponential Functions For a 5% growth rate: b = 1.05 or k = (1.05) 0.0488 You invest 10,000 in a fund that grows 8% per year. Using base-b form: A(t) = 10000 (1.08)^t Using natural form: A(t) = 10000 e^ 0.077t (since (1.08) 0.077 ) The growth factor b = 1 + r where r is the decimal growth rate: Investment growth over time with reinvested returns Revenue growing by a percentage each quarter Customer base growing through network effects Asset value declining by fixed percentage annually Content spread through exponential sharing Prices increasing by percentage over time A company car worth 30,000 depreciates 15% per year. After 5 years: V(5) = 30000 (0.85)^5 \ 13,311 The Power of Exponential Growth Exponential growth seems slow at first but becomes dramatically fast: To estimate how long it takes an investment to double: At 8% growth, money doubles in about 72/8 = 9 years! Exponential growth: constant percentage change per period Base-b form: f(t) = a b^t where b = 1 + r Natural form: f(t) = a e^ kt where k = (b) Rule of 72: Doubling time ≈ 72 ÷ (rate %) Next, we'll apply exponential growth to compound interest —the most important exponential model in finance. Then we'll explore what happens when compounding becomes continuous!

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