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Calculate population, investment, and bacteria growth with discrete or continuous models
Exponential growth occurs when the rate of increase is proportional to the current value. This creates a J-shaped curve where growth accelerates over time. Examples include population growth, compound interest, and bacteria reproduction.
Discrete growth (A = A₀(1 + r)ᵗ) occurs at specific intervals (like annual compound interest). Continuous growth (A = A₀eʳᵗ) occurs constantly without interruption (like bacterial reproduction). Continuous growth typically yields slightly higher values.
If you know the initial and final values, you can find the growth rate: r = (A/A₀)^(1/t) - 1 for discrete growth, or r = ln(A/A₀) / t for continuous growth. Express as a percentage by multiplying by 100.
e (Euler's number) ≈ 2.71828 is a mathematical constant that appears naturally in growth and decay processes. It's the base of natural logarithms and describes continuous compound growth.
Yes! For investments with compound interest, use discrete growth with A₀ as principal, r as interest rate (decimal), and t as years. For continuously compounded interest, use the continuous growth formula.
Population growth follows exponential patterns when resources are unlimited. A₀ is the initial population, r is the growth rate (births minus deaths), and t is time. Real populations eventually reach carrying capacity limits.
The time unit should match the rate period. If the rate is annual (per year), use years. If it's monthly, use months. For bacteria doubling every hour, use hours. Keep units consistent between rate and time.
Yes, but that represents exponential decay, not growth. A negative growth rate means the quantity decreases exponentially over time. Use the exponential decay calculator for clearer results with decay scenarios.