Exponential growth and decay describe quantities whose rate of change is proportional to the amount present. This idea appears in population growth, radioactive decay, compound interest, medicine levels in the body, and cooling. Calculus makes the model powerful because it connects a changing quantity to its derivative.
The result is a simple differential equation with solutions that curve upward or downward in a predictable way.
The core model is dQ/dt = kQ, where Q is the quantity, t is time, and k is the proportionality constant. If k is positive, the quantity grows exponentially, and if k is negative, it decays exponentially. The solution Q(t) = Q0e^(kt) uses the initial amount Q0 and shows how the whole function is determined by one starting value and one rate constant.
Half-life and doubling time are practical ways to describe the same process without always referring directly to k.
Key Facts
- Differential equation: dQ/dt = kQ.
- General solution: Q(t) = Q0e^(kt), where Q0 = Q(0).
- Growth occurs when k > 0, so Q(t) increases as t increases.
- Decay occurs when k < 0, so Q(t) decreases as t increases.
- Doubling time: T_d = ln(2)/k for k > 0.
- Half-life: T_1/2 = ln(2)/|k| for k < 0.
Vocabulary
- Exponential growth
- A process in which a quantity increases at a rate proportional to its current size.
- Exponential decay
- A process in which a quantity decreases at a rate proportional to its current size.
- Rate constant
- The constant k in dQ/dt = kQ that determines how quickly the quantity grows or decays.
- Half-life
- The time required for a decaying quantity to decrease to one half of its current value.
- Doubling time
- The time required for a growing quantity to increase to twice its current value.
Common Mistakes to Avoid
- Using a linear model instead of an exponential model, which is wrong when the rate depends on the current amount rather than staying constant.
- Forgetting the sign of k, which changes the meaning of the model because positive k gives growth and negative k gives decay.
- Using e^(kt) but not matching the time units, which is wrong because k must be measured per unit of the same time variable used in t.
- Confusing half-life with the time to reach zero, which is wrong because exponential decay approaches zero but never reaches it in the ideal model.
Practice Questions
- 1 A bacteria culture starts with 500 cells and follows dQ/dt = 0.35Q, with t in hours. Find Q(t) and estimate the population after 4 hours.
- 2 A radioactive sample has 80 grams and a half-life of 12 days. Write Q(t) in the form Q0e^(kt), then find the amount after 30 days.
- 3 Two substances both decay exponentially, but substance A has k = -0.08 and substance B has k = -0.20, with t in hours. Explain which substance loses a larger fraction of itself each hour and which has the shorter half-life.