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.
Understanding Calculus: Exponential Growth and Decay
The important feature is that the change is based on a fraction of the current amount, not a fixed number of units. A bank balance earning the same percentage each year gains more dollars later because the balance has become larger. A radioactive sample loses the same fraction in equal time intervals, so its decrease is large at first and smaller later.
This is why an exponential graph is curved. Its steepness changes continually. Linear change is different because it adds or removes an equal number of units during each equal time interval.
The constant called k has units tied to time. If time is measured in days, k is a rate per day. If time is measured in years, it is a rate per year.
This detail matters because changing the time unit changes the numerical value of k. A growth rate of about zero point zero three per day is not the same as zero point zero three per year. The size of k controls how quickly the curve changes.
Its sign gives the direction, while its magnitude gives the speed. The number e appears because it describes continuous compounding. It is the unique base that makes the rate of change of an exponential function match a constant multiple of the function itself.
Real data rarely follow an exponential model forever. A bacterial culture may grow nearly exponentially while food and space are plentiful. Later, crowding slows the growth.
This produces a different model called logistic growth. Compound interest can be exponential when money remains invested at a fixed rate, but deposits, withdrawals, fees, and changing interest rates alter the pattern. Cooling is often modeled by exponential decay only for the temperature difference between an object and its surroundings.
The object does not usually cool toward zero degrees. It cools toward room temperature or whatever temperature surrounds it.
Students often need to find a rate constant from measurements. First compare values separated by equal time intervals. A roughly constant ratio suggests an exponential pattern.
For decay, a repeated halving is especially useful. For growth, repeated doubling gives a clear clue. A graph with time on the horizontal axis and the logarithm of the quantity on the vertical axis becomes close to a straight line when the model fits.
Its slope is the rate constant. Always check units, the starting condition, and whether outside limits make the model unrealistic over long times. Exponential formulas can give sensible short term predictions while producing impossible long term results.
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.