Exponential Growth and Decay Models

What Exponential Growth and Decay Models Are For

Many real-world quantities change at a rate that is proportional to the amount currently present. Population growth, radioactive decay, continuously compounded interest, medication concentration, and the spread of certain quantities can all be modeled this way.

The key idea is simple: the more you have, the faster the quantity changes. If a quantity grows, its rate of change is positive and proportional to its current amount. If it decays, its rate of change is negative and proportional to its current amount.

That relationship is represented by the differential equation:

Here, is the quantity being modeled, is time, and is the proportionality constant. The sign of determines whether the model represents growth or decay.

The Exponential Model

The differential equation has a general solution of the form:

The constant represents the initial amount when . The constant controls how quickly the quantity changes.

If , the model represents exponential growth. If , the model represents exponential decay.

When an initial value is known, the model can be written especially conveniently as:

This form is extremely useful on the AP exam because it automatically incorporates the initial condition.

How to Recognize an Exponential Model

Look for a statement that says the rate of change is proportional to the current amount. Typical wording includes:

• "The population grows at a rate proportional to its population."
• "The amount of radioactive material decreases at a rate proportional to the amount remaining."
• "The concentration decreases at a rate proportional to its current concentration."
• "The rate of change is proportional to the amount present."

Mathematically, all of these statements point toward the same structure:

The word proportional is the major clue. It tells you that the quantity itself belongs on the right-hand side of the differential equation.

A Step-by-Step Method

1. Define the quantity. Identify what represents and what represents.
2. Translate the wording into a differential equation. If the rate is proportional to the amount, write .
3. Use the initial condition to determine the constant in the exponential model.
4. Use another data point, if one is provided, to determine the proportionality constant.
5. Answer the question. Substitute the requested time or quantity into the completed model.
6. Interpret the result in context. Include appropriate units and make sure the answer makes sense.

Worked Examples

Growth and Decay Rates

The constant tells you more than whether the quantity is growing or shrinking. Its magnitude controls how quickly the quantity changes.

For the model :

• : exponential growth.
• : exponential decay.
• Larger positive : faster growth.
• More negative : faster decay.

Because the derivative is:

the rate of change is itself proportional to the current amount. This is why exponential growth gets increasingly steep and exponential decay gradually flattens toward zero.

Continuous Growth vs. Discrete Growth

Be careful not to confuse continuous exponential growth with models based on a fixed number of compounding periods.

A discrete model often looks like:

whereas continuous growth is modeled by:

In AP Calculus, a statement involving a differential equation such as points directly toward the continuous exponential model.

Common Mistakes

Practice Problems