How to Use This Problem Set
These problems are designed to bring together the major ideas behind exponential and logistic models. Before looking at each answer, try to identify the model, determine what the differential equation tells you, and connect the mathematics to the situation described.
On AP Calculus questions, the most important skill is often not solving a complicated equation. It is interpreting what the equation says about the quantity, its rate of change, and its long-term behavior.
Key Models to Know
Exponential Growth and Decay
If a quantity changes at a rate proportional to its current size, its differential equation has the form:
When , the quantity increases exponentially. When , it decreases exponentially.
Logistic Growth
Logistic growth modifies exponential growth by including a limiting population, or carrying capacity. The standard logistic differential equation is:
Here, represents the carrying capacity. The factor slows the growth as the population approaches that capacity.
Worked Examples
A population is modeled by the differential equation . Find the rate of change of the population when .
A quantity satisfies and has initial value . Find the particular solution.
A population satisfies , where the carrying capacity is . Find the population's rate of change when .
For the logistic model , at what population size is the growth rate greatest?
Common Mistakes
A value such as describes the population itself. A value such as describes how quickly the population is changing. Keep and conceptually separate.
The logistic model is not simply . The additional factor is what causes growth to slow as the population approaches .
If the population is below carrying capacity, the standard logistic model has positive growth. If it is above carrying capacity, becomes negative, so the population decreases toward the carrying capacity.
An equilibrium solution is a value of the dependent variable for which the differential equation gives a rate of change of zero. For the standard logistic model, the equilibrium populations are and .
Practice Problems
A population satisfies . Find when .
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A quantity satisfies with . Find the particular solution.
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A population follows the exponential model . Determine the time required for the population to double.
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A population satisfies . Find the rate of change when .
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The population of a species is modeled by . Identify the carrying capacity and determine whether the population is increasing or decreasing when .
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For the differential equation , find all equilibrium solutions.
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A population follows the logistic equation . At what population size is the growth rate greatest?
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Suppose and the population currently satisfies . Is the population increasing or decreasing? Explain using the differential equation.
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Two populations have the same carrying capacity. Population A is at of its carrying capacity, while Population B is at . Which population has the greater logistic growth rate? Explain.
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A population is modeled by . Explain why the population initially behaves approximately like an exponential-growth model when the population is much smaller than the carrying capacity.
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For , find the value of when . What does your answer mean in context?
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Determine whether each differential equation represents exponential growth, exponential decay, logistic growth, or neither: , , and .
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What to Look for on the AP Exam
When you see a differential-equation model involving growth, immediately ask three questions:
- What is the current value of the quantity?
- What does the differential equation say about its instantaneous rate of change?
- What happens as the quantity gets very large or approaches a limiting value?
For logistic models, also look for the carrying capacity and the population level at which growth is fastest. These ideas frequently appear in qualitative questions even when you are never asked to solve the differential equation explicitly.
The logistic growth rate is greatest at half the carrying capacity. Below that point, the population is growing and accelerating; above it, the population is still growing but the growth rate is slowing. At the carrying capacity, the rate of change is zero.
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