Exponential Models with Differential Equations
Fit and interpret exponential growth or decay models, separating a continuous relative rate from interval percent change.
College Board Topic 7.8 · CED p. 139 (PDF p. 144) · CED effective Fall 2020; project course outline 2026–27.
When the rate of change is proportional to the quantity, y′=ky. With y(0)=y₀, the solution is y=y₀e^(kt). The constant k has inverse-time units. A positive k means growth and a negative k means decay for a positive quantity. The instantaneous relative rate y′/y is k; the change over a whole time interval is e^(kΔt)−1, which is not usually just kΔt.
Use measured values to find k: if y(T)=M, then k=ln(M/y₀)/T. Doubling time is ln2/k for growth; half-life is ln2/|k| for decay. In a motion context, integrate an acceleration or velocity equation and include the initial velocity or position. Check physical units and restrict a model to where its assumptions make sense. Newton cooling T′=−k(T−Troom) is exponential in the difference T−Troom, so shift the quantity before solving.
Let's work through it
- A culture starts with 100 cells and reaches 200 after 3 hours. Set 200=100e^(3k). Reason: A proportional growth model has exponential solutions.
- Solve k=(ln2)/3 h⁻¹. Reason: Take logarithms of the ratio 2.
- After 6 hours, y=100e^(2ln2)=400 cells. Reason: Evaluate the model with its fitted rate and retain units.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
For y′=−0.2y with t in days, what are the units of 0.2?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
For y′=−0.2y with t in days, what are the units of 0.2?
Worked answer and reason
Day⁻¹.
A quantity rate has units of quantity per day.
Retrieval 2
Is a continuous relative growth rate of 0.05 the same as a 5% increase after one unit of time?
Worked answer and reason
No; the one-unit increase is e^0.05−1, about 5.127%.
The differential rate acts continuously as the quantity changes.
Catch the tempting mistake
- Negative k makes a positive population negative.
- Exponential decay keeps y₀e^(kt) positive for finite t when y₀>0.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 7 problem 22 · No calculator · verbal
- Unit 7 problem 23 · No calculator · verbal
- Unit 7 problem 24 · Graphing calculator required · analytic
For the teacher
- Fit a model from two observations and ask students to interpret k before solving for a later time. Distinguish continuous relative rate from interval percent change.
Glossary and deeper reading · 3 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.
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Half-life
A half-life is the time required for a positive exponentially decaying quantity to fall to half its current amount. For y(t)=y₀eᵏᵗ with k<0, it equals ln 2 divided by −k. For example, a quantity with half-life 3 hours falls to one quarter of its initial amount after 6 hours. The same half-life applies from any starting time within the exponential model.
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Acceleration in calculus
Acceleration is the derivative of velocity with respect to time. In one-dimensional motion, it is also the second derivative of position. For example, s(t)=t² meters gives constant acceleration 2 meters per second squared. A negative acceleration can increase speed when the velocity is also negative.