AP® Calculus BC · Unit 7 · Topic 7.8
Exponential Models with Differential Equations
When a quantity's rate of change is proportional to its current amount, that quantity grows or decays exponentially, following y = y0 ekt, where k is the proportionality constant.
Think about why proportional growth has to look exponential. If dy/dt = ky, the bigger y gets, the faster it changes, and that faster change makes y even bigger. Nothing caps it and nothing straightens it out. A population with more members produces more births; a radioactive sample with more atoms has more decays per second. The rate feeds the amount, and the amount feeds the rate right back.
To get the formula, separate variables: dy/y = k dt. Integrate both sides: ln|y| = kt + C. Exponentiate: y = ekt+C = eC ekt. Since eC is just whatever y equals when t = 0, rename it y0, giving y = y0 ekt. k is positive for growth, negative for decay. This is the one differential equation behind every half-life, population, and cooling problem in this unit — the slope field for dy/dt = ky always looks like curves fanning away from y = 0, steeper wherever y is bigger.
This sits right after slope fields and Euler's method, which approximate solutions to differential equations without solving them. Exponential models are the payoff: a case where you can solve dy/dt = ky exactly, by hand, and check that the picture from the slope field matches the formula. Newton's Law of Cooling, dT/dt = k(T - R), is the same idea shifted so it levels off at room temperature R instead of 0.
Key terms
- Separation of variables
- A technique that rewrites a differential equation so all y terms are on one side and all t terms on the other before integrating.
- Half-life
- The fixed time it takes a decaying quantity to drop to half its previous amount.
- Newton's Law of Cooling
- The rule that an object's temperature changes at a rate proportional to the difference between its temperature and the surrounding temperature.
- Growth constant k
- The proportionality constant in dy/dt = ky, positive for growth and negative for decay.
- Asymptote
- A value a function approaches but never reaches, such as y = 0 for decay or y = R for cooling.
Watch it explained
- How it worksHow does dy/dt = ky integrate into the formula y = y0 times ekt?
- Worked exampleFind how long a 100 mg sample with a 12 year half-life takes to decay down to 25 mg.
- Worked exampleFind the temperature of 90 degree C coffee after 10 minutes in a 20 degree room with k = 0.05.
- Common mistakeWhy doesn't exponential decay ever actually reach zero?
Check yourself
End of topic.