Exponential models - Calculus AB AP Study Notes

Exponential models are differential equations where the rate of change of a quantity is proportional to the quantity itself. The fundamental form is dy/dt = ky, where y represents the quantity, t is time, and k is the growth constant (if k > 0) or decay constant (if k < 0).

General Solution Formula: The solution to dy/dt = ky is y = y₀e^(kt), where y₀ is the initial value at t = 0. This is derived through separation of variables: ∫(1/y)dy = ∫k dt, yielding ln|y| = kt + C, and solving for y.

Key Properties: • The doubling time (growth) or half-life (decay) can be found using t = ln(2)/|k| • The constant k has units of reciprocal time (e.g., per year, per hour) • When k > 0, exponential growth occurs (population, compound interest) • When k < 0, exponential decay occurs (radioactive decay, cooling)

Important Variations: • Limited growth model: dy/dt = k(L - y), where L is the limiting value • Logistic model: dy/dt = ky(1 - y/L), incorporating carrying capacity

Memory Aid - GRATE: Growth or decay, Rate proportional to amount, Always exponential function, Time is independent variable, Exponential base e

The exponential model appears frequently in AP Calculus AB free-response questions, typically requiring students to set up differential equations from word problems and solve them using initial conditions.

Population Growth: Bacteria in a petri dish exemplify exponential growth. If a colony has 1000 bacteria initially and grows at 15% per hour, the model is P(t) = 1000e^(0.15t). The rate of change is proportional to population size because more bacteria produce more offspring—each bacterium contributes to growth proportionally.

Radioactive Decay: Carbon-14 dating uses exponential decay. With a half-life of 5,730 years, archaeologists determine artifact ages. If k = -ln(2)/5730 ≈ -0.000121, then A(t) = A₀e^(-0.000121t). After 5,730 years, exactly half remains: A(5730) = A₀e^(-0.000121×5730) = A₀e^(-ln2) = A₀/2.

Newton's Law of Cooling: A cup of coffee at 90°C in a 20°C room cools following T(t) = 20 + 70e^(-kt). The temperature difference from ambient decreases exponentially. Think of it as the coffee "losing its extra heat" proportionally—the greater the temperature gap, the faster it cools.

Compound Interest: With continuous compounding at 5% annual rate, $1000 grows as A(t) = 1000e^(0.05t). This surpasses simple or periodic compounding because interest is constantly calculated on accumulating interest—the ultimate "money making money" scenario.

Real-World Analogy: Imagine a snowball rolling downhill. Its surface area determines how much snow it collects, but surface area depends on its current size. Bigger snowballs grow faster—this self-reinforcing mechanism mirrors exponential growth where the rate depends on the present amount. Similarly, debt with compounding interest "snowballs" exponentially.