Unit 7 Notes: Exponential Models via Differential Equations (AP Calculus AB)

Exponential model

A model where the rate of change of a quantity is proportional to the quantity itself, typically written as $\frac{dy}{dt} = ky$.

Proportional rate of change

A rate statement meaning there exists a constant k such that dy/dt = k·y (the derivative is always a constant multiple of the current amount).

Differential equation

An equation involving a function and its derivatives (e.g., $\frac{dy}{dt} = ky$) that describes how a quantity changes.

Proportionality constant (k)

The constant in $\frac{dy}{dt} = ky$; it equals the constant relative growth rate (per unit time) and determines growth ($k > 0$) or decay ($k < 0$).

Exponential growth

Behavior when $k > 0$ in $\frac{dy}{dt} = ky$; the quantity increases and the slope becomes steeper as the quantity gets larger.

Exponential decay

Behavior when $k < 0$ in $\frac{dy}{dt} = ky$; the quantity decreases quickly at first and then levels off toward 0 (for $y > 0$).

Instantaneous relative growth rate

$\frac{1}{y}\frac{dy}{dt}$; in an exponential model, it is constant and equals $k$ (an “instantaneous percent rate” as a decimal).

Constant percent change (continuous)

Another description of exponential behavior: the relative growth rate stays constant over time, so $\frac{1}{y}\frac{dy}{dt}=k$.

Separation of variables

A method for solving certain differential equations by rearranging so all y-terms are on one side and all t-terms are on the other (e.g., $\frac{1}{y}dy = k dt$).

Natural logarithm (ln)

The log base $e$; used when integrating $\frac{1}{y}$ (giving $\ln|y|$) and when solving exponential equations for $k$ or $t$.

Constant of integration (C)

The arbitrary constant that appears after integrating; for $\frac{dy}{dt} = ky$ it leads to the solution form $y = Ce^{kt}$.

General solution to $\frac{dy}{dt} = ky$

$y(t) = Ce^{kt}$, including the special case $y=0$ when $C=0$.

Initial condition

A given value of the function at a specific time (e.g., $y(0)=y_0$) used to determine the constant $C$ in the exponential solution.

Finding C from $y(t_0)=y_0$

If $y = Ce^{kt}$ and $y(t_0) = y_0$, then $C = y_0 \times e^{-kt_0}$ (so $C$ equals $y(0)$ only when $t_0=0$).

Using two points to find k (ratio method)

From $y(t)=Ce^{kt}$, using $y(t_1)=y_1$ and $y(t_2)=y_2$ gives $k = \frac{1}{(t_2-t_1)} \cdot \ln\left( \frac{y_2}{y_1} \right)$.

Finding k from instantaneous rate data

If $\frac{dy}{dt} = ky$ and at some time you know both $y$ and $\frac{dy}{dt}$, then $k = \frac{\frac{dy}{dt}}{y}$ (with units “per time”).

Doubling time (T_d)

The time it takes an exponential growth model to multiply by 2; $T_d = \frac{\ln(2)}{k}$ (for $k > 0$).

Half-life (T_{1/2})

The time it takes an exponential decay model to multiply by $\frac{1}{2}$; $T_{1/2} = \frac{-\ln(2)}{k}$ (positive when $k < 0$).

Time-to-reach formula

For $y(t)=Ce^{kt}$, the time to reach $y^*$ is $t = \frac{1}{k} \cdot \ln\left( \frac{y^*}{C} \right)$ (requires logarithms to solve).

Equilibrium (limiting) value (L)

The long-term value that a quantity approaches in models like $\frac{dy}{dt} = k(y−L)$; solutions level off at $y=L$.

Exponential approach to equilibrium

A model of the form $\frac{dy}{dt} = k(y−L)$ where the rate is proportional to the distance from equilibrium; typically $k < 0$ gives a stable approach toward $L$.

Shifted exponential solution

The solution to $\frac{dy}{dt} = k(y−L)$: $y(t)=L+Ce^{kt}$, meaning the exponential behavior occurs in the difference $y−L$.

Newton’s Law of Cooling/Heating

A temperature model stating $\frac{dT}{dt}$ is proportional to the difference between object temperature and ambient temperature: $\frac{dT}{dt} = k(T−T_s)$, usually with $k < 0$.

Ambient (surrounding) temperature (T_s)

The constant surrounding temperature in Newton’s Law of Cooling/Heating; the temperature difference $T−T_s$ decays toward 0.

Concavity of $y=Ce^{kt}$ (for $y > 0$)

Since $\frac{d^2y}{dt^2} = k^2y$, exponential solutions with $y > 0$ are concave up even during decay (the slope is negative but becomes less negative).