Study Notes: Exponential Growth and Decay in AP Calculus AB

Exponential Models

Mathematical models where the rate of change of a quantity is directly proportional to the current amount of that quantity.

Differential Equation

An equation that relates a function to its derivatives, used in exponential models as dy/dt = ky.

Constant of Proportionality

The constant 'k' in the equation dy/dt = ky, indicating the proportional relationship.

Separation of Variables

A technique used to solve differential equations by rearranging them into a form that can be integrated.

Exponential Growth

Occurs when k > 0, leading to unlimited increase in the quantity over time.

Exponential Decay

Occurs when k < 0, leading to a decrease in the quantity over time.

General Solution

The complete form of the solution to a differential equation, given as y(t) = y_0 e^(kt).

Initial Value

The value of y at time t = 0, represented as y_0 in the exponential model.

Half-Life

The time it takes for a quantity to reduce to half its initial value, related to the decay constant k.

Growth Constant

A positive constant 'k' in the context of exponential growth.

Decay Constant

A negative constant 'k' in the context of exponential decay.

Physical Quantity

A measurable entity, such as population or mass, represented by 'y' in an exponential model.

Instantaneous Rate of Change

The rate of change of y with respect to time, denoted as dy/dt.

Bacterial Growth

An example problem where the change in bacteria is modeled using the exponential growth equation.

Radioactive Decay

A scenario modeled by an exponential decay equation, illustrating how a substance loses mass over time.

Exponential Function

A function of the form y = Ce^(kt) that exhibits a constant proportional rate of growth or decay.

Proportional Rate

When the rate of change is directly related to the current amount of a quantity.

k > 0

Indicates exponential growth in a model.

k < 0

Indicates exponential decay in a model.

Algebraic Errors

Common mistakes made when manipulating exponential equations, particularly involving logarithms.

Natural Logarithm

The logarithm to the base e, often used in solving equations involving exponential functions.

Applications of Exponential Growth

Includes scenarios like unrestrained population growth and compounded interest.

Applications of Exponential Decay

Includes scenarios such as radioactive decay and drug elimination.

Integration Constant

The constant added during integration, representing the initial value in an exponential model.

Misreading Problems

A common mistake in identifying whether the rate of change is proportional to y or t.

Limit of y as t approaches infinity

The behavior of y in growth, leading to infinity, versus decay, leading to zero.

Graph of Exponential Functions

Visual representation comparing the curves of exponential growth and decay.