(11) PC I Chapter 3.5: Exponential Growth and Decay; Modeling Data

Exponential Growth Model

A mathematical model representing a quantity increasing at a rate proportional to its current value, commonly expressed as y = a₀ e^(kt).

Exponential Decay Model

A mathematical model representing a quantity decreasing at a rate proportional to its current value, typically in the form of y = a₀ e^(kt) where k is negative.

a₀ (Initial Value)

The starting value or amount in an exponential function at time t = 0.

k (Growth Rate Constant)

A constant in exponential equations that represents the rate of growth (k > 0) or decay (k < 0).

Half-Life

The time required for a quantity to reduce to half its initial value, often used in decay problems.

Logistic Growth Model

A model describing growth that approaches a maximum limit or carrying capacity, represented by the formula P(t) = c / (1 + ae^(-bt)).

Natural Exponential Constant (e)

An irrational number approximately equal to 2.718, used as the base for natural logarithms and exponential functions.

Newton's Law of Cooling

A principle describing the rate at which an object cools, expressed as T(t) = C + T₀ - C e^(kt), where C is the surrounding temperature.

Continuous Growth

Exponential growth occurring continuously over time, as opposed to discrete intervals.

Range of Exponential Functions

The set of possible output values for an exponential function, often from zero to infinity.

Horizontal Asymptote

A horizontal line that a graph approaches but does not touch, typically representing a value the function approaches over time.

Logarithmic Function

The inverse of the exponential function, often used in relation to problems involving exponential growth or decay.

Population Model

A mathematical representation of the growth or decline of a population over time, often using exponential functions.

Natural Logarithm (ln)

The logarithm to the base e, used frequently in calculus and exponential growth calculations.

Graphing Characteristics of Exponentials

Includes features such as passing the horizontal line test, having a horizontal asymptote at y = 0, and a y-intercept at (0, a₀).

Decay Rate

The rate at which a substance decreases over time, commonly expressed as a negative growth rate constant in decay models.