Glossary
Constant of Proportionality
The constant $k$ that relates two quantities in a proportional relationship, indicating the specific ratio or scaling factor between them.
Example:
If the rate at which a substance decays is proportional to the amount present, the decay constant of proportionality determines how quickly the substance diminishes.
Differential Equations
Equations that involve derivatives and represent the relationship between a function and its rate of change, helping to understand how functions change with respect to an independent variable.
Example:
The equation is a differential equation that models population growth, where the rate of change of the population is proportional to the current population size.
Directly Proportionality
A relationship where if quantity $a$ is directly proportional to quantity $b$, then $a = kb$, where $k$ is a constant.
Example:
The amount of simple interest earned on an investment is directly proportional to the principal amount, assuming a fixed interest rate and time period.
Inversely Proportionality
A relationship where if quantity $a$ is inversely proportional to quantity $b$, then $a = \frac{k}{b}$, where $k$ is a constant.
Example:
The intensity of light from a point source is inversely proportional to the square of the distance from the source.
Proportionality
The concept that two quantities vary in a consistent way with respect to each other, forming the basis for many differential equations.
Example:
In Hooke's Law, the force exerted by a spring exhibits proportionality to the distance it is stretched or compressed from its equilibrium position.
Rate of Change
How one quantity changes with respect to another, often represented by a derivative in calculus.
Example:
The rate of change of the volume of water in a tank with respect to time tells us how quickly the tank is filling or emptying.