Glossary
Constant of Integration ($C$)
An arbitrary constant that arises when finding the indefinite integral of a function, representing the family of all possible antiderivatives.
Example:
When you integrate , the result is , where C is the constant of integration that accounts for any constant term that would disappear upon differentiation.
Constant of Proportionality ($k$)
A constant in a differential equation like $dy/dt = ky$ that determines the rate and nature (growth or decay) of the change.
Example:
In a population growth model, a positive k indicates that the population is increasing, while a negative k would signify a decline.
Differential Equation
An equation that relates a function with its derivatives, describing how a quantity changes with respect to one or more independent variables.
Example:
The equation is a fundamental differential equation that models continuous exponential change, like the rate at which a rumor spreads.
Euler's Number ($e$)
An irrational mathematical constant approximately equal to 2.71828, fundamental in calculus for describing continuous growth and decay processes.
Example:
The formula for continuous compound interest, , directly uses Euler's number to model financial growth.
Exponential Growth/Decay Equation ($y = y_0 \cdot e^{kt}$)
The general solution to the differential equation $dy/dt = ky$, where $y_0$ is the initial amount, $k$ is the growth/decay constant, and $t$ is time.
Example:
If a radioactive substance has a half-life, its decay can be modeled by the exponential growth/decay equation where k would be negative.
Exponential Models
Mathematical functions used to describe quantities that grow or shrink at a rate proportional to their current size, often seen in scenarios like population growth or radioactive decay.
Example:
The spread of a new viral trend on social media can be accurately predicted using an exponential model, showing rapid initial growth.
Initial Condition ($y_0$)
A specific value of the dependent variable at a given starting point (often when the independent variable, like time, is zero), used to determine the unique solution to a differential equation.
Example:
If a bacterial culture starts with 100 cells, then 100 is the initial condition () for the population growth model.
Integration
The process of finding the antiderivative of a function, often used to solve differential equations by reversing the differentiation process.
Example:
After separating variables in a differential equation, you perform integration on both sides to find the original function.
Rate of Change ($dy/dt$)
Represents how quickly a quantity ($y$) is changing with respect to another variable, typically time ($t$).
Example:
If a car's speed is increasing, its acceleration is the rate of change of its velocity, expressed as .
Separation of Variables
A technique used to solve certain differential equations by rearranging the equation so that all terms involving one variable are on one side and all terms involving the other variable are on the opposite side.
Example:
To solve , you would use separation of variables to get before integrating.