| Term | Definition |
|---|---|
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| differential equation | An equation that relates a function to its derivatives, describing how a quantity changes in relation to one or more variables. |
| function | A mathematical relationship that assigns exactly one output value to each input value of an independent variable. |
| independent variable | The input variable of a function, typically represented as x, with respect to which the rate of change is measured. |
| Term | Definition |
|---|---|
| differential equation | An equation that relates a function to its derivatives, describing how a quantity changes in relation to one or more variables. |
| general solution | The complete family of solutions to a differential equation, containing arbitrary constants that represent all possible particular solutions. |
| solution | A function that satisfies a differential equation when substituted into it along with its derivatives. |
| verify | To confirm that a proposed function satisfies a differential equation by substituting it and its derivatives into the equation. |
| Term | Definition |
|---|---|
| differential equation | An equation that relates a function to its derivatives, describing how a quantity changes in relation to one or more variables. |
| first-order differential equations | Differential equations that involve only the first derivative of a function. |
| slope field | A graphical representation of a differential equation showing the slope of solution curves at a finite set of points in the plane. |
| solutions to differential equations | Functions that satisfy a given differential equation when substituted into it. |
| Term | Definition |
|---|---|
| differential equation | An equation that relates a function to its derivatives, describing how a quantity changes in relation to one or more variables. |
| solution | A function that satisfies a differential equation when substituted into it along with its derivatives. |
| Term | Definition |
|---|---|
| differential equation | An equation that relates a function to its derivatives, describing how a quantity changes in relation to one or more variables. |
| Euler's method | A numerical procedure for approximating solutions to differential equations by using tangent line segments to estimate values at successive points along a solution curve. |
| solution curve | A graph representing the solution to a differential equation, showing how the dependent variable changes with respect to the independent variable. |
| Term | Definition |
|---|---|
| antidifferentiation | The process of finding a function whose derivative is a given function; the reverse operation of differentiation, also known as integration. |
| differential equation | An equation that relates a function to its derivatives, describing how a quantity changes in relation to one or more variables. |
| general solution | The complete family of solutions to a differential equation, containing arbitrary constants that represent all possible particular solutions. |
| separation of variables | A method for solving 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 other side. |
| Term | Definition |
|---|---|
| differential equation | An equation that relates a function to its derivatives, describing how a quantity changes in relation to one or more variables. |
| domain restrictions | Limitations on the set of input values for which a solution to a differential equation is valid or defined. |
| general solution | The complete family of solutions to a differential equation, containing arbitrary constants that represent all possible particular solutions. |
| initial condition | Specified values of a function at particular points that determine which particular solution to a differential equation is selected. |
| particular solution | A specific solution to a differential equation obtained by using initial conditions to determine the values of arbitrary constants. |
| Term | Definition |
|---|---|
| differential equation | An equation that relates a function to its derivatives, describing how a quantity changes in relation to one or more variables. |
| exponential decay | A process in which a quantity decreases at a rate proportional to its current size, modeled by dy/dt = ky where k < 0. |
| exponential growth | A process in which a quantity increases at a rate proportional to its current size, modeled by dy/dt = ky where k > 0. |
| exponential growth and decay model | A differential equation of the form dy/dt = ky that models quantities that increase or decrease at a rate proportional to their current amount. |
| general solution | The complete family of solutions to a differential equation, containing arbitrary constants that represent all possible particular solutions. |
| initial condition | Specified values of a function at particular points that determine which particular solution to a differential equation is selected. |
| particular solution | A specific solution to a differential equation obtained by using initial conditions to determine the values of arbitrary constants. |
| proportional | A relationship between two quantities where one is a constant multiple of the other. |
| rate of change | The measure of how quickly a quantity changes with respect to another variable, often time. |
| Term | Definition |
|---|---|
| carrying capacity | The maximum value that a population or quantity can sustain in a logistic growth model, represented by the limiting value as time approaches infinity. |
| dependent variable | The variable in a differential equation whose value depends on the independent variable and whose rate of change is being described. |
| independent variable | The input variable of a function, typically represented as x, with respect to which the rate of change is measured. |
| initial condition | Specified values of a function at particular points that determine which particular solution to a differential equation is selected. |
| jointly proportional | A relationship where one quantity is proportional to the product of two or more other quantities. |
| limiting value | The value that a function approaches as the independent variable approaches infinity, representing the long-term behavior of the system. |
| logistic differential equation | A differential equation of the form dy/dt = ky(a - y) that models logistic growth, where the rate of change depends on both the current quantity and the difference from carrying capacity. |
| logistic growth model | A mathematical model describing population or quantity growth that accounts for limited resources, where growth rate depends on both the current size and the difference from carrying capacity. |
| rate of change | The measure of how quickly a quantity changes with respect to another variable, often time. |