| 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. |
| dx/dt | The derivative of x with respect to the parameter t; the rate of change of the x-coordinate as the parameter changes. |
| dy/dt | The derivative of y with respect to the parameter t; the rate of change of the y-coordinate as the parameter changes. |
| dy/dx | Leibniz notation for the derivative of y with respect to x. |
| parametric function | Functions where x and y coordinates are each expressed as separate functions of a third variable, typically time (t), rather than y as a function of x. |
| tangent line | A line that touches a curve at a single point and has a slope equal to the derivative of the function at that point. |
| 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. |
| parametric function | Functions where x and y coordinates are each expressed as separate functions of a third variable, typically time (t), rather than y as a function of x. |
| second derivative | The derivative of the first derivative, denoted f'', which describes the concavity of a function and indicates where it is concave up or concave down. |
| Term | Definition |
|---|---|
| arc length | The distance along a curve between two points, calculated using a definite integral. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| parametric function | Functions where x and y coordinates are each expressed as separate functions of a third variable, typically time (t), rather than y as a function of x. |
| 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. |
| vector-valued function | Functions that output vectors rather than scalar values, where each component is a function of the same independent variable. |
| Term | Definition |
|---|---|
| initial condition | Specified values of a function at particular points that determine which particular solution to a differential equation is selected. |
| parametric function | Functions where x and y coordinates are each expressed as separate functions of a third variable, typically time (t), rather than y as a function of x. |
| rate vector | A vector-valued function that describes the rate of change of position with respect to time, representing velocity or acceleration. |
| vector-valued function | Functions that output vectors rather than scalar values, where each component is a function of the same independent variable. |
| Term | Definition |
|---|---|
| acceleration | The derivative of the velocity function with respect to time, representing the rate of change of velocity for a moving particle. |
| displacement | The net change in position of a particle over a time interval, found by integrating the velocity vector. |
| parametric function | Functions where x and y coordinates are each expressed as separate functions of a third variable, typically time (t), rather than y as a function of x. |
| planar motion | The movement of a particle in a two-dimensional plane, described using parametric or vector-valued functions. |
| speed | The magnitude of the velocity vector, representing the rate at which a particle is moving without regard to direction. |
| total distance traveled | The total length of the path traveled by a particle over a time interval, found by integrating the speed. |
| vector-valued function | Functions that output vectors rather than scalar values, where each component is a function of the same independent variable. |
| velocity | The derivative of a position function with respect to time, representing the rate and direction of change of position for a moving particle. |
| 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. |
| polar coordinates | A coordinate system in which points are located by their distance from a fixed point (the pole) and an angle measured from a fixed direction (the polar axis). |
| polar equation | An equation that describes a curve using polar coordinates, typically in the form r = f(θ). |
| Term | Definition |
|---|---|
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| polar coordinates | A coordinate system in which points are located by their distance from a fixed point (the pole) and an angle measured from a fixed direction (the polar axis). |
| polar curve | Curves defined by equations in polar coordinates, where points are located by a distance r from the origin and an angle θ from the positive x-axis. |
| rectangular coordinates | A coordinate system in which points are located using perpendicular x and y axes, also known as Cartesian coordinates. |
| Term | Definition |
|---|---|
| areas of regions | The measure of the two-dimensional space enclosed by one or more curves. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| polar curve | Curves defined by equations in polar coordinates, where points are located by a distance r from the origin and an angle θ from the positive x-axis. |