Calculus Terms and Definitions
Coordinates And Geometry
Abscissa
Definition:
The horizontal coordinate of a point in a Cartesian plane, representing its distance from the vertical axis.
Cardioid
Definition:
A heart-shaped curve traced by a point on a circle rolling around another circle of equal radius.
Circle
Definition:
A set of all points in a plane equidistant from a fixed central point.
Coordinate System
Definition:
A framework used to locate points in space, typically defined by axes such as Cartesian, polar, or cylindrical coordinates.
Polar Coordinates
Definition:
A coordinate system where a point's location is determined by its distance from the origin and angle from a reference direction.
Functions And Rates Of Change
Absolute Value
Definition:
A measure of a number's magnitude regardless of its sign, denoting its distance from zero on a number line.
Average Rate Of Change
Definition:
The ratio of the change in a function's value to the change in the input variable, representing a slope over an interval.
Motion And Dynamics
Acceleration
Definition:
The rate at which an object's velocity changes with respect to time, expressed as a vector indicating direction and magnitude.
Angular Velocity
Definition:
The rate of change of angular displacement, indicating how quickly an object rotates around a specific axis.
Average Velocity
Definition:
The total displacement divided by the total time taken, representing the overall rate of motion over a time interval.
Rectilinear Motion
Definition:
Motion along a straight line, described by functions of time.
Velocity
Definition:
The rate of change of an object’s position with respect to time, described as a vector quantity.
Oscillatory Motion
Amplitude
Definition:
The maximum displacement of a periodic function or oscillation from its equilibrium position.
Harmonic Motion
Definition:
Oscillatory motion, such as that of a pendulum, described mathematically by sinusoidal functions.
Geometry And Trigonometry
Angle
Definition:
The rotational measurement between two intersecting lines or surfaces, often quantified in degrees or radians.
Catenary
Definition:
The curve formed by a hanging flexible chain or cable under its own weight, described mathematically as a hyperbolic cosine.
Integration
Antiderivative
Definition:
A function whose derivative equals the given function, often associated with indefinite integration.
Integrand
Definition:
The function being integrated in an integral expression.
Path Integral
Definition:
An integral that computes a quantity along a curve, often used in physics for work or line integrals.
Differentiation
Approximation By Differentials
Definition:
A technique for estimating the change in a function using its derivative and small increments in its variable.
Chain Rule
Definition:
A fundamental differentiation rule that allows the derivative of a composite function to be expressed in terms of the derivatives of its components.
Partial Derivative
Definition:
The derivative of a multivariable function with respect to one variable while keeping others constant.
Geometry And Calculus
Arclength
Definition:
The total length of a curve, calculated as the integral of the curve's infinitesimal segments.
Area
Definition:
A measure of the two-dimensional space enclosed within a boundary, often determined using integrals.
Axis Of Revolution
Definition:
A line about which a two-dimensional shape rotates to generate a three-dimensional solid.
Functions
Argument
Definition:
The independent variable of a function, especially in trigonometric or complex functions, which determines the function's output.
Composite Function
Definition:
A function formed by applying one function to the results of another, expressed as f(g(x)).
Continuous Function
Definition:
A function that has no breaks, jumps, or holes in its domain, allowing it to be drawn without lifting the pen.
Domain And Range
Definition:
The domain is the set of all input values for which a function is defined, and the range is the set of all resulting output values.
Graph Analysis
Asymptote
Definition:
A line that a curve approaches but never intersects or reaches as it extends infinitely in one or both directions.
Concavity
Definition:
A property describing the curvature of a graph, where a function is concave up if its slope increases and concave down if it decreases.
Inflection Point
Definition:
A point on a curve where the concavity changes direction.
Slope
Definition:
The ratio of the vertical change to the horizontal change between two points on a line or curve.
Tangent Line/Plane
Definition:
A line or plane that touches a curve or surface at a single point without crossing it locally.
Functions And Integration
Average Value Of A Function
Definition:
The mean of a function's values over a specific interval, calculated as the integral of the function divided by the interval's length.
Series And Convergence
Binomial Series
Definition:
An infinite expansion of expressions raised to a power, generalizing the binomial theorem for real or complex exponents.
Comparison Test
Definition:
A method for determining the convergence or divergence of a series by comparing it to another series with known behavior.
Convergence
Definition:
A property of a sequence or series approaching a finite limit as its terms progress to infinity.
Fourier Series
Definition:
A representation of periodic functions as an infinite sum of sines and cosines, each with specific coefficients.
Series
Definition:
The sum of terms in a sequence, which may converge to a finite value or diverge to infinity.
Sequences And Series
Bounded Sequence
Definition:
A sequence whose terms are confined within a finite range, having both upper and lower limits.
Geometry
Center Of Curvature
Definition:
The point at a given location on a curve where the osculating circle is centered, representing the curve's maximum bending at that point.
Centroid
Definition:
The geometric center of a plane figure or solid body, often corresponding to the average position of all points within it.
Conic Sections
Definition:
Curves obtained by intersecting a plane with a cone, resulting in shapes like circles, ellipses, parabolas, and hyperbolas.
Curvature
Definition:
A measure of how sharply a curve deviates from being a straight line at a given point.
Ellipse
Definition:
A closed curve in which the sum of the distances from any point on the curve to two fixed points (foci) is constant.
Hyperbola
Definition:
A curve formed by the intersection of a plane with a double cone, characterized by two separate branches.
Parabola
Definition:
A symmetric curve where any point is equidistant from a fixed focus and a directrix.
Radius Of Curvature
Definition:
The reciprocal of curvature, representing the radius of the osculating circle at a point on a curve.
Surface Area
Definition:
The total area covering the outer surface of a three-dimensional object, computed using integration for curved surfaces.
Intervals And Sets
Closed Interval
Definition:
A range of real numbers that includes its endpoints, denoted as [a, b].
Differential Equations
Differential Equations
Definition:
Equations involving derivatives that describe how a function changes in relation to its variables.
Direction Field
Definition:
A graphical representation showing the slope of solutions to a differential equation at various points.
Vector Calculus
Divergence
Definition:
A measure of how a vector field spreads out from a point, often computed as the dot product of the del operator with the field.
Gradient
Definition:
A vector indicating the direction of the steepest ascent of a scalar field, derived from its partial derivatives.
Vector Fields
Definition:
A function assigning a vector to every point in space, used to model physical phenomena like fluid flow or gravitational fields.
Limits
Limit
Definition:
The value a function or sequence approaches as the input or index approaches a specific point or infinity.
Functions And Graphs
Local Extremum
Definition:
The highest or lowest value of a function in a specific region, occurring at a local maximum or minimum.
Graphing
Parametric Equations
Definition:
Equations defining a curve using a parameter to express coordinates as functions of that parameter.