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