Limits
Limit
The value that a function approaches as the input approaches some value. Foundation of differential and integral calculus.
Read More →Calculus
The study of continuous change. The mathematics of motion, growth, and accumulation.
Browse Calculus Topics
All Calculus Topics
Limits
Continuity
A function is continuous if small changes in input result in small changes in output. No breaks, jumps, or holes in the graph.
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Limits
Infinite Limits
Understanding limits at infinity and infinite limits. Essential for analyzing function behavior over large domains.
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Derivatives
Derivative
Measures the rate at which a function changes at a given point. Represents the slope of the tangent line to a curve.
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Derivatives
Differentiation Rules
Power rule, product rule, quotient rule, and chain rule. Essential techniques for finding derivatives efficiently.
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Derivatives
Implicit Differentiation
Technique for finding derivatives when y is not explicitly defined as a function of x. Used for curves and relations.
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Derivatives
Applications of Derivatives
Optimization, related rates, curve sketching, and motion problems. Real-world applications of differential calculus.
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Derivatives
Higher Order Derivatives
Second, third, and nth derivatives. Applications to acceleration, concavity, and Taylor series.
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Integrals
Integral
A mathematical object that can be interpreted as an area or a generalization of area. Fundamental to calculus and analysis.
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Integrals
Definite Integral
Represents the signed area under a curve between two points. Evaluated using the Fundamental Theorem of Calculus.
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Integrals
Indefinite Integral
The antiderivative of a function. A family of functions whose derivative is the original function.
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Integrals
Integration Techniques
Substitution, integration by parts, partial fractions, and trigonometric integrals. Methods for evaluating complex integrals.
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Integrals
Applications of Integrals
Areas, volumes, arc lengths, work, and centroids. Real-world problems solved using integral calculus.
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Differential Equations
Differential Equation
An equation involving derivatives of a function. Models rates of change in physics, engineering, and biology.
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Differential Equations
Separable Equations
First-order equations where variables can be separated. Solved by integrating each side independently.
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Differential Equations
Linear Differential Equations
Equations linear in the unknown function and its derivatives. Solved using integrating factors or characteristic equations.
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Differential Equations
Second Order Equations
Equations involving second derivatives. Essential for modeling oscillations, vibrations, and electrical circuits.
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Series
Sequences and Series
Infinite sums and their convergence. Foundation for power series, Taylor series, and Fourier analysis.
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Series
Convergence Tests
Tests for determining if a series converges or diverges: comparison, ratio, root, and integral tests.
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Series
Power Series
Infinite series of the form Σaₙ(x-c)ⁿ. Used to represent functions as infinite polynomials with radius of convergence.
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Series
Taylor Series
Representation of a function as an infinite sum of terms calculated from its derivatives at a single point.
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Multivariable
Partial Derivatives
Derivatives of multivariable functions with respect to one variable while holding others constant.
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Multivariable
Multiple Integrals
Double and triple integrals for functions of several variables. Used to compute volumes, mass, and centroids.
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Multivariable
Gradient
A vector of partial derivatives pointing in the direction of steepest ascent. Essential for optimization.
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Multivariable
Vector Calculus
Line integrals, surface integrals, divergence, curl, and Stokes' theorem. Essential for physics and engineering.
Full article →Branches of Calculus
Calculus divides into differential calculus (rates of change, derivatives, optimization) and integral calculus (accumulation, areas, volumes). The Fundamental Theorem of Calculus unifies them: differentiation and integration are inverse operations.
Multivariable calculus extends both branches to functions of several variables. Partial derivatives, gradients, and directional derivatives generalize the derivative; double and triple integrals generalize the definite integral. Vector calculus adds line integrals, surface integrals, and the theorems of Green, Stokes, and Gauss.
Differential equations — ordinary and partial — apply calculus to model dynamic systems. From Newton's laws of motion to Maxwell's equations of electromagnetism, the language of science is differential equations built from calculus.
Numerical analysis provides algorithms for computing derivatives and integrals when closed forms are unavailable. Finite difference methods approximate derivatives; quadrature rules (Simpson's, Gaussian) approximate integrals; Runge–Kutta methods solve differential equations numerically.