Calculus

Introduction to Calculus

Calculus is a section of mathematics dealing with continuous change. It encompasses several fundamental concepts: limits, derivatives, integrals, and infinite series. These ideas work together to create a powerful mathematical framework.

The core components of calculus include:
Limits - examining the behavior of functions as they approach specific values
Differential calculus - studying rates of change through derivatives
Integral calculus - analyzing accumulation and total change
Infinite series - representing functions as sums of infinite terms

Differential calculus allows us to find instantaneous rates of change and optimize functions, while integral calculus provides tools for calculating areas, volumes, and accumulated quantities. The connection between these two branches, established by the Fundamental Theorem of Calculus, creates a unified system for analyzing continuous change.

Applications of calculus extend throughout science, engineering, and economics. In physics, it models motion and energy; in engineering, it optimizes designs and processes; in economics, it analyzes rates of growth and market behavior. The subject's precise mathematical framework makes it essential for understanding and describing natural phenomena.

Calculus Formulas

The Calculus Formulas page features fundamental laws and theorems across Limits, Derivatives, Integrals, and Integration Techniques. Each entry includes step-by-step explanations, key variables, worked examples, and real-world applications - from basic limit laws and differentiation rules to advanced integration methods and improper integrals.

Definition of a Limit

\lim_{x \to a} f(x) = L

Limit Laws - Sum Rule

\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)

Limit Laws - Difference Rule

\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)

Limit Laws - Product Rule

\lim_{x \to a} [f(x) \cdot g(x)] = \left(\lim_{x \to a} f(x)\right) \cdot \left(\lim_{x \to a} g(x)\right)

Limit Laws - Quotient Rule

\lim_{x \to a} \left( \frac{f(x)}{g(x)} \right) = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, provided \lim_{x \to a} g(x) \ne 0

Limit Laws - Constant Multiple Rule

\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)

Special Limit of Sine over x

\lim_{x \to 0} \frac{\sin x}{x} = 1

Special Exponential Limit

\lim_{x \to 0} \frac{e^x - 1}{x} = 1

Limit of (1 + 1/n)^n

\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e

Limit of a Constant Function

\lim_{x \to a} c = c

Limit of Identity Function

\lim_{x \to a} x = a

Definition of the Derivative

f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}

Power Rule

\frac{d}{dx}[x^n] = n x^{n-1}

Constant Rule

\frac{d}{dx}[c] = 0

Constant Multiple Rule

\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)

Sum and Difference Rules

\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)

Product Rule

\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)

Quotient Rule

\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}

Chain Rule

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Derivative of Sine Function

\frac{d}{dx}[\sin x] = \cos x

Derivative of Cosine Function

\frac{d}{dx}[\cos x] = -\sin x

Derivative of Tangent Function

\frac{d}{dx}[\tan x] = \sec^2 x

Derivative of Exponential Function

\frac{d}{dx}[e^{x}] = e^{x}

Derivative of a^x

\frac{d}{dx}[a^{x}] = a^{x} \ln a

Derivative of Natural Logarithm

\frac{d}{dx}[\ln x] = \frac{1}{x}

Derivative of Logarithm Base a

\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}

Derivative of Inverse Sine

\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1 - x^2}}

Derivative of Inverse Cosine

\frac{d}{dx}[\arccos x] = \frac{-1}{\sqrt{1 - x^2}}

Derivative of Inverse Tangent

\frac{d}{dx}[\arctan x] = \frac{1}{1 + x^2}

Indefinite Integral (Antiderivative)

\int f(x) \, dx = F(x) + C

Definite Integral

\int_a^b f(x) \, dx = F(b) - F(a)

Basic Integration Rule - Power Rule

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, where n \ne -1

Basic Integration Rule - Constant Multiple

\int c \cdot f(x) \, dx = c \cdot \int f(x) \, dx

Basic Integration Rule - Sum and Difference

\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx

Integral of Sine Function

\int \sin x \, dx = -\cos x + C

Integral of Cosine Function

\int \cos x \, dx = \sin x + C

Integral of Exponential Function

\int e^{x} \, dx = e^{x} + C

Integral of a^x

\int a^{x} \, dx = \frac{a^{x}}{\ln a} + C

Integral of Reciprocal Function

\int \frac{1}{x} \, dx = \ln |x| + C

Fundamental Theorem of Calculus - Part 1

\int_a^b f(x) \, dx = F(b) - F(a)

Fundamental Theorem of Calculus - Part 2

\frac{d}{dx} \left( \int_a^x f(t) \, dt \right) = f(x)

Integration by Substitution (Reverse Chain Rule)

If u = g(x), then \int f(g(x)) g'(x) \, dx = \int f(u) \, du

Integration by Parts

\int u \, dv = u v - \int v \, du

Partial Fractions Decomposition

Decompose \frac{P(x)}{Q(x)} into simpler fractions before integrating

Trigonometric Integrals

Use identities to simplify \int \sin^n x \cos^m x \, dx

Improper Integral

\int_a^{\infty} f(x) \, dx = \lim_{b \to \infty} \int_a^b f(x) \, dx

Average Value of a Function

f_{\text{avg}} = \frac{1}{b - a} \int_a^b f(x) \, dx

Definition of a Limit

\lim_{x \to a} f(x) = L

Limit Laws - Sum Rule

\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)

Limit Laws - Difference Rule

\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)

Limit Laws - Product Rule

\lim_{x \to a} [f(x) \cdot g(x)] = \left(\lim_{x \to a} f(x)\right) \cdot \left(\lim_{x \to a} g(x)\right)

Limit Laws - Quotient Rule

\lim_{x \to a} \left( \frac{f(x)}{g(x)} \right) = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, provided \lim_{x \to a} g(x) \ne 0

Limit Laws - Constant Multiple Rule

\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)

Special Limit of Sine over x

\lim_{x \to 0} \frac{\sin x}{x} = 1

Special Exponential Limit

\lim_{x \to 0} \frac{e^x - 1}{x} = 1

Limit of (1 + 1/n)^n

\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e

Limit of a Constant Function

\lim_{x \to a} c = c

Limit of Identity Function

\lim_{x \to a} x = a

Definition of the Derivative

f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}

Power Rule

\frac{d}{dx}[x^n] = n x^{n-1}

Constant Rule

\frac{d}{dx}[c] = 0

Constant Multiple Rule

\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)

Sum and Difference Rules

\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)

Product Rule

\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)

Quotient Rule

\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}

Chain Rule

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Derivative of Sine Function

\frac{d}{dx}[\sin x] = \cos x

Derivative of Cosine Function

\frac{d}{dx}[\cos x] = -\sin x

Derivative of Tangent Function

\frac{d}{dx}[\tan x] = \sec^2 x

Derivative of Exponential Function

\frac{d}{dx}[e^{x}] = e^{x}

Derivative of a^x

\frac{d}{dx}[a^{x}] = a^{x} \ln a

Derivative of Natural Logarithm

\frac{d}{dx}[\ln x] = \frac{1}{x}

Derivative of Logarithm Base a

\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}

Derivative of Inverse Sine

\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1 - x^2}}

Derivative of Inverse Cosine

\frac{d}{dx}[\arccos x] = \frac{-1}{\sqrt{1 - x^2}}

Derivative of Inverse Tangent

\frac{d}{dx}[\arctan x] = \frac{1}{1 + x^2}

Indefinite Integral (Antiderivative)

\int f(x) \, dx = F(x) + C

Definite Integral

\int_a^b f(x) \, dx = F(b) - F(a)

Basic Integration Rule - Power Rule

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, where n \ne -1

Basic Integration Rule - Constant Multiple

\int c \cdot f(x) \, dx = c \cdot \int f(x) \, dx

Basic Integration Rule - Sum and Difference

\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx

Integral of Sine Function

\int \sin x \, dx = -\cos x + C

Integral of Cosine Function

\int \cos x \, dx = \sin x + C

Integral of Exponential Function

\int e^{x} \, dx = e^{x} + C

Integral of a^x

\int a^{x} \, dx = \frac{a^{x}}{\ln a} + C

Integral of Reciprocal Function

\int \frac{1}{x} \, dx = \ln |x| + C

Fundamental Theorem of Calculus - Part 1

\int_a^b f(x) \, dx = F(b) - F(a)

Fundamental Theorem of Calculus - Part 2

\frac{d}{dx} \left( \int_a^x f(t) \, dt \right) = f(x)

Integration by Substitution (Reverse Chain Rule)

If u = g(x), then \int f(g(x)) g'(x) \, dx = \int f(u) \, du

Integration by Parts

\int u \, dv = u v - \int v \, du

Partial Fractions Decomposition

Decompose \frac{P(x)}{Q(x)} into simpler fractions before integrating

Trigonometric Integrals

Use identities to simplify \int \sin^n x \cos^m x \, dx

Improper Integral

\int_a^{\infty} f(x) \, dx = \lim_{b \to \infty} \int_a^b f(x) \, dx

Average Value of a Function

f_{\text{avg}} = \frac{1}{b - a} \int_a^b f(x) \, dx

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Calculus Terms and Definitions

Abscissa

The horizontal coordinate of a point in a Cartesian plane, representing its distance from the vertical axis.

Absolute Value

A measure of a number's magnitude regardless of its sign, denoting its distance from zero on a number line.

Acceleration

The rate at which an object's velocity changes with respect to time, expressed as a vector indicating direction and magnitude.

Amplitude

The maximum displacement of a periodic function or oscillation from its equilibrium position.

Angle

The rotational measurement between two intersecting lines or surfaces, often quantified in degrees or radians.

Angular Velocity

The rate of change of angular displacement, indicating how quickly an object rotates around a specific axis.

Antiderivative

A function whose derivative equals the given function, often associated with indefinite integration.

Approximation by Differentials

A technique for estimating the change in a function using its derivative and small increments in its variable.

Arclength

The total length of a curve, calculated as the integral of the curve's infinitesimal segments.

Area

A measure of the two-dimensional space enclosed within a boundary, often determined using integrals.

Argument

The independent variable of a function, especially in trigonometric or complex functions, which determines the function's output.

Asymptote

A line that a curve approaches but never intersects or reaches as it extends infinitely in one or both directions.

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.

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.

Average Velocity

The total displacement divided by the total time taken, representing the overall rate of motion over a time interval.

Axis of Revolution

A line about which a two-dimensional shape rotates to generate a three-dimensional solid.

Binomial Series

An infinite expansion of expressions raised to a power, generalizing the binomial theorem for real or complex exponents.

Bounded Sequence

A sequence whose terms are confined within a finite range, having both upper and lower limits.

Cardioid

A heart-shaped curve traced by a point on a circle rolling around another circle of equal radius.

Catenary

The curve formed by a hanging flexible chain or cable under its own weight, described mathematically as a hyperbolic cosine.

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.

Centroid

The geometric center of a plane figure or solid body, often corresponding to the average position of all points within it.

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.

Circle

A set of all points in a plane equidistant from a fixed central point.

Closed Interval

A range of real numbers that includes its endpoints, denoted as [a, b].

Comparison Test

A method for determining the convergence or divergence of a series by comparing it to another series with known behavior.

Composite Function

A function formed by applying one function to the results of another, expressed as f(g(x)).

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.

Conic Sections

Curves obtained by intersecting a plane with a cone, resulting in shapes like circles, ellipses, parabolas, and hyperbolas.

Continuous Function

A function that has no breaks, jumps, or holes in its domain, allowing it to be drawn without lifting the pen.

Convergence

A property of a sequence or series approaching a finite limit as its terms progress to infinity.

Coordinate System

A framework used to locate points in space, typically defined by axes such as Cartesian, polar, or cylindrical coordinates.

Curvature

A measure of how sharply a curve deviates from being a straight line at a given point.

Differential Equations

Equations involving derivatives that describe how a function changes in relation to its variables.

Direction Field

A graphical representation showing the slope of solutions to a differential equation at various points.

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.

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.

Ellipse

A closed curve in which the sum of the distances from any point on the curve to two fixed points (foci) is constant.

Fourier Series

A representation of periodic functions as an infinite sum of sines and cosines, each with specific coefficients.

Gradient

A vector indicating the direction of the steepest ascent of a scalar field, derived from its partial derivatives.

Harmonic Motion

Oscillatory motion, such as that of a pendulum, described mathematically by sinusoidal functions.

Hyperbola

A curve formed by the intersection of a plane with a double cone, characterized by two separate branches.

Inflection Point

A point on a curve where the concavity changes direction.

Integrand

The function being integrated in an integral expression.

Limit

The value a function or sequence approaches as the input or index approaches a specific point or infinity.

Local Extremum

The highest or lowest value of a function in a specific region, occurring at a local maximum or minimum.

Parabola

A symmetric curve where any point is equidistant from a fixed focus and a directrix.

Parametric Equations

Equations defining a curve using a parameter to express coordinates as functions of that parameter.

Partial Derivative

The derivative of a multivariable function with respect to one variable while keeping others constant.

Path Integral

An integral that computes a quantity along a curve, often used in physics for work or line integrals.

Polar Coordinates

A coordinate system where a point's location is determined by its distance from the origin and angle from a reference direction.

Radius of Curvature

The reciprocal of curvature, representing the radius of the osculating circle at a point on a curve.

Rectilinear Motion

Motion along a straight line, described by functions of time.

Series

The sum of terms in a sequence, which may converge to a finite value or diverge to infinity.

Slope

The ratio of the vertical change to the horizontal change between two points on a line or curve.

Surface Area

The total area covering the outer surface of a three-dimensional object, computed using integration for curved surfaces.

Tangent Line/Plane

A line or plane that touches a curve or surface at a single point without crossing it locally.

Velocity

The rate of change of an object’s position with respect to time, described as a vector quantity.

Vector Fields

A function assigning a vector to every point in space, used to model physical phenomena like fluid flow or gravitational fields.

Abscissa

The horizontal coordinate of a point in a Cartesian plane, representing its distance from the vertical axis.

Absolute Value

A measure of a number's magnitude regardless of its sign, denoting its distance from zero on a number line.

Acceleration

The rate at which an object's velocity changes with respect to time, expressed as a vector indicating direction and magnitude.

Amplitude

The maximum displacement of a periodic function or oscillation from its equilibrium position.

Angle

The rotational measurement between two intersecting lines or surfaces, often quantified in degrees or radians.

Angular Velocity

The rate of change of angular displacement, indicating how quickly an object rotates around a specific axis.

Antiderivative

A function whose derivative equals the given function, often associated with indefinite integration.

Approximation by Differentials

A technique for estimating the change in a function using its derivative and small increments in its variable.

Arclength

The total length of a curve, calculated as the integral of the curve's infinitesimal segments.

Area

A measure of the two-dimensional space enclosed within a boundary, often determined using integrals.

Argument

The independent variable of a function, especially in trigonometric or complex functions, which determines the function's output.

Asymptote

A line that a curve approaches but never intersects or reaches as it extends infinitely in one or both directions.

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.

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.

Average Velocity

The total displacement divided by the total time taken, representing the overall rate of motion over a time interval.

Axis of Revolution

A line about which a two-dimensional shape rotates to generate a three-dimensional solid.

Binomial Series

An infinite expansion of expressions raised to a power, generalizing the binomial theorem for real or complex exponents.

Bounded Sequence

A sequence whose terms are confined within a finite range, having both upper and lower limits.

Cardioid

A heart-shaped curve traced by a point on a circle rolling around another circle of equal radius.

Catenary

The curve formed by a hanging flexible chain or cable under its own weight, described mathematically as a hyperbolic cosine.

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.

Centroid

The geometric center of a plane figure or solid body, often corresponding to the average position of all points within it.

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.

Circle

A set of all points in a plane equidistant from a fixed central point.

Closed Interval

A range of real numbers that includes its endpoints, denoted as [a, b].

Comparison Test

A method for determining the convergence or divergence of a series by comparing it to another series with known behavior.

Composite Function

A function formed by applying one function to the results of another, expressed as f(g(x)).

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.

Conic Sections

Curves obtained by intersecting a plane with a cone, resulting in shapes like circles, ellipses, parabolas, and hyperbolas.

Continuous Function

A function that has no breaks, jumps, or holes in its domain, allowing it to be drawn without lifting the pen.

Convergence

A property of a sequence or series approaching a finite limit as its terms progress to infinity.

Coordinate System

A framework used to locate points in space, typically defined by axes such as Cartesian, polar, or cylindrical coordinates.

Curvature

A measure of how sharply a curve deviates from being a straight line at a given point.

Differential Equations

Equations involving derivatives that describe how a function changes in relation to its variables.

Direction Field

A graphical representation showing the slope of solutions to a differential equation at various points.

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.

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.

Ellipse

A closed curve in which the sum of the distances from any point on the curve to two fixed points (foci) is constant.

Fourier Series

A representation of periodic functions as an infinite sum of sines and cosines, each with specific coefficients.

Gradient

A vector indicating the direction of the steepest ascent of a scalar field, derived from its partial derivatives.

Harmonic Motion

Oscillatory motion, such as that of a pendulum, described mathematically by sinusoidal functions.

Hyperbola

A curve formed by the intersection of a plane with a double cone, characterized by two separate branches.

Inflection Point

A point on a curve where the concavity changes direction.

Integrand

The function being integrated in an integral expression.

Limit

The value a function or sequence approaches as the input or index approaches a specific point or infinity.

Local Extremum

The highest or lowest value of a function in a specific region, occurring at a local maximum or minimum.

Parabola

A symmetric curve where any point is equidistant from a fixed focus and a directrix.

Parametric Equations

Equations defining a curve using a parameter to express coordinates as functions of that parameter.

Partial Derivative

The derivative of a multivariable function with respect to one variable while keeping others constant.

Path Integral

An integral that computes a quantity along a curve, often used in physics for work or line integrals.

Polar Coordinates

A coordinate system where a point's location is determined by its distance from the origin and angle from a reference direction.

Radius of Curvature

The reciprocal of curvature, representing the radius of the osculating circle at a point on a curve.

Rectilinear Motion

Motion along a straight line, described by functions of time.

Series

The sum of terms in a sequence, which may converge to a finite value or diverge to infinity.

Slope

The ratio of the vertical change to the horizontal change between two points on a line or curve.

Surface Area

The total area covering the outer surface of a three-dimensional object, computed using integration for curved surfaces.

Tangent Line/Plane

A line or plane that touches a curve or surface at a single point without crossing it locally.

Velocity

The rate of change of an object’s position with respect to time, described as a vector quantity.

Vector Fields

A function assigning a vector to every point in space, used to model physical phenomena like fluid flow or gravitational fields.

See All Definitions

The Calculus Terms and Definitions page provides a comprehensive collection of essential calculus concepts organized across multiple categories including Functions, Differentiation, Integration, Geometry, Motion and Dynamics, and Vector Calculus. From fundamental concepts like derivatives and integrals to advanced topics in vector analysis and differential equations, each term is clearly defined to support understanding of calculus principles and their applications.

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Limits, Derivatives, and Integrals

Limits play a central role in defining both derivatives and integrals.They provide the precise language for handling values that are approached but not necessarily reached.
A derivative is defined as the limit of the average rate of change while the interval shrinks to zero and, in such a way, indicates how function changes at a specific point.
An integral, on the other hand, measures accumulation—such as area under a curve or total distance traveled. The concept of a limit is also used here, serving as the key tool for defining the integral through an infinite sum of infinitesimally small quantities.
What ties them all together is the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse processes: taking the derivative of an integral function returns the original function, and integrating a derivative recovers accumulated change.
In essence, limits allow us to rigorously define both change (via derivatives) and accumulation (via integrals), revealing a deep unity among these core ideas of calculus.

Calculus Symbols Reference

Our Calculus Symbols page offers a detailed catalog of notation used in differential and integral calculus. This comprehensive resource organizes symbols by their mathematical functions to help students and professionals navigate the language of calculus.
The reference covers essential notation across categories including differentiation (f'(x), df/dx, ∇f), integration (∫, ∬, ∮), limits (limₓ→c), and infinite series (∑). Advanced topics include vector calculus notation for divergence and curl, differential operators like the Laplacian (∇²f), and specialized notation for curvature and differential equations.
Each symbol is presented with its proper LaTeX representation and a concise explanation of its meaning, making this an essential resource for anyone working with calculus concepts in academic or professional settings.

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