Topic

Calculus

198 articles

Work Done by a Force: Integral Formula and Examples

Work done by a force that changes along the path is W = ∫ₐ b F(x),dx. This guide derives the integral, then solves springs, chains and pumping tanks step by step.

Work Done by a Force: Integral Formula and Examples

Surface Area of Revolution: Formula and Examples

Surface area of revolution is circumference times arc length, added along the curve: S = 2π∫ₐ b y,ds. This guide derives it, covers both axes and parametric curves, and handles integrals with no closed form.

Volumes by Slicing: Cross Sections and Examples

Volumes by slicing finds any solid's volume by integrating its cross-section area: V = ∫ₐ b A(x),dx. This guide covers square, triangle and semicircle slices, Cavalieri's principle, and six worked examples.

Sum and Difference Rule for Derivatives and Integrals

The sum and difference rule says the derivative of a sum is the sum of the derivatives. This guide proves it, extends it to integrals, works six examples, and shows where products break it.

Quadratic Approximation: Formula, Error, Examples

Quadratic approximation swaps a curve near x = a for the parabola matching its value, slope and bend. Inside: the formula, the reason for the 1/2, error size, and six checked examples.

One-Sided Limits: Left-Hand and Right-Hand Limits

A one-sided limit tracks f(x) from one side only, and the full limit exists exactly when the left and right values match. This guide covers one-sided limits notation, piecewise, absolute value and floor cases.

Notation for Differentiation: Leibniz, Lagrange, Newton

One derivative, four ways to write it. This guide to notation for differentiation compares Leibniz, Lagrange, Newton and Euler notation, with higher orders, partials, conversions and worked examples.

Modeling with Differential Equations: Steps and Examples

To model with a differential equation, write the rate of change as a rule, such as rate in minus rate out. This guide to modeling with differential equations builds, solves and checks five classic models.

List of Integrals of Trigonometric Functions: Full Table

Every row here reverses a derivative, an identity or a substitution. This list of integrals of trigonometric functions covers the six basics, squares, powers, products, reduction formulas and 1/(a + bsin x) forms.

List of Integrals of Logarithmic Functions With Proofs

Every integral of ln x traces back to xln x - x + C. This list of integrals of logarithmic functions groups the standard results by family, with domain conditions, derivation notes and verified examples.

List of Integrals of Rational Functions With Proofs

Every rational integral ends in polynomials, logarithms and arctangents. This list of integrals of rational functions groups the standard results by denominator, with derivations, conditions, partial-fraction patterns and verified examples.

List of Integrals of Inverse Trigonometric Functions

Two families: integrals of arcsin x and its relatives, and integrals that produce them. This list of integrals of inverse trigonometric functions gives both, with domains, derivations and completing-the-square forms.

List of Integrals of Irrational Functions With Examples

This list of integrals of irrational functions gives every standard square-root antiderivative with its conditions. Tables cover five radical families, with derivation notes, equivalent forms and six verified examples.

List of Integrals of Hyperbolic Functions: Full Tables

The integral of sinh x is cosh x + C, with no minus sign. This list of integrals of hyperbolic functions gives every standard form with its domain, derivation and worked examples.

Linear Approximation and Local Linearization Explained

Near a point, a smooth curve is almost a straight line, so its tangent line estimates nearby values. Linear approximation and local linearization: the formula, over or under, error size, and checked examples.

List of Integrals of Exponential Functions With Proofs

Every exponential integral grows from ∫ e x,dx = e x + C. This list of integrals of exponential functions groups the results by family, with derivations, conditions, common traps and verified examples.

L'Hôpital's Rule: Conditions, Forms and Examples

L'Hôpital's rule turns a 0/0 or (∞)/(∞) limit into the limit of (f'(x))/(g'(x)). This guide states the exact conditions, converts the other indeterminate forms and shows where the rule fails.

L'Hôpital's Rule: Conditions, Forms and Examples

Limits of Piecewise Functions: Steps and Examples

At a break point, compare the left and right pieces: equal one-sided limits mean the limit exists. This guide covers limits of piecewise functions, missing constants, absolute value cases and six examples.

Leibniz Newton Calculus Controversy: Timeline and Verdict

The Leibniz Newton calculus controversy was a priority fight that ended with independent discovery as the verdict. This guide dates every step, explains the 1712 report, and works examples in both notations.

Inverse Chain Rule Method: Reverse Chain Rule Integration

The inverse chain rule method integrates by spotting a chain-rule derivative and undoing it: ∫ g'(x),f'(g(x)),dx = f(g(x)) + C. This guide covers spotting, (ax+b)ⁿ, fixing constants and the u-substitution link.

Introduction To Limits: Meaning, Notation, Examples

A limit is the value a function heads toward as the input gets close to a point. This introduction to limits covers notation, tables, graphs, one-sided limits and when limits fail.

Introduction to Integration: Meaning, Rules, Examples

Integration adds up tiny pieces to find a total, and it undoes differentiation. This introduction to integration covers area, notation, indefinite and definite integrals, the first rules and a method roadmap.

Introduction to Differential Equations: Types, Examples

A differential equation links an unknown function to its derivatives, and its answer is a function, not a number. This introduction to differential equations covers reading one, the main types and checking a solution.

Introduction to Derivatives: Meaning, Rules, Examples

A derivative measures how fast a function changes at one exact point. This introduction to derivatives covers slope, average versus instantaneous rate, notation, the first four rules and worked examples.

Integration Using Reduction Formulas: Steps, Examples

A reduction formula rewrites an integral with power n as the same integral with a smaller power. This guide to integration using reduction formulas derives six standard formulas, proves Wallis' formula and works verified examples.

Integration of Logarithmic Functions With Examples

The integral of ln x is xln x - x + C, by integration by parts. This guide to integration of logarithmic functions extends that move to other bases, linear insides, powers and definite integrals.

Integration of Irrational Functions: Methods & Examples

Integration of irrational functions means removing the root by substitution, then integrating the rational result. This guide gives the method for each radical type, the standard forms, Euler substitutions and six verified examples.

Infinitesimal Calculus: Meaning, History and Examples

Infinitesimal calculus is the original name for calculus, built on quantities smaller than every positive number. This guide covers Leibniz's dx, why limits replaced infinitesimals, Robinson's revival, and examples worked both ways.

Infinite Limits: Definition, Asymptotes and Examples

An infinite limit means f(x) grows without bound as x approaches a, so the graph has a vertical asymptote there. This guide covers infinite limits, sign analysis, asymptotes and limits at infinity.

Higher-Order Derivatives: Notation, Patterns, Examples

Higher-order derivatives are what you get by differentiating again and again: f'', f''', up to f (n). This guide covers the notation, the nth-derivative patterns, their meaning in motion, and six worked examples.

General Leibniz Rule: Formula, Proof and Examples

The general Leibniz rule gives the nth derivative of a product in one line: (fg) (n) = ∑ k=0 ⁿnkf (n-k) g (k). This guide covers the formula, an induction proof, six examples and common mistakes.

Graphs And Limits: Reading Limits From A Graph

To read a limit from a graph, trace the curve toward the point from both sides and note the height it approaches. This guide to graphs and limits covers holes, jumps, asymptotes and oscillation.

General and Particular Solutions: Definitions, Examples

A general solution keeps its arbitrary constants; a particular solution fixes them. This guide shows how general and particular solutions connect through initial conditions, why constants match the order, and where singular solutions hide.

First-Order Differential Equations: Types and Methods

First-order differential equations contain only the first derivative, and five standard types cover most of them. This guide shows how to recognise each type, which method fits, and one checked example per type.

Evaluating Limits Algebraically: Methods and Examples

Evaluating limits algebraically means rewriting an expression until direct substitution works. This guide covers six techniques, a decision flow for choosing one, worked examples and the mistakes that cost marks.

Differential of a Function: Formula, Examples, Errors

The differential of a function is dy = f'(x),dx, the change measured along the tangent line. This guide compares it with the true change Δ y and uses it to estimate measurement errors.

Euler's Number (e): Definition, Value and Proofs

Euler's number is e ≈ 2.71828, the base of natural growth. This guide covers its limit, series and derivative definitions, the compound-interest origin, why e is irrational, and six worked examples.

Derivatives of Hyperbolic Functions: Formulas & Proofs

The derivative of sinh x is cosh x, and of cosh x is sinh x, with no minus sign. Here are the derivatives of hyperbolic functions and their inverses, derived from e x, with domains.

Derivatives of a Function in Parametric Form Explained

Derivatives of a function in parametric form come from one ratio: dy/dx = (dy/dt)/(dx/dt). This guide covers why it works, the second derivative trap, horizontal and vertical tangents, and six verified examples.

Derivative as Slope of a Tangent Line: Formula, Examples

The derivative at a point is the slope of the tangent line there. This guide to the derivative as slope of a tangent line covers secants becoming tangents, the tangent equation, and where it fails.

Convex and Concave Functions: Definition and Tests

A convex function's chords lie on or above its graph; a concave function's chords lie on or below it. This guide to convex and concave functions covers three equivalent tests, Jensen's inequality and naming clashes.

Convergent and Divergent Sequences: Tests and Examples

A sequence converges when its terms settle on one finite number; otherwise it diverges. This guide sorts convergent and divergent sequences into five behaviours and shows the four tests that decide which is which.

Convergence Tests: Which Series Test to Use and When

Convergence tests decide whether an infinite series adds up to a finite number. This guide ranks nine common tests in the order to try them, with a summary table, six examples and the inconclusive cases.

Continuity on an Interval: Endpoint Rules and Examples

Continuity on an interval means the function has no break at any point of that interval. Below: endpoint rules for open, closed and half-open intervals, six examples, and why IVT and EVT depend on it.

Constant Rule: Why The Derivative Of A Constant Is 0

The constant rule says the derivative of any constant is 0. This guide proves it from the limit definition, shows the flat-line picture, and covers integrating constants and the constant multiple mix-up.

Calculus with Python: SymPy and NumPy Guide

Calculus with Python means letting SymPy do exact calculus and NumPy do numerical calculus. This guide runs limits, derivatives, integrals, series and differential equations in code, with checked outputs.

Concavity and Points of Inflection: Steps and Examples

A point of inflection is where concavity flips, so the second derivative must change sign there. This guide to concavity and points of inflection builds the sign chart, rejects false candidates and works six examples.

Calculus Symbols and Notation: List, Meaning

Calculus symbols and notation are shorthand for limits, rates and sums. This reference gives every common symbol its meaning, a read-aloud phrase, an example, plus worked reading practice.

Branches of Calculus: Types, Uses and Examples

Calculus has two core branches, differential and integral, and at least seven major extensions. This guide to the branches of calculus shows what each studies, one worked example each, and the learning order.

Bounded Sequences: Definition, Sup, Inf and Examples

Bounded sequences never escape a fixed band of real numbers. This guide covers upper and lower bounds, the supremum and infimum, why a bound alone does not force convergence, and six worked examples.

Average vs Instantaneous Rate of Change Explained

The average rate of change measures change over an interval; the instantaneous rate measures it at one point. This guide to average vs instantaneous rate of change covers formulas, secant and tangent lines, and examples.

Archimedes and Early Infinitesimals: The Road to Calculus

Archimedes and early infinitesimals solved area and volume problems about 1,900 years before calculus. See exhaustion, the parabola's 4/3 rule, the sphere, and how Cavalieri, Fermat and Wallis carried the idea forward.

Applications of Calculus in Real Life, With Examples

Calculus is the math of rates, totals and best choices, and nearly every technical field uses it. These applications of calculus in real life come with seven fields, six worked examples and the common mistakes.

Center of Mass and Centroid: Formulas and Examples

The center of mass and centroid are balance points: the centroid uses shape alone, the center of mass also weighs density. This guide covers rods, regions between curves, laminas, symmetry and Pappus's theorem.

Volume Of Solids Of Revolution: Disk, Washer, Shell

The volume of solids of revolution is the volume you get when a flat region is rotated about a straight line, and integration adds up the thin pieces the spin creates. Three formulas do the whole job: the disk method V = π∫ₐ b [R(x)]²,dx when the solid is solid through the middle, the washer method V = π∫ₐ b ([R(x)]² - [r(x)]²),dx when there is a hole, and the cylindrical shell method V = 2π∫ₐ b x,h(x),dx when it is easier to slice parallel to the axis. Which one you pick depends on the axis of rotation and whether the region touches it.

Washer Method: Volume Of Revolution Formula

The Washer Method finds the volume of a solid of revolution that has a hole through it, by slicing the solid into thin rings (washers) perpendicular to the axis of rotation. Each ring has an outer radius R and an inner radius r, so its area is π(R² - r²), and the volume is V = π∫ₐ b (R² - r²),dx. Use it when the region being revolved does not touch the axis, so a gap is left in the middle.

Velocity and Acceleration: Calculus Derivatives

Velocity and acceleration are what you get when you differentiate a position function. If s(t) gives position, then velocity is the first derivative v(t) = s'(t) (a signed rate that carries direction), speed is its size |v(t)|, and acceleration is the second derivative a(t) = v'(t) = s''(t). Integration runs the chain backwards: v = ∫ a,dt and s = ∫ v,dt, each fixed by an initial condition.

History of Calculus: Timeline From Archimedes to Newton

The history of calculus is the story of how mathematicians learned to measure change and area, from Archimedes' method of exhaustion around 250 BCE, through Madhava's infinite series in 14th-century India, to the independent invention of the full subject by Isaac Newton and Gottfried Wilhelm Leibniz in the 1660s–1680s. Their tools worked but rested on shaky foundations, so a second wave, Cauchy, Bolzano, Weierstrass, and Riemann, spent the 1800s making calculus rigorous with the limit and the epsilon-delta definition.

Types Of Discontinuity: Removable, Jump & Infinite

The types of discontinuity are the ways a function can break at a point: removable (a single hole where the two-sided limit exists but does not match the function value), jump (the left and right limits are both finite but unequal), and infinite (the function shoots off to ±∞ at a vertical asymptote). A fourth, the essential or oscillating discontinuity such as sin(1/x) at x=0, has no limit at all. You tell them apart by comparing the one-sided limits, the two-sided limit, and the value f(a).

Taylors Theorem: Formula, Remainder & Examples

Taylors Theorem says any smooth function equals its degree-n Taylor polynomial plus a leftover remainder term: f(x) = ∑ k=0 ⁿ(f (k) (a))/(k factorial)(x-a) k + Rₙ(x). The Lagrange form of that remainder, Rₙ(x) = (f (n+1) (c))/((n+1) factorial)(x-a)ⁿ⁺¹ for some c between a and x, puts a hard number on the approximation error. It is the reason a Taylor series is trustworthy and the reason those series converge at all.

Transformations Of Functions: Rules & Examples

Transformations of functions are the six ways of moving or reshaping a graph without changing its basic form: shift it up or down with f(x) + k, shift it left or right with f(x - h), stretch or compress it vertically with a,f(x) or horizontally with f(bx), and reflect it with -f(x) (over the x-axis) or f(-x) (over the y-axis). The two rules that trip students up are horizontal: f(x - h) moves the graph right for positive h, and f(bx) scales it by 1/b, both the opposite of what the sign or number first suggests.

Tangent Line Equations: Formula, Steps & Examples

Tangent line equations give the straight line that just grazes a curve y = f(x) at a single point. At x = a the tangent is y = f(a) + f'(a)(x - a): its slope is the derivative f'(a), and it passes through the point (a, f(a)). Find the point, find the slope, and drop both into point-slope form.

Table of Integrals: Standard Formulas & Examples

A Table of Integrals is a grouped reference list of standard antiderivatives, one row for each function you meet in a first calculus course, from ∫ xⁿ,dx = (xⁿ⁺¹)/(n+1) + C to the trig, inverse-trig, and hyperbolic forms. Every indefinite entry carries a constant of integration +C, and every entry can be checked by differentiating the right-hand side back to the integrand. Use the table to look up a form quickly, then follow the linked method article when an integral does not match a row on sight.

Summation Notation: Sigma Σ Rules & Formulas

Summation notation uses the Greek capital letter sigma, ∑, to write a long sum in one compact symbol: ∑ i=1 ⁿ aᵢ means add the terms aᵢ as the index i runs from the lower limit 1 to the upper limit n. It obeys simple algebra, ∑(aᵢ + bᵢ) = ∑ aᵢ + ∑ bᵢ and ∑ c,aᵢ = c∑ aᵢ, and three closed-form power sums, ∑ i=1 ⁿ i = (n(n+1))/2, ∑ i=1 ⁿ i² = (n(n+1)(2n+1))/6, and ∑ i=1 ⁿ i³ = ((n(n+1))/2)², turn a long addition into one formula.

Table Of Derivatives: Rules, Trig & Formulas

A Table of Derivatives is the reference chart that gives the derivative of every standard function in one place: the algebraic rules (constant, power, constant-multiple, sum, product, quotient, chain) and the standard results for exponentials, logarithms, the six trigonometric functions, the six inverse trigonometric functions, and the hyperbolic functions. Read it top to bottom and every derivative you need for a first calculus course is a single lookup, with a chain-rule version for when the input is itself a function.

Riemann Sums: Left, Right And Midpoint Explained

Riemann sums approximate the area under a curve by slicing it into n thin rectangles and adding their areas: ∑ i=1 ⁿ f(xᵢ^*),Δ x, where Δ x = (b-a)/n. The left, right, and midpoint versions differ only in where you sample the height of each rectangle, and as n → ∞ every version converges to the exact definite integral ∫ₐ b f(x),dx.

Stationary Points: Maxima, Minima & Inflection

Stationary points are the points on a curve where the derivative is zero, f'(x) = 0, so the tangent line is horizontal. There are three kinds: a local maximum, a local minimum, and a point of inflection. You find them by solving f'(x) = 0, then classify each one with the first-derivative test (the sign change of f') or the second-derivative test (f''(x) < 0 means maximum, f''(x) > 0 means minimum, and f''(x) = 0 is inconclusive so you must check further).

Separable Differential Equations: Solve Step By Step

Separable differential equations are first-order equations that can be written as dy/dx = g(x),h(y), so that every y can be moved to one side and every x to the other. You solve them by separating the variables, integrating both sides, adding a single constant +C, and then solving for y or leaving the answer implicit. One extra step matters: each root of h(y) = 0 gives a constant singular solution that separation can quietly lose.

Standard Limits: Formulas, Proofs & Examples

Standard limits are a short list of limit results, such as lim x → 0 (sin x)/x = 1, that appear so often they are memorized rather than re-derived each time. This guide states all nine, proves the trigonometric ones from a unit-circle squeeze, derives the exponential and logarithmic ones, and shows why lim x → ∞ (1 + 1/x) x = e. Every result is a value that direct substitution leaves as the indeterminate form 0/0 or 1 ∞.

Second Derivative Test: Rules, Examples & When It Fails

The Second Derivative Test classifies a critical point c (a point where f'(c) = 0) by checking the sign of f''(c). If f''(c) < 0 the graph is concave down, so c is a local maximum; if f''(c) > 0 the graph is concave up, so c is a local minimum; if f''(c) = 0 the test is inconclusive and you fall back on the first derivative test. It is the fastest way to sort peaks from valleys when the second derivative is easy to compute.

Second Derivative: Concavity & Acceleration Explained

The second derivative f''(x) = d/dx[f'(x)] is the derivative of the derivative: it measures how fast the slope of a curve is changing. Its sign tells you the shape of the graph, with f'' > 0 meaning concave up and f'' < 0 meaning concave down, and a point where f'' changes sign is an inflection point. If f'(c) = 0 at a critical point, the second derivative test classifies it: f''(c) < 0 gives a local maximum and f''(c) > 0 a local minimum.

Root Test: Statement, Examples & When To Use It

The Root Test decides whether an infinite series ∑ aₙ converges by computing L = lim n → ∞ √[n]|aₙ| = lim n → ∞ |aₙ| 1/n. If L < 1 the series converges absolutely, if L > 1 (or L = ∞) it diverges, and if L = 1 the test is inconclusive and you need another test. It is the sharpest tool to reach for whenever the general term aₙ carries an nth power.

Removable Discontinuity: Definition & How To Find It

A removable discontinuity is a single-point gap, a "hole," where the limit of a function exists and is finite but the function either has no value there or has the wrong one. A function f has a removable discontinuity at x = a when lim x → a f(x) = L exists yet f(a) is undefined or f(a) ≠ L. It is called removable because redefining f(a) = L patches the hole and makes the function continuous.

Related Rates: Method, Formulas & Worked Examples

Related rates problems ask how fast one quantity changes when a second quantity, linked to it by an equation, is also changing with time. The method is fixed: draw the situation, name the variables, write the equation that ties them together, differentiate both sides with respect to time t using the chain rule, and only then substitute the numbers for the exact instant you care about. For an expanding circle, differentiating A = π r² gives dA/dt = 2π r dr/dt.

Ratio Test: Convergence Of Series Explained

The Ratio Test decides whether an infinite series ∑ aₙ converges or diverges by looking at the limit L = lim n→∞ |(a n+1)/aₙ|. If L < 1 the series converges absolutely, if L > 1 (or L = ∞) it diverges, and if L = 1 the test is inconclusive and tells you nothing. It is the fastest test to reach for when the terms contain factorials or exponentials.

Radius of Convergence: Formula, Examples & Interval

The radius of convergence of a power series ∑ cₙ(x-a)ⁿ is the number R such that the series converges whenever |x-a| < R and diverges whenever |x-a| > R. You find R with the ratio test, R = lim n → ∞ |cₙ/(c n+1)|, and there are only three outcomes: R = 0 (converges at one point), R = ∞ (converges everywhere), or a finite 0 < R < ∞. The interval of convergence is (a-R,, a+R) together with a separate check at each endpoint.

Properties Of Definite Integrals: Rules & Examples

The properties of definite integrals are shortcut rules that let you rewrite or simplify ∫ₐ b f(x),dx before computing it. The essential ones are reversing the limits flips the sign, a zero-width interval gives 0, an integral splits over any middle point, constants pull out, and symmetry over [-a, a] makes an odd function integrate to 0 and an even function integrate to twice its half. Used well, these turn a fearsome integral into a one-line answer.

Ordinary Differential Equations: Order, Degree & Solutions

Ordinary Differential Equations are equations that relate a function of a single variable to its derivatives, such as dy/dx = ky. The order is the highest derivative that appears, and the degree is the power of that highest derivative when the equation is polynomial in its derivatives. A general solution carries as many arbitrary constants as the order; a particular solution fixes those constants using initial or boundary conditions.

Order and Degree of a Differential Equation

The order and degree of a differential equation are the two numbers that classify it. The order is the order of the highest derivative that appears. The degree is the power of that highest-order derivative once the equation has been rewritten as a polynomial in its derivatives, meaning free of radicals and fractions that contain derivatives. If the equation can never be made polynomial in its derivatives, as when a derivative sits inside sin or an exponential, the degree is not defined.

Infinite Series: Convergence, Tests & Examples

An infinite series ∑ n=1 ∞ aₙ is the sum of infinitely many terms, and its value is defined as the limit of its partial sums S N = a₁ + a₂ + … + a N. If that limit is a finite number, the series converges; otherwise it diverges. Two families have closed rules: a geometric series ∑ arⁿ converges to a/(1-r) when |r| < 1, and a p-series ∑ 1/(n p) converges exactly when p > 1.

Constant Multiple Rule: Derivatives & Integrals

The Constant Multiple Rule says a constant factor slides straight through a derivative or an integral untouched. For differentiation, d/dx[c,f(x)] = c,f'(x); for integration, ∫ c,f(x),dx = c∫ f(x),dx. The reason is the same on both sides: a constant does not depend on the variable, so it factors out of the underlying limit.

Differential Calculus Vs Integral Calculus Explained

The whole of differential calculus vs integral calculus comes down to direction: differential calculus breaks a quantity apart to find its rate of change (the slope of a tangent line), while integral calculus adds infinitely many pieces back together to find a total (the area under a curve). Differentiation turns a position function into a velocity; integration turns velocity back into position. The Fundamental Theorem of Calculus proves the two are inverse processes, which is why they are two halves of one subject.

Derivatives Of Logarithmic Functions: Rules & Proof

The derivatives of logarithmic functions come down to three formulas: d/dxln x = 1/x, d/dxlogₐ x = 1/(xln a), and the chain-rule form d/dxln u = (u')/u. Each follows from the derivative of e x by implicit differentiation, and each reads geometrically as the slope of a logarithm curve, always positive and shrinking as x grows.

Derivative of a Function: Definition, Rules & Examples

The derivative of a function measures its instantaneous rate of change, and geometrically it is the slope of the tangent line to the graph at a point. It is defined by the limit f'(x) = lim h → 0 (f(x+h) - f(x))/h, and in practice you compute it with a small toolbox of rules: power, constant-multiple, sum, product, quotient, and chain. A function that is differentiable at a point is automatically continuous there.

Analytic Geometry: Formulas, Lines, Circles & Conics

Analytic geometry is the branch of mathematics that describes geometric shapes with algebra by placing them on a coordinate grid, so a point becomes a pair of numbers and a curve becomes an equation. Its core tools are the distance formula d = √((x₂ - x₁)² + (y₂ - y₁)²), the slope m = (y₂ - y₁)/(x₂ - x₁), the equation of a line, and the circle (x - h)² + (y - k)² = r². Because it turns curves into equations, it is the bridge that made calculus possible.

Optimization Problems in Calculus: Steps & Examples

Optimization problems in calculus ask for the largest or smallest value of a quantity, such as maximum area, maximum volume, or minimum cost. You write the quantity you care about (the objective) as a function of one variable using a constraint equation, differentiate it, solve f'(x) = 0 to find the critical points, and confirm whether each is a maximum or a minimum with the second-derivative test or by checking the endpoints of the domain.

Composition of Functions: fog, Domain & Examples

Composition of Functions feeds the output of one function into another: (f ° g)(x) = f(g(x)) means apply g first, then f. Order matters, so f ° g and g ° f are usually different functions. The domain of a composite is every input allowed by the inner function whose output is also allowed by the outer function, and being able to split a function into a composition is the exact skill the chain rule depends on.

Function Notation: Meaning, f(x), and Examples

Function notation is the way we name a rule and show what it does to an input: f(x) is read "f of x" and means the output of the function f when the input is x. It is not f multiplied by x. To evaluate f(a) you substitute a for every x; the same move on f(a+h) builds the difference quotient (f(x+h)-f(x))/h, which is the exact step that opens the door to calculus.

Calculus Formulas: Limits, Derivatives & Integrals

Calculus formulas fall into four working groups: standard limits, derivative rules, integral rules, and the theorems that connect them. The three you reach for most are the power rule d/dxxⁿ = nxⁿ⁻¹, its reverse the integral ∫ xⁿ,dx = (xⁿ⁺¹)/(n+1) + C (for n ≠ -1), and the Fundamental Theorem of Calculus ∫ₐ b f(x),dx = F(b) - F(a). This page groups every core formula into tables, links each group to its full article, and verifies each result by differentiating antiderivatives back.

Basic Integration Formulas: List, Rules & Examples

Basic integration formulas are the standard antiderivatives you reverse-engineer from the derivative rules: the power rule ∫ xⁿ,dx = (xⁿ⁺¹)/(n+1) + C for n ≠ -1, the exception ∫ 1/x,dx = ln|x| + C, the exponentials ∫ e x,dx and ∫ a x,dx, the six basic trigonometric integrals, and two forms that produce inverse-trig functions. Every one carries a +C when it is indefinite, and every one can be checked by differentiating the answer back to the integrand.

Differentiation And Integration: Inverse Operations

Differentiation and integration are the two central operations of calculus, and they are inverse operations: differentiation finds an instantaneous rate of change (the slope of a tangent), and integration finds an accumulated total (the area under a curve). The Fundamental Theorem of Calculus makes the link exact: differentiating an integral returns the original function, d/dx∫ₐ x f(t),dt = f(x), and integrating a rate recovers the net change, ∫ₐ b f'(x),dx = f(b) - f(a).

Numerical Integration: Methods, Formulas & Error

Numerical integration approximates the value of a definite integral by sampling the function at a few points instead of finding an antiderivative. You need it when no elementary antiderivative exists, as with ∫₀¹ e -x²,dx. The three standard rules build the area from simple shapes: the midpoint rule uses rectangles, the trapezoidal rule uses trapezoids, and Simpson's rule uses parabolas. Simpson's rule is by far the most accurate, with error shrinking like 1/n⁴ against 1/n² for the other two.

Monotonic Sequences: Increasing, Decreasing, Convergent

Monotonic sequences are sequences that head in one direction only: every term is greater than or equal to the one before (increasing) or less than or equal to the one before (decreasing). You test the direction by checking the sign of a n+1 - aₙ, comparing the ratio (a n+1)/aₙ to 1 for positive terms, or looking at f'(x) when aₙ = f(n). The payoff is the Monotone Convergence Theorem: a sequence that is both monotonic and bounded must converge.

Marginal Cost and Marginal Revenue With Calculus

Marginal cost and marginal revenue are the derivatives of the total cost and total revenue functions: MC = C'(x) and MR = R'(x). Each measures the rate of change of a total as one more unit is made or sold, so it closely approximates the cost or revenue of that next unit. Profit P(x) = R(x) - C(x) is largest where marginal profit is zero, which means MR = MC, confirmed by the second-order check P''(x) < 0.

Methods Of Integration: Techniques & Examples

The methods of integration are the standard techniques for finding an antiderivative: direct standard formulas, substitution (the reverse chain rule), integration by parts (the reverse product rule), partial fractions for rational functions, trigonometric substitution for square-root expressions, and trigonometric integrals handled with identities. Which one to use is decided by what the integrand looks like, and every result can be checked by differentiating the answer back to the integrand.

Logarithmic Differentiation: Formula, Steps & Examples

Logarithmic differentiation is a technique that takes the natural log of both sides of y = f(x), uses log laws to break products, quotients, and powers into sums, differences, and multiples, then differentiates implicitly to get (y')/y on the left. Solving gives y' = y·(the log-expanded derivative). Reach for it when the base and the exponent both vary, as in y = x x, or when a function is a large product or quotient of powers.

List of Integrals: Standard Integral Table

A List of Integrals is an organised table of standard antiderivatives: for each function you look up the formula whose derivative returns that function, then add the constant of integration +C. This page tabulates the power, exponential, logarithmic, trigonometric, inverse-trigonometric, and hyperbolic forms, plus the common patterns 1/(a²+x²) and 1/(√(a²-x²)), and every entry is checked by differentiating it back to the original function.

Linear Differential Equations: Integrating Factor Method

Linear differential equations are first-order equations of the form dy/dx + P(x),y = Q(x), where y and its derivative appear only to the first power and never multiplied together. You solve them with the integrating factor μ(x) = e ∫ P(x),dx: multiply through by μ, the left side collapses into d/dx[μ y], and one integration gives y = 1/(μ)∫ μ,Q,dx.

Limits at Infinity: Rules, Asymptotes & Examples

Limits at infinity describe the end behavior of a function: the value f(x) settles toward as x runs off to +∞ or -∞. When that value is a finite number L, the line y = L is a horizontal asymptote. The one fact that unlocks the topic is lim x → ∞ 1/(x p) = 0 for any p > 0; for a rational function you compare the degrees of the top and bottom, and the fastest way to see the answer is to divide every term by the highest power of x.

Limits and Derivatives: Class 11 Foundations

Limits and Derivatives are the two ideas that open calculus: a limit is the value a function heads toward as its input approaches a point, and a derivative is a particular limit that measures how fast the function is changing. Formally the derivative is f'(x) = lim h → 0 (f(x+h) - f(x))/h, the limit of the average rate of change as the step shrinks to nothing. Master the limit first, and the derivative, continuity, and every rule that follows fall out of it.

Limit of a Sequence: Definition, Rules & Examples

The limit of a sequence is the single value L that the terms aₙ settle toward as n grows without bound, written lim n → ∞ aₙ = L. A sequence that has such a finite limit is called convergent; one that does not is divergent. The formal test is the –N definition: aₙ → L when, for every tolerance > 0, all terms past some index N stay within of L.

Limit Of A Function: Definition & Examples

The limit of a function describes the single value f(x) gets arbitrarily close to as x approaches a point a from either side, written lim x → a f(x) = L. It is about the behaviour of f near a, not the value f(a) at the point, which is why a limit can exist even where the function is undefined. The two-sided limit exists only when the left-hand and right-hand limits agree.

Limit Laws: Rules, Proofs, And Examples

The limit laws are the rules that let you evaluate a limit by splitting it into simpler pieces: the limit of a sum, difference, product, quotient, power, or root is built from the limits of the parts. They hold whenever the limits of the parts exist, with one guard: the quotient law needs the denominator's limit to be non-zero. When direct substitution gives 0/0, the laws stall, and you first factor or rationalize before applying them.

Limit Formulas: Laws And Standard Limits With Examples

Limit formulas are the ready-made results that let you evaluate a limit without building it from scratch. They split into two groups: the limit laws, which say a limit distributes over sums, differences, products, quotients, constant multiples, and powers, and the standard limits, such as lim x → 0 (sin x)/x = 1 and lim x → 0 (e x - 1)/x = 1, which resolve the tricky 0/0 cases that direct substitution cannot.

Limit Definition of Derivative: First Principles

The Limit Definition of Derivative computes the exact slope of a curve at a single point as the limit of a difference quotient: f'(x) = lim h → 0 (f(x+h) - f(x))/h. Every differentiation rule you will ever use, including the power rule, is proved from this one limit. Computing a derivative straight from it is called differentiating "from first principles," and the limit is what safely resolves the 0/0 form you get if you simply set h = 0.

Limit Comparison Test: Statement, Rules & Examples

The Limit Comparison Test decides whether a positive-term series ∑ aₙ converges or diverges by comparing it to a simpler series ∑ bₙ whose behaviour you already know. Take the limit lim n → ∞ aₙ/bₙ = L: if L is a finite positive number (0 < L < ∞), the two series share the same fate, so both converge or both diverge. You pick bₙ by keeping only the dominant powers of aₙ, which usually leaves a p-series or a geometric series.

Jump Discontinuity: Definition, Examples & How To Find

A jump discontinuity occurs at a point where a function's left-hand limit and right-hand limit both exist and are finite, but they are not equal. Because the two one-sided limits disagree, the ordinary two-sided limit does not exist, and the graph shows a clean vertical gap, a step from one height to another. Step functions, the floor function, and the signum function all have jump discontinuities.

Integration Of Rational Functions: Full Method

Integration of rational functions is the routine for integrating any ratio (P(x))/(Q(x)) of two polynomials. Divide first if the fraction is improper (P ≥ Q), factor the denominator into linear and irreducible quadratic pieces, split it by partial fraction decomposition, and integrate each piece. Every such integral comes out in elementary form: a combination of logarithms, arctangents, and powers.

Integration of Exponential Functions: Rules & Examples

Integration of exponential functions reverses differentiation for bases like e and a. The base result is ∫ e x,dx = e x + C, because e x is its own derivative and its own antiderivative. When the exponent is scaled, divide by the scale: ∫ e kx,dx = 1/ke kx + C and ∫ a x,dx = (a x)/(ln a) + C. Every answer carries +C, and you can confirm each by differentiating it back.

Integration by Partial Fractions: Method & Examples

Integration by partial fractions rewrites a proper rational function (P(x))/(Q(x)) as a sum of simpler fractions, then integrates each piece, usually into logarithms and arctangents. If the fraction is improper (the degree of P is at least the degree of Q), you divide first, then decompose the leftover proper fraction. The denominator's factors decide the form: distinct linear factors give A/(x-a) terms, repeated linear factors add B/((x-a)²) terms, and an irreducible quadratic gives an (Ax+B)/(x²+bx+c) term.

Integrating Factor Method: Steps & Examples

The integrating factor method solves any first-order linear differential equation written in the standard form dy/dx + P(x),y = Q(x). You multiply the whole equation by the integrating factor μ(x) = e ∫ P(x),dx, which is chosen so the left side collapses into a single derivative d/dx[μ(x),y]. Integrating both sides then gives the general solution y = 1/(μ(x))(∫ μ(x),Q(x),dx + C).

Integral Test for Convergence: Rules & Examples

The Integral Test says that if f is continuous, positive, and decreasing on [1, ∞) and aₙ = f(n), then the series ∑ aₙ and the improper integral ∫₁^∞ f(x),dx either both converge or both diverge. It turns a question about an infinite sum into an area you can actually evaluate. The test proves the p-series rule, that ∑ 1/(n p) converges exactly when p > 1, and it shares convergence with the integral but not the numerical value.

Initial Value Problems: Solve IVPs Step by Step

Initial value problems (IVPs) are a differential equation paired with one or more initial conditions that fix the solution at a starting point. You solve one in two moves: find the general solution (which carries an arbitrary constant), then substitute the initial condition to solve for that constant and get the particular solution. The number of conditions you need equals the order of the equation, and a solution is guaranteed to exist and be unique when the right-hand side is continuous and Lipschitz in y.

Infinite Discontinuity: Definition, Examples & Graphs

An infinite discontinuity is a break in a function where at least one one-sided limit runs off to +∞ or -∞, so the graph shoots up or down a vertical asymptote instead of settling on a value. It is the kind of discontinuity you get from 1/x at x = 0, from 1/((x-a)²) at x = a, and from tan x at x = (π)/2. Because the limit is not a finite number, the break can never be patched, which is what separates it from a removable hole.

Indeterminate Forms: The 7 Types & L'Hôpital's Rule

Indeterminate forms are limit expressions like 0/0, (∞)/(∞), 0·∞, ∞-∞, 1 ∞, 0⁰, and ∞⁰ where plugging in the limit value gives no single answer, because two competing tendencies pull the result in opposite directions. They are a signal, not a verdict: the true limit can be any number until you resolve it. L'Hôpital's rule handles 0/0 and (∞)/(∞) by replacing lim f/g with lim (f')/(g'), and every other form is rewritten into one of those two before the rule applies.

Increasing And Decreasing Functions: Derivative Test

Increasing and decreasing functions are classified by the sign of the first derivative: where f'(x) > 0 the function is increasing, and where f'(x) < 0 it is decreasing. To find the increasing and decreasing intervals, find the critical points (where f'(x) = 0 or f'(x) is undefined), split the domain at those points, and test the sign of f' on each subinterval.

Indefinite Integrals: Rules, Formulas & Examples

Indefinite integrals reverse differentiation: ∫ f(x),dx = F(x) + C is the whole family of functions whose derivative is f(x), where F is any one antiderivative and C is the constant of integration. The answer is a function (with a +C), not a number, which is what separates it from the definite integral. The core tools are the constant-multiple rule, the sum and difference rules, and the power rule ∫ xⁿ,dx = (xⁿ⁺¹)/(n+1) + C for n ≠ -1.

Improper Integrals: Convergence, Types & Examples

Improper integrals are definite integrals where either a bound runs to infinity (Type 1) or the function blows up somewhere on the interval (Type 2). You never plug in the bad point directly; you replace it with a limit, ∫ₐ^∞ f(x),dx = lim t → ∞ ∫ₐ t f(x),dx, and the integral converges if that limit is a finite number and diverges if it is not. The key benchmark is the p-integral ∫₁^∞ 1/(x p),dx, which converges exactly when p > 1.

Homogeneous Differential Equations: How To Solve

Homogeneous differential equations are first-order equations that can be written as dy/dx = F(y/x), where the right side depends only on the ratio y/x. You solve them with one move: substitute v = y/x (so y = vx and dy/dx = v + xdv/dx), which turns the equation into a separable one in v and x. After integrating, you back-substitute v = y/x to recover y.

Harmonic Series: Why It Diverges (Proof & Examples)

The Harmonic Series is the infinite sum ∑ n=1 ∞ 1/n = 1 + 1/2 + 1/3 + 1/4 + …, and it diverges: its running total grows past every number, even though the individual terms 1/n shrink toward zero. Two short proofs settle it, Oresme's grouping of terms into blocks each larger than 1/2, and a comparison with the area under y = 1/x. It is the exact boundary case p = 1 of the p-series ∑ 1/(n p), which converges only when p > 1.

First Derivative Test: Find Maxima And Minima

The First Derivative Test classifies a critical point c by watching the sign of f' just left and right of it: if f' changes from positive to negative, c is a local maximum; if it changes from negative to positive, c is a local minimum; if the sign does not change, c is neither. You organise the signs on a sign chart of f', and the test stays conclusive even when the second-derivative test breaks down.

Extreme Value Theorem: Statement, Proof & Examples

The Extreme Value Theorem states that if a function f is continuous on a closed, bounded interval [a, b], then f attains an absolute maximum and an absolute minimum on [a, b], each at some point in [a, b]. It is an existence result: it guarantees the highest and lowest values are reached, but it does not tell you where. The closed-interval method it justifies finds them by comparing f at every critical point and at both endpoints.

Exponential Growth And Decay: Formula & Examples

Exponential growth and decay is the model in which a quantity changes at a rate proportional to its current size, written as the differential equation dy/dt = ky. Separating variables and integrating gives the solution y = y₀ e kt, where y₀ is the starting amount. When k > 0 the quantity grows, and when k < 0 it decays; the doubling time of growth and the half-life of decay both equal (ln 2)/( k), independent of where you start.

Exact Differential Equations: Test, Solve, Examples

Exact Differential Equations are first-order equations of the form M(x,y),dx + N(x,y),dy = 0 that come from differentiating a single hidden function F(x,y). The equation is exact exactly when ( M)/( y) = ( N)/( x), and when it is, there is a potential F with Fₓ = M and F y = N, so the whole solution is the family of level curves F(x,y) = C.

Divergence Test: Statement, Examples & Caveats

The Divergence Test, also called the nth-term test, says that if the terms of a series do not shrink to zero, the series diverges: when lim n→∞ aₙ ≠ 0 or that limit does not exist, ∑ aₙ diverges. The catch is one-directional. If lim n→∞ aₙ = 0, the test is inconclusive and proves nothing, since the harmonic series ∑ 1/n has terms going to zero yet still diverges. Use it as the first, fastest check on any series.

Disk Method: Volume Of A Solid Of Revolution

The disk method finds the volume of a solid of revolution that has no hole through its centre. You slice the solid perpendicular to the axis into thin circular disks of radius R and thickness dx, each with area π R², then add them up with an integral: V = π∫ₐ b [R(x)]²,dx about the x-axis, or V = π∫ c d [R(y)]²,dy about the y-axis. The radius R is the distance from the axis to the curve.

Direction Fields: Slope Fields For Differential Equations

Direction fields (also called slope fields) are pictures of a first-order differential equation dy/dx = f(x, y): at a grid of points you draw a short segment whose slope is f(x, y), and any solution curve threads through the field tangent to those segments everywhere. They let you read the qualitative behavior of solutions, where a constant equilibrium sits, whether it is stable, and where curves head in the long run, without ever solving the equation.

Differentiability vs Continuity: The Key Difference

Differentiability vs Continuity comes down to one direction of logic: every differentiable function is continuous, but a continuous function need not be differentiable. Differentiability is the stronger property, because it demands a single finite slope at a point, while continuity only demands an unbroken graph. A curve can be perfectly unbroken yet still have a corner, a cusp, or a vertical tangent where no slope exists.

Difference Quotient: Formula, Examples & Meaning

The difference quotient is the expression (f(x+h)-f(x))/h, and it measures the average rate of change of a function f over the interval from x to x+h. Geometrically it is the slope of the secant line joining two points on the curve. Letting the step shrink, lim h → 0 (f(x+h)-f(x))/h, turns that average slope into the derivative f'(x), the instantaneous rate of change and the slope of the tangent line.

Derivatives of Exponential Functions: Rules & Proof

The derivatives of exponential functions come down to two rules: d/dxe x =e x, the one function that equals its own derivative, and d/dxa x =a x ln a for any base a>0. Wrap an exponential around an inner function and the chain rule attaches its derivative: d/dxe u =e u,u' and d/dxa u =a u ln a· u'. Get the base right, keep the inner derivative, and never reach for the power rule, and every problem in this topic follows the same short recipe.

Derivative as Rate of Change: Meaning, Units, Examples

The derivative as rate of change is the instantaneous rate at which a function's output changes with respect to its input. It is the limit of the average rate of change (f(x+h)-f(x))/h as h → 0, so f'(x) answers "how fast is this quantity changing right now." Read as a rate, f'(x) becomes velocity (metres per second), marginal cost (dollars per unit), or a growth rate (people per year), and its units are always the units of the output divided by the units of the input.

Cylindrical Shell Method: Formula & Examples

The cylindrical shell method finds the volume of a solid of revolution by adding up thin nested tubes instead of flat slices. For a region revolved about the y-axis, the volume is V = ∫ₐ b 2π,(radius)(height),dx = ∫ₐ b 2π x,f(x),dx. Its advantage over the disk and washer methods is that you integrate along the axis perpendicular to the axis of revolution, so you rarely have to solve the curve for its inverse.

Curve Sketching: Step-By-Step Calculus Guide

Curve sketching is the calculus method for drawing an accurate graph of a function from its features rather than by plotting hundreds of points. You work a fixed checklist: domain, intercepts, symmetry, asymptotes, then the first derivative f'(x) for where the graph rises and falls and its turning points, then the second derivative f''(x) for where it bends and its inflection points, and finally you assemble the shape. The sign of f' tells you the tilt of the curve; the sign of f'' tells you the way it curves.

Continuity of a Function: Definition & Examples

Continuity of a function captures the idea of a graph with no breaks, holes, or jumps. A function f is continuous at x = a when three conditions all hold: f(a) is defined, lim x → a f(x) exists (the left and right limits agree), and lim x → a f(x) = f(a). Miss any one of the three and the function is discontinuous at that point.

Convergence and Divergence of Series: Tests & Examples

The convergence and divergence of series is decided by one thing: whether the sequence of partial sums S N = ∑ n=1 N aₙ settles on a finite number as N → ∞. If it does, the series converges to that number; if it does not, the series diverges. Two benchmark families anchor everything: a geometric series ∑ arⁿ converges exactly when |r| < 1, and a p-series ∑ 1/(n p) converges exactly when p > 1.

Constant Of Integration: Why Every Integral Needs +C

The constant of integration is the +C attached to every indefinite integral: ∫ f(x),dx = F(x) + C. It is there because differentiation erases constants, so an antiderivative is never a single curve but a whole family of parallel curves. An initial condition (a known point) picks out one curve from that family by fixing the value of C, and C disappears entirely from a definite integral because it cancels.

Continuity and Differentiability: Rules & Examples

Continuity and Differentiability are two levels of "smoothness" for a function. A function is continuous at x = a when three things hold: f(a) is defined, lim x → a f(x) exists, and the two are equal. A function is differentiable at x = a when its derivative f'(a) exists, which needs the left-hand and right-hand derivatives to agree. The key theorem: every differentiable function is continuous, but the converse is false, and f(x) = |x| is continuous at 0 yet not differentiable there.

Comparison Test for Series: Convergence & Examples

The Comparison Test decides whether a positive-term series converges or diverges by comparing it, term by term, to a series you already understand. If 0 ≤ aₙ ≤ bₙ and ∑ bₙ converges, then ∑ aₙ converges. If aₙ ≥ bₙ ≥ 0 and ∑ bₙ diverges, then ∑ aₙ diverges. The benchmarks you compare against are almost always a p-series or a geometric series.

Calculus in Physics: Derivatives & Integrals Explained

Calculus in physics is the use of derivatives to describe how quantities change and integrals to add those changes back up. Differentiating position gives velocity and differentiating again gives acceleration; integrating force over distance gives work, integrating force over time gives impulse, and integrating current over time gives electric charge. Every one of these is one operation, read in two directions: a derivative is the slope of a graph, and an integral is the area under it.

Calculus in Engineering: Uses, Formulas & Examples

Calculus in engineering is the everyday use of derivatives, integrals, and differential equations to design and analyse real systems. Derivatives handle rates of change and optimization (sizing a part for least material); integrals accumulate quantities (area, volume, a beam's total load, a centroid); differential equations model anything that evolves in time (an RC circuit charging, a control loop settling, heat spreading through a wall). Where physics uses calculus to state a natural law, engineering uses the same calculus to hit a specification.

Calculus In Economics: Marginal Analysis & Optimization

Calculus in economics turns totals into rates and rates back into totals: differentiate a cost, revenue, or profit function to get its marginal version, and integrate a marginal or demand function to recover a total or a surplus. A firm maximizes profit where marginal revenue equals marginal cost, R'(q) = C'(q), and consumer surplus is the area ∫₀ q^* (D(q) - p)dq between the demand curve and the market price. The same two operations also give price elasticity of demand and the continuous-compounding formula A = Pe rt.

Binomial Series: Formula, Convergence & Examples

The binomial series expands (1+x) k as an infinite power series ∑ n=0 ∞ kn xⁿ that works for any real power k, not just whole numbers. Each coefficient is the generalized binomial coefficient kn = (k(k-1)(k-2)…(k-n+1))/(n factorial), a falling product of n terms. The series converges to (1+x) k whenever |x| < 1, and when k is a non-negative integer it stops early and becomes the ordinary binomial theorem.

Average Value Of A Function: Formula & Examples

The average value of a function f on a closed interval [a, b] is f avg = 1/(b-a)∫ₐ b f(x),dx: integrate the function over the interval, then divide by the width. Geometrically it is the height of the rectangle over [a, b] whose area equals the area under the curve. When f is continuous, the Mean Value Theorem for Integrals guarantees at least one point c in [a, b] where f(c) = f avg.

Average Rate of Change: Formula & Examples

The average rate of change of a function f on an interval [a, b] is (f(b) - f(a))/(b - a): the change in output divided by the change in input. Geometrically it is the slope of the secant line through the points (a, f(a)) and (b, f(b)). Shrink the interval toward a single point and this average becomes the instantaneous rate of change, which is the derivative.

Area Under a Curve: Definite Integral & Examples

The area under a curve y = f(x) ≥ 0 between x = a and x = b equals the definite integral ∫ₐ b f(x),dx, which is itself the limit of Riemann sums, the total area of thinner and thinner rectangles under the curve. When part of the curve dips below the x-axis, the plain integral returns signed (net) area; for total area you split the interval where f crosses zero and integrate f(x). You evaluate either one with the antiderivative: ∫ₐ b f(x),dx = F(b) - F(a).

Arc Length of a Curve: Formula, Derivation & Examples

The arc length of a curve y = f(x) from x = a to x = b is L = ∫ₐ b √(1 + (dy/dx)²),dx. The formula comes from cutting the curve into tiny straight pieces, measuring each with the distance formula, and adding them up as the pieces shrink. For a curve given by parametric equations, the same idea gives L = ∫ t₁ t₂ √((dx/dt)² + (dy/dt)²),dt.

Area Between Two Curves: Formula & Examples

The area between two curves is the size of the region trapped between them, found by integrating the gap. When one curve stays above the other on [a, b], the area is ∫ₐ b (top - bottom),dx, and the two intersection points give the bounds a and b. If the curves are easier to write as functions of y, integrate ∫ c d (right - left),dy instead, and if the curves cross inside the interval, split the integral at each crossing (or integrate the absolute value f - g) so every piece stays positive.

Applications of Integration: Area, Volume & Work

The applications of integration are the geometric and physical quantities a definite integral ∫ₐ b f(x),dx can compute: area under a curve, area between two curves, volume of a solid of revolution, arc length, the average value of a function, work, and any accumulated change. Every one uses the same move: slice the quantity into infinitely thin pieces, write one integral for the piece, and add the pieces up. The integral is the total; the integrand is the size of one slice.

Applications of Derivatives: 7 Key Uses & Examples

The applications of derivatives are the practical uses of a single idea: the derivative measures how fast one quantity changes as another changes, and geometrically it is the slope of the tangent to a curve. That one tool finds rates of change, tangent and normal lines, where a function rises or falls, its highest and lowest points, its concavity, quick numerical approximations, and marginal cost in economics. This page is the umbrella over all of them, with a short worked example for each and a link to the dedicated article.

Alternating Series Test: Leibniz Rule & Examples

The Alternating Series Test, also called the Leibniz Test, says that a series whose signs flip, ∑ (-1)ⁿ bₙ or ∑ (-1)ⁿ⁺¹ bₙ with every bₙ > 0, converges when two things hold: the terms bₙ are eventually decreasing, and lim n→∞ bₙ = 0. When both are true, the sum settles on a finite value, and the error from stopping at the Nth term is never larger than the first term you left out, S - S N ≤ b N+1.

Alternating Series: Test, Convergence & Error Bound

An alternating series is a sum whose terms flip sign, written ∑ (-1)ⁿ bₙ or ∑ (-1)ⁿ⁺¹ bₙ with bₙ ≥ 0. The Alternating Series Test says such a sum converges when the sizes bₙ decrease and shrink to zero: b n+1 ≤ bₙ and lim n → ∞ bₙ = 0. The alternating harmonic series 1 - 12 + 13 - 14 + … converges to ln 2, and if you stop after N terms the error is no bigger than the first term you dropped, R N ≤ b N+1.

Absolute Value Functions: Graph, Derivative & Integral

Absolute Value Functions measure distance from zero, so f(x) = |x| equals x when x ≥ 0 and -x when x < 0. The graph is a V with its vertex at the origin, domain all real numbers, and range y ≥ 0. In calculus the surprise is that |x| is continuous everywhere yet not differentiable at x = 0, where the V has a sharp corner; away from that corner its derivative is sgn(x) and its integral is (x|x|)/2 + C.

What Is Calculus? Definition, Branches & Examples

What is calculus? It is the mathematics of continuous change. It has two branches that mirror each other: differential calculus, which measures how fast something changes (the derivative, the slope of a curve), and integral calculus, which measures how much accumulates (the integral, the area under a curve). Both rest on one idea, the limit, and the Fundamental Theorem of Calculus proves the two branches are inverses of each other.

Trigonometric Substitution: Rules & Examples

Trigonometric substitution is an integration method that replaces a square root of a quadratic with a trig function, so a Pythagorean identity can clear the root. Match the radical to its substitution: √(a²-x²) takes x=asinθ, √(a²+x²) takes x=atanθ, and √(x²-a²) takes x=asecθ. After integrating in θ, a reference triangle converts the answer back to x.

Trapezoidal Rule: Formula, Examples & Error

The Trapezoidal Rule estimates a definite integral ∫ₐ b f(x),dx by slicing [a,b] into n strips of width h=(b-a)/n and treating each strip as a trapezoid, giving Tₙ=h/2[f(x₀)+2f(x₁)+2f(x₂)+…+2f(x n-1)+f(xₙ)]. It is the simplest reliable way to find an area under a curve when the integral has no clean antiderivative, and its error shrinks like 1/(n²) as you add strips.

Taylor Series: Formula, Examples And Convergence

A Taylor series rewrites a function as an infinite sum of power terms built from the function's derivatives at a single point a: f(x) = ∑ n=0 ∞ (f (n) (a))/(n factorial)(x-a)ⁿ. When the centre is a = 0 the same formula is called a Maclaurin series. Truncating the sum after a few terms is how calculators evaluate e x, sin x, and cos x.

Squeeze Theorem: Statement, Proof & Examples

The Squeeze Theorem says that if a function f is trapped between two functions g and h near a point a, and both g and h approach the same limit L there, then f must approach L too. It is the standard tool for limits you cannot compute directly, and it is what proves the two cornerstone results lim x → 0 x² sin1/x = 0 and lim x → 0 (sin x)/x = 1.

Taylor Polynomials: Formula, Examples & Error Term

Taylor polynomials approximate a function f near a chosen point a by matching the function's value and its first n derivatives at that point. The degree-n Taylor polynomial is Pₙ(x)=∑ k=0 ⁿ (f (k) (a))/(k factorial)(x-a) k; when the center is a=0 it is called a Maclaurin polynomial. Near a the polynomial hugs the curve, and the Lagrange remainder measures how far off it can be.

Rolle's Theorem: Statement, Proof & Examples

Rolle's Theorem states that if a function f is continuous on the closed interval [a,b], differentiable on the open interval (a,b), and f(a) = f(b), then there is at least one point c in (a,b) where f'(c) = 0. Geometrically, a smooth curve that starts and ends at the same height must have a horizontal tangent somewhere in between. It is the special case of the Mean Value Theorem where the two endpoints sit at equal height.

Rational Functions: Asymptotes, Holes & Limits

Rational functions are ratios of two polynomials, f(x) = (P(x))/(Q(x)) with Q(x) ≠ 0. Their whole behaviour follows three checks: the domain excludes the zeros of Q; a zero of Q that does not cancel gives a vertical asymptote, while one that does cancel gives a hole; and comparing the degrees of P and Q gives the horizontal asymptote (or a slant one when P is exactly one degree higher). Limits describe each asymptote precisely, and the quotient rule differentiates the function.

Rate Of Change: Average vs Instantaneous

Rate of change measures how fast one quantity changes with respect to another. The average rate of change over an interval [a, b] is (f(b) - f(a))/(b - a), the slope of the secant line. The instantaneous rate of change at a point is the derivative f'(x) = lim h → 0 (f(x+h) - f(x))/h, the slope of the tangent line. Units always carry meaning: metres per second, dollars per unit, or square centimetres per second.

Quotient Rule: Formula, Proof & Examples

The quotient rule differentiates a ratio of two functions. For f(x) = (u(x))/(v(x)) with v(x) ≠ 0, the derivative is f'(x) = (u'v - uv')/(v²): the low (denominator) times the derivative of the high (numerator), minus the high times the derivative of the low, all over the low squared. The subtraction sign and the order of the two terms are what most students get wrong.

Product Rule: Formula, Proof, And Examples

The product rule gives the derivative of two functions multiplied together: if f(x) = u(x),v(x), then f'(x) = u'(x),v(x) + u(x),v'(x), often written (uv)' = u'v + uv'. The derivative of a product is not the product of the derivatives, and this article proves why from the limit definition and from the area of an expanding rectangle.

Power Series: Radius & Interval Of Convergence

A power series centered at a is an infinite sum ∑ n=0 ∞ cₙ (x-a)ⁿ, a polynomial with endlessly many terms. It converges to a finite value only on an interval around a, whose half-width is the radius of convergence R, found with the ratio test. The plainest example is the geometric power series ∑ n=0 ∞ xⁿ = 1/(1-x), which converges for |x| < 1 (so R = 1).

Power Rule in Calculus: Formula, Proof, Examples

The Power Rule is the calculus shortcut for differentiating a variable raised to a fixed power: d/dx[xⁿ] = n,xⁿ⁻¹. In words, multiply by the exponent, then drop the exponent by one. It works for every real n (positive, negative, and fractional), and it is a rule about the derivative of xⁿ, not the algebra rule for simplifying exponents.

Piecewise Functions: Graphs, Continuity & Examples

Piecewise functions use different sub-rules on different parts of the domain, written with a single brace as f(x) = cases … cases. To evaluate one, pick the rule whose condition matches the input. At a breakpoint x = c the function is continuous when the left limit, the right limit, and f(c) all agree, and differentiable when the one-sided derivatives also agree. Continuity does not guarantee differentiability: |x| is continuous at 0 but has a corner there.

Polynomial Functions: Degree, End Behavior & Calculus

Polynomial functions have the form f(x)=aₙ xⁿ + … + a₁ x + a₀, and their whole shape is fixed by two numbers: the degree n and the leading coefficient aₙ. Every polynomial is continuous and differentiable across all real numbers, so calculus reads it cleanly: the derivative f'(x) is a polynomial of degree n-1, its sign tells you where the graph rises or falls, and the second derivative tells you where it curves.

Parametric Equations: Slope, Arc Length & Examples

Parametric equations describe a curve by giving both coordinates as functions of a third variable, a parameter t: x = f(t) and y = g(t). You can often eliminate t to recover a Cartesian equation, and calculus works directly on the parameter: the slope is dy/dx = (dy/dt)/(dx/dt) (whenever dx/dt ≠ 0), and arc length is L = ∫ₐ b √((dx/dt)² + (dy/dt)²), dt.

Newtons Method: Formula, Steps & Examples

Newtons Method is an iterative root-finding technique: from a starting guess x₀, each step follows the tangent line of f down to the x-axis to produce a better estimate, using x n+1 = xₙ - (f(xₙ))/(f'(xₙ)). When it works it converges quadratically, roughly doubling the number of correct digits every step. It can fail when f'(xₙ) = 0, when the starting guess is poor, or when the estimates cycle or run off to infinity.

Natural Logarithm: Definition, Derivative & Rules

The natural logarithm, written ln x, is the logarithm with base e ≈ 2.71828, so ln x = log e x. In calculus it is defined as the area under the curve y = 1/t from t=1 to t=x, which is why ln 1 = 0 and ln e = 1. Its derivative is the cleanest in all of calculus, d/dx[ln x] = 1/x, and it is the antiderivative that fills the gap left by the power rule: ∫ 1/x,dx = ln|x| + C.

Monotonic Functions: Increasing And Decreasing

Monotonic Functions move in one direction across an interval: they are either entirely non-decreasing or entirely non-increasing, never both. You test monotonicity with the first derivative, where f'(x) > 0 means increasing, f'(x) < 0 means decreasing, and f'(x) = 0 marks a stationary candidate. A strictly monotonic function is one-to-one, which is exactly what makes it invertible.

Mean Value Theorem: Statement, Proof & Examples

The Mean Value Theorem states that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there is at least one point c in (a,b) where the instantaneous rate of change equals the average rate of change: f'(c) = (f(b) - f(a))/(b - a). Geometrically, some tangent line is parallel to the secant line joining the endpoints, and Rolle's theorem is the special case where f(a) = f(b).

Moments And Center Of Mass In Calculus

Moments measure how much a mass, or a spread-out region, tends to turn about an axis: the moment of a point mass about an axis is its mass times its distance from that axis. For a flat region under a curve f(x) on [a,b] with constant density, the moment about the y-axis is M y=∫ₐ b x,f(x),dx, the moment about the x-axis is Mₓ=∫ₐ b 1/2[f(x)]²,dx, the mass is m=∫ₐ b f(x),dx, and the center of mass sits at x=M y/m, y=Mₓ/m.

Maxima and Minima: Tests, Examples & Rules

Maxima and minima are the highest and lowest values a function reaches, either near a point (local) or across its whole domain (absolute). You find them at critical points, where the derivative f'(x) equals zero or is undefined, then classify each one with the first derivative test (sign change of f') or the second derivative test (f''(c)<0 gives a maximum, f''(c)>0 gives a minimum). For an absolute maximum or minimum on a closed interval, compare the function's values at every critical point against its values at the two endpoints.

Mathematical Induction: Proof Method & Examples

Mathematical induction is a method for proving that a statement P(n) is true for every integer n ≥ n₀. You prove two things: the base case P(n₀) is true, and the inductive step, that whenever P(k) is true, P(k+1) is true as well. Those two facts together force the statement to hold for the whole infinite chain, the way one falling domino topples an endless line.

Maclaurin Series: Formula And Key Expansions

The Maclaurin Series rewrites a function as an infinite polynomial built from its derivatives at 0: f(x)=∑ n=0 ∞ (f (n) (0))/(n factorial)xⁿ. It is the special case of the Taylor series centred at zero, and it is how a calculator turns sin x, cos x, and e x into additions it can actually perform, as long as x stays inside the interval of convergence.

Logarithmic Derivative: Formula & Examples

The logarithmic derivative of a function f is (f'(x))/(f(x)), which equals d/dx[ln f(x)] and measures the relative (percentage) rate of change of f. Logarithmic differentiation is the technique that puts this to work: take the natural log of both sides, use log laws to turn products, quotients, and powers into sums, differentiate implicitly, then solve for f'(x). It is the cleanest way to differentiate messy products and any function with a variable in both the base and the exponent, such as x x.

Inverse Functions and Differentiation Rule

Inverse Functions and Differentiation connects the slope of a function to the slope of its inverse: if b = f(a) and f'(a) ≠ 0, then (f⁻¹)'(b) = 1/(f'(a)). Geometrically, reflecting a graph across the line y = x swaps rise and run, so the two slopes are reciprocals. The rule needs f to be one-to-one and differentiable with a non-zero derivative at the matching point.

Intermediate Value Theorem: Statement & Examples

The Intermediate Value Theorem states that if f is continuous on a closed interval [a, b] and N is any value between f(a) and f(b), then there is at least one point c in [a, b] with f(c) = N. Its most-used corollary: if f is continuous and f(a) and f(b) have opposite signs, a root lies somewhere between a and b. The theorem promises a value exists, never where it is or how many there are.

Integration of Trigonometric Functions: Rules & Examples

Integration of trigonometric functions means finding the antiderivative of sin, cos, tan, and their relatives. The six building blocks are ∫ sin x,dx = -cos x + C, ∫ cos x,dx = sin x + C, ∫ sec² x,dx = tan x + C, ∫ csc² x,dx = -cot x + C, ∫ sec x tan x,dx = sec x + C, and ∫ csc x cot x,dx = -csc x + C. For higher powers you either split off one factor and substitute (odd powers) or use a power-reduction identity (even powers).

Integration by Substitution: Steps and Examples

Integration by Substitution is the reverse of the chain rule: when an integral has the shape ∫ f(g(x)),g'(x),dx, you set u = g(x) so that du = g'(x),dx, turning the whole thing into the simpler ∫ f(u),du. Integrate in u, substitute x back, and add +C. For a definite integral, change the limits to u as well, then evaluate.

Implicit Differentiation: Steps, Examples & Formula

Implicit differentiation finds dy/dx for an equation that is not solved for y, such as x² + y² = 25. You differentiate both sides with respect to x, treat y as a function of x so every y-term picks up a dy/dx through the chain rule, then solve for dy/dx. For the circle this gives dy/dx = -x/y, the slope of the tangent at any point.

Integration by Parts: Formula, LIATE & Examples

Integration by parts is the technique for integrating a product of two functions, using the formula ∫ u,dv = uv - ∫ v,du. It comes straight from reversing the product rule of differentiation. The whole skill is choosing which factor to call u (differentiate it) and which to call dv (integrate it), and the LIATE rule tells you how to pick.

Hyperbolic Functions: sinh, cosh, tanh Explained

Hyperbolic Functions are built from the exponential e x: x = (e x - e -x)/2, x = (e x + e -x)/2, and x = ( x)/( x). They obey the fundamental identity ² x - ² x = 1, which parametrizes the hyperbola x² - y² = 1 the way sine and cosine parametrize the circle. Their derivatives are clean: d/dx[ x] = x, d/dx[ x] = x, and d/dx[ x] = sech² x.

Harmonic Progression: Formula, Mean & Examples

A Harmonic Progression (HP) is a sequence of non-zero numbers whose reciprocals form an arithmetic progression. So a₁, a₂, a₃, … is an HP exactly when 1/a₁, 1/a₂, 1/a₃, … is an AP. To find any term, work in the reciprocal AP and flip the result back: aₙ = 1/(1/a₁ + (n-1)d). The harmonic mean of a and b is 2ab/(a+b), and the infinite harmonic series 1 + 12 + 13 + … grows without bound.

Graphing Functions: A Calculus Step-By-Step Guide

Graphing functions with calculus means building an accurate curve from what the equation tells you, in seven steps: domain, intercepts, symmetry, asymptotes, the first derivative f'(x) (where the curve rises or falls and where its peaks and valleys sit), the second derivative f''(x) (where it bends up or down and where it changes bend), then the assembled sketch. The first derivative gives the slope; the second derivative gives the concavity. Together they let you draw a curve you have never plotted, without guessing at points.

Geometric Series: Convergence, Sum & Formula

A geometric series adds the terms of a geometric sequence: a + ar + ar² + ar³ + …, where each term is the one before it times a fixed ratio r. Its finite sum is S N = (a(1 - r N+1))/(1 - r), and the infinite series converges to a/(1 - r) exactly when r < 1; for r ≥ 1 it diverges.

Fundamental Theorem of Calculus: Part 1 & Part 2

The Fundamental Theorem of Calculus says that differentiation and integration are inverse operations. Part 1 states that if g(x) = ∫ₐ x f(t),dt and f is continuous, then g'(x) = f(x): the derivative of an accumulated area is the function that built it. Part 2 states that ∫ₐ b f(x),dx = F(b) - F(a) whenever F' = f, which turns finding an area into a single subtraction.

Exponential Functions: Derivatives And Integrals

Exponential functions have the form f(x) = a x with a > 0 and a ≠ 1, and the star of calculus is the natural exponential f(x) = e x, the one function that is its own derivative: d/dx(e x) = e x. For a general base, d/dx(a x) = a x ln a and ∫ a x, dx = (a x)/(ln a) + C, while the growth-and-decay model y = y₀ e kt describes anything whose rate of change is proportional to its current size.

Elasticity Of Demand: Formula, Calculus & Examples

Elasticity of demand measures how sharply the quantity people buy responds to a change in price, and the calculus version is the point formula E = p/q/dp. Classify by size: |E|>1 is elastic, |E|<1 is inelastic, and |E|=1 is unit elastic. Because revenue is R = pq, the derivative dR/dp vanishes exactly at unit elasticity, so revenue is maximized where E = -1.

Domain And Range Of A Function: Rules & Examples

The domain and range of a function are, respectively, the set of allowable inputs x and the set of resulting outputs y. To find a domain you exclude the inputs that break the rule (division by zero, even roots of negatives, logs of non-positive numbers); to find a range you track the outputs algebraically, from the graph, or with calculus using extrema, asymptotes, and end behaviour.

Differentiability Of A Function: Definition & Rules

The differentiability of a function at a point a means the derivative f'(a) = lim h → 0 (f(a+h) - f(a))/h exists as a finite number, which requires the left-hand and right-hand difference-quotient limits to agree. Every differentiable function is continuous, but the reverse is false: a function can be continuous yet fail to be differentiable at a corner, a cusp, a vertical tangent, or a break.

Derivatives of Inverse Trigonometric Functions

The derivatives of inverse trigonometric functions turn six angle-finding functions into plain algebraic fractions: d/dx x=1/(√(1-x²)), d/dx x=1/(1+x²), and their four partners. Each one comes from implicit differentiation, and the three "co-function" pairs differ only by a minus sign. Get the chain-rule inner derivative and the domain right, and every problem in this topic follows the same short recipe.

Definition Of The Derivative: Limit Formula & Examples

The definition of the derivative is f'(x) = lim h → 0 (f(x+h) - f(x))/h: it measures the instantaneous rate of change of f at x, which equals the slope of the tangent line to the graph. You get it by shrinking a secant line between two nearby points until the two points merge, so the average rate of change turns into an instantaneous one. Every differentiation rule you will ever use is built from this single limit.

Definite Integrals: Definition, Properties & Examples

Definite integrals measure the signed area between a curve and the x-axis over a fixed interval [a, b], written ∫ₐ b f(x),dx and defined as the limit of Riemann sums. The Fundamental Theorem of Calculus evaluates one as F(b) - F(a), where F is any antiderivative of f. The result is a single number, with no +C, which is what separates it from the indefinite integral.

Antiderivatives: Rules, Formulas & Examples

An antiderivative of a function f is a function F whose derivative gives f back, that is F'(x) = f(x). Every function has a whole family of antiderivatives that differ only by a constant, written F(x) + C, and the indefinite integral ∫ f(x),dx = F(x) + C is the name for that family. Finding an antiderivative is differentiation run in reverse, so you can always check an answer by differentiating it and seeing the original function reappear.

Critical Points: Definition, How To Find & Examples

Critical Points of a function f are the interior domain points where f'(x) = 0 or where f'(x) does not exist. They matter because every local maximum and local minimum on the inside of a domain must occur at one of them, so they are the shortlist you search for peaks and valleys. A critical point is only a candidate, though: some are maxima, some are minima, and some (like x = 0 for f(x) = x³) are neither.

Chain Rule: Formula, Steps, and Examples

The chain rule differentiates a composite function, one function nested inside another, by the formula d/dxf(g(x)) = f'(g(x)),g'(x). In words, differentiate the outer function while holding the inner one fixed, then multiply by the derivative of the inner function. The single most common error is forgetting that final factor, the inner derivative g'(x).