{ "schema": "formula-validation-v2", "count": 42, "formulas": [ { "Id": "two-by-two-determinant", "Name": "2\u00D72 Matrix Determinant", "Category": "Linear Algebra", "Latex": "\\det\\begin{pmatrix}a\u0026b\\\\c\u0026d\\end{pmatrix}=ad-bc", "Description": "Calculates the determinant of a two-by-two matrix.", "Variables": [ "a, b, c, d: matrix entries", "det: scalar determinant" ], "Domain": "Applies to a 2 \u00D7 2 square matrix; a zero determinant indicates singularity and no two-sided inverse.", "Units": [ "determinant units are the product of one entry from each row and column" ], "BoundaryCases": [ "For 2\u00D72 Matrix Determinant, check zero, negative, and extreme input values before relying on the result.", "When using 2\u00D72 Matrix Determinant, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form \\det\\begin{pmatrix}a\u0026b\\\\c\u0026d\\end{pmatrix}=ad-bc for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For the matrix [[3, 2], [1, 4]], det(A) = 3\u00B74 \u2212 2\u00B71 = 12 \u2212 2 = 10.", "WorkedSteps": [ "Identify a = 3, b = 2, c = 1, and d = 4.", "Multiply the main diagonal entries: ad = 3 \u00D7 4 = 12.", "Multiply the other diagonal entries: bc = 2 \u00D7 1 = 2.", "Subtract to obtain det(A) = 12 \u2212 2 = 10." ], "Verification": "Because the determinant is nonzero, the matrix is invertible; direct multiplication by its computed inverse returns the identity matrix.", "Assumptions": [ "A zero determinant means the matrix is singular and has no inverse.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "CommonMistakes": [ "Do not reorder factors or mix incompatible dimensions when using 2\u00D72 Matrix Determinant; matrix operations are order-sensitive.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for 2\u00D72 Matrix Determinant.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry; NIST Digital Library of Mathematical Functions", "SourceIds": [ "openstax-algebra-trigonometry", "nist-dlmf" ], "url": "/formula/two-by-two-determinant/" }, { "Id": "circle-area", "Name": "Area of a Circle", "Category": "Geometry", "Latex": "A=\\pi r^2", "Description": "Calculates the area enclosed by a circle of radius r.", "Variables": [ "A: area", "r: radius", "\u03C0: circle constant, approximately 3.14159" ], "Domain": "Applies to a Euclidean circle with radius r \u2265 0; use the radius rather than the diameter unless you first divide the diameter by two.", "Units": [ "r uses a length unit", "A uses the corresponding squared length unit" ], "BoundaryCases": [ "For Area of a Circle, check zero, negative, and extreme input values before relying on the result.", "When using Area of a Circle, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form A=\\pi r^2 for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For a circle with radius 3 cm, A = \u03C0(3 cm)\u00B2 = 9\u03C0 cm\u00B2 \u2248 28.274 cm\u00B2.", "WorkedSteps": [ "Identify the radius as r = 3 cm.", "Square the radius: r\u00B2 = 9 cm\u00B2.", "Multiply by \u03C0 to obtain the exact area 9\u03C0 cm\u00B2.", "Round only at the end to obtain approximately 28.274 cm\u00B2." ], "Verification": "Dividing the calculated area by r\u00B2 gives \u03C0, and the area is positive with squared length units as required.", "Assumptions": [ "The radius is half the diameter.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "CommonMistakes": [ "Before substituting values into Area of a Circle, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Area of a Circle, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Area of a Circle.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/circle-area/" }, { "Id": "triangle-area", "Name": "Area of a Triangle", "Category": "Geometry", "Latex": "A=\\frac{1}{2}bh", "Description": "Calculates triangle area from a base and its perpendicular height.", "Variables": [ "A: area", "b: chosen base length", "h: perpendicular height to that base" ], "Domain": "Applies to any Euclidean triangle when h is measured perpendicular to the selected base, including when the altitude falls outside an obtuse triangle.", "Units": [ "b and h use compatible length units", "A uses squared length units" ], "BoundaryCases": [ "For Area of a Triangle, check zero, negative, and extreme input values before relying on the result.", "When using Area of a Triangle, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form A=\\frac{1}{2}bh for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For a triangle with base 10 cm and perpendicular height 6 cm, A = \u00BD(10)(6) = 30 cm\u00B2.", "WorkedSteps": [ "Choose a side as the base, b = 10 cm.", "Use the perpendicular distance to that base as h = 6 cm.", "Multiply the base and height: 10 \u00D7 6 = 60 cm\u00B2.", "Divide by two to obtain A = 30 cm\u00B2." ], "Verification": "A rectangle with the same base and height has area 60 cm\u00B2, so the triangle occupies exactly half that area.", "Assumptions": [ "The height must be perpendicular to the base.", "For the Area of a Triangle, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "CommonMistakes": [ "When copying Area of a Triangle, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Area of a Triangle, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Area of a Triangle.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/triangle-area/" }, { "Id": "arithmetic-sequence", "Name": "Arithmetic Sequence Formula", "Category": "Algebra", "Latex": "a_n=a_1\u002B(n-1)d", "Description": "Finds the nth term of a sequence with a constant difference.", "Variables": [ "a\u2099: nth term", "a\u2081: first term", "n: term number", "d: common difference" ], "Domain": "Applies to sequences with a constant additive difference between consecutive terms and integer term index n \u2265 1.", "Units": [ "a\u2081, d, and a\u2099 use the same units", "n is dimensionless" ], "BoundaryCases": [ "For Arithmetic Sequence Formula, check zero, negative, and extreme input values before relying on the result.", "When using Arithmetic Sequence Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form a_n=a_1\u002B(n-1)d for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For a sequence with first term 3 and common difference 4, the tenth term is a\u2081\u2080 = 3 \u002B (10 \u2212 1)4 = 39.", "WorkedSteps": [ "Identify a\u2081 = 3, d = 4, and n = 10.", "Compute n \u2212 1 = 9 intervals from the first term.", "Multiply the common difference: 9 \u00D7 4 = 36.", "Add the first term to obtain a\u2081\u2080 = 39." ], "Verification": "Listing the terms 3, 7, 11, \u2026 shows that adding 4 nine times reaches 39 at the tenth position.", "Assumptions": [ "The difference between consecutive terms must be constant.", "For the Arithmetic Sequence Formula, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "CommonMistakes": [ "Before substituting values into Arithmetic Sequence Formula, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Arithmetic Sequence Formula, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Arithmetic Sequence Formula.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/arithmetic-sequence/" }, { "Id": "arithmetic-series-sum", "Name": "Arithmetic Series Sum", "Category": "Algebra", "Latex": "S_n=\\frac n2(a_1\u002Ba_n)", "Description": "Computes the sum of the first n terms of an arithmetic sequence.", "Variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "Domain": "Apply Arithmetic Series Sum only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "Units": [], "BoundaryCases": [ "For Arithmetic Series Sum, check zero, negative, and extreme input values before relying on the result.", "When using Arithmetic Series Sum, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form S_n=\\frac n2(a_1\u002Ba_n) for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "The ten-term sequence 3,7,\u2026,39 has S\u2081\u2080=10(3\u002B39)/2=210.", "WorkedSteps": [ "Identify n=10, a\u2081=3, and a\u2099=39.", "Add the first and last terms to obtain 42.", "Multiply by n/2=5 to obtain 210." ], "Verification": "Pairing first with last gives five pairs of 42, also totaling 210.", "Assumptions": [ "For the Arithmetic Series Sum, verify assumptions and units before applying the expression.", "For the Arithmetic Series Sum, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Arithmetic Series Sum, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "CommonMistakes": [ "When copying Arithmetic Series Sum, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Arithmetic Series Sum, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Arithmetic Series Sum.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/arithmetic-series-sum/" }, { "Id": "bayes-theorem", "Name": "Bayes\u2019 Theorem", "Category": "Probability", "Latex": "P(A\\mid B)=\\frac{P(B\\mid A)P(A)}{P(B)}", "Description": "Updates the probability of an event after observing evidence.", "Variables": [ "P(A|B): posterior probability", "P(B|A): likelihood", "P(A): prior probability", "P(B): evidence probability" ], "Domain": "Requires defined prior and conditional probabilities with a nonzero probability for the observed evidence in the denominator.", "Units": [ "all probabilities are dimensionless values from 0 to 1 or equivalent percentages" ], "BoundaryCases": [ "For Bayes\u2019 Theorem, check zero, negative, and extreme input values before relying on the result.", "When using Bayes\u2019 Theorem, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form P(A\\mid B)=\\frac{P(B\\mid A)P(A)}{P(B)} for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "With 1% prevalence, 90% sensitivity, and 5% false-positive rate, P(condition|positive) = 0.009/(0.009 \u002B 0.0495) \u2248 0.1538, or 15.38%.", "WorkedSteps": [ "Compute the true-positive joint probability: 0.01 \u00D7 0.90 = 0.009.", "Compute the false-positive joint probability: 0.99 \u00D7 0.05 = 0.0495.", "Add both paths to get P(positive) = 0.0585.", "Divide 0.009 by 0.0585 to obtain approximately 0.1538." ], "Verification": "In a hypothetical 10,000-person group, about 90 true positives and 495 false positives produce 90/585 = 15.38%, matching the formula.", "Assumptions": [ "P(B) must be greater than zero.", "For the Bayes\u2019 Theorem, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "CommonMistakes": [ "When copying Bayes\u2019 Theorem, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not assume events are independent or mutually exclusive unless the problem states or proves that condition." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Bayes\u2019 Theorem.", "VerifiedAgainst": "OpenStax Introductory Statistics", "SourceIds": [ "openstax-intro-statistics" ], "url": "/formula/bayes-theorem/" }, { "Id": "binomial-probability", "Name": "Binomial Probability Formula", "Category": "Probability", "Latex": "P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}", "Description": "Finds the probability of exactly k successes in n independent Bernoulli trials.", "Variables": [ "n: number of trials", "k: number of successes", "p: success probability per trial", "X: success count" ], "Domain": "Apply Binomial Probability Formula only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "Units": [], "BoundaryCases": [ "For Binomial Probability Formula, check zero, negative, and extreme input values before relying on the result.", "When using Binomial Probability Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k} for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For three fair coin flips, the probability of exactly two heads is C(3,2)(0.5)\u00B2(0.5)=3/8=0.375.", "WorkedSteps": [ "Set n=3, k=2, and p=0.5.", "Compute C(3,2)=3 and p\u00B2(1-p)=0.125.", "Multiply 3\u00D70.125 to obtain 0.375." ], "Verification": "The three equally likely outcomes HHT, HTH, and THH out of eight total outcomes confirm 3/8.", "Assumptions": [ "Trials must be independent and use the same success probability.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "CommonMistakes": [ "Do not substitute sample and population quantities interchangeably in Binomial Probability Formula; map every symbol to its definition first.", "Do not assume events are independent or mutually exclusive unless the problem states or proves that condition." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Binomial Probability Formula.", "VerifiedAgainst": "OpenStax Introductory Statistics", "SourceIds": [ "openstax-intro-statistics" ], "url": "/formula/binomial-probability/" }, { "Id": "chain-rule", "Name": "Chain Rule", "Category": "Calculus", "Latex": "\\frac{d}{dx}f(g(x))=f\u0027(g(x))g\u0027(x)", "Description": "Differentiates a composite function by combining outer and inner derivatives.", "Variables": [ "f: outer function", "g: inner function", "x: independent variable" ], "Domain": "Applies when a differentiable outer function is composed with a differentiable inner function at the point being evaluated.", "Units": [ "derivative units are output units per input unit" ], "BoundaryCases": [ "For Chain Rule, check zero, negative, and extreme input values before relying on the result.", "When using Chain Rule, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form \\frac{d}{dx}f(g(x))=f\u0027(g(x))g\u0027(x) for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For y = (3x \u002B 1)\u2074, let u = 3x \u002B 1. Then dy/dx = 4u\u00B3\u00B73 = 12(3x \u002B 1)\u00B3.", "WorkedSteps": [ "Identify the outer function u\u2074 and the inner function u = 3x \u002B 1.", "Differentiate the outer function with respect to u: 4u\u00B3.", "Differentiate the inner function with respect to x: du/dx = 3.", "Multiply the derivatives and replace u to obtain 12(3x \u002B 1)\u00B3." ], "Verification": "Expanding (3x \u002B 1)\u2074 first and differentiating term by term produces the same polynomial as expanding 12(3x \u002B 1)\u00B3.", "Assumptions": [ "Differentiate the outer function while leaving the inner expression in place, then multiply by the inner derivative.", "For the Chain Rule, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "CommonMistakes": [ "When copying Chain Rule, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Chain Rule.", "VerifiedAgainst": "OpenStax Calculus Volume 1", "SourceIds": [ "openstax-calculus-volume-1" ], "url": "/formula/chain-rule/" }, { "Id": "circular-arc-length", "Name": "Circular Arc Length", "Category": "Geometry", "Latex": "s=r\\theta", "Description": "Calculates an arc length from a circle radius and a central angle measured in radians.", "Variables": [ "s: arc length", "r: radius", "\u03B8: central angle in radians" ], "Domain": "Apply Circular Arc Length only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "Units": [], "BoundaryCases": [ "For Circular Arc Length, check zero, negative, and extreme input values before relying on the result.", "When using Circular Arc Length, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form s=r\\theta for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For r=3 and \u03B8=\u03C0/2 radians, s=r\u03B8=3\u03C0/2\u22484.712.", "WorkedSteps": [ "Convert the central angle to radians if necessary.", "Multiply radius 3 by \u03C0/2.", "Report the result in the same length unit as the radius." ], "Verification": "A quarter of a circle with circumference 6\u03C0 has length 6\u03C0/4=3\u03C0/2.", "Assumptions": [ "Convert degrees to radians before using this form.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "CommonMistakes": [ "Before substituting values into Circular Arc Length, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Circular Arc Length, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Circular Arc Length.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/circular-arc-length/" }, { "Id": "circle-circumference", "Name": "Circumference of a Circle", "Category": "Geometry", "Latex": "C=2\\pi r=\\pi d", "Description": "Calculates the distance around a circle from its radius or diameter.", "Variables": [ "C: circumference", "r: radius", "d: diameter", "\u03C0: circle constant" ], "Domain": "Applies to a Euclidean circle with radius r \u2265 0; the equivalent form C = \u03C0d may be used when the diameter is known.", "Units": [ "r and C use the same length unit" ], "BoundaryCases": [ "For Circumference of a Circle, check zero, negative, and extreme input values before relying on the result.", "When using Circumference of a Circle, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form C=2\\pi r=\\pi d for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For a circle with radius 4 m, C = 2\u03C0(4 m) = 8\u03C0 m \u2248 25.133 m.", "WorkedSteps": [ "Identify the radius as r = 4 m.", "Multiply the radius by 2 to obtain the diameter 8 m.", "Multiply the diameter by \u03C0 to obtain the exact circumference 8\u03C0 m.", "Round at the final step to approximately 25.133 m." ], "Verification": "The ratio C/(2r) equals \u03C0, and the result has a length unit rather than a squared unit.", "Assumptions": [ "The diameter equals 2r.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "CommonMistakes": [ "Before substituting values into Circumference of a Circle, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Circumference of a Circle, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Circumference of a Circle.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/circle-circumference/" }, { "Id": "complex-polar-form", "Name": "Complex Number Polar Form", "Category": "Algebra", "Latex": "z=r(\\cos\\theta\u002Bi\\sin\\theta)=re^{i\\theta}", "Description": "Expresses a complex number by magnitude and argument.", "Variables": [ "r: nonnegative modulus", "\u03B8: argument in radians", "i: imaginary unit" ], "Domain": "Apply Complex Number Polar Form only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "Units": [], "BoundaryCases": [ "For Complex Number Polar Form, check zero, negative, and extreme input values before relying on the result.", "When using Complex Number Polar Form, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form z=r(\\cos\\theta\u002Bi\\sin\\theta)=re^{i\\theta} for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For z=1\u002Bi, r=\u221A(1\u00B2\u002B1\u00B2)=\u221A2 and \u03B8=atan2(1,1)=\u03C0/4.", "WorkedSteps": [ "Compute the modulus r=\u221A2.", "Determine the first-quadrant argument \u03B8=\u03C0/4.", "Write z=\u221A2(cos\u03C0/4\u002Bi sin\u03C0/4)=\u221A2e^(i\u03C0/4)." ], "Verification": "Converting back gives real and imaginary parts equal to 1.", "Assumptions": [ "The argument is defined modulo 2\u03C0.", "The angle unit and the domain or branch of any inverse trigonometric function must be explicit.", "For the Complex Number Polar Form, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "CommonMistakes": [ "Before substituting values into Complex Number Polar Form, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Complex Number Polar Form, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Complex Number Polar Form.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/complex-polar-form/" }, { "Id": "compound-interest", "Name": "Compound Interest Formula", "Category": "Applied Math", "Latex": "A=P\\left(1\u002B\\frac{r}{n}\\right)^{nt}", "Description": "Calculates a balance with interest compounded a fixed number of times per year.", "Variables": [ "A: final amount", "P: principal", "r: annual rate as a decimal", "n: compounding periods per year", "t: years" ], "Domain": "Assumes a fixed nominal annual rate, regular compounding, no deposits or withdrawals, and consistent time units for n and t.", "Units": [ "P and A use the same currency", "r is a decimal rate", "t is measured in years when r is annual" ], "BoundaryCases": [ "For Compound Interest Formula, check zero, negative, and extreme input values before relying on the result.", "When using Compound Interest Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form A=P\\left(1\u002B\\frac{r}{n}\\right)^{nt} for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "Invest $1,000 at a nominal annual rate of 5%, compounded monthly for 2 years: A = 1000(1 \u002B 0.05/12)\u00B2\u2074 \u2248 $1,104.94.", "WorkedSteps": [ "Set P = 1000, r = 0.05, n = 12, and t = 2.", "Compute the periodic rate: r/n = 0.05/12.", "Compute the number of periods: nt = 12 \u00D7 2 = 24.", "Evaluate 1000(1 \u002B 0.05/12)\u00B2\u2074 and round currency at the end." ], "Verification": "The balance exceeds the principal and is slightly above the annual-compounding result, which is consistent with positive monthly compounding at the same nominal rate.", "Assumptions": [ "The stated rate must be converted from percent to decimal.", "For the Compound Interest Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Map every symbol to the modeled quantity and verify units, domain restrictions, and simplifying assumptions." ], "CommonMistakes": [ "When copying Compound Interest Formula, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Compound Interest Formula with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Compound Interest Formula.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry; NIST Digital Library of Mathematical Functions", "SourceIds": [ "openstax-algebra-trigonometry", "nist-dlmf" ], "url": "/formula/compound-interest/" }, { "Id": "conditional-probability", "Name": "Conditional Probability", "Category": "Probability", "Latex": "P(A\\mid B)=\\frac{P(A\\cap B)}{P(B)}", "Description": "Calculates the probability of A when B is known to have occurred.", "Variables": [ "P(A|B): probability of A given B", "P(A \u2229 B): probability both occur", "P(B): probability of the condition" ], "Domain": "The conditioning event must have nonzero probability; all counts or probabilities must refer to the same population and sampling frame.", "Units": [ "probabilities are dimensionless", "counts use the same population base" ], "BoundaryCases": [ "For Conditional Probability, check zero, negative, and extreme input values before relying on the result.", "When using Conditional Probability, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form P(A\\mid B)=\\frac{P(A\\cap B)}{P(B)} for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "Among 18 students who study mathematics, 12 also study coding. P(coding|mathematics) = 12/18 = 2/3 \u2248 66.7%.", "WorkedSteps": [ "Restrict the sample space to the 18 students satisfying the condition.", "Count the 12 students who satisfy both mathematics and coding.", "Divide the intersection count by the condition count: 12/18.", "Simplify to 2/3 and convert to approximately 66.7%." ], "Verification": "Multiplying P(coding|mathematics) by P(mathematics) recovers the joint probability P(coding \u2229 mathematics) for the same population.", "Assumptions": [ "P(B) must be nonzero.", "For the Conditional Probability, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "CommonMistakes": [ "When copying Conditional Probability, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not assume events are independent or mutually exclusive unless the problem states or proves that condition." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Conditional Probability.", "VerifiedAgainst": "OpenStax Introductory Statistics", "SourceIds": [ "openstax-intro-statistics" ], "url": "/formula/conditional-probability/" }, { "Id": "cross-product-formula", "Name": "Cross Product Components", "Category": "Linear Algebra", "Latex": "\\mathbf a\\times\\mathbf b=\\begin{bmatrix}a_2b_3-a_3b_2\\\\a_3b_1-a_1b_3\\\\a_1b_2-a_2b_1\\end{bmatrix}", "Description": "Calculates the three-dimensional vector cross product.", "Variables": [ "a,b: 3D vectors", "result: the quantity produced after the stated inputs and conditions are applied" ], "Domain": "Apply Cross Product Components only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "Units": [], "BoundaryCases": [ "For Cross Product Components, check zero, negative, and extreme input values before relying on the result.", "When using Cross Product Components, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form \\mathbf a\\times\\mathbf b=\\begin{bmatrix}a_2b_3-a_3b_2\\\\a_3b_1-a_1b_3\\\\a_1b_2-a_2b_1\\end{bmatrix} for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For a=(1,0,0) and b=(0,1,0), a\u00D7b=(0,0,1).", "WorkedSteps": [ "Compute the first component 0\u00B70-0\u00B71=0.", "Compute the second component 0\u00B70-1\u00B70=0.", "Compute the third component 1\u00B71-0\u00B70=1." ], "Verification": "The result is perpendicular to both input vectors and has unit length.", "Assumptions": [ "The result is perpendicular to both vectors.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "CommonMistakes": [ "Do not reorder factors or mix incompatible dimensions when using Cross Product Components; matrix operations are order-sensitive.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Cross Product Components.", "VerifiedAgainst": "NIST Digital Library of Mathematical Functions", "SourceIds": [ "nist-dlmf" ], "url": "/formula/cross-product-formula/" }, { "Id": "derivative-definition", "Name": "Derivative Definition", "Category": "Calculus", "Latex": "f\u0027(x)=\\lim_{h\\to0}\\frac{f(x\u002Bh)-f(x)}{h}", "Description": "Defines the instantaneous rate of change of a function as a limit.", "Variables": [ "f(x): original function", "h: change in the input", "f\u2032(x): derivative at x" ], "Domain": "The derivative exists at x only when the difference quotient approaches a finite common limit from all required directions.", "Units": [ "derivative units are function-output units per x-unit" ], "BoundaryCases": [ "For Derivative Definition, check zero, negative, and extreme input values before relying on the result.", "When using Derivative Definition, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form f\u0027(x)=\\lim_{h\\to0}\\frac{f(x\u002Bh)-f(x)}{h} for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For f(x) = x\u00B2, [f(x\u002Bh) \u2212 f(x)]/h = [(x\u002Bh)\u00B2 \u2212 x\u00B2]/h = 2x \u002B h. Taking h \u2192 0 gives f\u2032(x) = 2x.", "WorkedSteps": [ "Substitute f(x) = x\u00B2 into the difference quotient.", "Expand (x \u002B h)\u00B2 and subtract x\u00B2 to obtain 2xh \u002B h\u00B2.", "Factor h and cancel it for h \u2260 0, leaving 2x \u002B h.", "Take the limit as h approaches 0 to obtain 2x." ], "Verification": "The result agrees with the power rule, and numerical secant slopes near a chosen x approach 2x as h becomes smaller.", "Assumptions": [ "The limit must exist for the derivative to exist.", "For the Derivative Definition, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "CommonMistakes": [ "When copying Derivative Definition, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Derivative Definition.", "VerifiedAgainst": "OpenStax Calculus Volume 1", "SourceIds": [ "openstax-calculus-volume-1" ], "url": "/formula/derivative-definition/" }, { "Id": "power-rule-derivative", "Name": "Derivative Power Rule", "Category": "Calculus", "Latex": "\\frac{d}{dx}x^n=nx^{n-1}", "Description": "Differentiates a power of x.", "Variables": [ "n: constant exponent", "x: variable" ], "Domain": "Apply the power rule on intervals where x^n is differentiable under the stated exponent and number-system assumptions.", "Units": [], "BoundaryCases": [ "For n=0, the derivative of the constant x^0=1 is zero even though the unsimplified pattern contains x^{-1}.", "For negative or fractional exponents, exclude points where the original real-valued power is not defined." ], "AlternativeForms": [ "For a monomial ax^n, linearity gives d(ax^n)/dx=anx^{n-1}." ], "WorkedExample": "For f(x)=3x^5, f\u2032(x)=3\u00B75x^4=15x^4.", "WorkedSteps": [ "Identify the coefficient and exponent of each power term before differentiating.", "Multiply the coefficient by the original exponent for the derivative coefficient.", "Reduce the exponent by one while preserving the variable and any constant multiplier.", "Verify the derivative at a sample point with a difference quotient or a trusted symbolic calculation." ], "Verification": "For a chosen value of x, compare the derivative with a small-step numerical difference quotient and confirm that the values converge as the step shrinks.", "Assumptions": [ "The exponent and domain must make the original function and resulting derivative well defined.", "Constant multiples should be retained and differentiated using linearity." ], "CommonMistakes": [ "Do not subtract one from the exponent without also multiplying by the original exponent.", "Do not use the elementary real-valued rule across points where a fractional or negative power is undefined." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked example, boundary cases, and independent verification method for Derivative Power Rule.", "VerifiedAgainst": "OpenStax Calculus Volume 1", "SourceIds": [ "openstax-calculus-volume-1" ], "url": "/formula/power-rule-derivative/" }, { "Id": "distance-formula", "Name": "Distance Formula", "Category": "Algebra", "Latex": "d=\\sqrt{(x_2-x_1)^2\u002B(y_2-y_1)^2}", "Description": "Calculates the straight-line distance between two points in a coordinate plane.", "Variables": [ "(x\u2081, y\u2081): first point", "(x\u2082, y\u2082): second point", "d: distance between the points" ], "Domain": "Applies to two points in a Euclidean Cartesian plane; the result is always nonnegative and independent of point order.", "Units": [ "both coordinate axes must use compatible length units" ], "BoundaryCases": [ "For Distance Formula, check zero, negative, and extreme input values before relying on the result.", "When using Distance Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form d=\\sqrt{(x_2-x_1)^2\u002B(y_2-y_1)^2} for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "The distance between (1, 2) and (4, 6) is \u221A[(4 \u2212 1)\u00B2 \u002B (6 \u2212 2)\u00B2] = \u221A(9 \u002B 16) = 5.", "WorkedSteps": [ "Find the horizontal change: \u0394x = 4 \u2212 1 = 3.", "Find the vertical change: \u0394y = 6 \u2212 2 = 4.", "Square and add the changes: 3\u00B2 \u002B 4\u00B2 = 25.", "Take the nonnegative square root to obtain a distance of 5." ], "Verification": "The coordinate differences form a 3\u20134\u20135 right triangle, so the computed straight-line distance of 5 is consistent with the Pythagorean theorem.", "Assumptions": [ "The order of the two points does not change the result.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "For the Distance Formula, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "CommonMistakes": [ "Do not drop the radical boundary in Distance Formula; verify exactly which terms are inside the root and whether the chosen root is valid.", "For the Distance Formula, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Distance Formula.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/distance-formula/" }, { "Id": "fundamental-theorem-calculus", "Name": "Fundamental Theorem of Calculus", "Category": "Calculus", "Latex": "\\int_a^b f(x)\\,dx=F(b)-F(a),\\quad F\u0027=f", "Description": "Connects definite integration with antiderivatives.", "Variables": [ "f: integrand", "F: any antiderivative of f", "a, b: integration limits" ], "Domain": "Apply Fundamental Theorem of Calculus only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "Units": [], "BoundaryCases": [ "For Fundamental Theorem of Calculus, check zero, negative, and extreme input values before relying on the result.", "When using Fundamental Theorem of Calculus, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form \\int_a^b f(x)\\,dx=F(b)-F(a),\\quad F\u0027=f for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "Evaluate \u222B\u2080\u00B92x dx. An antiderivative is F(x)=x\u00B2, so F(1)-F(0)=1.", "WorkedSteps": [ "Choose F(x)=x\u00B2 because F\u2032(x)=2x.", "Evaluate the upper endpoint: F(1)=1.", "Evaluate the lower endpoint and subtract: 1-0=1." ], "Verification": "A geometric check gives the area of a triangle with base 1 and height 2, which is also 1.", "Assumptions": [ "The function must satisfy the theorem\u2019s continuity conditions on the interval.", "Integration limits, the differential, and any constant of integration must be included where required.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "CommonMistakes": [ "Keep the integration bounds and differential attached to Fundamental Theorem of Calculus, and include a constant of integration for an indefinite integral.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Fundamental Theorem of Calculus.", "VerifiedAgainst": "OpenStax Calculus Volume 1", "SourceIds": [ "openstax-calculus-volume-1" ], "url": "/formula/fundamental-theorem-calculus/" }, { "Id": "geometric-sequence", "Name": "Geometric Sequence Formula", "Category": "Algebra", "Latex": "a_n=a_1r^{n-1}", "Description": "Finds the nth term of a sequence with a constant ratio.", "Variables": [ "a\u2099: nth term", "a\u2081: first term", "r: common ratio", "n: term number" ], "Domain": "Applies to sequences with a constant multiplicative ratio between consecutive nonzero terms and integer index n \u2265 1.", "Units": [ "a\u2081 and a\u2099 use the same units", "r and n are dimensionless" ], "BoundaryCases": [ "For Geometric Sequence Formula, check zero, negative, and extreme input values before relying on the result.", "When using Geometric Sequence Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form a_n=a_1r^{n-1} for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For a sequence with first term 2 and common ratio 3, the fifth term is a\u2085 = 2\u00B73\u2074 = 162.", "WorkedSteps": [ "Identify a\u2081 = 2, r = 3, and n = 5.", "Compute the exponent n \u2212 1 = 4.", "Evaluate the repeated factor: 3\u2074 = 81.", "Multiply by the first term: 2 \u00D7 81 = 162." ], "Verification": "The sequence 2, 6, 18, 54, 162 multiplies by 3 at every step and confirms the fifth term.", "Assumptions": [ "Each term is obtained by multiplying the previous term by r.", "For the Geometric Sequence Formula, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "CommonMistakes": [ "Before substituting values into Geometric Sequence Formula, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Geometric Sequence Formula, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Geometric Sequence Formula.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/geometric-sequence/" }, { "Id": "haversine-distance", "Name": "Haversine Distance", "Category": "Geometry", "Latex": "d=2R\\arcsin\\sqrt{\\sin^2\\frac{\\Delta\\varphi}{2}\u002B\\cos\\varphi_1\\cos\\varphi_2\\sin^2\\frac{\\Delta\\lambda}{2}}", "Description": "Approximates great-circle distance between two latitude\u2013longitude points on a sphere.", "Variables": [ "R: sphere radius", "\u03C6: latitude in radians", "\u03BB: longitude in radians" ], "Domain": "Apply Haversine Distance only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "Units": [], "BoundaryCases": [ "For Haversine Distance, check zero, negative, and extreme input values before relying on the result.", "When using Haversine Distance, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form d=2R\\arcsin\\sqrt{\\sin^2\\frac{\\Delta\\varphi}{2}\u002B\\cos\\varphi_1\\cos\\varphi_2\\sin^2\\frac{\\Delta\\lambda}{2}} for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For two equatorial points separated by 90\u00B0 longitude, a=sin\u00B2(\u03C0/4)=1/2 and d=2Rasin(1/\u221A2)=\u03C0R/2.", "WorkedSteps": [ "Convert 90\u00B0 to \u0394\u03BB=\u03C0/2 and set both latitudes to zero.", "Compute the haversine term a=1/2.", "Evaluate 2Rasin(\u221Aa)=\u03C0R/2." ], "Verification": "The points are one quarter of a great circle apart, so the distance must be one quarter of 2\u03C0R.", "Assumptions": [ "Earth is not a perfect sphere; ellipsoidal methods are more accurate.", "For the Haversine Distance, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "The angle unit and the domain or branch of any inverse trigonometric function must be explicit." ], "CommonMistakes": [ "When copying Haversine Distance, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Haversine Distance, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Haversine Distance.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/haversine-distance/" }, { "Id": "integration-by-parts", "Name": "Integration by Parts", "Category": "Calculus", "Latex": "\\int u\\,dv=uv-\\int v\\,du", "Description": "Transforms an integral of a product using the product rule in reverse.", "Variables": [ "u: selected function to differentiate", "dv: selected differential to integrate", "du: derivative of u", "v: antiderivative of dv" ], "Domain": "Applies to integrals where the integrand can be expressed as a product u\u00B7dv and both required derivatives and antiderivatives exist.", "Units": [ "the antiderivative has integrand units multiplied by the input unit" ], "BoundaryCases": [ "For Integration by Parts, check zero, negative, and extreme input values before relying on the result.", "When using Integration by Parts, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form \\int u\\,dv=uv-\\int v\\,du for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "Evaluate \u222Bx e\u02E3 dx. Choose u = x and dv = e\u02E3dx, giving du = dx and v = e\u02E3. The result is xe\u02E3 \u2212 e\u02E3 \u002B C = e\u02E3(x \u2212 1) \u002B C.", "WorkedSteps": [ "Choose u = x because it simplifies when differentiated, and choose dv = e\u02E3 dx.", "Compute du = dx and integrate dv to obtain v = e\u02E3.", "Apply \u222Bu dv = uv \u2212 \u222Bv du to get xe\u02E3 \u2212 \u222Be\u02E3 dx.", "Integrate the remaining term and include the constant: e\u02E3(x \u2212 1) \u002B C." ], "Verification": "Differentiating e\u02E3(x \u2212 1) gives e\u02E3(x \u2212 1) \u002B e\u02E3 = xe\u02E3, which recovers the original integrand.", "Assumptions": [ "Choose u so that differentiating it simplifies the expression.", "Integration limits, the differential, and any constant of integration must be included where required.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "CommonMistakes": [ "Keep the integration bounds and differential attached to Integration by Parts, and include a constant of integration for an indefinite integral.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Integration by Parts.", "VerifiedAgainst": "OpenStax Calculus Volume 1", "SourceIds": [ "openstax-calculus-volume-1" ], "url": "/formula/integration-by-parts/" }, { "Id": "laplace-transform", "Name": "Laplace Transform", "Category": "Differential Equations", "Latex": "F(s)=\\int_0^{\\infty}f(t)e^{-st}\\,dt", "Description": "Maps a time-domain function to a complex-frequency function useful for initial-value problems.", "Variables": [ "f(t): time-domain function", "s: complex frequency" ], "Domain": "Apply Laplace Transform only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "Units": [], "BoundaryCases": [ "For Laplace Transform, check zero, negative, and extreme input values before relying on the result.", "When using Laplace Transform, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form F(s)=\\int_0^{\\infty}f(t)e^{-st}\\,dt for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For f(t)=1 and Re(s)\u003E0, F(s)=\u222B\u2080\u221Ee^(-st)dt=[-e^(-st)/s]\u2080\u221E=1/s.", "WorkedSteps": [ "Require Re(s)\u003E0 so e^(-st) tends to zero.", "Integrate e^(-st) to -e^(-st)/s.", "Evaluate the endpoints to obtain 1/s." ], "Verification": "Differentiating the antiderivative and checking the endpoint limit confirms the result.", "Assumptions": [ "Convergence requires a suitable region in the complex plane.", "Integration limits, the differential, and any constant of integration must be included where required.", "State the domain together with any initial or boundary conditions, and confirm the regularity assumptions required by the method." ], "CommonMistakes": [ "Keep the integration bounds and differential attached to Laplace Transform, and include a constant of integration for an indefinite integral.", "Do not present a general solution as the requested solution before applying all initial or boundary conditions." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Laplace Transform.", "VerifiedAgainst": "NIST Digital Library of Mathematical Functions", "SourceIds": [ "nist-dlmf" ], "url": "/formula/laplace-transform/" }, { "Id": "law-of-cosines", "Name": "Law of Cosines", "Category": "Geometry", "Latex": "c^2=a^2\u002Bb^2-2ab\\cos C", "Description": "Relates the sides of any triangle to the cosine of one included angle.", "Variables": [ "a, b, c: side lengths", "C: angle opposite side c" ], "Domain": "Use the law of cosines for a Euclidean triangle when the paired side lengths and included or opposite angle are identified consistently.", "Units": [], "BoundaryCases": [ "At C=0\u00B0 or 180\u00B0, the configuration becomes degenerate and should be interpreted as a limiting case rather than an ordinary triangle.", "For C=90\u00B0, the law of cosines must reduce exactly to the Pythagorean theorem." ], "AlternativeForms": [ "Solving for the angle gives C=arccos((a\u00B2\u002Bb\u00B2\u2212c\u00B2)/(2ab)) when a and b are nonzero and the ratio lies in [\u22121,1]." ], "WorkedExample": "For a=3, b=4, and C=90\u00B0, c\u00B2=9\u002B16\u221224\u00B70=25, so c=5.", "WorkedSteps": [ "Label each side opposite its corresponding angle and identify the known side-angle relationship.", "Substitute a, b, and C into c\u00B2=a\u00B2\u002Bb\u00B2\u22122ab cos C with the correct angle mode.", "Evaluate the squared length before taking the nonnegative square root for the physical side length.", "Verify that the three side lengths satisfy the triangle inequality and that the result reduces to the Pythagorean theorem when C=90\u00B0." ], "Verification": "Check the triangle inequalities and, for a right angle, confirm that the cosine term vanishes and the equation becomes c\u00B2=a\u00B2\u002Bb\u00B2.", "Assumptions": [ "Side lengths are positive and form a valid triangle.", "Angle C is opposite side c, and all angle measurements use a consistent degree or radian mode." ], "CommonMistakes": [ "Do not pair an angle with the wrong opposite side.", "Check calculator angle mode and preserve the negative sign before 2ab cos C." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked example, boundary cases, and independent verification method for Law of Cosines.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry 2e", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/law-of-cosines/" }, { "Id": "law-of-sines", "Name": "Law of Sines", "Category": "Geometry", "Latex": "\\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C}", "Description": "Relates each side of a triangle to the sine of its opposite angle.", "Variables": [ "a, b, c: side lengths", "A, B, C: opposite angles" ], "Domain": "Apply Law of Sines only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "Units": [], "BoundaryCases": [ "For Law of Sines, check zero, negative, and extreme input values before relying on the result.", "When using Law of Sines, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form \\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C} for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "With A=30\u00B0, a=5, and B=45\u00B0, b=5sin45\u00B0/sin30\u00B0=5\u221A2\u22487.071.", "WorkedSteps": [ "Write b/sin45\u00B0=5/sin30\u00B0.", "Solve for b by multiplying by sin45\u00B0.", "Use sin45\u00B0=\u221A2/2 and sin30\u00B0=1/2." ], "Verification": "The larger angle B=45\u00B0 is opposite the larger side b\u22487.071, which is consistent.", "Assumptions": [ "The ambiguous SSA case can produce zero, one, or two triangles.", "For the Law of Sines, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The angle unit and the domain or branch of any inverse trigonometric function must be explicit.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "CommonMistakes": [ "When copying Law of Sines, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Law of Sines, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Law of Sines.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/law-of-sines/" }, { "Id": "midpoint-formula", "Name": "Midpoint Formula", "Category": "Geometry", "Latex": "M=\\left(\\frac{x_1\u002Bx_2}{2},\\frac{y_1\u002By_2}{2}\\right)", "Description": "Finds the point halfway between two coordinate points.", "Variables": [ "(x\u2081,y\u2081),(x\u2082,y\u2082): endpoints", "M: midpoint" ], "Domain": "Use the midpoint formula for two points expressed in the same Cartesian coordinate system and compatible units.", "Units": [], "BoundaryCases": [ "If the endpoints are identical, the midpoint must equal that same point.", "If one coordinate pair is symmetric about zero, the corresponding midpoint coordinate should be zero." ], "AlternativeForms": [ "In vector form, the midpoint can be written as M=(P_1\u002BP_2)/2 when vector addition and scalar division are defined." ], "WorkedExample": "For endpoints (\u22122, 4) and (6, 10), the midpoint is ((\u22122\u002B6)/2, (4\u002B10)/2) = (2, 7).", "WorkedSteps": [ "List the two endpoints in the same coordinate order and confirm that their axes and units match.", "Add the two x-coordinates and divide that sum by two to obtain the midpoint x-coordinate.", "Add the two y-coordinates and divide that sum by two to obtain the midpoint y-coordinate.", "Verify the result by checking that the displacement from the first endpoint to the midpoint equals the displacement from the midpoint to the second endpoint." ], "Verification": "Subtract the first endpoint from the midpoint and the midpoint from the second endpoint; the two displacement vectors should be identical.", "Assumptions": [ "Both endpoints use the same coordinate axes, scale, and units.", "The midpoint is obtained by averaging corresponding coordinates, not by averaging distances." ], "CommonMistakes": [ "Do not pair x\u2081 with y\u2082 or otherwise mix coordinate positions.", "Keep the factor of 1/2 on both coordinate sums and preserve negative signs inside parentheses." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked example, boundary cases, and independent verification method for Midpoint Formula.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry 2e", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/midpoint-formula/" }, { "Id": "percentage-change", "Name": "Percentage Change Formula", "Category": "Applied Math", "Latex": "\\%\\text{ change}=\\frac{\\text{new}-\\text{old}}{|\\text{old}|}\\times100\\%", "Description": "Measures relative increase or decrease from an original nonzero value.", "Variables": [ "old: original value", "new: later value", "% change: signed relative difference" ], "Domain": "The original value must be nonzero; the sign distinguishes an increase from a decrease, and context determines whether percentage change is meaningful.", "Units": [ "old and new values must use the same unit", "the final result is a percentage" ], "BoundaryCases": [ "For Percentage Change Formula, check zero, negative, and extreme input values before relying on the result.", "When using Percentage Change Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form \\%\\text{ change}=\\frac{\\text{new}-\\text{old}}{|\\text{old}|}\\times100\\% for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "A value rises from 80 to 100. Percentage change = (100 \u2212 80)/80 \u00D7 100% = 25%.", "WorkedSteps": [ "Identify the original value as 80 and the new value as 100.", "Calculate the change: 100 \u2212 80 = 20.", "Divide by the original value: 20/80 = 0.25.", "Multiply by 100% to report a 25% increase." ], "Verification": "Applying a 25% increase to 80 gives 80 \u00D7 1.25 = 100, recovering the stated new value.", "Assumptions": [ "The original value must be nonzero.", "For the Percentage Change Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Map every symbol to the modeled quantity and verify units, domain restrictions, and simplifying assumptions." ], "CommonMistakes": [ "When copying Percentage Change Formula, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Percentage Change Formula with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Percentage Change Formula.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry; NIST Digital Library of Mathematical Functions", "SourceIds": [ "openstax-algebra-trigonometry", "nist-dlmf" ], "url": "/formula/percentage-change/" }, { "Id": "point-slope-form", "Name": "Point-Slope Form", "Category": "Algebra", "Latex": "y-y_1=m(x-x_1)", "Description": "Defines a line from one point and its slope.", "Variables": [ "m: slope", "(x\u2081,y\u2081): known point" ], "Domain": "Apply Point-Slope Form only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "Units": [], "BoundaryCases": [ "For Point-Slope Form, check zero, negative, and extreme input values before relying on the result.", "When using Point-Slope Form, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form y-y_1=m(x-x_1) for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "A line through (2,3) with slope 4 is y-3=4(x-2), which simplifies to y=4x-5.", "WorkedSteps": [ "Insert m=4, x\u2081=2, and y\u2081=3.", "Distribute 4 to obtain y-3=4x-8.", "Add 3 to both sides to obtain y=4x-5." ], "Verification": "The simplified line gives y=3 at x=2 and has slope 4.", "Assumptions": [ "Convert to slope-intercept form when a y-intercept is needed.", "For the Point-Slope Form, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "CommonMistakes": [ "Before substituting values into Point-Slope Form, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Point-Slope Form, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Point-Slope Form.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/point-slope-form/" }, { "Id": "product-rule", "Name": "Product Rule", "Category": "Calculus", "Latex": "(fg)\u0027=f\u0027g\u002Bfg\u0027", "Description": "Differentiates the product of two differentiable functions.", "Variables": [ "f, g: differentiable functions", "f\u2032, g\u2032: their derivatives" ], "Domain": "Applies where both factor functions are differentiable; it is not equivalent to multiplying the two derivatives.", "Units": [ "derivative units are product-output units per input unit" ], "BoundaryCases": [ "For Product Rule, check zero, negative, and extreme input values before relying on the result.", "When using Product Rule, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form (fg)\u0027=f\u0027g\u002Bfg\u0027 for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For f(x) = x\u00B2 sin x, f\u2032(x) = (2x)sin x \u002B x\u00B2 cos x.", "WorkedSteps": [ "Separate the product into u(x) = x\u00B2 and v(x) = sin x.", "Differentiate each factor: u\u2032(x) = 2x and v\u2032(x) = cos x.", "Apply (uv)\u2032 = u\u2032v \u002B uv\u2032 without dropping either term.", "Substitute the factors to obtain 2x sin x \u002B x\u00B2 cos x." ], "Verification": "At x = 0, both the symbolic derivative and a local numerical slope are 0, providing a simple consistency check.", "Assumptions": [ "Differentiate one factor at a time while keeping the other factor unchanged.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "CommonMistakes": [ "Before substituting values into Product Rule, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Product Rule.", "VerifiedAgainst": "OpenStax Calculus Volume 1", "SourceIds": [ "openstax-calculus-volume-1" ], "url": "/formula/product-rule/" }, { "Id": "pythagorean-theorem", "Name": "Pythagorean Theorem", "Category": "Geometry", "Latex": "a^2\u002Bb^2=c^2", "Description": "Relates the side lengths of a right triangle.", "Variables": [ "a, b: perpendicular leg lengths", "c: hypotenuse length" ], "Domain": "Applies only to Euclidean right triangles; c denotes the side opposite the right angle and must be the longest side.", "Units": [ "a, b, and c must use the same length unit" ], "BoundaryCases": [ "For Pythagorean Theorem, check zero, negative, and extreme input values before relying on the result.", "When using Pythagorean Theorem, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form a^2\u002Bb^2=c^2 for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For a right triangle with legs 3 and 4, calculate the hypotenuse: c = \u221A(3\u00B2 \u002B 4\u00B2) = \u221A25 = 5.", "WorkedSteps": [ "Confirm that the triangle is right-angled and that a and b are the perpendicular legs.", "Square the leg lengths: 3\u00B2 = 9 and 4\u00B2 = 16.", "Add the squared lengths: 9 \u002B 16 = 25.", "Take the nonnegative square root to obtain c = 5." ], "Verification": "The result forms the known 3\u20134\u20135 right triangle, and 5\u00B2 = 25 equals 3\u00B2 \u002B 4\u00B2 = 25.", "Assumptions": [ "The theorem applies only to right triangles.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "CommonMistakes": [ "Before substituting values into Pythagorean Theorem, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Pythagorean Theorem, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Pythagorean Theorem.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/pythagorean-theorem/" }, { "Id": "quadratic-discriminant", "Name": "Quadratic Discriminant", "Category": "Algebra", "Latex": "\\Delta=b^2-4ac", "Description": "Determines the number and type of roots of a quadratic equation.", "Variables": [ "a, b, c: coefficients of ax\u00B2\u002Bbx\u002Bc", "\u0394: discriminant" ], "Domain": "Apply Quadratic Discriminant only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "Units": [], "BoundaryCases": [ "For Quadratic Discriminant, check zero, negative, and extreme input values before relying on the result.", "When using Quadratic Discriminant, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form \\Delta=b^2-4ac for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For x\u00B2-4x\u002B4=0, \u0394=(-4)\u00B2-4(1)(4)=16-16=0.", "WorkedSteps": [ "Identify a=1, b=-4, and c=4.", "Square b and compute 4ac.", "Subtract to obtain \u0394=0, indicating one repeated real root." ], "Verification": "Factoring gives (x-2)\u00B2=0, confirming the repeated root x=2.", "Assumptions": [ "\u0394\u003E0 gives two distinct real roots; \u0394=0 gives one repeated real root.", "For the Quadratic Discriminant, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "CommonMistakes": [ "Before substituting values into Quadratic Discriminant, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Quadratic Discriminant, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Quadratic Discriminant.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/quadratic-discriminant/" }, { "Id": "quadratic-formula", "Name": "Quadratic Formula", "Category": "Algebra", "Latex": "x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}", "Description": "Finds the roots of a quadratic equation ax\u00B2 \u002B bx \u002B c = 0 when a is nonzero.", "Variables": [ "a: coefficient of x\u00B2 (a \u2260 0)", "b: coefficient of x", "c: constant term", "x: solution or root" ], "Domain": "Use for equations in the form ax\u00B2 \u002B bx \u002B c = 0 with a \u2260 0; complex roots are allowed when the discriminant is negative.", "Units": [ "a, b, c, and x use compatible problem-specific units" ], "BoundaryCases": [ "For Quadratic Formula, check zero, negative, and extreme input values before relying on the result.", "When using Quadratic Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a} for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "Solve x\u00B2 \u2212 5x \u002B 6 = 0. Here a = 1, b = \u22125, and c = 6. The discriminant is 1, so the two branches give x = 3 and x = 2.", "WorkedSteps": [ "Identify the coefficients from standard form: a = 1, b = \u22125, and c = 6.", "Compute the discriminant: b\u00B2 \u2212 4ac = 25 \u2212 24 = 1.", "Substitute into the formula: x = (5 \u00B1 \u221A1) / 2.", "Evaluate both branches to obtain x = 3 and x = 2." ], "Verification": "Substituting 3 gives 9 \u2212 15 \u002B 6 = 0, and substituting 2 gives 4 \u2212 10 \u002B 6 = 0, so both roots satisfy the original equation.", "Assumptions": [ "The discriminant b\u00B2 \u2212 4ac determines the number and type of real roots.", "For the Quadratic Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "The plus-minus sign represents separate branches; evaluate and verify both when both are admissible." ], "CommonMistakes": [ "Do not collapse the \u00B1 in Quadratic Formula into one value; calculate each admissible branch and check it in the original problem.", "For the Quadratic Formula, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Quadratic Formula.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/quadratic-formula/" }, { "Id": "vertex-form", "Name": "Quadratic Vertex Form", "Category": "Algebra", "Latex": "y=a(x-h)^2\u002Bk", "Description": "Expresses a parabola using its vertex and vertical scale.", "Variables": [ "(h,k): vertex", "a: vertical scale and opening direction" ], "Domain": "Apply Quadratic Vertex Form only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "Units": [], "BoundaryCases": [ "For Quadratic Vertex Form, check zero, negative, and extreme input values before relying on the result.", "When using Quadratic Vertex Form, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form y=a(x-h)^2\u002Bk for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For y=2(x-3)\u00B2\u002B1, the vertex is (3,1). At x=5, y=2(2\u00B2)\u002B1=9.", "WorkedSteps": [ "Read h=3 and k=1 from a(x-h)\u00B2\u002Bk.", "Use the positive coefficient a=2 to identify an upward-opening parabola.", "Substitute x=5 to obtain y=9." ], "Verification": "Substituting x=3 gives the minimum y=1, confirming the stated vertex.", "Assumptions": [ "a\u003E0 opens upward; a\u003C0 opens downward.", "For the Quadratic Vertex Form, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "CommonMistakes": [ "Before substituting values into Quadratic Vertex Form, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Quadratic Vertex Form, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Quadratic Vertex Form.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/vertex-form/" }, { "Id": "rectangle-area", "Name": "Rectangle Area", "Category": "Geometry", "Latex": "A=lw", "Description": "Calculates the area of a rectangle.", "Variables": [ "l: length", "w: width", "A: area" ], "Domain": "Apply Rectangle Area only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "Units": [], "BoundaryCases": [ "For Rectangle Area, check zero, negative, and extreme input values before relying on the result.", "When using Rectangle Area, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form A=lw for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "A rectangle 5 cm long and 3 cm wide has area A=5\u00D73=15 cm\u00B2.", "WorkedSteps": [ "Identify l=5 cm and w=3 cm.", "Multiply the side lengths.", "Report the result in square centimetres." ], "Verification": "Counting a 5-by-3 grid gives 15 unit squares.", "Assumptions": [ "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "CommonMistakes": [ "Before substituting values into Rectangle Area, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Rectangle Area, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Rectangle Area.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/rectangle-area/" }, { "Id": "rectangle-perimeter", "Name": "Rectangle Perimeter", "Category": "Geometry", "Latex": "P=2(l\u002Bw)", "Description": "Calculates the perimeter of a rectangle.", "Variables": [ "l: length", "w: width", "P: perimeter" ], "Domain": "Apply Rectangle Perimeter only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "Units": [], "BoundaryCases": [ "For Rectangle Perimeter, check zero, negative, and extreme input values before relying on the result.", "When using Rectangle Perimeter, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form P=2(l\u002Bw) for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For l=5 cm and w=3 cm, P=2(5\u002B3)=16 cm.", "WorkedSteps": [ "Add one length and one width: 5\u002B3=8 cm.", "Double the sum to include opposite sides.", "Report 16 cm as a length." ], "Verification": "Adding all four sides, 5\u002B3\u002B5\u002B3, also gives 16.", "Assumptions": [ "Perimeter uses linear units.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "CommonMistakes": [ "Before substituting values into Rectangle Perimeter, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Rectangle Perimeter, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Rectangle Perimeter.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/rectangle-perimeter/" }, { "Id": "sample-mean", "Name": "Sample Mean", "Category": "Statistics", "Latex": "\\bar{x}=\\frac{1}{n}\\sum_{i=1}^{n}x_i", "Description": "Calculates the arithmetic average of n observed values.", "Variables": [ "x\u0304: sample mean", "x\u1D62: ith observation", "n: number of observations" ], "Domain": "Applies to quantitative observations with sample size n \u003E 0; sensitivity to extreme values should be considered for skewed data.", "Units": [ "the sample mean has the same unit as the observations" ], "BoundaryCases": [ "For Sample Mean, check zero, negative, and extreme input values before relying on the result.", "When using Sample Mean, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form \\bar{x}=\\frac{1}{n}\\sum_{i=1}^{n}x_i for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For the sample 2, 4, 6, 8, the mean is (2 \u002B 4 \u002B 6 \u002B 8)/4 = 20/4 = 5.", "WorkedSteps": [ "Add all sample observations to obtain a total of 20.", "Count the observations: n = 4.", "Divide the total by the sample size: 20/4.", "Report the sample mean as 5 in the original measurement unit." ], "Verification": "The deviations from 5 are \u22123, \u22121, 1, and 3, which sum to zero as deviations from an arithmetic mean should.", "Assumptions": [ "Every observation contributes equally.", "For the Sample Mean, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Sample Mean, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Sample Mean, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "CommonMistakes": [ "When copying Sample Mean, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Sample Mean, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Sample Mean.", "VerifiedAgainst": "OpenStax Introductory Statistics", "SourceIds": [ "openstax-intro-statistics" ], "url": "/formula/sample-mean/" }, { "Id": "sample-variance", "Name": "Sample Variance", "Category": "Statistics", "Latex": "s^2=\\frac{1}{n-1}\\sum_{i=1}^{n}(x_i-\\bar{x})^2", "Description": "Measures sample spread using squared deviations and Bessel\u2019s correction.", "Variables": [ "s\u00B2: sample variance", "x\u1D62: observation", "x\u0304: sample mean", "n: sample size" ], "Domain": "Uses the n \u2212 1 denominator for a sample estimate when n \u003E 1; population variance uses a different denominator.", "Units": [ "variance uses the square of the observation unit" ], "BoundaryCases": [ "For Sample Variance, check zero, negative, and extreme input values before relying on the result.", "When using Sample Variance, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form s^2=\\frac{1}{n-1}\\sum_{i=1}^{n}(x_i-\\bar{x})^2 for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For 2, 4, 6, 8, the mean is 5. Squared deviations sum to 20, so s\u00B2 = 20/(4 \u2212 1) = 20/3 \u2248 6.667.", "WorkedSteps": [ "Calculate the sample mean: x\u0304 = 5.", "Compute deviations \u22123, \u22121, 1, and 3, then square them.", "Add the squared deviations: 9 \u002B 1 \u002B 1 \u002B 9 = 20.", "Divide by n \u2212 1 = 3 to obtain approximately 6.667." ], "Verification": "The variance is nonnegative, and taking its square root gives a sample standard deviation of about 2.582 in the original unit.", "Assumptions": [ "The denominator n \u2212 1 is used when estimating population variance from a sample.", "For the Sample Variance, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Sample Variance, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Sample Variance, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "CommonMistakes": [ "When copying Sample Variance, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Sample Variance, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Sample Variance.", "VerifiedAgainst": "OpenStax Introductory Statistics", "SourceIds": [ "openstax-intro-statistics" ], "url": "/formula/sample-variance/" }, { "Id": "slope-intercept-form", "Name": "Slope-Intercept Form", "Category": "Algebra", "Latex": "y=mx\u002Bb", "Description": "Expresses a straight line using its slope and vertical intercept.", "Variables": [ "m: slope of the line", "b: y-intercept", "x: horizontal coordinate", "y: vertical coordinate" ], "Domain": "Represents nonvertical straight lines in a Cartesian plane; vertical lines cannot be written with a finite slope m.", "Units": [ "m has y-units per x-unit", "b and y use the same units" ], "BoundaryCases": [ "For Slope-Intercept Form, check zero, negative, and extreme input values before relying on the result.", "When using Slope-Intercept Form, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form y=mx\u002Bb for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For y = 2x \u002B 3 and x = 4, the output is y = 2(4) \u002B 3 = 11. The line crosses the y-axis at 3 and rises 2 units per 1 unit of run.", "WorkedSteps": [ "Read the slope and intercept from the equation: m = 2 and b = 3.", "Substitute x = 4 into y = mx \u002B b.", "Multiply 2 \u00D7 4 = 8, then add the intercept 3.", "Report the point on the line as (4, 11)." ], "Verification": "Using the intercept point (0, 3), the slope between (0, 3) and (4, 11) is (11 \u2212 3)/(4 \u2212 0) = 2, matching the equation.", "Assumptions": [ "A positive slope rises from left to right; a negative slope falls.", "For the Slope-Intercept Form, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "CommonMistakes": [ "Before substituting values into Slope-Intercept Form, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Slope-Intercept Form, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Slope-Intercept Form.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/slope-intercept-form/" }, { "Id": "standard-error-mean", "Name": "Standard Error of the Mean", "Category": "Statistics", "Latex": "SE_{\\bar{x}}=\\frac{s}{\\sqrt{n}}", "Description": "Estimates the standard deviation of the sample mean\u2019s sampling distribution.", "Variables": [ "SE: standard error", "s: sample standard deviation", "n: sample size" ], "Domain": "Assumes independent observations and a sampling setting where the sample mean has a meaningful standard error; n must be positive.", "Units": [ "SEM has the same unit as the measured variable" ], "BoundaryCases": [ "For Standard Error of the Mean, check zero, negative, and extreme input values before relying on the result.", "When using Standard Error of the Mean, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form SE_{\\bar{x}}=\\frac{s}{\\sqrt{n}} for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For sample standard deviation s = 12 and sample size n = 36, SEM = 12/\u221A36 = 12/6 = 2.", "WorkedSteps": [ "Identify the sample standard deviation as s = 12.", "Identify the sample size as n = 36.", "Compute the square root of the sample size: \u221A36 = 6.", "Divide s by \u221An to obtain a standard error of 2." ], "Verification": "Quadrupling the sample size from 36 to 144 would halve the standard error from 2 to 1, consistent with the inverse square-root relationship.", "Assumptions": [ "The standard error decreases as sample size increases.", "For the Standard Error of the Mean, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "For the Standard Error of the Mean, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "CommonMistakes": [ "When copying Standard Error of the Mean, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Standard Error of the Mean, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Standard Error of the Mean.", "VerifiedAgainst": "OpenStax Introductory Statistics", "SourceIds": [ "openstax-intro-statistics" ], "url": "/formula/standard-error-mean/" }, { "Id": "cylinder-surface-area", "Name": "Total Surface Area of a Cylinder", "Category": "Geometry", "Latex": "S=2\\pi r^2\u002B2\\pi rh", "Description": "Adds the areas of two circular bases and the curved lateral surface.", "Variables": [ "S: total surface area", "r: radius", "h: height" ], "Domain": "Apply Total Surface Area of a Cylinder only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "Units": [], "BoundaryCases": [ "For Total Surface Area of a Cylinder, check zero, negative, and extreme input values before relying on the result.", "When using Total Surface Area of a Cylinder, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form S=2\\pi r^2\u002B2\\pi rh for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For r=2 and h=5, S=2\u03C0(2\u00B2)\u002B2\u03C0(2)(5)=8\u03C0\u002B20\u03C0=28\u03C0\u224887.965.", "WorkedSteps": [ "Compute the two circular ends: 2\u03C0r\u00B2=8\u03C0.", "Compute the lateral area: 2\u03C0rh=20\u03C0.", "Add the components to obtain 28\u03C0." ], "Verification": "The net consists of two circles and a rectangle of width 2\u03C0r and height h, giving the same total.", "Assumptions": [ "For lateral area only, omit the 2\u03C0r\u00B2 term.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "CommonMistakes": [ "Before substituting values into Total Surface Area of a Cylinder, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Total Surface Area of a Cylinder, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Total Surface Area of a Cylinder.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/cylinder-surface-area/" }, { "Id": "dot-product", "Name": "Vector Dot Product", "Category": "Linear Algebra", "Latex": "\\mathbf{a}\\cdot\\mathbf{b}=\\sum_{i=1}^{n}a_ib_i=\\|\\mathbf{a}\\|\\|\\mathbf{b}\\|\\cos\\theta", "Description": "Combines two vectors into a scalar and relates their components to the angle between them.", "Variables": [ "a, b: vectors", "a\u1D62, b\u1D62: vector components", "\u03B8: angle between vectors" ], "Domain": "Defined for equal-dimensional real vectors in a Euclidean inner-product space; the result is a scalar rather than a vector.", "Units": [ "the result uses the product of the two vectors\u2019 component units" ], "BoundaryCases": [ "For Vector Dot Product, check zero, negative, and extreme input values before relying on the result.", "When using Vector Dot Product, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form \\mathbf{a}\\cdot\\mathbf{b}=\\sum_{i=1}^{n}a_ib_i=\\|\\mathbf{a}\\|\\|\\mathbf{b}\\|\\cos\\theta for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For vectors (1, 2, 3) and (4, \u22121, 2), the dot product is 1\u00B74 \u002B 2\u00B7(\u22121) \u002B 3\u00B72 = 4 \u2212 2 \u002B 6 = 8.", "WorkedSteps": [ "Confirm both vectors have the same number of components.", "Multiply corresponding components: 4, \u22122, and 6.", "Add the component products: 4 \u2212 2 \u002B 6.", "Report the scalar dot product as 8." ], "Verification": "Using a\u00B7b = |a||b|cos \u03B8 gives the same scalar when the vector magnitudes and included angle are evaluated.", "Assumptions": [ "Perpendicular nonzero vectors have dot product zero.", "For the Vector Dot Product, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "The angle unit and the domain or branch of any inverse trigonometric function must be explicit." ], "CommonMistakes": [ "Do not omit the index or bounds in Vector Dot Product; changing either one changes which terms are included.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Vector Dot Product.", "VerifiedAgainst": "OpenStax Calculus Volume 1; NIST Digital Library of Mathematical Functions", "SourceIds": [ "openstax-calculus-volume-1", "nist-dlmf" ], "url": "/formula/dot-product/" }, { "Id": "sphere-volume", "Name": "Volume of a Sphere", "Category": "Geometry", "Latex": "V=\\frac{4}{3}\\pi r^3", "Description": "Calculates the volume enclosed by a sphere.", "Variables": [ "V: volume", "r: sphere radius", "\u03C0: circle constant" ], "Domain": "Applies to a Euclidean sphere with radius r \u2265 0; do not substitute the diameter directly for r.", "Units": [ "r uses a length unit", "V uses the corresponding cubic length unit" ], "BoundaryCases": [ "For Volume of a Sphere, check zero, negative, and extreme input values before relying on the result.", "When using Volume of a Sphere, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form V=\\frac{4}{3}\\pi r^3 for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For a sphere with radius 3 cm, V = (4/3)\u03C0(3\u00B3) = 36\u03C0 cm\u00B3 \u2248 113.097 cm\u00B3.", "WorkedSteps": [ "Identify the radius as r = 3 cm.", "Cube the radius: r\u00B3 = 27 cm\u00B3.", "Multiply 27 by 4/3 to obtain 36.", "Multiply by \u03C0 for the exact volume 36\u03C0 cm\u00B3, then approximate if needed." ], "Verification": "Dividing the result by r\u00B3 gives 4\u03C0/3, and the answer uses cubic length units as a volume must.", "Assumptions": [ "Volume is measured in cubic units.", "For the Volume of a Sphere, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "CommonMistakes": [ "When copying Volume of a Sphere, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Volume of a Sphere, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Volume of a Sphere.", "VerifiedAgainst": "OpenStax Algebra and Trigonometry", "SourceIds": [ "openstax-algebra-trigonometry" ], "url": "/formula/sphere-volume/" }, { "Id": "z-score", "Name": "Z-Score Formula", "Category": "Statistics", "Latex": "z=\\frac{x-\\mu}{\\sigma}", "Description": "Expresses how many population standard deviations a value lies from the mean.", "Variables": [ "z: standardized score", "x: observed value", "\u03BC: population mean", "\u03C3: population standard deviation" ], "Domain": "The standard deviation must be positive; interpretation as a normal-distribution percentile additionally requires an appropriate distribution model.", "Units": [ "x, \u03BC, and \u03C3 use the same unit", "z is dimensionless" ], "BoundaryCases": [ "For Z-Score Formula, check zero, negative, and extreme input values before relying on the result.", "When using Z-Score Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "AlternativeForms": [ "Keep the canonical LaTeX form z=\\frac{x-\\mu}{\\sigma} for copying; rearrange only after preserving equivalence and domain restrictions." ], "WorkedExample": "For x = 85, mean \u03BC = 70, and standard deviation \u03C3 = 10, z = (85 \u2212 70)/10 = 1.5.", "WorkedSteps": [ "Subtract the mean from the observation: 85 \u2212 70 = 15.", "Confirm the standard deviation is positive: \u03C3 = 10.", "Divide the deviation by the standard deviation: 15/10.", "Report z = 1.5, meaning the value is 1.5 standard deviations above the mean." ], "Verification": "Rearranging x = \u03BC \u002B z\u03C3 gives 70 \u002B 1.5(10) = 85, recovering the original observation.", "Assumptions": [ "A positive z-score lies above the mean; a negative one lies below.", "For the Z-Score Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Z-Score Formula, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "CommonMistakes": [ "When copying Z-Score Formula, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Z-Score Formula, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "ReviewedAt": "2026-07-26", "ReviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Z-Score Formula.", "VerifiedAgainst": "OpenStax Introductory Statistics", "SourceIds": [ "openstax-intro-statistics" ], "url": "/formula/z-score/" } ] }