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• Algebra• Trigonometry• Precalculus• CalculusIakaDifferential• CalculusIIakaIntegral• CalculusIIIakaMultiorSeveralVariable• LinearAlgebra• DifferentialEquationswithLinearAlgebra• MathematicalPhysicsOperations

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BooknotationsNote:Somesymbolsmayhavedifferentmeaningsindifferentcoursesi.e.neverassume.And ∧Or ∨In ∈Manipulationorrowreductionoccurred ~Implies ⇒Becomes ⇐Ifandonlyif ⇔Therefore ∴Because ∵Equivalent/definedas ≡Euler’snumber𝑒

𝑒 =1𝑛!

∞

012

= lim0→∞

1 +1𝑛

0

Anyothervector(scalarnotbold/hat/vec) 𝑣 ≡ 𝐯

Copyright©WESOLVETHEMLLC|WeSolveThem.com 3

TableofContentsBooknotations...................................................................................................................................2

ALGEBRA..........................................................................................................................................17GeneralSymbolsandNotations.......................................................................................................................................................17Typesofnumbers...................................................................................................................................................................................18Properties...................................................................................................................................................................................................18Meanings....................................................................................................................................................................................................18Complementationofsets.....................................................................................................................................................................18SetLaws......................................................................................................................................................................................................19DeMorgan’sLaws...................................................................................................................................................................................19NumberofElementsinaSet..............................................................................................................................................................19Axioms.........................................................................................................................................................................................................19Arithmetic..................................................................................................................................................................................................20Exponents..................................................................................................................................................................................................20Radicals.......................................................................................................................................................................................................20ComplexNumbers..................................................................................................................................................................................21AddingandSubtractingFractions...................................................................................................................................................21Logarithmic...............................................................................................................................................................................................21Log“Base”Notation................................................................................................................................................................................21Log“Natural”Notation.........................................................................................................................................................................21

*Factoring...................................................................................................................................................................................................22Note:...............................................................................................................................................................................................................22

LongDivision............................................................................................................................................................................................22CompleteTheSquare............................................................................................................................................................................22Example1:Solvingforx(Formula1)..............................................................................................................................................23Example2:Solvingforx(Formula2)..............................................................................................................................................23

Compositions............................................................................................................................................................................................24Functions....................................................................................................................................................................................................24VerticalLineTest......................................................................................................................................................................................24Even/OddFunction..................................................................................................................................................................................24AverageRateofChange........................................................................................................................................................................24SecantLine..................................................................................................................................................................................................24DifferenceQuotient.................................................................................................................................................................................24

DistanceFormula....................................................................................................................................................................................24MidpointFormula...................................................................................................................................................................................25QuadraticFormula.................................................................................................................................................................................25Proof:.............................................................................................................................................................................................................25Discriminant:..............................................................................................................................................................................................25

GraphingaLine........................................................................................................................................................................................26PointSlopeForm:.....................................................................................................................................................................................26SlopeInterceptForm:.............................................................................................................................................................................26StandardorGeneralForm...................................................................................................................................................................27ParallelLine(equalslopes).................................................................................................................................................................27PerpendicularLine(productofslopesare-1).............................................................................................................................27

*DomainRestrictions............................................................................................................................................................................27Polynomial..................................................................................................................................................................................................27Fraction........................................................................................................................................................................................................27Radical,ifniseven..................................................................................................................................................................................27Radical,ifnisodd....................................................................................................................................................................................27FractionwithRadicalindenominator...........................................................................................................................................27NaturalLog.................................................................................................................................................................................................27


Exponential.................................................................................................................................................................................................27InverseFunctions...................................................................................................................................................................................27Asymptotes,HolesandGraphs.........................................................................................................................................................28HoleinaGraph..........................................................................................................................................................................................28ThreeGeneralCasesforHorizontalAsymptotes........................................................................................................................28Ex.1HorizontalandVertical..............................................................................................................................................................29

..................................................................................................................................................................29Ex.2Oblique...............................................................................................................................................................................................29

..............................................................................................................................................................29Ex.3HorizontalandVertical..............................................................................................................................................................29

................................................................................................................................................................29Inequalities................................................................................................................................................................................................29InterestFormulas...................................................................................................................................................................................29PhysicsFormulas....................................................................................................................................................................................29Symmetry...................................................................................................................................................................................................29ByPoint.........................................................................................................................................................................................................30Testing...........................................................................................................................................................................................................30

Variations(Proportionality)..............................................................................................................................................................30CommonGraphsandFormulas........................................................................................................................................................30EquationofaLine....................................................................................................................................................................................33EquationofParabola.............................................................................................................................................................................33EquationofCircle.....................................................................................................................................................................................33EquationofEllipse...................................................................................................................................................................................33EquationofHyperbola(1)...................................................................................................................................................................33EquationofHyperbola(2)...................................................................................................................................................................33Areas..............................................................................................................................................................................................................34SurfaceAreas.............................................................................................................................................................................................34Volumes........................................................................................................................................................................................................34BusinessFunctions...................................................................................................................................................................................34

AverageRateofChangeof𝒇andSlopeofSecantLine...........................................................................................................35DifferenceQuotient.................................................................................................................................................................................35

Functions....................................................................................................................................................................................................35GraphShiftsandCompressions........................................................................................................................................................35Systemsofequations.............................................................................................................................................................................36


Rankofmatrixandpivots...................................................................................................................................................................36Determinate’sofa(2x2)matrix.......................................................................................................................................................37Determinateofa(3x3)andhighermatrices...............................................................................................................................37CofactorExpansion.................................................................................................................................................................................37

TRIGONOMETRY...............................................................................................................................38*Note:...........................................................................................................................................................................................................38RadianandDegreeConversion........................................................................................................................................................38BasicGraphs..............................................................................................................................................................................................39UsingPythagorean’sTheorem..........................................................................................................................................................41ReciprocalIdentities.............................................................................................................................................................................43PythagoreanIdentities.........................................................................................................................................................................44EvenandOddFunctions......................................................................................................................................................................44Example........................................................................................................................................................................................................45

DoubleAngleFormulas........................................................................................................................................................................46HalfAngleFormulas..............................................................................................................................................................................47SumandDifferenceFormulas...........................................................................................................................................................48ProducttoSumFormulas....................................................................................................................................................................48SumtoProductFormulas....................................................................................................................................................................49HyperbolicFunctions........................................................................................................................50Notation......................................................................................................................................................................................................50Graphs..........................................................................................................................................................................................................50Identities.....................................................................................................................................................................................................51DIFFERENTIALCALCULUS(CALCI).....................................................................................................52Translation-...............................................................................................................................................................................................52NotationsforLimits...............................................................................................................................................................................52Theactuallimit.........................................................................................................................................................................................52Lefthandlimit...........................................................................................................................................................................................52Righthandlimit........................................................................................................................................................................................52Limitexists..................................................................................................................................................................................................52Limitdoesnotexists(DNE).................................................................................................................................................................52Continuousfunction................................................................................................................................................................................52LeftContinuousfunction.......................................................................................................................................................................52RightContinuousfunction....................................................................................................................................................................52Non-continuousfunction......................................................................................................................................................................52

TypesofDiscontinuity..........................................................................................................................................................................53Jump...............................................................................................................................................................................................................53Removable...................................................................................................................................................................................................53Infinite...........................................................................................................................................................................................................53

LimitLawsandProperties..................................................................................................................................................................54LimitofaConstant..................................................................................................................................................................................54LimitofSingleVariable.........................................................................................................................................................................54IfTheFunctionisContinuous.............................................................................................................................................................54TheConstantMultipleLaw..................................................................................................................................................................54TheSumandDifferenceLaw..............................................................................................................................................................54TheProductLaw......................................................................................................................................................................................54TheQuotientLaw.....................................................................................................................................................................................54ThePowerLaw..........................................................................................................................................................................................54TheRootLaw.............................................................................................................................................................................................54ExponentialLaw.......................................................................................................................................................................................54

InfiniteLimits...........................................................................................................................................................................................54Case1:...........................................................................................................................................................................................................54


Case2:...........................................................................................................................................................................................................54Case3:...........................................................................................................................................................................................................54

PreciseDefinitionofaLimit𝜺, 𝜹.......................................................................................................................................................55Limit...............................................................................................................................................................................................................55LeftHandLimit.........................................................................................................................................................................................55RightHandLimit......................................................................................................................................................................................55

Derivationof“TheDifferenceQuotient”.......................................................................................................................................55SlopeofSecantLineorDifferenceQuotient.................................................................................................................................55IntermediateValueTheorem..............................................................................................................................................................55

CommonLimits........................................................................................................................................................................................56InfiniteLimits.............................................................................................................................................................................................56

Derivatives.......................................................................................................................................57TheLimitDefinitionofaDerivative.................................................................................................................................................57

Notations....................................................................................................................................................................................................57TimeDerivatives.......................................................................................................................................................................................57TheSlopeNotationforCalculus........................................................................................................................................................58

TangentLine.............................................................................................................................................................................................58PhysicsNotation......................................................................................................................................................................................58DerivativeRules(operatornotations)..........................................................................................................................................59DerivativeofaConstant........................................................................................................................................................................59SumandDifference.................................................................................................................................................................................59PowerRule..................................................................................................................................................................................................59ConstantMultipleRule...........................................................................................................................................................................59ProductRule...............................................................................................................................................................................................59QuotientRule.............................................................................................................................................................................................59ChainRule....................................................................................................................................................................................................59

DerivativeRules(primenotations)................................................................................................................................................60DerivativeofaConstant........................................................................................................................................................................60PowerRule..................................................................................................................................................................................................60ConstantMultipleRule...........................................................................................................................................................................60ProductRule...............................................................................................................................................................................................60QuotientRule.............................................................................................................................................................................................60ChainRule....................................................................................................................................................................................................60

ExponentialandLogarithmic............................................................................................................................................................60exp{u}............................................................................................................................................................................................................60NaturalLog.................................................................................................................................................................................................60BaseLog.......................................................................................................................................................................................................60Exponential.................................................................................................................................................................................................60

InverseFunctionDerivative...............................................................................................................................................................60TrigDerivatives.......................................................................................................................................................................................61Standard.......................................................................................................................................................................................................61Inverse...........................................................................................................................................................................................................61

CommonDerivatives.............................................................................................................................................................................61Operator.......................................................................................................................................................................................................61Prime..............................................................................................................................................................................................................62

ImplicitDifferentiation.....................................................................................................................62TangentLine.............................................................................................................................................................................................62RelatedRates............................................................................................................................................................................................63HyperbolicFunctions........................................................................................................................63Notation......................................................................................................................................................................................................63Graphs..........................................................................................................................................................................................................64


Identities.....................................................................................................................................................................................................65Derivatives.................................................................................................................................................................................................65Standard.......................................................................................................................................................................................................65Inverse...........................................................................................................................................................................................................65

Extrema............................................................................................................................................66GraphingProcess....................................................................................................................................................................................66CriticalNumbers.......................................................................................................................................................................................66Max/Min.......................................................................................................................................................................................................66Increasinganddecreasing...................................................................................................................................................................67Concavity......................................................................................................................................................................................................67Pointsofinflection...................................................................................................................................................................................67

Theorems...................................................................................................................................................................................................67Rolle’sTheorem.........................................................................................................................................................................................67MeanValueTheorem..............................................................................................................................................................................67

First&SecondDerivativeTest.........................................................................................................................................................67L’Hospital’sRule...............................................................................................................................68IndeterminateForms............................................................................................................................................................................68Rule...............................................................................................................................................................................................................68Process..........................................................................................................................................................................................................68

Optimization....................................................................................................................................69

BusinessFormulas............................................................................................................................69

Antiderivatives&Integration...........................................................................................................70BasicRules.................................................................................................................................................................................................70RiemannSumforAreaApproximation.........................................................................................................................................70AreaApproximationRules.................................................................................................................................................................71MidpointRule.............................................................................................................................................................................................71TrapezoidRule..........................................................................................................................................................................................71SimpsonRule..............................................................................................................................................................................................71

TheIntegralNotation∫ ...................................................................................................................71DefiniteIntegralProperties...............................................................................................................................................................71FundamentalTheorems.......................................................................................................................................................................72LimitDefinitionofaDefiniteIntegral............................................................................................................................................72DifferentialEquation(1storder).......................................................................................................................................................72CommonIntegrals....................................................................................................................................................................................73

DefiniteIntegralRules..........................................................................................................................................................................73Substitution.................................................................................................................................................................................................74IntegrationbyParts................................................................................................................................................................................74

INTEGRALCALCULUS(CALCLII).........................................................................................................74

ParametricandPolarOperations......................................................................................................74Notations....................................................................................................................................................................................................74FirstDerivative..........................................................................................................................................................................................74SecondDerivative.....................................................................................................................................................................................74

Trigonometric..........................................................................................................................................................................................75Circle..............................................................................................................................................................................................................75Ellipse............................................................................................................................................................................................................75PolarDerivative........................................................................................................................................................................................75PolarEquationsforEllipse...................................................................................................................................................................75PolarEquationsforHyperbola..........................................................................................................................................................75


PolarEquationsforParabola.............................................................................................................................................................76Antiderivatives&Integration...........................................................................................................76BasicRules.................................................................................................................................................................................................76RiemannSumforAreaApproximation.........................................................................................................................................77AreaApproximationRules.................................................................................................................................................................77MidpointRule.............................................................................................................................................................................................77TrapezoidRule..........................................................................................................................................................................................77

TheIntegralNotation∫ ...................................................................................................................77DefiniteIntegralProperties...............................................................................................................................................................77FundamentalTheorems.......................................................................................................................................................................78LimitDefinitionofaDefiniteIntegral............................................................................................................................................78DifferentialEquation(1storder).......................................................................................................................................................79CommonIntegrals....................................................................................................................................................................................79

DefiniteIntegralRules..........................................................................................................................................................................80Substitution.................................................................................................................................................................................................80IntegrationbyParts................................................................................................................................................................................80

TrigSubstitution.....................................................................................................................................................................................80TrigIdentity..............................................................................................................................................................................................80PartialFractions......................................................................................................................................................................................80IntegrationSteps..............................................................................................................................81ImproperIntegration............................................................................................................................................................................81InfiniteBounds..........................................................................................................................................................................................81UndefinedBounds....................................................................................................................................................................................81

Areas,Volumes,andCurveLength...................................................................................................82Areawithrespecttoanaxis...............................................................................................................................................................82Cartesian......................................................................................................................................................................................................82

Areabetweencurves.............................................................................................................................................................................82PolarArea...................................................................................................................................................................................................82Volumeaboutanaxis(DiskMethod).............................................................................................................................................82Volumebetweencurves(WasherMethod).................................................................................................................................82CylindricalShellMethod......................................................................................................................................................................83ArcLength..................................................................................................................................................................................................83SurfaceArea..............................................................................................................................................................................................83PhysicsApplications..............................................................................................................................................................................83CenterofMasswithConstantDensity.............................................................................................................................................83

SequencesvsSeries..........................................................................................................................84

SequenceTests.................................................................................................................................84

SeriesTests......................................................................................................................................84Taylorseries.............................................................................................................................................................................................85MaclaurinSeries......................................................................................................................................................................................85PowerSeries.............................................................................................................................................................................................85Radius/IntervalofConverges...........................................................................................................................................................853DCalculus.......................................................................................................................................86Magnitude..................................................................................................................................................................................................86UnitVectors...............................................................................................................................................................................................86Dot/CrossProduct.................................................................................................................................................................................87Dot...................................................................................................................................................................................................................87Properties....................................................................................................................................................................................................87


Cross...............................................................................................................................................................................................................87Properties....................................................................................................................................................................................................87

AnglesBetweenVectors......................................................................................................................................................................87Projections.................................................................................................................................................................................................87Areas/Volume..........................................................................................................................................................................................88Triangle........................................................................................................................................................................................................88Parallelogram............................................................................................................................................................................................88Parallelepiped............................................................................................................................................................................................88

Line...............................................................................................................................................................................................................88Linefromtiptotip...................................................................................................................................................................................88

EquationofaPlane................................................................................................................................................................................88VectorFunctions.....................................................................................................................................................................................88Limit...............................................................................................................................................................................................................89Derivative....................................................................................................................................................................................................89DefiniteIntegral........................................................................................................................................................................................89IndefiniteIntegral....................................................................................................................................................................................89

DifferentiationRules.............................................................................................................................................................................89Arclength...................................................................................................................................................................................................89Tangents.....................................................................................................................................................................................................89UnitTangentVector................................................................................................................................................................................90Curvature1.................................................................................................................................................................................................90Curvature2(vectorfunction).............................................................................................................................................................90Curvature3(singlevariable).............................................................................................................................................................90Curvature4(parametric).....................................................................................................................................................................90NormalVector...........................................................................................................................................................................................90BinormalVector........................................................................................................................................................................................90

TangentialandNormalComponents(acceleration)...............................................................................................................91PhysicsNotations...................................................................................................................................................................................91Position.........................................................................................................................................................................................................91Velocity.........................................................................................................................................................................................................91Speed..............................................................................................................................................................................................................91Acceleration................................................................................................................................................................................................91Curvature.....................................................................................................................................................................................................91TangentialComponent(acceleration)...........................................................................................................................................91NormalComponent(acceleration)..................................................................................................................................................91Acceleration................................................................................................................................................................................................91Note:...............................................................................................................................................................................................................91DotProductofVelocityandAcceleration......................................................................................................................................91TangentialAcceleration........................................................................................................................................................................91NormalAcceleration...............................................................................................................................................................................91Frenet-SerretFormulas.........................................................................................................................................................................92

PartialDerivatives............................................................................................................................92MixedPartial.............................................................................................................................................................................................92TangentPlane...........................................................................................................................................................................................92ChainRule..................................................................................................................................................................................................92MULTIVARIABLECALCULUS(CALCIII)...............................................................................................93Magnitude..................................................................................................................................................................................................93UnitVectors...............................................................................................................................................................................................93Dot/CrossProduct.................................................................................................................................................................................93Dot...................................................................................................................................................................................................................93Properties....................................................................................................................................................................................................93


Cross...............................................................................................................................................................................................................93Properties....................................................................................................................................................................................................94

AnglesBetweenVectors......................................................................................................................................................................94Projections.................................................................................................................................................................................................94Areas/Volume..........................................................................................................................................................................................94Triangle........................................................................................................................................................................................................94Parallelogram............................................................................................................................................................................................94Parallelepiped............................................................................................................................................................................................94

Line...............................................................................................................................................................................................................94Linefromtiptotip...................................................................................................................................................................................94

EquationofaPlane................................................................................................................................................................................95VectorFunctions.....................................................................................................................................................................................95Limit...............................................................................................................................................................................................................95Derivative....................................................................................................................................................................................................95DefiniteIntegral........................................................................................................................................................................................95IndefiniteIntegral....................................................................................................................................................................................95

DifferentiationRules.............................................................................................................................................................................95Arclength...................................................................................................................................................................................................95Tangents.....................................................................................................................................................................................................96UnitTangentVector................................................................................................................................................................................96Curvature1.................................................................................................................................................................................................96Curvature2(vectorfunction).............................................................................................................................................................96Curvature3(singlevariable).............................................................................................................................................................96Curvature4(parametric).....................................................................................................................................................................96NormalVector...........................................................................................................................................................................................96BinormalVector........................................................................................................................................................................................96

TangentialandNormalComponents(acceleration)...............................................................................................................97PhysicsNotations...................................................................................................................................................................................97Position.........................................................................................................................................................................................................97Velocity.........................................................................................................................................................................................................97Speed..............................................................................................................................................................................................................97Acceleration................................................................................................................................................................................................97Curvature.....................................................................................................................................................................................................97TangentialComponent(acceleration)...........................................................................................................................................97NormalComponent(acceleration)..................................................................................................................................................97Acceleration................................................................................................................................................................................................97Note:...............................................................................................................................................................................................................97DotProductofVelocityandAcceleration......................................................................................................................................97TangentialAcceleration........................................................................................................................................................................97NormalAcceleration...............................................................................................................................................................................97Frenet-SerretFormulas.........................................................................................................................................................................98

PartialDerivatives............................................................................................................................98MixedPartial.............................................................................................................................................................................................98EquationofaPlane................................................................................................................................................................................98NormalVector...........................................................................................................................................................................................98

Distance/VectorBetweenPoints.....................................................................................................................................................98Vectorfromtwopoints..........................................................................................................................................................................98

TangentPlane...........................................................................................................................................................................................98Equationofasphere..............................................................................................................................................................................99ChainRule..................................................................................................................................................................................................99Gradient𝜵𝒇...............................................................................................................................................................................................99DirectionalDerivative...........................................................................................................................................................................99


Differentials...............................................................................................................................................................................................99ImplicitDifferentiation.........................................................................................................................................................................99Extrema..........................................................................................................................................100LagrangeMultipliers...........................................................................................................................................................................100TwoConstraints.....................................................................................................................................................................................101

MultipleIntegrals...........................................................................................................................101Double........................................................................................................................................................................................................101AverageValue.........................................................................................................................................................................................101TypeI.........................................................................................................................................................................................................101TypeII........................................................................................................................................................................................................101Polar...........................................................................................................................................................................................................102TypeIII......................................................................................................................................................................................................102Moments&CenterofMass...............................................................................................................................................................102Moments....................................................................................................................................................................................................102Centerofmass.........................................................................................................................................................................................102MomentofInertia.................................................................................................................................................................................102

SurfaceArea............................................................................................................................................................................................103TripleIntegrals......................................................................................................................................................................................103Moments&CenterofMass...............................................................................................................................................................103Moments....................................................................................................................................................................................................103CenterofMass.........................................................................................................................................................................................104MomentsofInertia...............................................................................................................................................................................104

CylindricalCoordinates......................................................................................................................................................................104SphericalCoordinates.........................................................................................................................................................................104ChangeofVariables.............................................................................................................................................................................1042DJacobian..............................................................................................................................................................................................1043DJacobian..............................................................................................................................................................................................105

LineIntegrals..................................................................................................................................105General......................................................................................................................................................................................................105Smooth.......................................................................................................................................................................................................105NotSmooth..............................................................................................................................................................................................105

𝒙, 𝒚Derivatives......................................................................................................................................................................................105Vectorform..............................................................................................................................................................................................106Respectto𝒛.............................................................................................................................................................................................106MultipleFunctions𝑷,𝑸, 𝑹................................................................................................................................................................106Work...........................................................................................................................................................................................................106GradientLineIntegral.........................................................................................................................................................................107ConservativeVectorField.................................................................................................................................................................107Green’sTheorem...................................................................................................................................................................................107Curl𝜵.........................................................................................................................................................................................................107Divergence...............................................................................................................................................................................................107StokesTheorem.....................................................................................................................................................................................107DivergenceTheorem...........................................................................................................................................................................108PreCalculusReview........................................................................................................................108Arithmetic................................................................................................................................................................................................108Exponential..............................................................................................................................................................................................108Radicals.....................................................................................................................................................................................................108Fractions...................................................................................................................................................................................................108Logarithmic.............................................................................................................................................................................................109OtherFormulas/Equations..............................................................................................................................................................109


Areas...........................................................................................................................................................................................................111SurfaceAreas..........................................................................................................................................................................................111Volumes....................................................................................................................................................................................................111DomainRestrictions............................................................................................................................................................................111RightTriangle.........................................................................................................................................................................................112ReciprocalIdentities............................................................................................................................................................................112

DoubleAngleFormulas......................................................................................................................................................................113HalfAngleFormulas............................................................................................................................................................................113SumandDifferenceFormulas.........................................................................................................................................................113ProducttoSumFormulas..................................................................................................................................................................113SumtoProductFormulas..................................................................................................................................................................113

UnitCircle......................................................................................................................................114

Pre-CALCIIIReference....................................................................................................................115DerivativeRules(primenotations)..............................................................................................................................................115DerivativeofaConstant.....................................................................................................................................................................115PowerRule...............................................................................................................................................................................................115ConstantMultipleRule........................................................................................................................................................................115ProductRule............................................................................................................................................................................................115QuotientRule..........................................................................................................................................................................................115ChainRule.................................................................................................................................................................................................115

ExponentialandLogarithmic..........................................................................................................................................................115exp{u}.........................................................................................................................................................................................................115NaturalLog..............................................................................................................................................................................................115BaseLog....................................................................................................................................................................................................115Exponential..............................................................................................................................................................................................115

InverseFunctionDerivative.............................................................................................................................................................115TrigDerivatives.....................................................................................................................................................................................116Standard....................................................................................................................................................................................................116Inverse........................................................................................................................................................................................................116

CommonDerivatives...........................................................................................................................................................................116Operator....................................................................................................................................................................................................116Prime...........................................................................................................................................................................................................117

ImplicitDifferentiation...................................................................................................................117TangentLine...........................................................................................................................................................................................117RelatedRates..........................................................................................................................................................................................118HyperbolicFunctions......................................................................................................................118Notation....................................................................................................................................................................................................118Identities...................................................................................................................................................................................................118Derivatives...............................................................................................................................................................................................119Standard....................................................................................................................................................................................................119Inverse........................................................................................................................................................................................................119

Antiderivatives&Integration.........................................................................................................119BasicRules...............................................................................................................................................................................................119RiemannSumforAreaApproximation.......................................................................................................................................119AreaApproximationRules...............................................................................................................................................................120MidpointRule..........................................................................................................................................................................................120TrapezoidRule.......................................................................................................................................................................................120

TheIntegralNotation∫ .................................................................................................................121DefiniteIntegralProperties.............................................................................................................................................................121


FundamentalTheorems.....................................................................................................................................................................121LimitDefinitionofaDefiniteIntegral..........................................................................................................................................122DifferentialEquation(1storder)....................................................................................................................................................122CommonIntegrals.................................................................................................................................................................................122

DefiniteIntegralRules........................................................................................................................................................................123Substitution..............................................................................................................................................................................................123IntegrationbyParts.............................................................................................................................................................................123

TrigSubstitution...................................................................................................................................................................................123TrigIdentity............................................................................................................................................................................................123PartialFractions....................................................................................................................................................................................123PHYSICSINFO.................................................................................................................................124Basicsymbols.........................................................................................................................................................................................124Derivingformulas.................................................................................................................................................................................124Units..............................................................................................................................................125SystemInternationalUnits(S.I.Units)........................................................................................................................................125Unitconversion.....................................................................................................................................................................................125Example.....................................................................................................................................................................................................125

Vectors...........................................................................................................................................125Notation....................................................................................................................................................................................................125Addition/Subtraction..........................................................................................................................................................................125Visually.......................................................................................................................................................................................................125DotProduct..............................................................................................................................................................................................126CrossProduct..........................................................................................................................................................................................126MagnitudeorLengthofavector....................................................................................................................................................127Unitizingavector..................................................................................................................................................................................127ResultantVector....................................................................................................................................................................................127

Summingitup........................................................................................................................................................................................128FreeBodyDiagram..............................................................................................................................................................................129Averagevelocity(straight-line)......................................................................................................................................................130Instantaneousvelocity(Calculus).................................................................................................................................................130ParametricEquationGraphingExample...................................................................................................................................131

AverageAcceleration(straight-line)............................................................................................................................................131InstantaneousAcceleration(Calculus).......................................................................................................................................131Formulas(one-dimensional)...........................................................................................................................................................131Velocity:.....................................................................................................................................................................................................132AverageVelocity:...................................................................................................................................................................................132Distance:....................................................................................................................................................................................................132Velocity:.....................................................................................................................................................................................................132Distance:....................................................................................................................................................................................................132

IntegrationDerivations(Calculus)................................................................................................................................................132VectorNotations...................................................................................................................................................................................132VectorDerivatives................................................................................................................................................................................132Magnitudeofvector.............................................................................................................................................................................133ProjectileMotion...................................................................................................................................................................................133CircularMotion..............................................................................................................................133

Force..............................................................................................................................................134Resultantvector𝑹(thesumofallvectors)...............................................................................................................................134

Newton’sFirstLawofMotion.........................................................................................................................................................134Newtonunit𝑵........................................................................................................................................................................................134

Newton’sSecondLawofMotion....................................................................................................................................................134


Formulas...................................................................................................................................................................................................134LINEARALGEBRA............................................................................................................................134

Rankofmatrixandpivots...............................................................................................................135

Lengthofavectorandtheunitvector............................................................................................135

SolutionsofAugmentedMatrices...................................................................................................136CoefficientMatrix.................................................................................................................................................................................136UniqueSolution.....................................................................................................................................................................................137InfiniteSolution.....................................................................................................................................................................................137NoSolution..............................................................................................................................................................................................137SolvingSystemofEquations...........................................................................................................137

GaussJordanAugmentedMatrix....................................................................................................138

RowOperationRulesandGuidelinesforSolveaSystemofMatrices..............................................139

EchelonForms:EF,REF,RREF.........................................................................................................141EchelonForm.........................................................................................................................................................................................141ReducedEchelonForm......................................................................................................................................................................141ReducedRowEchelonForm............................................................................................................................................................141LinearDependence.........................................................................................................................142Linearcombination..............................................................................................................................................................................142Ex1:Setu,v,wLinearlyDependent..............................................................................................................................................142Ex2:Setu,v,wLinearlyIndependent...........................................................................................................................................143Ex3:VectorsLinearlyIndependent.............................................................................................................................................143Ex4:VectorsLinearlyD.....................................................................................................................................................................143ependent...................................................................................................................................................................................................143Ex5:Polynomials..................................................................................................................................................................................144Ex6:(M_(2x2))......................................................................................................................................................................................144ColumnSpace-RowSpace-NullSpace-Kernel.............................................................................145IdentifyRowSpace...............................................................................................................................................................................145IdentifyColumnSpace........................................................................................................................................................................145NullSpace(Kernel)..............................................................................................................................................................................145LUDDecompositionandElementaryMatrices................................................................................146

Transpose.......................................................................................................................................147

SymmetricmatrixforA=LDU=LDL^T...............................................................................................148

Matrixadditionandsubtraction.....................................................................................................149

Multiplythematrices(2x2)(2x3).....................................................................................................150

MatrixMultiplication(mxn)(nxp)...................................................................................................150

Idempotentmatrix.........................................................................................................................152

RotationandTranslate...................................................................................................................153Ex.1.............................................................................................................................................................................................................153Ex.2.............................................................................................................................................................................................................154Rotateaboutapoint𝒄, 𝒅................................................................................................................155

Nilpotentmatrix(eigenvaluesarezero)..........................................................................................155


Determinantrules..........................................................................................................................156

Proofs............................................................................................................................................157

Determinate’sofa(2x2)matrix......................................................................................................158

Determinateofa(3x3)andhighermatrices....................................................................................159CofactorExpansion..............................................................................................................................................................................159VectorSpace,SubspaceandSubset................................................................................................161

Cramer’srules................................................................................................................................162

Basiscoordinatevector..................................................................................................................162Ex.1..............................................................................................................................................................................................................162Ex.2.............................................................................................................................................................................................................163Adjugateofamatrix.......................................................................................................................164ComputetheAdjugate........................................................................................................................................................................165Inverseofa2x2Matrix...................................................................................................................165

Inverseof3x3.................................................................................................................................167

Trace..............................................................................................................................................168

CholeskyDecomposition................................................................................................................169

Eigenvalues....................................................................................................................................170

Eigenvectors...................................................................................................................................170

DiagonlizeaMatrix........................................................................................................................171

SingularValueDecomposition........................................................................................................172

Systemofdifferentialequations.....................................................................................................173

LinearProgramming:SimplexMethod............................................................................................174

DIFFERENTIALEQUATIONS.............................................................................................................175Introtothefirst-orderdifferentialequation............................................................................................................................175Homogeneous..........................................................................................................................................................................................176Nonhomogeneous..................................................................................................................................................................................176Linear.........................................................................................................................................................................................................176Non-linear................................................................................................................................................................................................176

1stOrderSolutionMethods............................................................................................................177SeparableVariable...............................................................................................................................................................................177Scenario.....................................................................................................................................................................................................177Ex.1(Explicitvs.Implicit).................................................................................................................................................................177Ex.2(SeparableVariable)................................................................................................................................................................178Ex.3*(IVPProblem)............................................................................................................................................................................178

1stOrderLinearNon-homogeneousi.e.y’+P(x)y=Q(x)........................................................................................................179Process.......................................................................................................................................................................................................179Ex.1.............................................................................................................................................................................................................179Ex.2.............................................................................................................................................................................................................179

ExactDifferentialEquation..............................................................................................................................................................180Situation....................................................................................................................................................................................................180Ex.1.............................................................................................................................................................................................................180


Note:............................................................................................................................................................................................................181General,ParticularandSuperpositionSolutions....................................................................................................................181GeneralSolution....................................................................................................................................................................................181ParticularSolution...............................................................................................................................................................................181SuperpositionSolution........................................................................................................................................................................181

LinearHomogenouswithConstantCoefficients.....................................................................................................................181Scenario.....................................................................................................................................................................................................181Auxiliaryequation................................................................................................................................................................................181Solution(s)................................................................................................................................................................................................181GenerallySpeaking...............................................................................................................................................................................182Ex.1.............................................................................................................................................................................................................182Ex.2.............................................................................................................................................................................................................183Ex.3.............................................................................................................................................................................................................183Ex.4.............................................................................................................................................................................................................183Ex.5IVPy(0)=1,y’(0)=2,y’’(0)=3,y’’’(0)=4...............................................................................................................................184

ReductionofOrder...............................................................................................................................................................................184Process.......................................................................................................................................................................................................184Solution......................................................................................................................................................................................................184Ex.1.............................................................................................................................................................................................................185

Substitution.............................................................................................................................................................................................185GeneralSituation...................................................................................................................................................................................185SubstitutionSolutionMethod..........................................................................................................................................................185

IntegratingFactors...............................................................................................................................................................................187Ex.1.............................................................................................................................................................................................................187

SECONDORDERDIFFERENTIALEQUATIONS....................................................................................188Auxiliaryequation................................................................................................................................................................................188Solution(s)...............................................................................................................................................................................................188ReductionofOrder...............................................................................................................................................................................191Bessel’sEquationofOrder𝒗............................................................................................................................................................193SolutiontoFirstKindBessel(𝒗 =fraction)..............................................................................................................................193SolutiontoSecondKindBessel(𝒗 =integer)...........................................................................................................................194SolutiontoThirdKindBessel(𝜶𝒙 = 𝒕).......................................................................................................................................194

Variationofparameters.....................................................................................................................................................................194Methodofundeterminedcoefficients..........................................................................................................................................195SecondSolutionforReductionofOrder.....................................................................................................................................196UnitCircle......................................................................................................................................198


ALGEBRAGeneralSymbolsandNotationsSymbol Meaning Example= Equal 0 = 0≠ Notequal 1 ≠ 0± PlusorMinus 𝑥 = ±𝑎 ⇒ 𝑥 = 𝑎𝑜𝑟𝑥 = −𝑎∓ MinusorPlus 𝑥 = ∓𝑎 ⇒ 𝑥 = −𝑎𝑜𝑟𝑥 = 𝑎𝑖𝑓𝑓,⇔ Ifandonlyif 𝑝 ⇒ 𝑞and𝑞 ⇒ 𝑝then𝑝 ⇔ 𝑞⇒ Implies 𝑝 ⇒ 𝑞< Lessthan 𝑥 − 𝑎 < 0 ⇒ 𝑥 < 𝑎≤ Lessthanequal 𝑥 − 𝑎 ≤ 0 ⇒ 𝑥 ≤ 𝑎≥ Greaterthanequal 𝑥 − 𝑎 ≥ 0 ⇒ 𝑥 ≥ 𝑎> Greaterthan 𝑥 − 𝑎 > 0 ⇒ 𝑥 > 𝑎× Times 2×3 = 6∗ or ∙ Multiplication 2 ∗ 3 = 6or2 ⋅ 3 = 6… … Multiplication 2 3 = 6… … Multiplication 2 3 = 6… … Multiplication 2 3 = 6… … … … Exponential

Multiplication2 3 opq 3 − 2 = 6 r 1

= 6∞ Infinity Neverends𝛥 Displacementorchange

of𝛥𝑥 = 𝑥 − 𝑥2

∑ Summation𝑎0𝑥0

o

01r

= 𝑎r𝑥r + 𝑎q𝑥q + 𝑎o𝑥o

𝜃 Theta–reservedforangles

𝜃 =𝜋4 = 45°

𝑓 𝑥 Functionof𝑥 𝑓 𝑥 = 𝑥0 +⋯𝑓 𝑥, 𝑦 Functionof𝑥and𝑦 𝑓 𝑥, 𝑦 = 𝑥𝑦 0 +⋯


∈ Inorelementof 𝑥 ∈ 𝑎, 𝑏 means𝑎 ≤ 𝑥 < 𝑏∀ Forall ∀�(𝑓𝑜𝑟𝑎𝑙𝑙𝑥)∴ Therefore 𝑥 − 𝑎 = 0 ⇔ 𝑥 = 𝑎 ∴ 𝑥 = 𝑎∵ Because ∵ 𝑥 − 𝑎 = 0, 𝑥 = 𝑎≡ Equivalent −2, 3 ≡ −2 < 𝑥 < 3, Openinterval −2, 3 ≡ −2 < 𝑥 < 3, Closedinterval 2, 3 ≡ 2 ≤ 𝑥 ≤ 3⊂ ProperSubset 𝐴 ⊂ 𝐵 ⇒ 𝐵 ⊄ 𝐴⊆ Subset(equal) 𝐴 ⊆ 𝐵 ⇒ 𝐴 = 𝐵, Halfopen/closed 1, 4 ≡ 1 ≤ 𝑥 < 4… ,… Setofnumbers 1,3,5,7 ∪ Union 𝐷 = −∞, 0 ∪ 0,∞

1,2,3 ∪ 3,4,5 = 1, 2, 3, 4, 5 ∩ Intersection 1,2,3 ∩ 3,4,5 = 3 ℝ Realnumbers 𝐷 = −∞,∞ 𝑃 𝑥2, 𝑦2 Point 1, 𝑓 1 TypesofnumbersIntegers

Rational

Irrational

Complex

… ,−3,−2,−1,0, 1,2,3, … ��, 𝑏 ≠ 0and𝑎, 𝑏areintegers

Anumberthatcannotbeexpressedasafractione.g.𝜋

𝑥 = 𝑎 + 𝑏𝑖where𝑎and𝑏areanynumber

PropertiesReflexive Symmetric Transitive Substitution𝑎 = 𝑎 𝑎 = 𝑏then𝑏 = 𝑎 𝑎 = 𝑏and𝑏 = 𝑐then𝑎 = 𝑐 𝑎 = 𝑏thenbcanreplace𝑎MeaningsBoth𝐴and𝐵havethesameelements 𝐴 = 𝐵

Subset:Ifeveryelementofaset𝐴isin𝐵

𝐴 ⊆ 𝐵 ⇒ 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ⇔ 𝐴 = 𝐵

ProperSubset:IfeveryelementinAisalsoinBbut𝐴 ≠ 𝐵:

𝐴 ⊂ 𝐵

Intersection:Theelementsthatarebothin𝐴andin𝐵

𝐴 ∩ 𝐵 = 𝑥 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵

Union:Allelementsfrom𝐴and𝐵arein𝐴union𝐵

𝐴 ∪ 𝐵 = 𝑥 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐴 ∩ 𝐵

Compliment:If𝐴 ⊂ 𝑈,and𝑈istheuniversalset

𝐴 = 𝐴� = 𝑥 𝑥 ∈ 𝑈 ∧ 𝑥 ∉ 𝐴

Complementationofsets


a. 𝑈� = ∅

b. ∅� = 𝑈

c. 𝐴� � = 𝐴

d. 𝐴 ∪ 𝐴� = 𝑈

e. 𝐴 ∩ 𝐴� = ∅

SetLaws𝐴 ∪ 𝐵 = 𝐵 ∪ 𝐴

Commutativelawforunion

𝐴 ∩ 𝐵 = 𝐵 ∩ 𝐴 Commutativelawforintersection

𝐴 ∪ 𝐵 ∪ 𝐶 = 𝐴 ∪ 𝐵 ∪ 𝐶

Associativelawforunion

𝐴 ∩ 𝐵 ∩ 𝐶 = 𝐴 ∩ 𝐵 ∩ 𝐶

Associativelawforintersection

𝐴 ∪ 𝐵 ∩ 𝐶 = 𝐴 ∪ 𝐵 ∩ 𝐴 ∪ 𝐶

Distributivelawforunion

𝐴 ∩ 𝐵 ∪ 𝐶 = 𝐴 ∩ 𝐵 ∪ 𝐴 ∩ 𝐶 DistributivelawforintersectionDeMorgan’sLaws

i. 𝐴 ∪ 𝐵 � = 𝐴� ∩ 𝐵� ii. 𝐴 ∩ 𝐵 � = 𝐴� ∪ 𝐵� NumberofElementsinaSetNote:𝐴 ∧ 𝐵arefinitesets𝑛 𝐴 ∪ 𝐵 = 𝑛 𝐴 + 𝑛 𝐵 − 𝑛 𝐴 ∩ 𝐵 𝑛 𝐴 ∩ 𝐵 = 𝑛 𝐴 + 𝑛 𝐵 − 𝑛 𝐴 ∪ 𝐵 𝑛 𝐴 ∪ 𝐵 ∪ 𝐶 = 𝑛 𝐴 + 𝑛 𝐵 + 𝑛 𝐶 − 𝑛 𝐴 ∩ 𝐵 − 𝑛 𝐴 ∩ 𝐶 − 𝑛 𝐵 ∩ 𝐶 + 𝑛(𝐴 ∩ 𝐵 ∩ 𝐶)AxiomsSubstitutionPrinciple If𝑎 = 𝑏,then𝑎canbesubstitutedfor𝑏

Commutative–Addition 𝑎 + 𝑏 = 𝑏 + 𝑎

Commutative–Multiplication 𝑎𝑏 = 𝑏𝑎

Associativity–Addition 𝑎 + 𝑏 + 𝑐 = 𝑎 + 𝑏 + 𝑐

Associativity–Multiplication 𝑎 𝑏𝑐 = 𝑎𝑏 𝑐

Reflexive 𝑎 = 𝑎

Symmetric If𝑎 = 𝑏then𝑏 = 𝑎

Transitive If𝑎 = 𝑏and𝑏 = 𝑐then𝑎 = 𝑐


DistributionProperty 𝑎 𝑏 + 𝑐 = 𝑎𝑏 + 𝑎𝑐and 𝑎 + 𝑏 𝑐 = 𝑎𝑐 + 𝑏𝑐

CancellationProperty − −𝑎 = 𝑎

Identity–Addition 𝑎 + 0 = 𝑎and0 + 𝑎 = 𝑎

AdditiveInverse 𝑎 + −𝑎 = 0and– 𝑎 + 𝑎 = 0

Identity–Multiplication 𝑎 1 = 𝑎and 1 𝑎 = 𝑎

MultiplicativeProperty–Zero 𝑎 0 = 0and 0 𝑎 = 0

MultiplicativePropertyfor-1 𝑎 −1 = −𝑎and −1 𝑎 = −𝑎

MultiplicativeInverse 𝑎 𝑎pr = 1and 𝑎pr 𝑎 = 1

Arithmetic𝑎𝑏 ± 𝑎𝑐 = 𝑎 𝑏 ± 𝑐 = 𝑏 ± 𝑐 𝑎

𝑎𝑏𝑐 =

𝑎𝑏𝑐

𝑎𝑏 ±

𝑐𝑑 =

𝑎𝑑 ± 𝑏𝑐𝑏𝑑

𝑎 − 𝑏𝑐 − 𝑑 =

𝑏 − 𝑎𝑑 − 𝑐

𝑎𝑏 + 𝑎𝑐𝑎 = 𝑏 + 𝑐, 𝑎 ≠ 0

𝑎𝑏𝑐 =

𝑎𝑏𝑐

𝑎𝑏𝑐=

𝑎1 ∙

𝑐𝑏 =

𝑎𝑐𝑏

𝑎 ± 𝑏𝑐 =

𝑎𝑐 ±

𝑏𝑐

𝑎𝑏𝑐𝑑

=𝑎𝑏 ∙𝑑𝑐 =

𝑎𝑑𝑏𝑐

𝑖𝑓𝑎 ± 𝑏 = 0𝑡ℎ𝑒𝑛𝑎 = ∓𝑏 Exponents𝑎r = 𝑎

𝑎2 = 1

𝑎p0 =1𝑎0

1𝑎p0 = 𝑎0

𝑎0𝑎� = 𝑎0��

𝑎0

𝑎� = 𝑎0p�

𝑎𝑏

0=𝑎0

𝑏0

𝑎𝑏

p0=𝑎p0

𝑏p0 =𝑏0

𝑎0

𝑎0r� = 𝑎

0� = 𝑎

r�

0

𝑎0 � = 𝑎0� = 𝑎�0 = 𝑎� 0Radicals


𝑎 = 𝑎� = 𝑎r� = 𝑎rq

𝑎��= 𝑎�� = 𝑎

r�0

𝑎0� = 𝑎, 𝑛𝑖𝑠𝑜𝑑𝑑

𝑎𝑏

�=

𝑎�

𝑏� =𝑎r0

𝑏r0=

𝑎𝑏

r0

𝑎�� = 𝑎�0

𝑎0� = 𝑎 , 𝑛𝑖𝑠𝑒𝑣𝑒𝑛

𝑥q = 𝑥 , −∞ < 𝑥 <∞ 𝑥q= 𝑥, 𝑥 ≥ 0

ComplexNumbers𝑥 = 𝑎 ± 𝑖𝑏

Conjugate𝑥 = 𝑎 ∓ 𝑏𝑖 𝑎 + 𝑏𝑖 𝑐 + 𝑑𝑖 = 𝑎𝑐 − 𝑏𝑑 + 𝑎𝑑 + 𝑏𝑐 𝑖

𝑖 = −1

𝑖q = −1

−𝑎 = 𝑖 𝑎,𝑎 ≥ 0𝑥𝑥 = 𝑎q + 𝑏q

AddingandSubtractingFractions

𝑎𝑏 ±

𝑐𝑑 =


𝑔 𝑥𝑓 𝑥 ±

ℎ 𝑥𝑟 𝑥 =

𝑔 𝑥 𝑟 𝑥 ± 𝑓 𝑥 ℎ 𝑥𝑓 𝑥 𝑟 𝑥

LogarithmicLog“Base”NotationNote:log 𝑥 = logr2 𝑥 oritmaybe log 𝑥 = ln 𝑥 = log¤ 𝑥;log xisthegeneralnotationforln xbutinsomebooksorcalculatorslog x = logr2 xandvice-versa.ln 𝑏ln 𝑎 = log� 𝑏

𝑦 = log� 𝑥 ⇒ 𝑥 = 𝑏¦

𝑒 = 2.718281828…

log� 𝑎 = 1

log� 1 = 0

log� 𝑎� = 𝑥

log¤ 𝑥 = ln 𝑥

log� 𝑥� = 𝑏 log� 𝑥

log� 𝑥𝑦 = log� 𝑥 + log� 𝑦

log�𝑥𝑦 = log� 𝑥 − log� 𝑦

𝑒 =

1𝑛!

∞

012

𝑒�© =𝑎0𝑡0

𝑛!

∞

012

Log“Natural”Notation*Itisunlikelythatthenotationinvolving“log”willbeusedthroughoutthecourse;youmayseeitinthebeginningofthecourse,asareviewofsomesortbutthatshouldbeaboutallyou’llsee.The“ln 𝑢”notationwillbethestandardasitiseasiertomanipulate.

log� 𝑏 =ln 𝑏ln 𝑎

𝑦 = ln 𝑥 ⇒ 𝑥 = 𝑒¦

𝑦 = 𝑒� ⇒ 𝑥 = ln 𝑦


𝑒 =1𝑛!

«

012

ln 𝑎 = undefined, 𝑎 ≤ 0

ln 1 = 0

ln 𝑒� = 𝑥 ⇒ 𝑒®¯ � = 𝑥

ln 𝑒r = 1 ⇒ 𝑒®¯ r = 1

ln 𝑥� = 𝑏 ln 𝑥

ln 𝑥𝑦 = ln 𝑥 + ln 𝑦

ln𝑥𝑦 = ln 𝑥 − ln 𝑦

ln 𝑥pr = ln1𝑥 = − ln 𝑥

Domains: ln 𝑥 , 𝐷 = 0,∞

ln 𝑥 , 𝐷 = 𝑥 𝑥 > 0, 𝑥 < 0

*Factoring𝑥0 + 𝑥� = 𝑥0 1 + 𝑥�p0 = 𝑥� 𝑥0p� + 1

𝑥q − 𝑎q = 𝑥 + 𝑎 𝑥 − 𝑎

𝑥q + 2𝑎𝑥 + 𝑎q = 𝑥 + 𝑎 q 𝑥q + 𝑎 + 𝑏 𝑥 + 𝑎𝑏 = (𝑥 + 𝑎)(𝑥 + 𝑏)

𝑥o + 3𝑎𝑥q + 3𝑎q𝑥 + 𝑎o = 𝑥 + 𝑎 o

𝑥o − 3𝑎𝑥q + 3𝑎q𝑥 − 𝑎o = 𝑥 − 𝑎 o

𝑥o + 𝑎o = (𝑥 + 𝑎)(𝑥q − 𝑎𝑥 + 𝑎q)

𝑥o − 𝑎o = 𝑥 − 𝑎 𝑎q + 𝑎𝑥 + 𝑥q

𝑥 + 𝑎 o = 𝑥o + 3𝑎𝑥q + 3𝑎q𝑥 + 𝑎o

𝑥 − 𝑎 o = 𝑥o − 3𝑎𝑥q + 3𝑎q𝑥 − 𝑎o

Note:*Commonmistakestudentsmakewhen

solvingforx:

Thesolutionof𝑥 = 0waslost,thus:

𝑥q − 𝑥 = 0 ⇒ 𝑥q = 𝑥 ⇒ 𝑥 = 1 𝑥q − 𝑥 = 0

⇒ 𝑥 𝑥 − 1 = 0 ⇔ 𝑥 = 0𝑜𝑟𝑥 = 1LongDivision(quotient)(divisor)+(remainder)=dividend

P=Divisor Q=Dividend R=Quotient

CompleteTheSquare


𝑦 = 𝑎𝑥q + 𝑏𝑥 + 𝑐

= 𝑎 𝑥q +𝑏𝑎 𝑥 + 𝑐 = 𝑎 𝑥q +

𝑏𝑎 𝑥 +

𝑏2𝑎

q

−𝑏2𝑎

q

+ 𝑐

= 𝑎 𝑥q +𝑏𝑎 𝑥 +

𝑏2𝑎

q

− 𝑎𝑏2𝑎

q

+ 𝑐 = 𝑎 𝑥 +𝑏2𝑎

q

− 𝑎𝑏q

4𝑎q + 𝑐

= 𝑎 𝑥 +𝑏2𝑎

q

−𝑏q

4𝑎 + 𝑐

∴ 𝑦 = 𝑎 𝑥 +𝑏2𝑎

q

+ 𝑐 −𝑏q

4𝑎Example1:Solvingforx(Formula1)

𝑎𝑥q + 𝑏𝑥 = 0

⇒ 𝑥q +𝑏𝑎 𝑥 =

0𝑎 ⇒ 𝑥q +

𝑏𝑎 𝑥 + 0 = 0

⇒ 𝑥q +𝑏𝑎 𝑥 +

𝑏2𝑎

q

−𝑏2𝑎

q

= 0 ⇒ 𝑥q +𝑏𝑎 𝑥 +

𝑏2𝑎

q

=𝑏2𝑎

q

⇒ 𝑥 +𝑏2𝑎

q

=𝑏q

4𝑎q ⇒ 𝑥 +𝑏2𝑎 = ±

𝑏q

4𝑎q ⇒ 𝑥 = −𝑏2𝑎 ±

𝑏2𝑎

∴ 𝑥 = 0or𝑥 = −𝑏𝑎

Example2:Solvingforx(Formula2)

𝑎𝑥q + 𝑏𝑥 + 𝑐 = 0


𝑐𝑎 =

0𝑎 ⇒ 𝑥q +

𝑏𝑎 𝑥 +

𝑐𝑎 +

𝑏2𝑎

q

−𝑏2𝑎

q

= 0


𝑏2𝑎

q

=𝑏q

2q𝑎q −𝑐𝑎 ⇒ 𝑥 +

𝑏2𝑎

q

=𝑏q − 4𝑎𝑐4𝑎q

⇒ 𝑥 +𝑏2𝑎 = ±

𝑏q − 4𝑎𝑐2𝑎 ⇒ 𝑥 = −

𝑏2𝑎 ±

𝑏q − 4𝑎𝑐2𝑎


∴ 𝑥 =−𝑏 ± 𝑏q − 4𝑎𝑐

2𝑎

Compositions𝑓 ∘ 𝑔 𝑥 = 𝑓 𝑔 𝑥

𝑓 ± 𝑔 𝑥 = 𝑓 𝑥 ± 𝑔 𝑥

𝑓 ∙ 𝑔 𝑥 = 𝑓 𝑥 𝑔 𝑥

𝑓𝑔 𝑥 =

𝑓 𝑥𝑔 𝑥 , 𝑔 𝑥 ≠ 0

FunctionsVerticalLineTest𝑓 𝑥 isafunctionifitpassestheverticallinetesti.e.ifyoudrawaverticallineanywhereonthegraph,andthegraphof𝑓onlycrossesitonce.Even/OddFunctionEven: 𝑓 −𝑥 = 𝑓 𝑥

(symmetricwithrespectto𝑦-axis)

Odd: 𝑓 −𝑥 = −𝑓 𝑥 (symmetricwithrespecttoorigin)

AverageRateofChange

𝛥𝑦𝛥𝑥 =

𝑓 𝑥 − 𝑓 𝑥2𝑥 − 𝑥2

, 𝑥 ≠ 𝑥2

SecantLineTheslopeofthesecantlineisthesameastheaveragerateofchangei.e.𝑚 = ² � p² �³

�p�³youthen

takeoneofthetwopointsandplugtheitinto𝑦 − 𝑦2 =² � p² �³

�p�³𝑥 − 𝑥2 andsimplify.

DifferenceQuotient

𝑚 =𝑓 𝑥 + 𝛥𝑥 − 𝑓 𝑥

𝛥𝑥 =𝑓 𝑥 + ℎ − 𝑓 𝑥

ℎ

DistanceFormulaDistancebetweentwopointsonanumberline

DistancebetweentwopointsinaCartesiancoordinatesystemi.e.xvs.ygraph

𝑃 𝑥2 = 𝑃 𝑥r = 𝑃 𝑎 , 𝑄 = 𝑄 𝑥 = 𝑄 𝑥q = 𝑄 𝑏

𝑃 𝑥2, 𝑦2 , 𝑄 𝑥, 𝑦


𝑑 𝑃, 𝑄 = 𝑥 − 𝑥2 q = 𝑥 − 𝑥2

= 𝑏 − 𝑎 q = 𝑏 − 𝑎 = 𝑥q − 𝑥r q = 𝑥q − 𝑥r

𝑑 𝑃, 𝑄 = 𝑥 − 𝑥2 q + 𝑦 − 𝑦2 q

MidpointFormula𝑃 𝑥r, 𝑦r &𝑄 𝑥q, 𝑦q 𝑚 𝑃,𝑄 =

𝑥q + 𝑥r2 ,

𝑦q + 𝑦r2

QuadraticFormula

𝑎𝑥q + 𝑏𝑥 + 𝑐 = 0 ⇔ 𝑥 =−𝑏 ± 𝑏q − 4𝑎𝑐

2𝑎 Proof:

𝑎𝑥q + 𝑏𝑥 + 𝑐 = 0


𝑐𝑎 =

0𝑎 ⇒ 𝑥q +

𝑏𝑎 𝑥 +

𝑐𝑎 +

𝑏2𝑎

q

−𝑏2𝑎

q

= 0


𝑏2𝑎

q

=𝑏q

2q𝑎q −𝑐𝑎 ⇒ 𝑥 +

𝑏2𝑎

q

=𝑏q − 4𝑎𝑐4𝑎q

⇒ 𝑥 +𝑏2𝑎 = ±

𝑏q − 4𝑎𝑐2𝑎 ⇒ 𝑥 = −

𝑏2𝑎 ±

𝑏q − 4𝑎𝑐2𝑎

∴ 𝑥 =−𝑏 ± 𝑏q − 4𝑎𝑐

2𝑎

Discriminant:

i)Tworealsolutionsif𝑏q − 4𝑎𝑐 > 0ii)Repeatedsolutionsif𝑏q − 4𝑎𝑐 = 0iii)Twocomplexsolutions𝑖𝑓𝑏q − 4𝑎𝑐 < 0


GraphingaLineFromtheform𝑦 = 𝑚𝑥 + 𝑏youcaneasilygraphalinebyidentifyingtwopointsandthenconnectingthem.Theequationwillmoregenerallyappearas𝑦 = �

±»𝑥 + 𝑏where𝑚 = �

±»,𝑐istheriseand

±𝑑istherun(𝑐alwaysgoesupand𝑑goeseitherleftorright.)

Thefirstpointis𝑃r 0, 𝑏 Thesecondpointis𝑃q(±𝑑, 𝑏 + 𝑐)Plotthesetwopointsandconnectalinethroughthem.

PointSlopeForm:𝑦 − 𝑦2 = 𝑚 𝑥 − 𝑥2

𝑚 = 𝑠𝑙𝑜𝑝𝑒 =𝛥𝑦𝛥𝑥

⇒ 𝑚 =𝛥𝑦𝛥𝑥 =

𝑦 − 𝑦2𝑥 − 𝑥2

⇒ 𝑚 =

𝑦 − 𝑦2𝑥 − 𝑥2

⇒ 𝑥 − 𝑥2 𝑚 = 𝑦 − 𝑦2

∴ 𝑦 − 𝑦2 = 𝑚(𝑥 − 𝑥2)

SlopeInterceptForm:𝑦 = 𝑚𝑥 + 𝑏

𝑚 = 𝑠𝑙𝑜𝑝𝑒 =𝛥𝑦𝛥𝑥 ⇒ 𝑚 =

𝛥𝑦𝛥𝑥 =

𝑦 − 𝑦2𝑥 − 𝑥2

⇒ 𝑚 =

𝑦 − 𝑦2𝑥 − 𝑥2

⇒ 𝑥 − 𝑥2 𝑚 = 𝑦 − 𝑦2 ⇒ 𝑚𝑥 − 𝑚𝑥2 = 𝑦 − 𝑦2⇒ 𝑦 = 𝑚𝑥 −𝑚𝑥2 + 𝑦2


⇒ 𝑦 = 𝑚𝑥 + 𝑦2 − 𝑚𝑥2⇒ 𝑦 = 𝑚𝑥 + 𝑦2 − 𝑚𝑥2 , setting𝑏 = 𝑦2 − 𝑚𝑥2

∴ 𝑦 = 𝑚𝑥 + 𝑏StandardorGeneralForm

𝐴𝑥 + 𝐵𝑦 = 𝐶ParallelLine(equalslopes)

𝑦r = 𝑚𝑥 + 𝑏r ∥ 𝑦q = 𝑚𝑥 + 𝑏qPerpendicularLine(productofslopesare-1)

𝑦r = 𝑚𝑥 + 𝑏r ⊥ 𝑦q = −1𝑚𝑥 + 𝑏q

*DomainRestrictionsForthefollowing,𝑓 𝑥 , 𝑔 𝑥 , ℎ 𝑥 areassumedtobecontinuousforallrealnumbers.Polynomial 𝑥 = 𝑎2𝑥0 ± 𝑎r𝑥0pr ± 𝑎q𝑥0pq ± ⋯± 𝑎0𝑥0p0 NoRestrictions

Fraction

ℎ 𝑥 =𝑓 𝑥𝑔 𝑥

𝒈 𝒙 ≠ 𝟎

Radical,ifniseven

𝑓 𝑥 = 𝑔(𝑥)� 𝒈 𝒙 ≥ 𝟎

Radical,ifnisodd

𝑓 𝑥 = 𝑔(𝑥)� NoRestrictions

FractionwithRadicalindenominator

ℎ 𝑥 =𝑓 𝑥𝑔 𝑥�

𝐈𝐟𝐧𝐢𝐬𝐞𝐯𝐞𝐧𝒈 𝒙> 𝟎𝐢𝐟𝐧𝐢𝐬𝐨𝐝𝐝𝒈 𝒙≠ 𝟎

NaturalLog

𝑓 𝑥 = ln 𝑔 𝑥 𝒈 𝒙 > 𝟎

Exponential

ℎ(𝑥) = 𝑓 𝑥 È � 𝐍𝐨𝐑𝐞𝐬𝐭𝐫𝐢𝐜𝐭𝐢𝐨𝐧𝐬

InverseFunctions𝑦 = 𝑓 𝑥 ⇒ 𝑥 = 𝑓 𝑦pr = 𝑓 𝑓pr 𝑥 If𝑓 𝑥 isone-to-oneithasaninverseThedomainof𝑓 𝑥 istherangeof𝑓pr 𝑥 Therangeof𝑓 𝑥 isthedomainof𝑓pr 𝑥


𝑦 = 𝑓 𝑥 ⇒ 𝑥 = 𝑓 𝑦 ⇒ 𝑦 = 𝑓pr 𝑥 Asymptotes,HolesandGraphs

Anasymptoteoccurswherethefunctionisgettinginfinitelyclosetoalineonthegraphbutnevertouchestheline.Horizontalasymptotesmaycrossthelinefromtime-to-time;itistheendbehaviorweareconcernedwith.Therearethreetypesofasymptotes:Horizontal,VerticalandOblique.Obliqueasymptotes,willmostlikely,notbeusedinyourcalculuscoursebutverticalandhorizontalwillbeusedfrequentlyinordertographfunctions.HoleinaGraph 𝑓 𝑥 =

𝑥q − 4𝑥 − 2 ⇒ 𝑥 ≠ 2

ThreeGeneralCasesforHorizontalAsymptotesSincetherearesomanyconditionsandsituationsforasymptotesandthemethodslearnedinalgebraaresominimaltowhatisusedincalculus,wewillcomebacktothislater.Case1

𝑓 𝑥 =𝑥� + 𝑥�pr + ⋯𝑥0 + 𝑥0pr + ⋯

𝑛 > 𝑚 ⇒ 𝐻𝐴:𝑦 = 0

Case2𝑓 𝑥 =

𝑥� + 𝑥�pr + ⋯𝑥0 + 𝑥0pr + ⋯

𝑛 < 𝑚 ⇒ 𝐻𝐴:𝑛𝑜𝑛𝑒

Case3𝑓 𝑥 =

a𝑥� + 𝑥�pr + ⋯b𝑥0 + 𝑥0pr + ⋯ 𝑛 = 𝑚 ⇒ 𝐻𝐴:𝑦 =

𝑎𝑏


Ex.1HorizontalandVertical 𝑓 𝑥 =

𝑥q + 𝑥 + 1𝑥o + 𝑥q + 𝑥 + 1

𝐻𝐴:𝑦 = 0, 𝑉𝐴:𝑥 = −2

Ex.2Oblique𝑓 𝑥 =

𝑥o + 𝑥q + 𝑥 + 1𝑥q + 𝑥 + 1

𝑁𝑜𝐻𝐴, 𝑂𝐴:𝑦 = 𝑥

Ex.3HorizontalandVertical 𝑓 𝑥 =

3𝑥o + 𝑥2𝑥o + 1

𝐻𝐴:𝑦 =32 , 𝑉𝐴:𝑥 = −

12Ó

Inequalities𝑓 𝑥 < 𝑎 ⇒ −𝑎 < 𝑓 𝑥 < 𝑎or𝑓 𝑥 < 𝑎and𝑓 𝑥 > −𝑎𝑓 𝑥 ≤ 𝑎 ⇒ −𝑎 ≤ 𝑓 𝑥 ≤ 𝑎or𝑓 𝑥 ≤ 𝑎and𝑓 𝑥 ≥ −𝑎InterestFormulas

𝐴 = 𝐴2𝑒Ô©

𝑃 = 𝑃2

𝑟12

1 − 1 + 𝑟12

pr L=Loan𝑃 =MonthlyPayment𝑟 =Interestrateforannual𝑡 =Loanlengthinmonths

PhysicsFormulas(rate)(time)=distance𝑟𝑡 = 𝑑Symmetry


ByPoint𝑥-axis Foreverypoint 𝑥, 𝑦 thereisa 𝑥,−𝑦 𝑦-axis Foreverypoint 𝑥, 𝑦 thereisa −𝑥, 𝑦 origin Foreverypoint 𝑥, 𝑦 thereisa −𝑥,−𝑦 Testing𝑥-axis:Replaceeach𝑦witha– 𝑦,ifthesameequationresults,itissymmetric.𝑦-axis:Replaceeach𝑥witha– 𝑥,ifthesameequationresults,itissymmetric.Origin:Replaceeach𝑥, 𝑦witha−𝑥, – 𝑦,ifthesameequationresults,itissymmetric.Variations(Proportionality)𝑘istheconstantofproportionality

𝑦isproportionaltox:y=𝑘𝑥

𝑦isinverselyproportionalto𝑥:𝑦 = Ö

�

CommonGraphsandFormulas

𝑦 = 𝑥q𝑦 = 𝑥o

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𝑦 = 𝑥𝑦 =1𝑥


_________________________________________________________________________________________________________

𝑦 = 𝑒�𝑦 = ln 𝑥

_________________________________________________________________________________________________________

𝑥q + 𝑦q = 1𝑥q − 𝑦q = 1

_________________________________________________________________________________________________________

𝑦q − 𝑥q = 1𝑦 = 𝑥q − 1


EquationofaLine

𝑠𝑙𝑜𝑝𝑒 = 𝑚 =𝑦q − 𝑦r𝑥q − 𝑥r

𝑦 = 𝑚𝑥 + 𝑏

𝑦q − 𝑦r = 𝑚 𝑥q − 𝑥r

𝐴𝑥 + 𝐵𝑦 = 𝐶

EquationofParabola

Vertex: ℎ, 𝑘 = − �q�, 𝑓 − �

q�


𝑦 = 𝑎 𝑥 − ℎ q + 𝑘

EquationofCircleCenter: ℎ, 𝑘 Radius:𝑟

𝑥q + 𝑦q + 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 ⇒

𝑥 − ℎ q + 𝑦 − 𝑘 q = 𝑟q

EquationofEllipse

RightPoint: ℎ + 𝑎, 𝑘

LeftPoint: ℎ − 𝑎, 𝑘

TopPoint: ℎ, 𝑘 + 𝑏

BottomPoint: ℎ, 𝑘 − 𝑏

𝑥 − ℎ q

𝑎q +𝑦 − 𝑘 q

𝑏q = 1

EquationofHyperbola(1)Center: ℎ, 𝑘 Slope:± �

�

Asymptotes:𝑦 = ± ��𝑥 − ℎ + 𝑘

Vertices: ℎ + 𝑎, 𝑘 , ℎ − 𝑎, 𝑘

𝑥 − ℎ q

𝑎q −𝑦 − 𝑘 q

𝑏q = 1

EquationofHyperbola(2)Center: ℎ, 𝑘 Slope:± �

�


Vertices: ℎ, 𝑘 + 𝑏 , ℎ, 𝑘 − 𝑏

𝑦 − 𝑘 q

𝑎q −𝑥 − ℎ q

𝑏q = 1


AreasSquare:𝐴 = 𝐿q = 𝑊qRectangle:𝐴 = 𝐿 ∙ 𝑊Circle:𝐴 = 𝜋 ∙ 𝑟qEllipse:𝐴 = 𝜋 ∙ 𝑎𝑏Triangle:𝐴 = r

q𝑏 ∙ ℎTrapezoid:𝐴 = r

q𝑎 + 𝑏 ∙ ℎ

Parallelogram:𝑏 ∙ ℎRhombus:𝐴 = ÙÚ

q,𝑝and𝑞arethediagonals

SurfaceAreas

Cube:𝐴Û = 6𝐿q = 6𝑊qBox:𝐴Û = 2(𝐿𝑊 +𝑊𝐻 +𝐻𝐿)Sphere:𝐴Û = 4𝜋𝑟qCone:𝐴Û = 𝜋𝑟 𝑟 + ℎq + 𝑟q Cylinder:2𝜋𝑟ℎ + 2𝜋𝑟q

VolumesCube:𝑉 = 𝐿o = 𝑊oBox:𝑉 = 𝐿 ∙ 𝑊 ∙ 𝐻Sphere:𝑉 = Ü

o𝜋 ∙ 𝑟o

Cone:𝑉 = r

o𝜋 ∙ 𝑟qℎEllipsoid:𝑉 = Ü

o𝜋 ∙ 𝑎𝑏𝑐,𝑎, 𝑏, 𝑐aretheradii

BusinessFunctionsCostFunction𝐶 𝑥

RevenueFunction𝑅 𝑥

ProfitFunction𝑃 𝑥 = 𝑅 𝑥 − 𝐶 𝑥

MarginalCostFunction𝐶Þ 𝑥

MarginalRevenueFunction𝑅Þ 𝑥

MarginalProfitFunction𝑃Þ 𝑥 = 𝑅Þ 𝑥 =𝐶Þ 𝑥

AverageCostFunction 𝐶 𝑥 =

𝐶 𝑥𝑥

AverageRevenueFunction𝑅 𝑥 = ß �

�

AverageProfitFunction𝑃 𝑥 = à �

�= ß � pá �

�


AverageRateofChangeof𝒇andSlopeofSecantLine

â¦â�= ² � p² �

�p�= 𝑚Û¤��0©from𝑃r(𝑎, 𝑓 𝑎 )and𝑃q 𝑏, 𝑓 𝑏

DifferenceQuotient𝑚Û¤��0© =

² â��³ p² �³â�

,𝛥𝑥 = ℎ ⇒ 𝑚Û¤��0© =² ��ã p² �

ã, ℎ ≠ 0

FunctionsConstantFunction 𝑦 = 𝑐IdentityFunction 𝑦 = 𝑥SquareFunction 𝑦 = 𝑥qCubeFunction 𝑦 = 𝑥oSquareRootFunction 𝑦 = 𝑥CubeRootFunction 𝑦 = 𝑥Ó ReciprocalFunction 𝑦 =

1𝑥

AbsoluteValueFunction 𝑦 = 𝑥 GreatestIntegerFunction 𝑦 = int 𝑥 ∗PiecewiseFunction 𝑓 𝑥 = 𝑔 𝑥 , 𝑥 ∈ 𝐷r

ℎ 𝑥 , 𝑥 ∈ 𝐷q

PowerFunction 𝑦 = 𝑎𝑥0RationFunction

𝑓 𝑥 =𝑔 𝑥ℎ 𝑥 , ℎ 𝑥 ≠ 0

GraphShiftsandCompressionsVerticallyupfor𝑓 𝑥 𝑓 𝑥 + 𝑘Verticallydownfor𝑓 𝑥 𝑓 𝑥 − 𝑘Horizontallyleftfor𝑓 𝑥 𝑓 𝑥 + ℎ Horizontallyrightfor𝑓 𝑥 𝑓 𝑥 − ℎ 𝑎𝑓 𝑥 multiplyeachy-coordinatebyaVerticallyCompressed:0 < 𝑎 < 1VerticallyStretched:𝑎 > 1

𝑥, 𝑎𝑦

𝑓 𝑎𝑥 Multiplyeachx-coordinatebyr�

HorizontalCompression:𝑎 > 1HorizontalStretch:0 < 𝑎 < 1

1𝑎 𝑥, 𝑦

Reflectionaboutx-axis −𝑓 𝑥 Reflectionabouty-axis 𝑓 −𝑥


Systemsofequations

𝑎𝑥 + 𝑏𝑦 = 𝑒𝑐𝑥 + 𝑑𝑦 = 𝑓 ⇒ 𝑎 𝑏

𝑐 𝑑𝑥𝑦 =

𝑒𝑓 ⇒ 𝑎 𝑏

𝑐 𝑑𝑒𝑓

𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑𝑒𝑥 + 𝑓𝑦 + 𝑔𝑦 = ℎ𝑖𝑥 + 𝑗𝑦 + 𝑘𝑧 = 𝑙

⇒ 𝑎 𝑏 𝑐𝑒 𝑓 𝑔𝑖 𝑗 𝑘

𝑥𝑦𝑧=

𝑑ℎ𝑙⇒

𝑎 𝑏 𝑐𝑒 𝑓 𝑔𝑖 𝑗 𝑘

𝑑ℎ𝑙

TheCoefficientMatrix=𝑎 𝑏 𝑐𝑒 𝑓 𝑔𝑖 𝑗 𝑘

Rankofmatrixandpivots

𝟏 1 , 𝑟𝑎𝑛𝑘 𝐴r = 1

𝟏 1 , 𝑟𝑎𝑛𝑘 𝐴ç = 1

𝟏1 , 𝑟𝑎𝑛𝑘 𝐴q = 1

𝟏0 , 𝑟𝑎𝑛𝑘 𝐴è = 1

𝟏 1 1 , 𝑟𝑎𝑛𝑘 𝐴o = 1

𝟏 1 1 , 𝑟𝑎𝑛𝑘 𝐴r2 = 1

𝟏11, 𝑟𝑎𝑛𝑘 𝐴Ü = 1

𝟏00, 𝑟𝑎𝑛𝑘 𝐴rr = 1

𝟏 00 𝟏 , 𝑟𝑎𝑛𝑘 𝐴é = 2

𝟏 1 11 1 11 1 1

, 𝑟𝑎𝑛𝑘 𝐴rq = 1

𝟏 0 00 𝟏 1 , 𝑟𝑎𝑛𝑘 𝐴ê = 2

𝟏 1 11 1 −𝟏1 1 1

, 𝑟𝑎𝑛𝑘 𝐴ro = 2

𝟏 00 00 𝟏

, 𝑟𝑎𝑛𝑘 𝐴ë = 2

𝟏 1 10 𝟏 10 0 𝟏

, 𝑟𝑎𝑛𝑘 𝐴rÜ = 3


Determinate’sofa(2x2)matrixVariouswaystocheckdeterminant(2x2):

𝐴 = 𝑎 𝑏

𝑐 𝑑 ⇒ det 𝐴 = 𝐴 = 𝑎 𝑏𝑐 𝑑 = 𝑎 𝑑 − 𝑏 𝑐

𝐴 = 𝑎 𝑏

0 𝑐 ⇒ det 𝐴 = 𝐴 = 𝑎 𝑏0 𝑐 = 𝑎 𝑐

𝐴 = 𝑎 0

0 𝑏 ⇒ det 𝐴 = 𝐴 = 𝑎 00 𝑏 = 𝑎 𝑏

Determinateofa(3x3)andhighermatricesCofactorExpansionNote:

𝑎 𝑏𝑐 𝑑 = 𝑎 𝑑 − 𝑏 𝑐

𝑎 𝑏 𝑐𝑑 𝑒 𝑓𝑔 ℎ 𝑖

= +𝑎 𝑒 𝑓ℎ 𝑖

− 𝑏 𝑑 𝑓𝑔 𝑖 + 𝑐 𝑑 𝑒

𝑔 ℎ

𝑎 𝑏𝑒 𝑓

𝑐 𝑑𝑔 ℎ

𝑖 𝑗𝑚 𝑛

𝑘 𝑙𝑜 𝑝

= +𝑎𝑓 𝑔 ℎ𝑗 𝑘 𝑙𝑛 𝑜 𝑝

− 𝑏𝑒 𝑔 ℎ𝑖 𝑘 𝑙𝑚 0 𝑝

+ 𝑐𝑒 𝑓 ℎ𝑖 𝑗 𝑙𝑚 𝑛 𝑝

− 𝑑𝑒 𝑓 𝑔𝑖 𝑗 𝑘𝑚 𝑛 𝑜


TRIGONOMETRYIntrigonometry,muchislearned,buttobeeffectiveinCalculus,thereisonlyasmallportionoftrigonometrythatmustbemasteredandmemorizedinordertosolveproblems.Mainly:howtoevaluatetheunitcircle,righttriangles,performtrigonometricsubstitutionsandafewothers.Forthisportionofthebookwewillfocusonthesetopics.

*Note:

cospr 𝜃 = arccos 𝜃 ≠1

cos 𝜃 , cos 𝜃 pr =1

cos 𝜃 = sec 𝜃 ≠ cospr 𝜃

Thisisacommonreasonwhy arccos θ pronouncedarccosine isusedinplaceof cospr θ (pronouncedcosineinverse)

Thisistrueforallfunctionsandoperatorsi.e.𝑓pr 𝑥 istheinverseof𝑓where 𝑓 𝑥 pristhereciprocalof𝑓i.e. 𝑓 𝑥 pr = r

² �

RadianandDegreeConversion

𝜃îïðñïï = 𝜃ñòîóò¯180°

𝜋 𝜃ñòîóò¯ = 𝜃îïðñïï𝜋

180°

i. e.

𝜃îïðñïï = 45° ⇒ 𝜃ñòîóò¯ = 45°𝜋

180° =45180 (1°p°)𝜋 =

14 12 𝜋 =

𝜋4

Noticethatthedegreecancelsoutjustlikeavariableandthattheabsenceofthedegree

symbolsimpliesradians.

𝜃ñòîóò¯ =𝜋4 ⇒ 𝜃îïðñïï =

𝜋4

180°

𝜋 =1804

° 𝜋𝜋 = 45° 1 = 45°


BasicGraphs

𝑦 = sin 𝑥 𝑦 = cos 𝑥

_________________________________________________________________________________________________________________

_

𝑦 = csc 𝑥 𝑦 = sec 𝑥

_________________________________________________________________________________________________________________

_

𝑦 = tan 𝑥 𝑦 = cot 𝑥

_________________________________________________________________________________________________________________

_


𝑦 = arcsin 𝑥 = sinpr 𝑥 𝑦 = arccos 𝑥 = cospr 𝑥

_________________________________________________________________________________________________________________

_𝑦 = arctan 𝑥 = tanpr 𝑥


UsingPythagorean’sTheorem

𝑥q + 𝑦q = 𝑟q ⇔ 𝑟 = 𝑥q + 𝑦q

AngleFromTheHorizontalAngleFromTheVertical_________________________________________________________________________________________________________________

_

cos 𝛼 =𝑥𝑟 cos 𝛽 =

𝑦𝑟

__________________________________________________________________________________________________________________

tan 𝛼 =

𝑦𝑥 tan 𝛽 =

𝑥𝑦

__________________________________________________________________________________________________________________

sin 𝛼 =

𝑦𝑟 sin 𝛽 =

𝑥𝑟

__________________________________________________________________________________________________________________

𝑥 = 𝑟 cos 𝛼 𝑦 = 𝑟 cos 𝛽

__________________________________________________________________________________________________________________

𝑦 = 𝑟 sin 𝛼 𝑥 = 𝑟 sin 𝛽

__________________________________________________________________________________________________________________

𝛼 = arctan

𝑦𝑥 = tanpr

𝑦𝑥 𝛽 = arctan

𝑥𝑦 = tanpr

𝑥𝑦

__________________________________________________________________________________________________________________

Pleasenotethatthepreviousandfollowingevaluationofarighttriangleusingfunctionsisnotaformaldefinitionoratheorem–itissimplyatechniquethatcanbeusedforsimplifyingaproblemanditshouldbeknownthatwhenfindingangles,thedomainmustbeconsideredaswell.


𝑓 𝑥 q + 𝑔 𝑥 q = ℎ 𝑥 q

AngleFromTheHorizontalAngleFromTheVertical_________________________________________________________________________________________________________________

_

cos 𝛼 =𝑓 𝑥ℎ 𝑥 cos 𝛽 =

𝑔 𝑥ℎ 𝑥

__________________________________________________________________________________________________________________

sin 𝛼 =𝑔 𝑥ℎ 𝑥 sin 𝛽 =

𝑓 𝑥ℎ 𝑥

__________________________________________________________________________________________________________________

tan 𝛼 =𝑔 𝑥𝑓 𝑥 tan 𝛽

𝑓 𝑥𝑔 𝑥

__________________________________________________________________________________________________________________

α = arctan𝑔 𝑥𝑓 𝑥 𝛽 = arctan

𝑓 𝑥𝑔 𝑥

__________________________________________________________________________________________________________________

𝑓 𝑥 = ℎ 𝑥 cos 𝛼 𝑔 𝑥 = ℎ 𝑥 cos 𝛽

__________________________________________________________________________________________________________________

𝑔 𝑥 = ℎ 𝑥 sin 𝛼 𝑓 𝑥 = h x sin 𝛽


________________________________________________________________________________________________________________

*IMPORTANT:Makesureeverythingaboutevaluatingrighttriangles,theunitcircleandtrigonometricidentitiesisfullyunderstoodasthiswillbeusedindetailthroughoutCalculusand

Physics.

TrigonometricFormulasandIdentities

Becausetherearequiteafewtrigonometricformulasandidentitiesanditisquitedifficulttomemorizeallofthem,itisextremelyimportanttounderstandhowtoderivetheseformulasand

identitiesfromknownformulasandidentities.

Whenworkingincalculusonewillfrequentlyreplaceatrigonometricstatementwithanothertrigonometricstatement.Thisisoneofthemostdifficultpartsofcalculusforpeopletograsp,itisnotbecauseitishard,itisbecausestudentsflythroughtrigonometryandneverreallyunderstandwhattheyweredoinganddonothaveenoughpracticewithsymbolrecognitionforreplacement.

ReciprocalIdentities

sin 𝜃 =1

csc 𝜃 csc 𝜃 =1

sin 𝜃 tan 𝜃 =1

cot 𝜃

csc 𝜃 =1

sec 𝜃 sec 𝜃 =1

cos 𝜃 cot 𝜃 =1

tan 𝜃

tan 𝜃 =sin 𝜃cos 𝜃 cot 𝜃 =

cos 𝜃sin 𝜃

Important:Thisisagoodplacetostartgettingacquaintedwithhowtousetrig-substitution,asthiswillappearfrequentlythroughoutcalculus.

tan 𝜃 =sin 𝜃cos 𝜃 = sin 𝜃

1cos 𝜃 ⇒ sin 𝜃

1cos 𝜃 = sin 𝜃 sec 𝜃

⇒ sin 𝜃 sec 𝜃 =1

csc 𝜃 sec 𝜃 ⇒ 1

csc 𝜃 sec 𝜃 =sec 𝜃csc 𝜃

tan 𝜃 =sec 𝜃csc 𝜃 ⇒ tan 𝜃 q =

sec 𝜃csc 𝜃

q

⇒ tanq 𝜃 =secq 𝜃cscq 𝜃 =

tanq 𝜃 + 11 + cotq 𝜃

Asseeninthepreviousexamples,thepossibilitiesfortrig-substitutionareendless.Often,in

calculus,onejusthastokeeptryingdifferentformsuntilaformthatworksisfound.


PythagoreanIdentities

Oftenstudentscannotrememberalltheidentitiesbutwithsinq 𝜃 + cosq 𝜃 = 1andafewsimple

concepts,alltheidentitiescaneasilybefound.

Derivationsinq 𝜃 + cosq 𝜃 = 1:

𝑥 = 𝑟 cos 𝜃 , 𝑦 = 𝑟 sin 𝜃 , 𝑥q + 𝑦q = 𝑟q

𝑥q + 𝑦q = 𝑟 cos 𝜃 q + 𝑟 sin 𝜃 q = 𝑟q cosq 𝜃 + 𝑟q sinq 𝜃

= 𝑟q cosq 𝜃 + sinq 𝜃 = 𝑟q 1 = 𝑟q

∴ 𝑥q + 𝑦q = 𝑟q

__________________________________________________________________________________________________________________

Derivationfortanq 𝜃 + 1 = secq 𝜃:

1cosq 𝜃 sinq 𝜃 + cosq 𝜃 = 1 ⇒

sinq 𝜃cosq 𝜃 +

cosq 𝜃cosq 𝜃 =

1cosq 𝜃 ⇒

sin 𝜃cos 𝜃

q

+ 1 =1

cos 𝜃

q

∴ tanq 𝜃 + 1 = secq 𝜃_________________________________________________________________________________________________________________

_

Derivationfor1 + cotq 𝜃 = cscq 𝜃:

1sinq 𝜃 sinq 𝜃 + cosq 𝜃 = 1 ⇒

sinq 𝜃sinq 𝜃 +

cosq 𝜃sinq 𝜃 =

1sinq 𝜃 ⇒ 1 +

cos 𝜃sin 𝜃

q

=1

sin 𝜃

q

∴ 1 + cotq 𝜃 = cscq 𝜃

EvenandOddFunctions

Even ⇔ 𝑓 −𝑥 = 𝑓 𝑥

Odd ⇔ 𝑓 −𝑥 = −𝑓 𝑥

Odd sin −𝜃 = −sin 𝜃

Even cos −𝜃 = cos 𝜃


Odd tan −𝜃 = − tan 𝜃

Odd csc −𝜃 = −csc 𝜃

Even sec −𝜃 = sec 𝜃

Odd cot 𝜃 = −cot 𝜃

Example

TheremaybeatimeinCalculusi.e.IntegralCalculusortowardstheendofthefirstsemesterofCalculusorduringthesecondsemester,depending…Youwillneedtobeabletoeasilyidentifyan

oddfunction.Hereisanexamplewhenthismaybenecessary:

sin 𝑥 cos 𝑥ln |𝑥| − sinq 𝑒�

q

pq𝑑𝑥

Thisiscalleda‘DefiniteIntegral,’trynotbescaredofit,itisactuallyquitesimpletoevaluateinthiscasebecause 𝑓 𝑥�

p� 𝑑𝑥 = 0if𝑓 𝑥 isanoddfunction.Wemustnowshow𝑓 𝑥 isanoddfunction.

𝑓 𝑥 =sin 𝑥 cos 𝑥ln 𝑥 − sinq 𝑒� ⇒ 𝑓 −𝑥 =

sin −𝑥 cos −𝑥ln −𝑥 − sinq 𝑒 p� �

sin −𝑥 = −sin 𝑥 𝑜𝑑𝑑

cos −𝑥 = cos 𝑥 𝑒𝑣𝑒𝑛

ln −𝑥 = ln 𝑥 𝑒𝑣𝑒𝑛

sinq 𝑒 p� � = sinq 𝑒�� 𝑒𝑣𝑒𝑛

Plugeverythingbackin

∴ 𝑓 −𝑥 = −sin 𝑥 cos 𝑥ln 𝑥 − sinq 𝑒� = −𝑓 𝑥

Inotherwords,inordertoshowafunctionisodd,simplypluganegativesigninwitheveryxandthenevaluateeachindividualfunctionandseeifyouhaveanevenoroddnumberofnegative

signs.


DoubleAngleFormulas

*Important

ThehalfangleanddoubleangleformulasalongwiththePythagoreanidentitiesareusedfrequentlythroughout

calculus.Itisamustthatyoumemorizetheunderstandingandderivationsisfullycomprehended.

Foradetailedlistofallidentities,seethereferencesheetsinthebackofthebook.

Derivationforsin 2𝜃 = 2 sin 𝜃 cos 𝜃:

sin 2𝜃 = sin 𝜃 + 𝜃 = sin 𝜃 cos 𝜃 + sin 𝜃 cos 𝜃 = 2 sin 𝜃 cos 𝜃_________________________________________________________________________________________________________________

_

Derivationforcos 2𝜃 = 1 − 2 sinq 𝜃:

cos(2𝜃) = cosq 𝜃 − sinq 𝜃 = 2 cosq 𝜃 − 1 = 1 − 2 sinq 𝜃_________________________________________________________________________________________________________________

_

Asonecansee,theseformulasareallderivedfromthePythagoreanidentitiesandtherearemanywaystofindthem.Ifthiscanbeunderstoodproperlythenmemorizingthemisnotentirelynecessary.

OtherDerivations:

cos 2𝜃 = cos(𝜃 + 𝜃) = cos 𝜃 cos 𝜃 − sin 𝜃 sin 𝜃 = cosq 𝜃 − sinq 𝜃

__________________________________________________________________________________________________________________

cos 2𝜃 = cos(𝜃 + 𝜃) = cos 𝜃 cos 𝜃 − sin 𝜃 sin 𝜃 = cosq 𝜃 − sinq 𝜃 = cosq 𝜃 − (1 − cosq 𝜃)

= cosq −1 + cosq 𝜃 = 2 cosq 𝜃 − 1

__________________________________________________________________________________________________________________

cos 2𝜃 = cos(𝜃 + 𝜃) = cos 𝜃 cos 𝜃 − sin 𝜃 sin 𝜃 = cosq 𝜃 − sinq 𝜃

= 1 − sinq 𝜃 − sinq 𝜃 = 1 − 2 sinq 𝜃

__________________________________________________________________________________________________________________

tan 2𝜃 = tan 𝜃 + 𝜃 =tan 𝜃 + tan 𝜃1 − tan 𝜃 tan 𝜃 =

2 tan 𝜃1 − tanq 𝜃


HalfAngleFormulas

sinq 𝜃 =12 1 − cos 2𝜃

Derivation:

sinq 𝜃 = 1 − cosq 𝜃 = 1 − cos 𝜃 cos 𝜃 = 1 −12 cos 𝜃 − 𝜃 + cos 𝜃 + 𝜃

= 1 −12 cos 0 + cos 2𝜃 = 1 −

12 1 + cos 2𝜃 = 1 −

12 −

12 cos 2𝜃

=12 −

12 cos 2𝜃 =

12 [1 − cos(2𝜃)]

__________________________________________________________________________________________________________________

cosq 𝜃 =12 [1 + 𝑐𝑜𝑠 2𝜃 ]

Derivation:

cosq 𝜃 = 1 − sinq 𝜃 = 1 − sin 𝜃 sin 𝜃 = 1 −12 cos(𝜃 − 𝜃 − cos 𝜃 + 𝜃 ]

= 1 −12 cos 0 − cos 2𝜃 = 1 −

12 1 − cos 2𝜃 = 1 −

12 +

12 cos 2𝜃

=12 +

12 cos 2𝜃 =

12 1 + cos 2𝜃

__________________________________________________________________________________________________________________

tanq 𝜃 =1 − cos(2𝜃)1 + cos(2𝜃)

Derivation:

tanq 𝜃 = secq 𝜃 − 1 =1

cos 𝜃

q

− 1 =1

cos 𝜃 cos 𝜃 − 1 =1

12 cos 𝜃 − 𝜃 + cos 𝜃 + 𝜃

− 1


=2

1 + cos 2𝜃 − 1 =2

1 + cos 2𝜃 −1 + cos 2𝜃1 + cos 2𝜃 =

2 − 1 + cos 2𝜃1 + cos 2𝜃 =

1 − cos 2𝜃1 + cos 2𝜃

SumandDifferenceFormulas

sin 𝛼 ± 𝛽 = sin 𝛼 cos 𝛽 ± cos 𝛼 sin 𝛽

__________________________________________________________________________________________________________________

cos(𝛼 ± 𝛽) = cos 𝛼 cos 𝛽 ∓ sin 𝛼 cos 𝛽_________________________________________________________________________________________________________________

_

tan 𝛼 ± 𝛽 =tan𝛼 ± tan𝛽1 ∓ tan𝛼 𝑡𝑎𝑛𝛽

__________________________________________________________________________________________________________________

Thederivationsofthesum&difference,producttosumandsumtoproductformulasareabitmorecomplicated.Trytoshowtheyaretruewithoutreferencinganything.Thiswillprovetobeanexcellentpractice.Remember,gettingitcorrectisnotalwaysthepointofpractice.Onemust

sometimesgointhewrongdirectiontolearnthattheyarenotontherightpath.

ProducttoSumFormulas

sin 𝛼 sin 𝛽 =12 [cos 𝛼 − 𝛽 − cos(𝛼 + 𝛽)]

__________________________________________________________________________________________________________________

cos 𝛼 cos 𝛽 =12 [cos 𝛼 − 𝛽 + cos(𝛼 + 𝛽)]

__________________________________________________________________________________________________________________

sin 𝛼 cos 𝛽 =12 [sin 𝛼 + 𝛽 + sin 𝛼 − 𝛽 ]

__________________________________________________________________________________________________________________

cos 𝛼 sin 𝛽 =12 sin 𝛼 + 𝛽 − sin 𝛼 − 𝛽


SumtoProductFormulas

sin 𝛼 + sin 𝛽 = 2 sin𝛼 + 𝛽2 cos

𝛼 − 𝛽2

__________________________________________________________________________________________________________________

sin 𝛼 − sin 𝛽 = 2 cos𝛼 + 𝛽2 sin

𝛼 − 𝛽2

__________________________________________________________________________________________________________________

cos 𝛼 + cos 𝛽 = 2 cos𝛼 + 𝛽2 cos

𝛼 − 𝛽2

__________________________________________________________________________________________________________________

cos 𝛼 − cos 𝛽 = −2 sin𝛼 + 𝛽2 sin

𝛼 − 𝛽2


HyperbolicFunctions

Irarelyseehyperbolicfunctionsintheaveragecalculuscoursebuteveryonceinawhilethetopicpopsupanditseemsliketheteacherisobsessedwiththemwhenitdoes.Itisgoodtoknowhowtousethem.Nottoodifferentfromworkingwithtrigonometricoperations,justalittlemore

involved.

Notation

sinh 𝑥 =𝑒� − 𝑒p�

2 csch 𝑥 =2

𝑒� + 𝑒p�

cosh 𝑥 =𝑒� + 𝑒p�

2 sech 𝑥 =2

𝑒� + 𝑒p�

tanh 𝑥 =𝑒� − 𝑒p�

𝑒� + 𝑒p� coth 𝑥 =𝑒� + 𝑒p�

𝑒� − 𝑒p�

Graphs


2 csch 𝑥 =2

𝑒� + 𝑒p�

_


cosh 𝑥 =𝑒� + 𝑒p�

2 sech 𝑥 =2

𝑒� + 𝑒p�

__________________________________________________________________________________________________________________

tanh 𝑥 =𝑒� − 𝑒p�


𝑒� − 𝑒p�

Identities

sinh −𝑥 = −sinh 𝑥 cosh −𝑥 = cosh 𝑥

coshq 𝑥 − sinhq 𝑥 = 11 − tanhq 𝑥 = sechq 𝑥

sinh 𝑥 + 𝑦 = sinh 𝑥 cosh 𝑦 + cosh 𝑥 sinh 𝑦

cosh 𝑥 + 𝑦 = cosh 𝑥 cosh 𝑦 + sinh 𝑥 sinh 𝑦

sinhpr 𝑥 = ln 𝑥 + 𝑥q + 1 , −∞ ≤ 𝑥 ≤∞

coshpr 𝑥 = ln 𝑥 + 𝑥q − 1 , 𝑥 ≥ 1

tanhpr 𝑥 =12 ln

1 + 𝑥1 − 𝑥 , −1 < 𝑥 < 1


DIFFERENTIALCALCULUS(CALCI)Translation-Thelimitoffofxasxgoestoa lim

�→�𝑓(𝑥)

fofxapproachesthelimitasxapproachesa 𝑓 𝑥 → 𝐿as𝑥 → 𝑎

NotationsforLimitsTheactuallimit lim

�→�𝑓(𝑥) = 𝐿 ⇔ 𝐿p = 𝐿�

Lefthandlimit lim�→�ü

𝑓 𝑥 = 𝐿p

Righthandlimit lim�→�ý

𝑓 𝑥 = 𝐿�

Limitexists lim�→�ü

𝑓 𝑥 = lim�→�ý

𝑓(𝑥)

Limitdoesnotexists(DNE) lim�→�ü

𝑓 𝑥 ≠ lim�→�ý

𝑓 𝑥

Continuousfunction 𝑓 𝑎 = 𝐿 = 𝐿±

LeftContinuousfunction 𝑓 𝑎 = 𝐿p

RightContinuousfunction 𝑓 𝑎 = 𝐿�

Non-continuousfunction 𝑓 𝑎 = 𝑈𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 ≠ 𝐿


TypesofDiscontinuity

Jump

Occurswithpiecewisefunctionsi.e.

𝑓 𝑥 = −𝑥,𝑥 < 1𝑥 + 1,𝑥 ≥ 1

Removable

𝑓 𝑥 =𝑥q − 4𝑥 − 2 =

𝑥 − 2 𝑥 + 2𝑥 − 2 ⇒

𝑔 𝑥 = 𝑥 + 2 ⇒ 𝑥 ≠ 2 ∵ 𝑓 2 =00

Occursatholesinthegraph

Infinite

𝑓 𝑥 =𝑥q + 𝑥 + 1

𝑥o + 𝑥q + 𝑥 + 1∴ 𝐻𝐴:𝑦 = 0, 𝑉𝐴:𝑥 = −2

Occursatasymptotes


LimitLawsandPropertiesLimitofaConstant lim

�→�𝑐 = 𝑐

LimitofSingleVariable lim�→�

𝑥 = 𝑎

IfTheFunctionisContinuous lim�→�

𝑓(𝑥) = 𝑓(𝑎)

TheConstantMultipleLaw lim�→�

[𝑐𝑓 𝑥 ] = 𝑐 lim�→�

𝑓(𝑥)

TheSumandDifferenceLaw lim�→�

[𝑓 𝑥 ± 𝑔 𝑥 ] = lim�→�

𝑓(𝑥) ± lim�→�

𝑔(𝑥)

TheProductLaw lim�→�

[𝑓 𝑥 𝑔 𝑥 ] = lim�→�

𝑓(𝑥) ∙ lim�→�

𝑔(𝑥)

TheQuotientLawlim�→�

𝑓 𝑥𝑔 𝑥 =

lim�→�

𝑓(𝑥)

lim�→�

𝑔(𝑥) , lim�→�

𝑔(𝑥) ≠ 0

ThePowerLaw lim�→�

𝑓 𝑥 0 = lim�→�

𝑓 𝑥0, 𝑛 ∈ ℕ

TheRootLaw lim�→�

𝑓(𝑥)� = lim�→�

𝑓(𝑥)� , 𝑛 ∈ ℕ

ExponentialLaw lim�→�

𝑎² � = 𝑎®óÿ!→"² �

InfiniteLimits

Therearethreebasiccasesforevaluatingnon-trig/logfunctionsatinfinity.Thisiswherethehorizontalasymptoteformulasarise--usedinAlgebra.Case1:

lim�→∞

𝑥� + 𝑥�pr + ⋯𝑥0 + 𝑥0pr + ⋯ = 0

𝑛 > 𝑚, 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦𝑏𝑦𝑥−𝑛

𝑥−𝑛

Ratioofpolynomialsofdegreem&n

Case2:

lim�→∞

𝑥� + 𝑥�pr + ⋯𝑥0 + 𝑥0pr + ⋯ =∞

𝑛 < 𝑚, 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦𝑏𝑦𝑥−𝑛

𝑥−𝑛


Case3:

lim�→∞

a𝑥� + 𝑥�pr + ⋯b𝑥0 + 𝑥0pr + ⋯ =

ab

𝑛 = 𝑚, 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦𝑏𝑦𝑥−𝑛

𝑥−𝑛



PreciseDefinitionofaLimit𝜺, 𝜹Thelimitoffofxasxgoestoa lim

�→�𝑓(𝑥)

fofxapproachesthelimitasxapproachesa

𝑓 𝑥 → 𝐿as𝑥 → 𝑎

LimitForevery𝜖 > 0,thereisa𝛿 > 0suchthat0 < 𝑥 − 𝑎 < 𝛿and 𝑓 𝑥 − 𝐿 < 𝜖LeftHandLimitForevery𝜖 > 0,thereisa𝛿 > 0suchthat𝑎 − 𝛿 < 𝑥 < 𝑎and 𝑓 𝑥 − 𝐿 < 𝜖RightHandLimitForevery𝜖 > 0,thereisa𝛿 > 0suchthat𝑎 < 𝑥 < 𝑎 + 𝛿and 𝑓 𝑥 − 𝐿 < 𝜖Derivationof“TheDifferenceQuotient”

𝑚 =𝛥𝑦𝛥𝑥 ⇒

𝛥𝑦𝛥𝑥 =

𝑦 − 𝑦2𝑥 − 𝑥2

, 𝑦 = 𝑓(𝑥)

⇒𝑦 − 𝑦2𝑥 − 𝑥2

=𝑓 𝑥 − 𝑓 𝑥2

𝑥 − 𝑥2, 𝛥𝑥 = 𝑥 − 𝑥2 ⇔ 𝑥 = 𝛥𝑥 + 𝑥2

⇒𝑓 𝑥 − 𝑓(𝑥2)

𝑥 − 𝑥2=𝑓 𝛥𝑥 + 𝑥2 − 𝑓(𝑥2)

𝛥𝑥 ≡𝑓 𝑥 + ℎ − 𝑓(𝑥)

ℎ

SlopeofSecantLineorDifferenceQuotient

𝑚 =𝑓 𝑥 + ℎ − 𝑓 𝑥

ℎ ≡𝑓 𝑥 + 𝛥𝑥 − 𝑓 𝑥

𝛥𝑥 ⇔ ℎ = 𝛥𝑥

IntermediateValueTheoremIf𝑓iscontinuouson 𝑎, 𝑏 ,𝑓 𝑎 < 𝑁 < 𝑓 𝑏 and𝑓 𝑎 ≠ 𝑓 𝑏 ,thenthereisa𝑐 ∈ 𝑎, 𝑏 ∋ 𝑓 𝑐 =𝑁.(∋means“suchthat”)


CommonLimitsInfiniteLimitslim�→∞

𝑥0 =∞, 𝑛 > 0 lim�→∞

1𝑥0 = 0 , 𝑛 > 0

lim�→∞

𝑥0 = 0, 𝑛 < 0

lim�→∞

1𝑥0 =∞, 𝑛 < 0 lim

�→∞𝑒� =∞

lim�→∞

1𝑒� = 0

lim�→∞

𝑒p� = 0 lim�→∞

1𝑒p� =∞

lim�→∞

ln 𝑥 =∞

lim�→∞

1ln 𝑥 = 0

lim�→∞

ln 𝑥pr = −∞

lim�→∞

1ln 𝑥pr = 0

lim�→∞

sin 𝑥 = −1to1, 𝐷𝑁𝐸 lim�→∞

sin1𝑥 = 0 lim

�→∞cos 𝑥 = −1to1

lim�→∞

tan 𝑥 = −∞to∞, 𝐷𝑁𝐸 lim�→∞

cos1𝑥 = 1 lim

�→∞csc 𝑥

= −∞to − 1&1to∞,DNE

lim�→∞

tan1𝑥 = 0 lim

�→∞csc

1𝑥 =∞ lim

�→∞sec 𝑥

= −∞to − 1&1to∞,DNE

lim�→∞

sec1𝑥 = 1 lim

�→∞cot 𝑥 = −∞to∞ lim

�→∞cot

1𝑥 =∞


DerivativesTheLimitDefinitionofaDerivative

𝑓Þ 𝑥 = limã→2

𝑓 𝑥 + ℎ − 𝑓(𝑥)ℎ ≡ lim

�→�³

𝑓 𝑥 + 𝛥𝑥 − 𝑓 𝑥𝛥𝑥 ⇔ ℎ = 𝛥𝑥

Theapostrophein𝑓Þ 𝑥 or𝑦′denotesderivative.

Notations0thDerivative 𝑦 = 𝑓 𝑥 =

𝑑𝐹𝑑𝑥 =

𝑑𝑌𝑑𝑥

1stDerivative 𝑦Þ = 𝑓Þ 𝑥 =𝑑𝑦𝑑𝑥

2ndDerivative𝑦ÞÞ = 𝑓ÞÞ 𝑥 =

𝑑q𝑦𝑑𝑦q

3rdDerivative𝑦ÞÞÞ = 𝑓ÞÞÞ 𝑥 =

𝑑o𝑦𝑑𝑦o

4thDerivative𝑦 Ü = 𝑓 Ü 𝑥 =

𝑑Ü𝑦𝑑𝑦Ü

𝑛-.Derivative 𝑦 0 = 𝑓 0 𝑥 =𝑑0𝑦𝑑𝑦0

Note:

1. Anyderivativeafterthe3rdiswrittenas𝑓(0) 𝑥 or𝑦(0)nottobeconfusedwithapower𝑦 Ü =4thderivative≠ 𝑦Ü = 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦

2. »»[… ]

iscalled“TheDerivativeOperator”itsimplymeanstotakethederivativeofwhateverfollowswithrespecttowhateverisin[…].

TimeDerivatives𝑑𝑑𝑡 𝑦 = 𝑦Þ 𝑡 = 𝑦,1/-derivativewithrespecttotime

𝑑q

𝑑𝑡q 𝑥 = 𝑥ÞÞ 𝑡 = 𝑥,2¯îderivativewithrespecttotime


TheSlopeNotationforCalculus

𝑚 = lim�→�

𝑓 𝑥 − 𝑓 𝑎𝑥 − 𝑎 ≡ lim

ã→2

𝑓 𝑎 + ℎ − 𝑓 𝑎ℎ = 𝑚

Slopeoffunctionakaderivative:

𝑓Þ 𝑥 = limã→2

𝑓 𝑥 + ℎ − 𝑓(𝑥)ℎ

𝑦 = 𝑓 𝑥 ⇒ 𝑦2 = 𝑓 𝑥2

∴ 𝑓 𝑥 − 𝑓 𝑥2 = 𝑓Þ 𝑥 (𝑥 − 𝑥2)

TangentLine

𝑓 𝑥 , 𝑥 = 𝑎 𝑦0 = 𝑓Þ 𝑎 𝑥 − 𝑎 + 𝑓 𝑎

PhysicsNotation

𝑠 = 𝑠 𝑡 , Distance

𝑣 = 𝑣 𝑡 = 𝑠Þ = 𝑠Þ 𝑡 =𝑑𝑠𝑑𝑡 = 𝑠, 1/-DerivativeVelocity

𝑎 = 𝑎 𝑡 =𝑑𝑣𝑑𝑡 = 𝑣Þ 𝑡 = 𝑣Þ = 𝑣 =

𝑑q𝑠𝑑𝑡q = 𝑠ÞÞ 𝑡 = 𝑠ÞÞ = 𝑠, 2ndDerivativeAcceleration


DerivativeRules(operatornotations)DerivativeofaConstant

𝑑𝑑𝑥 𝑐 = 0

SumandDifference 𝑑𝑑𝑥 𝑓 𝑥 + 𝑔 𝑥 =

𝑑𝑑𝑥 𝑓 𝑥 +

𝑑𝑑𝑥 𝑔 𝑥

PowerRule 𝑑

𝑑𝑥 𝑥0 = 𝑛𝑥0pr

ConstantMultipleRule

𝑑𝑑𝑥 𝑐𝑓 𝑥 = 𝑐

𝑑𝑑𝑥 𝑓 𝑥

ProductRule 𝑑

𝑑𝑥 𝑓 𝑥 𝑔 𝑥 = 𝑓 𝑥𝑑𝑑𝑥 𝑔 𝑥 + 𝑔 𝑥

𝑑𝑑𝑥 𝑓 𝑥

QuotientRule 𝑑

𝑑𝑥𝑓 𝑥𝑔 𝑥 =

𝑔 𝑥 𝑑𝑑𝑥 𝑓 𝑥 − 𝑓 𝑥 𝑑

𝑑𝑥 𝑔 𝑥𝑔 𝑥 q

ChainRule

𝑑𝑑𝑥 𝑓 ∘ 𝑔 𝑥 =

𝑑𝑑𝑥 𝑓 𝑔 𝑥 =

𝑑𝑓𝑑𝑔 ⋅

𝑑𝑔𝑑𝑥 = 𝑓′ 𝑔 𝑥 ∙ 𝑔′(𝑥)

_


DerivativeRules(primenotations)

DerivativeofaConstant

𝑐 Þ = 0

PowerRule 𝑥0 ′ = 𝑛𝑥0pr


𝑐𝑢 Þ = 𝑐𝑢′

ProductRule 𝑢𝑣 Þ = 𝑢𝑣Þ + 𝑣𝑢′

QuotientRule

𝑢𝑣

Þ=𝑣𝑢Þ − 𝑢𝑣′

𝑣q

ChainRule

[𝑢 𝑣 ]′ = 𝑢Þ 𝑣 ∙ 𝑣′

ExponentialandLogarithmic

Operator Primeexp{u} 𝑑

𝑑𝑥 𝑒² � = 𝑒² � ∙ 𝑓Þ 𝑥

𝑒3 Þ = 𝑒3 ⋅ 𝑢′

NaturalLog 𝑑𝑑𝑥 ln 𝑓 𝑥 =

𝑓Þ 𝑥𝑓 𝑥

ln 𝑢 Þ =𝑢Þ

𝑢

BaseLogNote:log� 𝑎 ≡

®¯ �®¯ �

𝑑𝑑𝑥 log� 𝑓 𝑥 =

1ln 𝑏 ⋅


log� 𝑢 Þ =1ln 𝑏 ⋅

𝑢Þ

𝑢

Exponential 𝑑𝑑𝑥 𝑎

² � = 𝑎² � 𝑓Þ 𝑥 ln 𝑎

𝑎3 Þ = 𝑎3𝑢Þ ln 𝑎

InverseFunctionDerivative

𝑑𝑑𝑥 𝑓

pr 𝑥�=

1𝑓Þ 𝑓pr 𝑎

, 𝑓pr 𝑎 = 𝑏 ⇔ 𝑓 𝑏 = 𝑎


TrigDerivativesStandard

sin 𝑢 Þ = cos 𝑢 ∙ 𝑢Þ cos 𝑢 Þ = − sin 𝑢 ∙ 𝑢Þ tan 𝑢 Þ = secq 𝑢 ∙ 𝑢Þ

csc 𝑢 Þ = − csc 𝑢 cot 𝑢 ∙ 𝑢Þ sec 𝑢 Þ = sec 𝑢 tan 𝑢 ∙ 𝑢Þ cot 𝑢 Þ = − cscq 𝑢 ∙ 𝑢′

Inverse

sinpr 𝑢 Þ =𝑢′1 − 𝑢q

cospr 𝑢 Þ = −𝑢′1 − 𝑢q

tanpr 𝑢 Þ =𝑢′

1 + 𝑢q

cscpr 𝑢 Þ = −𝑢′

𝑢 𝑢q − 1 secpr 𝑢 Þ =

𝑢′𝑢 𝑢q − 1

cotpr 𝑢 Þ = −𝑢′

1 + 𝑢q

CommonDerivativesOperator𝑑𝑑𝑥 𝑦 =

𝑑𝑦𝑑𝑥

𝑑𝑑𝑥 𝑥

0 = 𝑛𝑥0pr𝑑𝑑𝑥 𝑦

0 = 𝑛𝑦0pr𝑑𝑦𝑑𝑥

𝑑𝑑𝑥 𝑒

� = 𝑒�𝑑𝑑𝑥 𝑒

² � = 𝑒² � 𝑓Þ 𝑥 𝑑𝑑𝑥 ln 𝑥 =

1𝑥

𝑑𝑑𝑥 ln 𝑓 𝑥 =


𝑑𝑑𝑥 𝑎

� = 𝑎� ln 𝑎𝑑𝑑𝑥 𝑎

² � = 𝑎² � 𝑓Þ 𝑥 ln 𝑎

𝑑𝑑𝑥 sin 𝑥 = cos 𝑥

𝑑𝑑𝑥 csc 𝑥 = −csc 𝑥 cot 𝑥

𝑑𝑑𝑥 cos 𝑥 = −sin 𝑥

𝑑𝑑𝑥 (sec 𝑥) = sec 𝑥 tan 𝑥

𝑑𝑑𝑥 tan 𝑥 = secq 𝑥

𝑑𝑑𝑥 cot 𝑥 = −cscq 𝑥

𝑑𝑑𝑥 sin

pr 𝑥 =1

1 − 𝑥q

𝑑𝑑𝑥 csc

pr 𝑥 =−1

𝑥 𝑥q − 1

𝑑𝑑𝑥 cos

pr 𝑥 =−11 − 𝑥q

𝑑𝑑𝑥 sec

pr 𝑥 =1

𝑥 𝑥q − 1

𝑑𝑑𝑥 tan

pr 𝑥 =1

1 + 𝑥q𝑑𝑑𝑥 cot

pr 𝑥 =−1

1 + 𝑥q

𝑑𝑑𝑥 sinh 𝑥 = cosh 𝑥

𝑑𝑑𝑥 csch 𝑥 = −csch 𝑥 coth 𝑥

𝑑𝑑𝑥 cosh 𝑥 = sinh 𝑥

𝑑𝑑𝑥 sech 𝑥 = −sech 𝑥 tanh 𝑥

𝑑𝑑𝑥 tanh 𝑥 = sechq 𝑥

𝑑𝑑𝑥 coth 𝑥 = −cschq 𝑥


Prime𝑒3 Þ = 𝑢Þ𝑒3

ln 𝑢 Þ =𝑢Þ

𝑢 𝑎3 Þ = 𝑢Þ𝑎3 ln 𝑎

sin 𝑢 Þ = 𝑢Þ cos 𝑢 cos 𝑢 Þ = −𝑢Þ sin 𝑢 tan 𝑢 Þ = 𝑢Þ secq 𝑢

csc 𝑢 Þ = −𝑢Þ csc 𝑢 cot 𝑢 sec 𝑢 Þ = 𝑢Þ sec 𝑢 tan 𝑢 cot 𝑢 Þ = −𝑢Þ cscq 𝑢

arcsin 𝑢 Þ =𝑢Þ

1 − 𝑢q arccos 𝑢 Þ =

−𝑢Þ

1 − 𝑢q arctan 𝑢 Þ =

𝑢Þ

1 + 𝑢q

arccsc 𝑢 Þ =−𝑢Þ

𝑢 𝑢q − 1 arcsec 𝑢 Þ =

𝑢Þ

𝑢 𝑢q − 1 arccot 𝑢 Þ =

−𝑢Þ

1 + 𝑢q

ImplicitDifferentiation𝑑𝑑 𝒙 𝒚

Alwayspayattentiontothevariables

𝑑𝑦𝑑𝑥 = 𝑦Þ

𝑑𝑑𝑥 𝑦

q 2 𝑦 qpr 𝑑𝑑𝑥 𝑦 = 2𝑦𝑦′

Chain/PowerRule 𝑑𝑑𝑥 𝑦

0 = 𝑛𝑦0pr𝑑𝑦𝑑𝑥 ≡ 𝑛𝑦0pr𝑦′

Chain/Product 𝑑𝑑𝑥 𝑥𝑦 = 𝑥

𝑑𝑦𝑑𝑥 + 𝑦

𝑑𝑥𝑑𝑥 ≡ 𝑥𝑦Þ + 𝑦

Chain/Quotient𝑑𝑑𝑥

𝑥𝑦 =

𝑦 𝑑𝑥𝑑𝑥 − 𝑥𝑑𝑦𝑑𝑥

𝑦q ≡𝑦 − 𝑥𝑦′𝑦q

Logarithmic 𝑑𝑑𝑥 ln 𝑦 =

𝑦Þ

𝑦


¦ = 𝑦Þ𝑎¦ ln 𝑎

Euler’sNumber 𝑑𝑑𝑥 𝑒

¦ = 𝑦Þ𝑒¦

Trigonometric 𝑑𝑑𝑥 sin 𝑦 = cos 𝑦 ⋅

𝑑𝑦𝑑𝑥 = cos 𝑦 ⋅ 𝑦′

TangentLine


𝑓 𝑥, 𝑦 = 0, 𝑃 𝑎, 𝑏 ⇒ 𝑦0 = 𝑓Þ 𝑎, 𝑏 𝑥 − 𝑎 + 𝑏RelatedRatesTheideaforrelatedrates,ingeneral,istofindtheequationthatrelatesgeometricallytothequestion,implicitlydifferentiateit,andthenpluginthegivenvariablesandsolvefortheunknown.Hereareafewexamplesi.e.justusetheequation/formulathatmimicstheobjectinquestion.Righttriangle 𝑎q + 𝑏q = 𝑐q ⇒ 𝑎𝑎Þ 𝑡 + 𝑏𝑏Þ 𝑡 = 𝑐𝑐Þ 𝑡

Circle 𝐴 = 𝜋𝑟q ⇒ 𝑑𝐴𝑑𝑡 = 2𝜋𝑟𝑟Þ𝑟 𝑡

Sphere 𝑉 =43𝜋𝑟

o ⇒ 𝑉Þ 𝑡 = 4𝜋𝑟q𝑑𝑟𝑑𝑡

HyperbolicFunctions

Notation


2 csch 𝑥 =2

𝑒� + 𝑒p� tanh 𝑥 =𝑒� − 𝑒p�

𝑒� + 𝑒p�

sech 𝑥 =2

𝑒� + 𝑒p� cosh 𝑥 =𝑒� + 𝑒p�

2 coth 𝑥 =𝑒� + 𝑒p�

𝑒� − 𝑒p�


Graphs


2 csch 𝑥 =2

𝑒� + 𝑒p�

cosh 𝑥 =

𝑒� + 𝑒p�

2 sech 𝑥 =2

𝑒� + 𝑒p�

tanh 𝑥 =

𝑒� − 𝑒p�


𝑒� − 𝑒p�


Identities





sinhpr 𝑥 = ln 𝑥 + 𝑥q + 1 , −∞ ≤ 𝑥 ≤∞


tanhpr 𝑥 =12 ln

1 + 𝑥1 − 𝑥 , −1 < 𝑥 < 1

DerivativesStandardsinh 𝑢 Þ = 𝑢′ cosh 𝑢 cosh 𝑢 Þ = 𝑢Þ sinh 𝑢 tanh 𝑢 Þ = 𝑢Þ sechq 𝑢

csch 𝑢 Þ = −𝑢Þ csch 𝑢 coth 𝑢 sech 𝑢 Þ = −𝑢Þ sech 𝑢 tanh 𝑢 coth 𝑢 Þ = −𝑢Þ cschq 𝑢

Inverse

sinhpr 𝑢 Þ =𝑢Þ

1 + 𝑢q coshpr 𝑢 Þ =

𝑢Þ

𝑢q − 1 tanhpr 𝑢 Þ =

𝑢Þ

1 − 𝑢q

cschpr 𝑢 Þ = −𝑢Þ

𝑢 1 + 𝑢q sechpr 𝑢 Þ = −

𝑢Þ

𝑢 1 − 𝑢q cothpr 𝑢 Þ =

𝑢Þ

1 − 𝑢q


ExtremaGraphingProcess

i) Identifythedomainofthefunction,asymptotes,andintercepts.

ii) Computethefirstderivative,setitequaltozeroandsolvefor𝑦Þ = 0, 𝑦Þ = undefined(criticalnumbers).

iii) Identifywhetherthefirstderivativeispositiveornegativetotheleftandrightofeachcriticalnumber.Ifitispositive,itisincreasing.Ifitisnegative,itisdecreasing.

iv) Computethesecondderivative,setitequaltozeroandsolve𝑦ÞÞ = 0, 𝑦Þ′ = undefined(criticalnumbers).

v) Identifywhetherthesecondderivativeispositiveornegativetotheleftandrightofeachcriticalnumber.Ifitispositive,itisconcaveup.Ifitisnegative,itisconcavedown.

vi) Verifythattheintervalsofincreasing,decreasingandconcavitylineupwiththedomainandthenidentifywhetherthecriticalnumbersaremaximums,minimumsorpointsofinflection.

vii) Usethisinformationtographthefunction.

CriticalNumbersCriticalnumbersoccurwherethederivative(s)isequaltozeroandorundefined.Max/MinAbsoluteMaximum

AbsoluteMinimum

LocalMax

LocalMin

𝑓 𝑐 ≥ 𝑓 𝑥 ∀�∈ 𝐷 𝑓 𝑐 ≤ 𝑓 𝑥 ∀�∈ 𝐷 𝑓 𝑐 ≥ 𝑓 𝑥 ∀�𝑥 → 𝑐 𝑓 𝑐 ≥ 𝑓 𝑥 ∀�𝑥 → 𝑐


Note:Absolutemax/mincanoccuratlocalsi.e.ifthelocalisthehighest/lowestpointonthegraph,itisalsoabsolute.Themax/minoccuronlyif𝑓 𝑐 isdefinedi.e.thefunctionmustbecontinuousatthecriticalnumberorendofintervals.IncreasinganddecreasingWhereverthefirstderivativeispositive,thefunctionisincreasing.Whereverthefirstderivativeisnegative,thefunctionisdecreasing.ConcavityWhereverthesecondderivativeispositive,thefunctionisconcaveup.Whereverthesecondderivativeisnegative,thefunctionisconcavedown.PointsofinflectionApointofinflectionoccurswhentotheleft/rightofthecriticalnumberhaveoppositeconcavity,and𝑓 𝑐 isdefined.TheoremsRolle’sTheoremIfthefollowingthreeconditionshold,thereisavalueintheinterval 𝑎, 𝑏 suchthat𝑓′ 𝑣𝑎𝑙𝑢𝑒 = 0

1) 𝑓iscontinuouson 𝑎, 𝑏 2) 𝑓isdifferentiableon 𝑎, 𝑏 3) 𝑓 𝑎 = 𝑓 𝑏

MeanValueTheoremIfthefollowingtwoconditionsholdtrue,thenthereisavaluein 𝑎, 𝑏 suchthat

𝑓Þ 𝑐 =𝑓 𝑏 − 𝑓 𝑎

𝑏 − 𝑎

1) 𝑓iscontinuouson 𝑎, 𝑏 2) 𝑓isdifferentiableon 𝑎, 𝑏 3) Simplyverifythefirsttwoconditions,andthesolvefor𝑐intheaboveequations,andthen

verify𝑐isin 𝑎, 𝑏

First&SecondDerivativeTest1st:Thefirsttestistoosimplytesttheleftandrightsideofthecriticalnumber(s)toseeifthefunctionisincreasing/decreasingandthenif𝑓isdefinedatthatcriticalnumberitisamax/min.


2nd:Forthesecondderivativetest,thereisamaximumoraminimumifthefollowingaretrue.Minimum Maximum TestFails𝑓Þ 𝑐 = 0 ∧ 𝑓ÞÞ 𝑐 > 0 𝑓Þ 𝑐 = 0 ∧ 𝑓ÞÞ 𝑐 < 0 𝑓Þ 𝑐 = 0 ∧ 𝑓ÞÞ 𝑐 = 0

L’Hospital’sRuleIndeterminateForms

lim�→�

𝑓 𝑥 =00

lim�→�

𝑓 𝑥 =±∞±∞

lim�→�

𝑓 𝑥 = ±∞

∞ lim

�→�𝑓 𝑥 = 0 ⋅∞

lim�→�

𝑓 𝑥 =∞−∞ lim�→�

𝑓 𝑥 = 02 lim�→�

𝑓 𝑥 = 1∞ lim�→�

𝑓 𝑥 =∞2

RuleIfthelimitisoneofthefollowingforms:

lim�→�

𝑓 𝑥 = 0 ⋅∞,∞−∞, 02, 1∞,∞2

andcanthebemanipulatedintoonofthefollowingforms:

lim�→�

𝑓 𝑥 ~ lim�→�

𝑔 𝑥ℎ 𝑥 =

00 ,±∞±∞ , ±

∞

∞

Then

lim�→�

𝑔 𝑥ℎ 𝑥 = lim

�→�

𝑔′ 𝑥ℎ′ 𝑥 = lim

�→�

𝑔′′ 𝑥ℎ′′ 𝑥 = ⋯ lim

�→�

𝑔 0 𝑥ℎ 0 𝑥

ProcessYouwillneedtoperformmanipulationstothefunctionsinordertousethisrule(ingeneral).ThemostcommonscenarioisapplyingalogarithmicrulewhenyouhaveaexponentialfunctionLet𝑢 = 𝑓 𝑥 ,and𝑣 = 𝑔 𝑥

lim�→�

𝑔 𝑥 ² � = lim�→�

𝑣3 = lim�→�

𝑒®¯ 45 = lim�→�

𝑒3 ®¯ 4 = lim�→�

𝑒®¯ 4r3 = 𝑒

®óÿ!→"

®¯ 4r3


= 𝑒22 = 𝑒

∞∞ =ℋ 𝑒

®óÿ!→"

»»� ®¯ 4»»�r3

Nowyoumaygettheexponentintotheappropriateindeterminateform,andtaketheratioofderivatives.Note:Thepreviousproblems,isacommonprobleminschool.Therearemanydifferentmanipulationsfordifferentfunctions.Tolistthemallwouldbeimpossible,anditwouldtakeawayfromthepurposeoflearningproblemsolvingskills.Youwillneedtousetheentirealgebraic,limit,andderivativerulestogetherinordertosuccessfullysolvetheproblems.

Optimization(Ingeneral)Optimizationistosimplyfindtwofunctionsthatfitageometricshapei.e.onethatrepresentsthegeometry,andtheothertofitthenumbergivenintheprobleme.g.areaofrectanglewithperimeterP.Rearrangetoplugonefunctionintheotherandthenusederivativeteststofindthemax/min(s).GeneralIdea:Maxareofrectanglewithperimeter𝑃

𝐴 𝑥, 𝑦 = 𝑥𝑦, 2𝑥 + 2𝑦 = 𝑃

∴ 𝐴 𝑥 = 𝑥𝑃 − 2𝑥2

Onceyoufindthefunctionthatyouaretryingtooptimize,inthiscase“maxarea”usethederivativeteststofindthevaluesinquestion.

BusinessFormulasCostFunction 𝐶 𝑥

MarginalCostFunction 𝐶Þ 𝑥 =𝑑𝑑𝑥 𝐶 𝑥

Demand/Pricefunction(priceperunit) 𝑝 𝑥

Revenue 𝑅 𝑥 = 𝑝 𝑥 ⋅ 𝑥

MarginalRevenue 𝑅Þ 𝑥 =𝑑𝑑𝑥 𝑅 𝑥 = 𝑝 𝑥 + 𝑥𝑝Þ 𝑥

ProfitFunction 𝑃 𝑥 = 𝑅 𝑥 − 𝐶 𝑥

MarginalProfitFunction 𝑃Þ 𝑥 = 𝑅Þ 𝑥 − 𝐶Þ 𝑥

AverageProfitFunction𝑃 𝑥 =

𝑃 𝑥𝑥


Antiderivatives&IntegrationBasicRulesPowerRuleforantiderivatives 𝑦Þ = 𝑥0 ⇒ 𝑦 =

1𝑛 + 1𝑥

0 + 𝐶 ⇔ 𝑛 ≠ −1

Exponential𝑦Þ = 𝑎� ⇒ 𝑦 =

𝑎�

ln 𝑎 + 𝐶

NaturalLog(case1) 𝑦Þ =1𝑥 ⇒ 𝑦 = ln 𝑥 + 𝐶

NaturalLog(case2) 𝑦Þ =1

𝑎𝑥 + 𝑏 ⇒ 𝑦 =1𝑎 ln 𝑎𝑥 + 𝑏 + 𝐶

NaturalLog(case3)𝑦Þ =

𝑢Þ 𝑥𝑢 𝑥 ⇒ 𝑦 = ln 𝑢 𝑥 + 𝐶

Euler’sNumber(case1) 𝑦Þ = 𝑒�� ⇒ 𝑦 =1𝑎 𝑒

�� + 𝐶


�� + 𝐶

Euler’sNumber(case3) 𝑦Þ = 𝑢Þ 𝑥 𝑒3 � ⇒ 𝑦 = 𝑒3 � + 𝐶

Anti-Chain-RuleSubstitutionMethod 𝑦Þ = 𝑓Þ 𝑔 𝑥 𝑔Þ 𝑥 ⇒ 𝑦 = 𝑓 𝑔 𝑥 + 𝐶

RiemannSumforAreaApproximation

𝐴 ≈ lim0→∞

𝑓 𝑥8∗0

81r

𝛥𝑥 , 𝛥𝑥 =𝑏 − 𝑎𝑛 , 𝑥8 = 𝑎 + 𝑖 ∙ 𝛥𝑥

𝑐0

81r

= 𝑐𝑛 𝑖0

81r

=𝑛 𝑛 + 1

2

𝑐𝑓 𝑥8

0

81r

= 𝑐 𝑓 𝑥8

0

81r

𝑖q0

81r

=𝑛 𝑛 + 1 2𝑛 + 1

6

𝑓 𝑥8 ± 𝑔 𝑥8

0

81r

= 𝑓 𝑥8

0

81r

± 𝑔 𝑥8

0

81r

𝑖o0

81r

=𝑛 𝑛 + 1

2

q


AreaApproximationRulesMidpointRule

𝑓 𝑥�

�𝑑𝑥 ≈

𝑏 − 𝑎𝑛 𝑓

𝑥r + 𝑥q2 + 𝑓

𝑥q + 𝑥o2 +⋯

TrapezoidRule

𝑓 𝑥�

�𝑑𝑥 ≈

𝑏 − 𝑎2𝑛 𝑓 𝑥r + 2𝑓 𝑥q + 2𝑓 𝑥o + ⋯+ 2𝑓 𝑥0pr + 𝑓 𝑥0

SimpsonRule

𝑓 𝑥�

�𝑑𝑥 ≈

𝑏 − 𝑎3𝑛 𝑓 𝑥r + 4𝑓 𝑥q + 2𝑓 𝑥o + 4𝑓 𝑥Ü + ⋯+ 2𝑓 𝑥0pq + 4𝑓 𝑥0pr + 𝑓 𝑥0

TheIntegralNotation∫

lim0→∞

𝑓(𝑥8∗)0

81r

𝛥𝑥 ≡ 𝑓(𝑥)�

�𝑑𝑥

DefiniteIntegralProperties

𝑓 𝑥�

�𝑑𝑥 = 𝐹 𝑏 − 𝐹 𝑎 𝑐

�

�𝑑𝑥 = 𝑐 𝑏 − 𝑎

𝑓 𝑥�

�𝑑𝑥 = 0 𝑐𝑓 𝑥

�

�𝑑𝑥 = 𝑐 𝑓 𝑥

�

�𝑑𝑥

𝑓 𝑥�

p�𝑑𝑥 = 0

⇔ 𝑓 −𝑥 = −𝑓 𝑥 odd

𝑓 𝑥 ± 𝑔 𝑥�

�𝑑𝑥 = 𝑓 𝑥

�

�𝑑𝑥 ± 𝑔 𝑥

�

�𝑑𝑥

𝑓 𝑥�

p�𝑑𝑥 = 2 𝑓 𝑥

�

2

⇔ 𝑓 −𝑥 = 𝑓 𝑥 even

𝑓 𝑥�


𝒌

�𝑑𝑥 + 𝑓 𝑥

�

𝒌𝑑𝑥


NOTE:

𝑓 𝑥 ⋅ 𝑔 𝑥 𝑑𝑥 ≠ 𝑔 𝑥 𝑑𝑥 ⋅ 𝑓 𝑥 𝑑𝑥

𝑓 𝑥�

�𝑑𝑥 = − 𝑓 𝑥

�

�𝑑𝑥

FundamentalTheorems

Let𝑓 𝑥 = 𝑢and𝑔 𝑥 = 𝑣forthefollowing:

𝑖)𝑦 = 𝑓 𝑡 𝑑𝑡4

3⇒ 𝑦Þ = 𝑓 𝑣 ∙ 𝑣Þ − 𝑓 𝑢 ∙ 𝑢′

𝑦 = 𝑓 𝑡 𝑑𝑡4

�⇒ 𝑦Þ = 𝑓 𝑣 ∙ 𝑣Þ − 𝑓 𝑎 ∙ 𝑎Þ = 𝑓 𝑣 ∙ 𝑣Þ − 0 = 𝑓 𝑣 ∙ 𝑣Þ

𝑦 = 𝑓 𝑡 𝑑𝑡�

3⇒ 𝑦Þ = 𝑓 𝑏 ∙ 𝑏Þ − 𝑓 𝑢 ∙ 𝑢Þ = 0 − 𝑓 𝑢 ∙ 𝑢Þ = −𝑓 𝑢 ∙ 𝑢′

LimitDefinitionofaDefiniteIntegral

𝑖𝑖) lim0→∞

𝑓(𝑥8∗)0

81r

𝛥𝑥 = 𝑓(𝑥)�

�𝑑𝑥 = 𝐹 𝑏 − 𝐹 𝑎

𝛥𝑥 =𝑏 − 𝑎𝑛 , 𝑥8 = 𝑎 + 𝑖 ∙ 𝛥𝑥

DifferentialEquation(1storder)

𝑦Þ = 𝑓Þ 𝑥 ⇒ 𝑑𝑦𝑑𝑥 = 𝑓Þ 𝑥 ⇒ 𝑑𝑦 = 𝑓Þ 𝑥 𝑑𝑥 ⇒ 𝑑𝑦 = 𝑓Þ 𝑥 𝑑𝑥

⇒ 𝑦 + 𝑐r = 𝑓 𝑥 + 𝑐q ⇒ 𝑦 = 𝑓 𝑥 + 𝑐q − 𝑐r = 𝑓 𝑥 + 𝑐o ≡ 𝑓 𝑥 + 𝐶


CommonIntegrals

𝑑𝑥 = 𝑥 + 𝐶 𝑘 𝑑𝑥 = 𝑘𝑥 + 𝐶 𝑥 𝑑𝑥 =12 𝑥

q + 𝐶

𝑥q 𝑑𝑥 =13 𝑥

o + 𝐶 𝑥0 𝑑𝑥 =1

𝑛 + 1𝑥0�r + 𝐶

⇔ 𝑛 ≠ −1

1𝑥 𝑑𝑥 = ln |𝑥| + 𝐶

𝑒� 𝑑𝑥 = 𝑒� + 𝐶 𝑒�� 𝑑𝑥 =1𝑎 𝑒

�� + 𝐶 𝑒�� 𝑑𝑥 =1𝑎 𝑒

�� + 𝐶

1𝑥 + 1𝑑𝑥 = ln 𝑥 + 1 + 𝐶

1𝑎𝑥 + 𝑏 𝑑𝑥 =

1𝑎 ln 𝑎𝑥 + 𝑏 + 𝐶 𝑓 𝑢 𝑢′𝑑𝑢 = 𝐹 𝑢 + 𝐶

𝑒3𝑢′𝑑𝑢 = 𝑒3 + 𝐶 𝑢Þ

𝑢 𝑑𝑢 = ln 𝑢 + 𝐶 𝑓 𝑥�

�= 𝐹 𝑏 − 𝐹 𝑎

𝑢Þ cos 𝑢 𝑑𝑢 = sin 𝑢 + 𝐶 𝑢Þ sin 𝑢 𝑑𝑢 = −cos 𝑢 + 𝐶 𝑢Þ secq 𝑢 𝑑𝑢 = tan 𝑢 + 𝐶

𝑢Þ csc 𝑢 sec 𝑢 𝑑𝑢 = −csc 𝑢 + 𝐶 𝑢Þ sec 𝑢 tan 𝑢 𝑑𝑢 = sec 𝑢 + 𝐶 𝑢Þ cscq 𝑢 𝑑𝑢 = −cot 𝑢 + 𝐶

𝑢Þ

1 − 𝑢q𝑑𝑢 = arcsin 𝑢 + 𝐶

−𝑢Þ

1 − 𝑢q𝑑𝑢 = arccos 𝑢 + 𝐶

𝑢Þ

1 + 𝑢q 𝑑𝑢 = arctan 𝑢 + 𝐶

DefiniteIntegralRules


Substitution𝑓 𝑔 𝑥 𝑔Þ 𝑥�

�𝑑𝑥 = 𝑓 𝑢

È �

È �𝑑𝑢

IntegrationbyParts

𝑓 𝑥 𝑔Þ 𝑥�

�𝑑𝑥 = 𝑓 𝑥 𝑔 𝑥 �

� − 𝑔 𝑥 𝑓Þ 𝑥�

�𝑑𝑥

Let𝑢 = 𝑓 𝑥 𝑑𝑣 = 𝑔Þ 𝑥 𝑑𝑥𝑑𝑢 = 𝑓Þ 𝑥 𝑑𝑥 𝑣 = 𝑔 𝑥 Then

𝑢�

�𝑑𝑣 = 𝑢𝑣 �

� − 𝑣�

�𝑑𝑢

INTEGRALCALCULUS(CALCLII)

ParametricandPolarOperationsNotations

𝑥 = 𝑥 𝑡 , 𝑡 ∈ 𝑎, 𝑏 𝑦 = 𝑦 𝑡 , 𝑡 ∈ 𝑎, 𝑏

𝑥Þ 𝑡 =𝑑𝑥𝑑𝑡 ≡ 𝑥 𝑦Þ 𝑡 =

𝑑𝑦𝑑𝑡 ≡ 𝑦

FirstDerivative

𝑦Þ 𝑡𝑥Þ 𝑡 =

𝑑𝑦𝑑𝑡𝑑𝑥𝑑𝑡

=𝑑𝑦𝑑𝑡 ⋅

𝑑𝑡𝑑𝑥 =

𝑑𝑦𝑑𝑥

SecondDerivative

𝑑q𝑦𝑑𝑥q =

𝑑𝑑𝑥𝑑𝑦𝑑𝑥 =

𝑑𝑑𝑥𝑦Þ 𝑡𝑥Þ 𝑡 =

𝑥Þ 𝑡 𝑑𝑑𝑥 𝑦

Þ 𝑡 − 𝑦Þ 𝑡 𝑑𝑑𝑥 𝑥

Þ 𝑡𝑥Þ 𝑡 q =

𝑥Þ 𝑡 𝑑𝑑𝑥𝑑𝑦𝑑𝑡 – 𝑦

Þ 𝑡 𝑑𝑑𝑥𝑑𝑥𝑑𝑡

𝑥Þ 𝑡 q

=𝑥Þ 𝑡 𝑑

𝑑𝑡𝑑𝑦𝑑𝑥 − 𝑦

Þ 𝑡 𝑑𝑑𝑡𝑑𝑥𝑑𝑥

𝑥Þ 𝑡 q =𝑥Þ 𝑡 𝑑

𝑑𝑡𝑑𝑦𝑑𝑥 − 𝑦

Þ 𝑡 𝑑𝑑𝑡 1

𝑥Þ 𝑡 q =𝑥Þ 𝑡 𝑑

𝑑𝑡𝑑𝑦𝑑𝑥

𝑥Þ 𝑡 q =𝑑𝑑𝑡𝑑𝑦𝑑𝑥

𝑥Þ 𝑡


∴𝑑q𝑦𝑑𝑥q =

𝑑𝑑𝑡𝑑𝑦𝑑𝑥𝑑𝑥𝑑𝑡

Trigonometric𝑥q + 𝑦q = 𝑟q 𝑥 = 𝑟 cos 𝜃 𝑦 = 𝑟 sin 𝜃 𝜃 = arctan

𝑦𝑥

Circle𝑥 − ℎ𝑟

q

+𝑦 − 𝑘𝑟

q

= 1 = cos 𝜃 q + sin 𝜃 q ⇒ 𝑥 − ℎ𝑟 = cos 𝜃 ∧

𝑦 − 𝑘𝑟 = sin 𝜃 , 𝜃 ∈ 0, 2𝜋

Ellipse𝑥 − ℎ𝑎

q

+𝑦 − 𝑘𝑏

q

= 1 = cos 𝜃 q + sin 𝜃 q ⇒ 𝑥 − ℎ𝑎 = cos 𝜃 ∧

𝑦 − 𝑘𝑏 = sin 𝜃 , 𝜃 ∈ 0, 2𝜋

PolarDerivative

𝑑𝑦𝑑𝑥 =

𝑑𝑦𝑑𝜃𝑑𝑟𝑑𝜃

=𝑑𝑑𝜃 𝑟 sin 𝜃𝑑𝑑𝜃 𝑟 cos 𝜃

=𝑟 𝜃 cos 𝜃 + 𝑟Þ 𝜃 sin 𝜃𝑟Þ 𝜃 cos 𝜃 − 𝑟 𝜃 sin 𝜃

PolarEquationsforEllipse𝑥q

𝑎q +𝑦q

𝑏q = 10 ≤ 𝑎 < 𝑏 𝑐q = 𝑎q − 𝑏q

Foci ±𝑐, 0 Vertices ±𝑎, 0

𝑥q

𝑏q +𝑦q

𝑎q = 10 ≤ 𝑎 < 𝑏 𝑐q = 𝑎q − 𝑏q

Foci 0,±𝑐, Vertices 0,±𝑎,

𝑒 < 1 𝑒 =eccentricity,𝑑 =diretrix

𝑟 𝜃 =𝑒𝑑

𝑎 ± 𝑒 cos 𝜃 𝑟 𝜃 =𝑒𝑑

𝑎 ± 𝑒 sin 𝜃𝑐q = 𝑎q − 𝑏q𝑒 =

𝑐𝑎

PolarEquationsforHyperbola


𝑥q

𝑎q −𝑦q

𝑏q = 1𝑐q = 𝑎q + 𝑏qFoci ±𝑐, 0 Vertices ±𝑎, 0 Asymptotes𝑦 = ± �

�𝑥

𝑦q

𝑎q −𝑥q

𝑏q = 1𝑐q = 𝑎q + 𝑏qFoci 0,±𝑐, Vertices 0,±𝑎, Asymptotes𝑦 = ±�

�𝑥

𝑒 > 11 𝑒 =eccentricity,𝑑 =diretrix

𝑟 𝜃 =𝑒𝑑

𝑎 ± 𝑒 cos 𝜃 𝑟 𝜃 =𝑒𝑑

𝑎 ± 𝑒 sin 𝜃𝑐q = 𝑎q + 𝑏q𝑒 =

𝑐𝑎

PolarEquationsforParabola𝑒 = 1 𝑒 =eccentricity,𝑑 =diretrix 𝑦q = 4𝑝𝑥, 𝑑 = −𝑝

𝑥q = 4𝑝𝑦, 𝑑 = −𝑝

𝑟 𝜃 =𝑑

𝑎 ± cos 𝜃 𝑟 𝜃 =𝑑

𝑎 ± sin 𝜃


1𝑛 + 1𝑥

0 + 𝐶 ⇔ 𝑛 ≠ −1


𝑎�

ln 𝑎 + 𝐶



𝑎𝑥 + 𝑏 ⇒ 𝑦 =1𝑎 ln 𝑎𝑥 + 𝑏 + 𝐶




�� + 𝐶


�� + 𝐶





𝐴 ≈ lim0→∞

𝑓 𝑥8∗0

81r

𝛥𝑥 , 𝛥𝑥 =𝑏 − 𝑎𝑛 , 𝑥8 = 𝑎 + 𝑖 ∙ 𝛥𝑥

𝑐0

81r

= 𝑐𝑛 𝑖0

81r

=𝑛 𝑛 + 1

2

𝑐𝑓 𝑥8

0

81r

= 𝑐 𝑓 𝑥8

0

81r

𝑖q0

81r

=𝑛 𝑛 + 1 2𝑛 + 1

6

𝑓 𝑥8 ± 𝑔 𝑥8

0

81r

= 𝑓 𝑥8

0

81r

± 𝑔 𝑥8

0

81r

𝑖o0

81r

=𝑛 𝑛 + 1

2

q


𝑓 𝑥�

�𝑑𝑥 ≈



𝑥q + 𝑥o2 +⋯

TrapezoidRule

𝑓 𝑥�

�𝑑𝑥 ≈



lim0→∞

𝑓(𝑥8∗)0

81r


�𝑑𝑥


𝑓 𝑥�


�

�𝑑𝑥 = 𝑐 𝑏 − 𝑎


𝑓 𝑥�


�


�

�𝑑𝑥

𝑓 𝑥�

p�𝑑𝑥 = 0




�


�

�𝑑𝑥

𝑓 𝑥�


�

2


𝑓 𝑥�


𝒌


�

𝒌𝑑𝑥

NOTE:


𝑓 𝑥�

�𝑑𝑥 = − 𝑓 𝑥

�

�𝑑𝑥

FundamentalTheorems



3⇒ 𝑦Þ = 𝑓 𝑣 ∙ 𝑣Þ − 𝑓 𝑢 ∙ 𝑢′







𝑓(𝑥8∗)0

81r



𝛥𝑥 =𝑏 − 𝑎𝑛 , 𝑥8 = 𝑎 + 𝑖 ∙ 𝛥𝑥





CommonIntegrals


q + 𝐶


o + 𝐶 𝑥0 𝑑𝑥 =1

𝑛 + 1𝑥0�r + 𝐶

⇔ 𝑛 ≠ −1

1𝑥 𝑑𝑥 = ln |𝑥| + 𝐶

𝑒� 𝑑𝑥 = 𝑒� + 𝐶 𝑒�� 𝑑𝑥 =1𝑎 𝑒

�� + 𝐶 𝑒�� 𝑑𝑥 =1𝑎 𝑒

�� + 𝐶

1𝑥 + 1𝑑𝑥 = ln 𝑥 + 1 + 𝐶





�= 𝐹 𝑏 − 𝐹 𝑎



𝑢Þ


−𝑢Þ


𝑢Þ



DefiniteIntegralRulesSubstitution

𝑓 𝑔 𝑥 𝑔Þ 𝑥�


È �

È �𝑑𝑢

IntegrationbyParts



� − 𝑔 𝑥 𝑓Þ 𝑥�

�𝑑𝑥


𝑢�


� − 𝑣�

�𝑑𝑢

TrigSubstitution𝑎q − 𝑥q 𝑎q + 𝑥q 𝑥q − 𝑎q

1 − sinq 𝜃 = cosq 𝜃 1 + tanq 𝜃 = secq 𝜃 secq 𝜃 − 1 = tanq 𝜃

𝑥 = 𝑎 sin 𝜃 𝑥 = 𝑎 tan 𝜃 𝑥 = 𝑎 sec 𝜃

𝜃 ∈ −𝜋2 ,𝜋2 𝜃 ∈ −

𝜋2 ,𝜋2 𝜃 ∈ 0,

𝜋2 ∨ 𝜃 ∈ 𝜋,

3𝜋2

TrigIdentity

tan 𝑥 𝑑𝑥 =sin 𝑥cos 𝑥 𝑑𝑥 = −

1cos 𝑥 ⋅ − sin 𝑥 𝑑𝑥,

𝑑𝑑𝑥 ln 𝑢 𝑥 =

1𝑢𝑑𝑢𝑑𝑥

= − ln cos 𝑥 + 𝐶 = ln1

cos 𝑥 + 𝐶 = ln sec 𝑥 + 𝐶

PartialFractions𝑝 𝑥

𝑥 𝑥 + 1 =𝐴𝑥 +

𝐵𝑥 + 1

𝑝 𝑥𝑥q 𝑥 + 1 =

𝐴𝑥 +

𝐵𝑥q +

𝐶𝑥 + 1


𝑝 𝑥𝑥 𝑥q + 1 =

𝐴𝑥 +

𝐵𝑥 + 𝐶𝑥q + 1

𝑝 𝑥𝑥 𝑥q + 1 q =

𝐴𝑥 +

𝐵𝑥 + 𝐶𝑥q + 1 +

𝐷𝑥 + 𝐸𝑥q + 1 q

IntegrationSteps

Askyourselfthefollowingquestions:

1. Istheintegrandinintegratableform?2. CanIperformafunctionortrig-identitymanipulation?3. ShouldIuseU-SubstitutionorTrig-Substitution?4. IntegrationbyParts?5. Partialfractiondecomposition?

Foradefiniteintegralalwayschecktoseeifthefunctionisdefinedonthebounds

ImproperIntegrationInfiniteBounds

𝑓 𝑥�∞

p∞𝑑𝑥 = 𝑓 𝑥

2

p∞𝑑𝑥 + 𝑓 𝑥

�∞

2𝑑𝑥 = lim

©:→p∞𝑓 𝑥2

©:𝑑𝑥 + lim

©�→∞𝑓 𝑥

©�

2𝑑𝑥

UndefinedBounds

𝑓 𝑥�

�𝑑𝑥, 𝑥 ∈ 𝑎, 𝑏 ⇒ lim

©:→ ! ü𝑓 𝑥2

©:𝑑𝑥 + lim

©�→�ý𝑓 𝑥

©�

2𝑑𝑥


Areas,Volumes,andCurveLengthAreawithrespecttoanaxisCartesian𝑥 − 𝑎𝑥𝑖𝑠 𝑦 − 𝑎𝑥𝑖𝑠

𝐴 = 𝑓 𝑥�

�𝑑𝑥 ⇔ 𝑓 𝑥 ≥ 0∀�∈ 𝑎, 𝑏 𝐴 = 𝑔 𝑦

»

�𝑑𝑦 ⇔ 𝑔 𝑦 ≥ 0∀¦∈ 𝑐, 𝑑

AreabetweencurvesGiventwocurves𝑓 ∧ 𝑔setthemequaltoeachothertofindallx-coordinatesofintersection.

𝐴 = 𝑓 𝑥 − 𝑔 𝑥�

�𝑑𝑥 ⇔ 𝑓 𝑥 ≥ 𝑔 𝑥 ∀�∈ 𝑎, 𝑏

or

𝐴 = 𝑓 𝑥 − 𝑔 𝑥�;ý:

�;𝑑𝑥 = 𝑓 𝑥 − 𝑔 𝑥

��

�:𝑑𝑥 + 𝑓 𝑥 − 𝑔 𝑥

�Ó

��𝑑𝑥 +⋯

PolarArea

𝐴 =12 𝑟 𝜃 q

<�

<:𝑑𝜃 ∧ 𝐴 =

12 𝑅 𝜃 q − 𝑟 𝜃 q

<�

<:𝑑𝜃

Volumeaboutanaxis(DiskMethod)𝑥 − 𝑎𝑥𝑖𝑠 𝑦 − 𝑎𝑥𝑖𝑠

𝑉 = 𝜋 𝑓 𝑥 q�

�𝑑𝑥 ⇔ 𝑓 𝑥 ≥ 0∀�∈ 𝑎, 𝑏 𝑉 = 𝜋 𝑔 𝑦 q

»

�𝑑𝑦 ⇔ 𝑔 𝑦 ≥ 0∀¦∈ 𝑐, 𝑑

Volumebetweencurves(WasherMethod)Giventwocurves𝑓 ∧ 𝑔setthemequaltoeachothertofindallx-coordinatesofintersection.

𝑉 = 𝜋 𝑓 𝑥 q − 𝑔 𝑥 q�

�𝑑𝑥 ⇔ 𝑓 𝑥 ≥ 𝑔 𝑥 ∀�∈ 𝑎, 𝑏


CylindricalShellMethodRotateabout𝑦 − 𝑎𝑥𝑖𝑠 Rotateabout𝑥 − 𝑎𝑥𝑖𝑠

𝑉 = 2𝜋𝑥𝑓 𝑥�

�𝑑𝑥 𝑉 = 2𝜋𝑦𝑔 𝑦

»

�𝑑𝑦

ArcLengthCartesian Polar Parametric

𝐿 = 1 − 𝑓Þ 𝑥 q�

�𝑑𝑥

𝐿 = 𝑟 𝜃 q − 𝑟Þ 𝜃 q<:

<:𝑑𝜃

𝐿 = 𝑥Þ 𝑡 q − 𝑦Þ 𝑡 q©�

©:𝑑𝑡

SurfaceAreaCartesian Polar Parametric

𝑆�p��8Û = 2𝜋𝑓 𝑥�

�𝑑𝑙,

𝑑𝑙 = 1 − 𝑓Þ 𝑥 q𝑑𝑥

𝑆¦p��8Û = 2𝜋𝑔 𝑦�

�𝑑𝑙,

𝑑𝑙 = 1 − 𝑔Þ 𝑦 q𝑑𝑦

𝑆�p��8Û = 2𝜋𝑟 𝜃 cos 𝜃<:

<:𝑑𝑙

𝑑𝑙 = 𝑟 𝜃 q − 𝑟Þ 𝜃 q𝑑𝜃

𝑆¦p��8Û = 2𝜋𝑟 𝜃 sin 𝜃<:

<:𝑑𝑙

𝑑𝑙 = 𝑟 𝜃 q − 𝑟Þ 𝜃 q𝑑𝜃

𝑆�p��8Û = 2𝜋𝑦 𝑡©�

©:𝑑𝑙,

𝑑𝑙 = 𝑥Þ 𝑡 q − 𝑦Þ 𝑡 q𝑑𝜃

𝑆¦p��8Û = 2𝜋𝑥 𝑡©�

©:𝑑𝑙,

𝑑𝑙 = 𝑥Þ 𝑡 q − 𝑦Þ 𝑡 q𝑑𝜃

PhysicsApplicationsCenterofMasswithConstantDensity𝒙-coordinate 𝒚-coordinate


𝑥 =𝑀¦

𝑚 𝑦 =𝑀�

𝑚

𝑀¦ = 𝜌 𝑥𝑓 𝑥�

�𝑑𝑥 𝑀� =

𝜌2 𝑓 𝑥 q

�

�𝑑𝑥

𝑚 = 𝜌𝐴 = 𝜌 𝑓 𝑥�

�𝑑𝑥, 𝑓 𝑥 ≥ 0 ∈ 𝑎, 𝑏 𝑚 = 𝜌𝐴 = 𝜌 𝑓 𝑥

�

�𝑑𝑥, 𝑓 𝑥 ≥ 0 ∈ 𝑎, 𝑏

∴ 𝑥 =1𝐴 𝑥𝑓 𝑥

�

�𝑑𝑥 ∴ 𝑦 =

12𝐴 𝑓 𝑥 q

�

�𝑑𝑥

𝑥 =1𝐴 𝑥 𝑓 𝑥 − 𝑔 𝑥

�

�𝑑𝑥, 𝑓 ≥ 𝑔 ∈ 𝑎, 𝑏 𝑀� =

12𝐴 𝑓 𝑥 q − 𝑔 𝑥 q

�

�𝑑𝑥, 𝑓 ≥ 𝑔 ∈ 𝑎, 𝑏

SequencesvsSeriesSequence Series

𝑎0 = 𝑎2, 𝑎r, 𝑎q, …

𝑎0

∞

012

= 𝑎2 + 𝑎r + 𝑎q + ⋯

SequenceTests𝑎0Converges 𝑎0Diverges

lim0→∞

𝑎0 = 𝐿 lim0→∞

𝑎0 = ±∞ ∨ 𝐷𝑁𝐸

SeriesTestsTest Form Condition Diverges ConvergesGeometric

𝑎𝑟0pr∞

01r

𝒓 ≥ 𝟏 𝒓 < 𝟏𝑺 =

𝒂𝟏 − 𝒓

P-Series 1𝑛Ù

∞

01r

𝒑 ≤ 𝟏 𝒑 > 𝟏

IntegralTest𝑎0

∞

01r

𝒂𝒏ispositiveanddecreasingon[𝟏,∞) 𝒇 𝒙 𝒅𝒙 =∞

∞

𝟏

𝒆. 𝒈.𝒂𝒏 =𝟏𝒏𝟐

⇒ 𝒇 𝒙 =𝟏𝒙𝟐

𝒇 𝒙 𝒅𝒙 = 𝒌∞

𝟏

Comparison𝑎0

∞

01r

𝒂𝒏,𝒃𝒏𝐚𝐫𝐞𝐩𝐨𝐬𝐢𝐭𝐢𝐯𝐞 𝒂𝒏 ≥ 𝒃𝒏∀𝒏

⇔ 𝒃𝒏𝐝𝐢𝐯𝐞𝐫𝐠𝐞𝐬

𝒂𝒏 ≤ 𝒃𝒏∀𝒏⇔ 𝒃𝒏𝐜𝐨𝐧𝐯𝐞𝐫𝐠𝐞𝐬


LimitComparison 𝑎0

∞

01r

𝒂𝒏,𝒃𝒏𝐚𝐫𝐞𝐩𝐨𝐬𝐢𝐭𝐢𝐯𝐞𝐥𝐢𝐦𝒏→∞

𝒂𝒏𝒃𝒏

= 𝒌, 𝒌 > 𝟎

𝜮𝒃𝒏𝐃𝐢𝐯𝐞𝐫𝐠𝐞𝐬 𝜮𝒃𝒏𝐂𝐨𝐧𝐯𝐞𝐫𝐠𝐞𝐬

AlternatingSeries −1 0pr𝑐0

∞

01r

𝒄𝒏 > 𝟎

Doesnotshowdivergence

𝒄𝒏�𝟏 ≤ 𝒄𝒏∀𝒏&𝐥𝐢𝐦𝒏→∞

𝒄𝒏 = 𝟎Ratio

𝑎0

∞

01r

𝐥𝐢𝐦

𝒏→∞

𝒂𝒏�𝟏𝒂𝒏

> 𝟏

𝐨𝐫 =∞

𝐥𝐢𝐦𝒏→∞

𝒂𝒏�𝟏𝒂𝒏

< 𝟏

Root𝑎0

∞

01r

𝐥𝐢𝐦

𝒏→∞𝒂𝒏

𝒏 > 𝟏𝐨𝐫 =∞

𝐥𝐢𝐦𝒏→∞

|𝒂𝒏|𝒏 < 𝟏

TestforAbsolute/ConditionalConvergence

AbsolutelyConvergent

ConditionallyConvergent

If 𝑎0

∞

01r

𝐂𝐨𝐧𝐯𝐞𝐫𝐠𝐞𝐬𝐭𝐡𝐞𝐧 |𝑎0

∞

01r

|Converges |𝑎0

∞

01r

|Diverges

Ifallelsefails,perform“ThenthTermforDivergenceTest”i.e.if lim

0→∞𝑎0 ≠ 0or𝐷𝑁𝐸thenthesum

diverges—doesnotshowconvergence.Taylorseries

𝑓 𝑥 ≈𝑓 0 𝑎𝑛! 𝑥 − 𝑎 0 = 𝑓 𝑥 + 𝑓Þ 𝑥 𝑥 − 𝑎 +

𝑓ÞÞ 𝑥2! 𝑥 − 𝑎 q +

𝑓ÞÞÞ 𝑥3! 𝑥 − 1 o + ⋯

MaclaurinSeries

𝑓 𝑥 ≈𝑓 0 0𝑛! 𝑥 0 = 𝑓 0 + 𝑓Þ 0 𝑥 +

𝑓ÞÞ 02! 𝑥q +

𝑓ÞÞÞ 03! 𝑥o + ⋯

PowerSeries

11 − 𝑢 = 𝑢 0

∞

012

Radius/IntervalofConvergesTheROCandintervalofconvergenceforafunctionisfoundbyputting𝑓intoit’spowerseriesrepresentation,andthenapplying,ingeneral,either“geometricseriestest”,“ratiotest”,andor


“roottest”.Note:Theratio/roottestrequireyoutoplugtheintervalendsbackintotheseries,andusewhatevertestisnecessarytofindiftheseriesisdivergent/convergentatthatend-point.From 𝑥 − 𝑎 < 𝑅𝐼 = 𝑎 − 𝑅, 𝑎 + 𝑅 𝐼 = 𝑎 − 𝑅, 𝑎 + 𝑅 𝐼 = 𝑎 − 𝑅, 𝑎 + 𝑅 𝐼 = (𝑎 − 𝑅, 𝑎 + 𝑅)

3DCalculusMagnitude

𝑣 = 𝐯 = 𝑣r, 𝑣q, 𝑣o ⇒ 𝑣 = 𝐯 = 𝑣rq + 𝑣qq + 𝑣oq

UnitVectors

𝑣 =𝑣𝑣 ≡ 𝐯 =

𝐯𝐯

𝚤 ≡ 𝐢 𝚥 ≡ 𝐣 𝑘 ≡ 𝐤𝚤 = 1, 0, 0 𝚥 = 0, 1, 0 𝑘 = 1, 0, 0 Note:𝑣 = 𝑣r, 𝑣q, 𝑣o = 𝑣r 1, 0, 0 + 𝑣q 0, 1, 0 + 𝑣o 0, 0, 1 = 𝑣r𝚤 + 𝑣q𝚥 + 𝑣o𝑘 = 𝑣r𝐢 + 𝑣q𝐣+ 𝑣o𝐤


Dot/CrossProductDot𝑎 ⋅ 𝑏 = 𝐚 ⋅ 𝐛

= 𝑎r, 𝑎q, 𝑎o ⋅ 𝑏r, 𝑏q, 𝑏o

= 𝑎r𝑏r + 𝑎q𝑏q + 𝑎o𝑏o

Properties𝐚 ⋅ 𝐚 = 𝐚 q 𝐚 ⋅ 𝐛 = 𝐛 ⋅ 𝐚

𝐚 ⋅ 𝐛+ 𝐜 = 𝐚 ⋅ 𝐛+ 𝐚 ⋅ 𝐜 k𝐚 ⋅ 𝐛 = k 𝐚 ⋅ 𝐛 = 𝐚 ⋅ k𝐛

Cross𝑎×𝑏 = 𝐚×𝐛 = 𝑎r, 𝑎q, 𝑎o × 𝑏r, 𝑏q, 𝑏o

=𝚤 𝚥 𝑘𝑎r 𝑎q 𝑎o𝑏r 𝑏q 𝑏o

=

𝑎q 𝑎o𝑏q 𝑏o 𝚤 −

𝑎r 𝑎o𝑏r 𝑏o 𝚥 +

𝑎r 𝑎q𝑏r 𝑏q 𝑘

= 𝑎q𝑏o − 𝑏q𝑎o 𝚤 − 𝑎r𝑏o − 𝑏r𝑎o 𝚥 + 𝑎r𝑏q − 𝑏r𝑎q 𝑘

Properties𝐚×𝐛 = −𝐛×𝐚

k𝐚 ×𝐛 = k 𝐚×𝐛 = 𝐚× k𝐛

𝐚 ⋅ 𝐛×𝐜 = 𝐚×𝐛 ⋅ 𝐜

𝐚+ 𝐛 ×𝐜 = 𝐚×𝐜 + 𝐛×𝐜

𝐚× 𝐛+ 𝐜 = 𝐚×𝐛+ 𝐚×𝐜 𝐚× 𝐛×𝐜 = 𝐚 ⋅ 𝐜 𝐛− 𝐚 ⋅ 𝐛 𝐜

AnglesBetweenVectors𝑎 ⋅ 𝑏 = 𝑎 𝑏 cos 𝜃 𝑎×𝑏 = 𝑎 𝑏 sin 𝜃

⇒ 𝜃 = arccos𝑎 ⋅ 𝑏𝑎 𝑏

⇒ 𝜃 = arcsin𝑎×𝑏𝑎 𝑏

ProjectionsScalar Vector

compò𝑏 =𝑎 ⋅ 𝑏𝑎 proj𝐚𝑏 =

𝑎 ⋅ 𝑏𝑎 q 𝑎


Areas/VolumeTriangle

Parallelogram

Parallelepiped

𝐴 =12 𝑎×𝑏 𝐴 = 𝑎×𝑏 𝑉 = 𝑎 ⋅ 𝑏×𝑐

Lineℒ 𝑡 = 𝑃2 + 𝑡𝑣

= 𝑥2, 𝑦2, 𝑧2 + 𝑡 𝑎, 𝑏, 𝑐

= 𝑥2 + 𝑎𝑡, 𝑦2 + 𝑏𝑡, 𝑧2 + 𝑐𝑡 = 𝑥2, 𝑦2, 𝑧2 + 𝑡 𝑥 − 𝑥2, 𝑦 − 𝑦2, 𝑧 − 𝑧2

𝑣 = 𝑃r𝑃q = 𝑃q − 𝑃r

= 𝑥, 𝑦, 𝑧 − 𝑥2, 𝑦2, 𝑧2 = 𝑥 − 𝑥2, 𝑦 − 𝑦2, 𝑧 − 𝑧2

= 𝑎, 𝑏, 𝑐

LinefromtiptotipAlinesegmentfromthetipstwovectorsbeginningfromtheoriginto𝑣r → 𝑣qis

ℒ 𝑡 = 1 − 𝑡 𝑣r + 𝑡𝑣q, 𝑡 ∈ 0, 1

EquationofaPlane𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑 ⇒ 𝑛 = 𝑎, 𝑏, 𝑐 ⊥ surface𝑛isperpendiculartothesurface 𝑣isintheplane,𝑃2 = 𝑥2, 𝑦2, 𝑧2 (pointinplane)𝑛 ⊥ 𝑣 ⇒ 𝑛 ⋅ 𝑣 = 𝑎, 𝑏, 𝑐 ⋅ 𝑥 − 𝑥2, 𝑦 − 𝑦2, 𝑧 − 𝑧2 = 𝑎 𝑥 − 𝑥2 + 𝑏 𝑦 − 𝑦2 + 𝑐 𝑧 − 𝑧2 = 0VectorFunctions𝑟 𝑡 = 𝑟r 𝑡 , 𝑟q 𝑡 , 𝑟o 𝑡 = 𝑓 𝑡 , 𝑔 𝑡 , ℎ 𝑡


Limit lim©→�

𝑟 𝑡 = lim©→�

𝑓 𝑡 , lim©→�

𝑔 𝑡 , lim©→�

ℎ 𝑡

Derivative 𝑑𝑟𝑑𝑡 = 𝑓Þ 𝑡 , 𝑔Þ 𝑡 , ℎÞ 𝑡

DefiniteIntegral𝑟 𝑡

©�

©:𝑑𝑡 = 𝑟r 𝑡

©�

©:𝑑𝑡 𝚤 + 𝑟q 𝑡

©�

©:𝑑𝑡 𝚥 + 𝑟o 𝑡

©�

©:𝑑𝑡 𝑘

IndefiniteIntegral 𝑟 𝑡 𝑑𝑡 = 𝑟r 𝑡 𝑑𝑡 𝚤 + 𝑟q 𝑡 𝑑𝑡 𝚥 + 𝑟o 𝑡 𝑑𝑡 𝑘 + 𝐶

DifferentiationRulesNote:𝑣 𝑡 , 𝑢 𝑡 , 𝑓 𝑡 FunctiondotVector VectorcrossVector

𝑑𝑑𝑡 𝑓 𝑡 ⋅ 𝑢 𝑡 = 𝑢 𝑡

𝑑𝑓𝑑𝑡 + 𝑓 𝑡

𝑑𝑢𝑑𝑡

𝑑𝑑𝑡 𝑢 𝑡 ×𝑣 𝑡 =

𝑑𝑢𝑑𝑡 ×𝑣 𝑡 + 𝑢 𝑡 ×

𝑑𝑣𝑑𝑡

VectordotVector ChainRule

𝑑𝑑𝑡 𝑢 𝑡 ⋅ 𝑣 𝑡 = 𝑣 𝑡 ⋅

𝑑𝑢𝑑𝑡 + 𝑢 𝑡 ⋅

𝑑𝑣𝑑𝑡

𝑑𝑑𝑡 𝑢 𝑓 𝑡 = 𝑢Þ 𝑓 𝑡 𝑓Þ 𝑡

Arclength

𝐿 =𝑑𝑟r𝑑𝑡

q

+𝑑𝑟q𝑑𝑡

q

+𝑑𝑟o𝑑𝑡

q©�

©:𝑑𝑡 = 𝑓Þ 𝑡 q + 𝑔Þ 𝑡 q + ℎÞ 𝑡 q

©�

©:𝑑𝑡 =

𝑑𝑟𝑑𝑡

©�

©:𝑑𝑡

Tangents


UnitTangentVector𝐓 𝑡 =

𝐫Þ 𝑡𝐫Þ 𝑡 , 𝐫Þ 𝑡 =

𝑑𝑠𝑑𝑡

Curvature1𝜅(𝑡) =

𝑑𝐓𝑑𝑠 =

𝑑𝐓𝑑𝑡𝑑𝑡𝑑𝑠 =

𝑑𝐓𝑑𝑡𝑑𝑠𝑑𝑡

=𝐓Þ 𝑡𝐫Þ 𝑡

Curvature2(vectorfunction)𝜅(𝑡) =

𝐫Þ 𝑡 ×𝐫ÞÞ 𝑡𝐫Þ 𝑡 o

Curvature3(singlevariable)𝜅(𝑥) =

𝑓ÞÞ 𝑥

1 + 𝑓Þ 𝑥 qoq

Curvature4(parametric)𝜅 𝑡 =

𝑥Þ 𝑡 𝑦ÞÞ 𝑡 − 𝑦Þ 𝑡 𝑥ÞÞ 𝑡

𝑥Þ 𝑡 q + 𝑦Þ 𝑡 qoq

NormalVector𝐍 𝑡 =

𝐓Þ 𝑡𝐓Þ 𝑡

BinormalVector 𝐁 𝑡 = 𝐓 𝑡 ×𝐍 𝑡


TangentialandNormalComponents(acceleration)PhysicsNotationsPosition 𝑟 𝑡 ≡ 𝐫 𝑡

Velocity 𝑣 𝑡 = 𝑟Þ 𝑡 =

𝑑𝑟𝑑𝑡 =

𝑑𝐫𝑑𝑡 = 𝐫Þ 𝑡

Speed

𝑣 = 𝑣 𝑡 = 𝑟Þ 𝑡

Acceleration 𝑎 𝑡 = 𝑣Þ 𝑡 = 𝑟ÞÞ 𝑡

𝐓 𝑡 =𝐫Þ 𝑡𝐫Þ 𝑡 =

𝑣 𝑡𝑣 𝑡 =

𝑣𝑣

𝑣 = 𝑣𝐓 ⇒ 𝑑𝑣𝑑𝑡 = 𝑎 = 𝑣Þ𝐓+ 𝑣𝐓′

Curvature𝜅 =

𝐓Þ

𝐫Þ =𝐓Þ

𝑣 ⇒ 𝜅𝑣 = 𝐓′

TangentialComponent(acceleration) 𝑎^ =𝑑𝑑𝑡 𝑟

Þ =𝑑𝑣𝑑𝑡 = 𝑣Þ, 𝑣 = 𝑣 = 𝑟′ ≡ 𝐫′

NormalComponent(acceleration) 𝑎_ = 𝜅𝑣q

Acceleration

𝒂 = 𝑣Þ𝐓+ 𝜅𝑣q𝐍 = 𝑎^𝐓+ 𝑎_𝐍

Note:

𝐓 ⋅ 𝐓 = 1 ∧ 𝐓 ⋅ 𝐍 = 0

DotProductofVelocityandAcceleration

𝑣 ⋅ 𝑎 = 𝑣𝐓 ⋅ 𝑣Þ𝐓+ 𝜅𝑣q𝐍 = 𝑣𝑣Þ𝐓 ⋅ 𝐓+ 𝜅𝑣o𝐓 ⋅ 𝐍 = 𝑣𝑣′

TangentialAcceleration 𝑎^ = 𝑣Þ =

𝑣 ⋅ 𝑎𝑣 =

𝐫Þ 𝑡 ⋅ 𝐫ÞÞ 𝑡𝐫Þ 𝑡

NormalAcceleration𝑎_ = 𝜅𝑣q =

𝐫Þ 𝑡 ×𝐫ÞÞ 𝑡𝐫Þ 𝑡


Frenet-SerretFormulas𝑑𝐓𝑑𝑠 = 𝜅𝐍

𝑑𝐍𝑑𝑠 = −𝜅𝐓+ 𝜏𝐁

𝑑𝐁𝑑𝑥 = −𝜏𝐍

PartialDerivativesGivenamultivariablefunctione.g.𝑓 𝑥, 𝑦, 𝑧 ,thenapartialderivativeisthederivativewithrespecttoavariablewheretheothervariablesaretreatingasconstantsi.e.donotimplicitlydifferentiate.𝜕𝑓𝜕𝑥 = 𝑓� = 𝑓� 𝑥, 𝑦, 𝑧

𝜕𝑓𝜕𝑦 = 𝑓¦ = 𝑓¦ 𝑥, 𝑦, 𝑧

𝜕𝑓𝜕𝑧 = 𝑓b = 𝑓b 𝑥, 𝑦, 𝑧

𝜕q𝑓𝜕𝑥q = 𝑓��

𝜕q𝑓𝜕𝑦q = 𝑓¦¦

𝜕q𝑓𝜕𝑧q = 𝑓bb

MixedPartial

𝜕q𝑓𝜕𝑥𝜕𝑦 = 𝑓�¦,

𝜕q𝑓𝜕𝑦𝜕𝑥 = 𝑓¦�

TangentPlane

𝑧 − 𝑧2 = 𝑓� 𝑥2, 𝑦2 𝑥 − 𝑥2 + 𝑓¦ 𝑥2, 𝑦2 𝑦 − 𝑦2

ChainRule𝑑𝑧𝑑𝑡 =

𝜕𝑧𝜕𝑥𝑑𝑥𝑑𝑡 +

𝜕𝑧𝜕𝑦𝑑𝑦𝑑𝑡 , 𝑥 = 𝑥 𝑡 ∧ 𝑦 = 𝑦 𝑡

𝜕𝑧𝜕𝑠 =

𝜕𝑧𝜕𝑥𝜕𝑥𝜕𝑠 +

𝜕𝑧𝜕𝑦𝜕𝑦𝜕𝑠 ,

𝜕𝑧𝜕𝑡 =

𝜕𝑧𝜕𝑥𝜕𝑥𝜕𝑡 +

𝜕𝑧𝜕𝑦𝜕𝑦𝜕𝑡 , 𝑥 = 𝑥 𝑠, 𝑡 ∧ 𝑦 = 𝑦 𝑠, 𝑡


MULTIVARIABLECALCULUS(CALCIII)Magnitude

𝑣 = 𝐯 = 𝑣r, 𝑣q, 𝑣o ⇒ 𝑣 = 𝐯 = 𝑣rq + 𝑣qq + 𝑣oq

UnitVectors

𝑣 =𝑣𝑣 ≡ 𝐯 ≡ 𝐮 =

𝐯𝐯

𝚤 ≡ 𝐢 𝚥 ≡ 𝐣 𝑘 ≡ 𝐤𝚤 = 1, 0, 0 𝚥 = 0, 1, 0 𝑘 = 1, 0, 0 Note:𝑣 = 𝑣r, 𝑣q, 𝑣o = 𝑣r 1, 0, 0 + 𝑣q 0, 1, 0 + 𝑣o 0, 0, 1 = 𝑣r𝚤 + 𝑣q𝚥 + 𝑣o𝑘 = 𝑣r𝐢 + 𝑣q𝐣+ 𝑣o𝐤Dot/CrossProductDot𝑎 ⋅ 𝑏 = 𝐚 ⋅ 𝐛

= 𝑎r, 𝑎q, 𝑎o ⋅ 𝑏r, 𝑏q, 𝑏o

= 𝑎r𝑏r + 𝑎q𝑏q + 𝑎o𝑏o

Properties𝐚 ⋅ 𝐚 = 𝐚 q 𝐚 ⋅ 𝐛 = 𝐛 ⋅ 𝐚

𝐚 ⋅ 𝐛+ 𝐜 = 𝐚 ⋅ 𝐛+ 𝐚 ⋅ 𝐜 k𝐚 ⋅ 𝐛 = k 𝐚 ⋅ 𝐛 = 𝐚 ⋅ k𝐛

Cross𝑎×𝑏 = 𝐚×𝐛 = 𝑎r, 𝑎q, 𝑎o × 𝑏r, 𝑏q, 𝑏o

=𝚤 𝚥 𝑘𝑎r 𝑎q 𝑎o𝑏r 𝑏q 𝑏o

=

𝑎q 𝑎o𝑏q 𝑏o 𝚤 −

𝑎r 𝑎o𝑏r 𝑏o 𝚥 +

𝑎r 𝑎q𝑏r 𝑏q 𝑘

= 𝑎q𝑏o − 𝑏q𝑎o 𝚤 − 𝑎r𝑏o − 𝑏r𝑎o 𝚥 + 𝑎r𝑏q − 𝑏r𝑎q 𝑘


Properties𝐚×𝐛 = −𝐛×𝐚

k𝐚 ×𝐛 = k 𝐚×𝐛 = 𝐚× k𝐛

𝐚 ⋅ 𝐛×𝐜 = 𝐚×𝐛 ⋅ 𝐜

𝐚+ 𝐛 ×𝐜 = 𝐚×𝐜 + 𝐛×𝐜

𝐚× 𝐛+ 𝐜 = 𝐚×𝐛+ 𝐚×𝐜 𝐚× 𝐛×𝐜 = 𝐚 ⋅ 𝐜 𝐛− 𝐚 ⋅ 𝐛 𝐜

AnglesBetweenVectors𝑎 ⋅ 𝑏 = 𝑎 𝑏 cos 𝜃 𝑎×𝑏 = 𝑎 𝑏 sin 𝜃

⇒ 𝜃 = arccos𝑎 ⋅ 𝑏𝑎 𝑏

⇒ 𝜃 = arcsin𝑎×𝑏𝑎 𝑏

ProjectionsScalar Vector

compò𝑏 =𝑎 ⋅ 𝑏𝑎 proj𝐚𝑏 =

𝑎 ⋅ 𝑏𝑎 q 𝑎

Areas/VolumeTriangle

Parallelogram

Parallelepiped

𝐴 =12 𝑎×𝑏 𝐴 = 𝑎×𝑏 𝑉 = 𝑎 ⋅ 𝑏×𝑐

Lineℒ 𝑡 = 𝑃2 + 𝑡𝑣

= 𝑥2, 𝑦2, 𝑧2 + 𝑡 𝑎, 𝑏, 𝑐

= 𝑥2 + 𝑎𝑡, 𝑦2 + 𝑏𝑡, 𝑧2 + 𝑐𝑡 = 𝑥2, 𝑦2, 𝑧2 + 𝑡 𝑥 − 𝑥2, 𝑦 − 𝑦2, 𝑧 − 𝑧2

𝑣 = 𝑃r𝑃q = 𝑃q − 𝑃r

= 𝑥, 𝑦, 𝑧 − 𝑥2, 𝑦2, 𝑧2 = 𝑥 − 𝑥2, 𝑦 − 𝑦2, 𝑧 − 𝑧2

= 𝑎, 𝑏, 𝑐

LinefromtiptotipAlinesegmentfromthetipstwovectorsbeginningfromtheoriginto𝑣r → 𝑣qis

ℒ 𝑡 = 1 − 𝑡 𝑣r + 𝑡𝑣q, 𝑡 ∈ 0, 1


EquationofaPlane𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑 ⇒ 𝑛 = 𝑎, 𝑏, 𝑐 ⊥ surface𝑛isperpendiculartothesurface 𝑣isintheplane,𝑃2 = 𝑥2, 𝑦2, 𝑧2 (pointinplane)𝑛 ⊥ 𝑣 ⇒ 𝑛 ⋅ 𝑣 = 𝑎, 𝑏, 𝑐 ⋅ 𝑥 − 𝑥2, 𝑦 − 𝑦2, 𝑧 − 𝑧2 = 𝑎 𝑥 − 𝑥2 + 𝑏 𝑦 − 𝑦2 + 𝑐 𝑧 − 𝑧2 = 0VectorFunctions𝑟 𝑡 = 𝑟r 𝑡 , 𝑟q 𝑡 , 𝑟o 𝑡 = 𝑓 𝑡 , 𝑔 𝑡 , ℎ 𝑡

Limit lim©→�

𝑟 𝑡 = lim©→�

𝑓 𝑡 , lim©→�

𝑔 𝑡 , lim©→�

ℎ 𝑡

Derivative 𝑑𝑟𝑑𝑡 = 𝑓Þ 𝑡 , 𝑔Þ 𝑡 , ℎÞ 𝑡

DefiniteIntegral𝑟 𝑡

©�

©:𝑑𝑡 = 𝑟r 𝑡

©�

©:𝑑𝑡 𝚤 + 𝑟q 𝑡

©�

©:𝑑𝑡 𝚥 + 𝑟o 𝑡

©�

©:𝑑𝑡 𝑘

IndefiniteIntegral 𝑟 𝑡 𝑑𝑡 = 𝑟r 𝑡 𝑑𝑡 𝚤 + 𝑟q 𝑡 𝑑𝑡 𝚥 + 𝑟o 𝑡 𝑑𝑡 𝑘 + 𝐶

DifferentiationRulesNote:𝑣 𝑡 , 𝑢 𝑡 , 𝑓 𝑡 FunctiondotVector VectorcrossVector

𝑑𝑑𝑡 𝑓 𝑡 ⋅ 𝑢 𝑡 = 𝑢 𝑡

𝑑𝑓𝑑𝑡 + 𝑓 𝑡

𝑑𝑢𝑑𝑡

𝑑𝑑𝑡 𝑢 𝑡 ×𝑣 𝑡 =

𝑑𝑢𝑑𝑡 ×𝑣 𝑡 + 𝑢 𝑡 ×

𝑑𝑣𝑑𝑡

VectordotVector ChainRule

𝑑𝑑𝑡 𝑢 𝑡 ⋅ 𝑣 𝑡 = 𝑣 𝑡 ⋅

𝑑𝑢𝑑𝑡 + 𝑢 𝑡 ⋅

𝑑𝑣𝑑𝑡

𝑑𝑑𝑡 𝑢 𝑓 𝑡 = 𝑢Þ 𝑓 𝑡 𝑓Þ 𝑡

Arclength

𝐿 =𝑑𝑟r𝑑𝑡

q

+𝑑𝑟q𝑑𝑡

q

+𝑑𝑟o𝑑𝑡

q©�

©:𝑑𝑡 = 𝑓Þ 𝑡 q + 𝑔Þ 𝑡 q + ℎÞ 𝑡 q

©�

©:𝑑𝑡 =

𝑑𝑟𝑑𝑡

©�

©:𝑑𝑡


TangentsUnitTangentVector

𝐓 𝑡 =𝐫Þ 𝑡𝐫Þ 𝑡 , 𝐫Þ 𝑡 =

𝑑𝑠𝑑𝑡

Curvature1𝜅(𝑡) =

𝑑𝐓𝑑𝑠 =

𝑑𝐓𝑑𝑡𝑑𝑡𝑑𝑠 =

𝑑𝐓𝑑𝑡𝑑𝑠𝑑𝑡

=𝐓Þ 𝑡𝐫Þ 𝑡

Curvature2(vectorfunction)𝜅(𝑡) =

𝐫Þ 𝑡 ×𝐫ÞÞ 𝑡𝐫Þ 𝑡 o

Curvature3(singlevariable)𝜅(𝑥) =

𝑓ÞÞ 𝑥

1 + 𝑓Þ 𝑥 qoq

Curvature4(parametric)𝜅 𝑡 =

𝑥Þ 𝑡 𝑦ÞÞ 𝑡 − 𝑦Þ 𝑡 𝑥ÞÞ 𝑡

𝑥Þ 𝑡 q + 𝑦Þ 𝑡 qoq

NormalVector𝐍 𝑡 =

𝐓Þ 𝑡𝐓Þ 𝑡

BinormalVector 𝐁 𝑡 = 𝐓 𝑡 ×𝐍 𝑡


TangentialandNormalComponents(acceleration)PhysicsNotationsPosition 𝑟 𝑡 ≡ 𝐫 𝑡

Velocity 𝑣 𝑡 = 𝑟Þ 𝑡 =

𝑑𝑟𝑑𝑡 =

𝑑𝐫𝑑𝑡 = 𝐫Þ 𝑡

Speed

𝑣 = 𝑣 𝑡 = 𝑟Þ 𝑡

Acceleration 𝑎 𝑡 = 𝑣Þ 𝑡 = 𝑟ÞÞ 𝑡

𝐓 𝑡 =𝐫Þ 𝑡𝐫Þ 𝑡 =

𝑣 𝑡𝑣 𝑡 =

𝑣𝑣

𝑣 = 𝑣𝐓 ⇒ 𝑑𝑣𝑑𝑡 = 𝑎 = 𝑣Þ𝐓+ 𝑣𝐓′

Curvature𝜅 =

𝐓Þ

𝐫Þ =𝐓Þ

𝑣 ⇒ 𝜅𝑣 = 𝐓′

TangentialComponent(acceleration) 𝑎^ =𝑑𝑑𝑡 𝑟

Þ =𝑑𝑣𝑑𝑡 = 𝑣Þ, 𝑣 = 𝑣 = 𝑟′ ≡ 𝐫′

NormalComponent(acceleration) 𝑎_ = 𝜅𝑣q

Acceleration

𝒂 = 𝑣Þ𝐓+ 𝜅𝑣q𝐍 = 𝑎^𝐓+ 𝑎_𝐍

Note:

𝐓 ⋅ 𝐓 = 1 ∧ 𝐓 ⋅ 𝐍 = 0

DotProductofVelocityandAcceleration

𝑣 ⋅ 𝑎 = 𝑣𝐓 ⋅ 𝑣Þ𝐓+ 𝜅𝑣q𝐍 = 𝑣𝑣Þ𝐓 ⋅ 𝐓+ 𝜅𝑣o𝐓 ⋅ 𝐍 = 𝑣𝑣′

TangentialAcceleration 𝑎^ = 𝑣Þ =

𝑣 ⋅ 𝑎𝑣 =

𝐫Þ 𝑡 ⋅ 𝐫ÞÞ 𝑡𝐫Þ 𝑡

NormalAcceleration𝑎_ = 𝜅𝑣q =

𝐫Þ 𝑡 ×𝐫ÞÞ 𝑡𝐫Þ 𝑡


Frenet-SerretFormulas𝑑𝐓𝑑𝑠 = 𝜅𝐍

𝑑𝐍𝑑𝑠 = −𝜅𝐓+ 𝜏𝐁

𝑑𝐁𝑑𝑥 = −𝜏𝐍

PartialDerivativesGivenamultivariablefunctione.g.𝑓 𝑥, 𝑦, 𝑧 ,thenapartialderivativeisthederivativewithrespecttoavariablewheretheothervariablesaretreatingasconstantsi.e.donotimplicitlydifferentiate.𝜕𝑓𝜕𝑥 = 𝑓� = 𝑓� 𝑥, 𝑦, 𝑧

𝜕𝑓𝜕𝑦 = 𝑓¦ = 𝑓¦ 𝑥, 𝑦, 𝑧

𝜕𝑓𝜕𝑧 = 𝑓b = 𝑓b 𝑥, 𝑦, 𝑧

𝜕q𝑓𝜕𝑥q = 𝑓��

𝜕q𝑓𝜕𝑦q = 𝑓¦¦

𝜕q𝑓𝜕𝑧q = 𝑓bb

MixedPartial

𝜕q𝑓𝜕𝑥𝜕𝑦 = 𝑓�¦,

𝜕q𝑓𝜕𝑦𝜕𝑥 = 𝑓¦�

EquationofaPlane

𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑NormalVectorThenormalvector𝑛 = 𝑎, 𝑏, 𝑐 ,isextractedfromtheequationofaplane,andthenormalvectorisperpendiculartothesurface.Distance/VectorBetweenPointsVectorfromtwopoints

𝑃r 𝑎, 𝑏, 𝑐 ∧ 𝑃q 𝑑, 𝑒, 𝑓 ⇒ 𝑃r𝑃q = 𝑃q − 𝑃r = 𝑑 − 𝑎, 𝑒 − 𝑏, 𝑓 − 𝑐

𝑃r𝑃q = 𝑑 − 𝑎 q + 𝑒 − 𝑏 q + 𝑓 − 𝑐 qTangentPlane

𝑧 − 𝑧2 = 𝑓� 𝑥2, 𝑦2 𝑥 − 𝑥2 + 𝑓¦ 𝑥2, 𝑦2 𝑦 − 𝑦2


Equationofasphere

𝑥 − ℎ q + 𝑦 − 𝑘 q + 𝑧 − 𝑙 q = 𝑟q, center: ℎ, 𝑘, 𝑙 radius: 𝑟

ChainRule𝑑𝑧𝑑𝑡 =

𝜕𝑧𝜕𝑥𝑑𝑥𝑑𝑡 +

𝜕𝑧𝜕𝑦𝑑𝑦𝑑𝑡 , 𝑥 = 𝑥 𝑡 ∧ 𝑦 = 𝑦 𝑡

𝜕𝑧𝜕𝑠 =

𝜕𝑧𝜕𝑥𝜕𝑥𝜕𝑠 +

𝜕𝑧𝜕𝑦𝜕𝑦𝜕𝑠 ,

𝜕𝑧𝜕𝑡 =

𝜕𝑧𝜕𝑥𝜕𝑥𝜕𝑡 +

𝜕𝑧𝜕𝑦𝜕𝑦𝜕𝑡 , 𝑥 = 𝑥 𝑠, 𝑡 ∧ 𝑦 = 𝑦 𝑠, 𝑡

Gradient𝜵𝒇Thesymbol𝛻iscallednablaordel;𝜕iscalledpartialordel.Itwouldbeappropriatetouse“del”asdelisforpartialderivativesjustasnablais.Thegradientof𝑓isnotedas𝛻𝑓,andisequalthevectorfunctionofpartialsi.e.

𝛻𝑓 =𝜕𝑓𝜕𝑥 𝐢 +

𝜕𝑓𝜕𝑦 𝐣+

𝜕𝑓𝜕𝑧 𝐤

DirectionalDerivativeGiven𝑓 𝑥, 𝑦, 𝑧 ,𝑣 = 𝑣r, 𝑣q, 𝑣o ,and𝑃 𝑥2, 𝑦2, 𝑧2

𝐷𝐮𝑓 ≡ 𝛻𝑓 𝑥2, 𝑦2, 𝑧e ⋅ 𝐮 ≡ 𝛻𝑓 𝑥2, 𝑦2, 𝑧e ⋅𝑣𝑣

𝐷𝐮𝑓 =1𝑣 𝑓� 𝑥2, 𝑦2, 𝑧e , 𝑓¦ 𝑥2, 𝑦2, 𝑧e , 𝑓b 𝑥2, 𝑦2, 𝑧e ⋅ 𝑣r, 𝑣q, 𝑣o

Differentials

𝑑𝑓 = 𝑓� 𝑥, 𝑦 𝛥𝑥 + 𝑓¦ 𝑥, 𝑦 𝛥𝑦 + 𝑓b 𝑥, 𝑦 𝛥𝑧ImplicitDifferentiation

𝜕𝑧𝜕𝑥 = −

𝜕𝐹𝜕𝑥𝜕𝐹𝜕𝑧

∧ 𝜕𝑧𝜕𝑦 = −

𝜕𝐹𝜕𝑦𝜕𝐹𝜕𝑧


ExtremaGivenathree-dimensionalfunction𝑓,wecanfindtheextremabyusingpartialderivatives,andderivativetests.Process:Set𝑓� = 0 Set𝑓¦ = 0 Solvefor

𝑥, 𝑦 = 𝑐r, 𝑐q (criticalpoint)Evaluate𝑓�� 𝑐r, 𝑐q 𝑓¦¦ 𝑓�¦ 𝑓¦� True:𝑓�¦ = 𝑓¦�

𝐷 =𝑓�� 𝑓�¦𝑓¦� 𝑓¦¦

= 𝑓��𝑓¦¦ − 𝑓�¦q

LocalMin:𝐷 > 0and𝑓�� 𝑐r, 𝑐q > 0LocalMax:𝐷 > 0and𝑓�� 𝑐r, 𝑐q < 0Saddle:𝐷 < 0LagrangeMultipliersThesearelikepuzzlesi.e.thesetupisprettystraightforward,butyoumayneedtomakemultipleattemptstofindtherightpattern.2DGiven𝑓 𝑥, 𝑦 (function)and𝑔 𝑥, 𝑦 = 𝑘(constraint)then𝛻𝑓 𝑥, 𝑦 = 𝜆𝛻𝑔 𝑥, 𝑦 Solvethefollowingsystem:𝑓� = 𝜆𝑔� 𝑓¦ = 𝜆𝑔¦ 𝑔 𝑥, 𝑦 = 𝑘3DGiven𝑓 𝑥, 𝑦, 𝑧 (function)and𝑔 𝑥, 𝑦, 𝑧 = 𝑘(constraint)then𝛻𝑓 𝑥, 𝑦, 𝑧 = 𝜆𝛻𝑔 𝑥, 𝑦, 𝑧 Solvethefollowingsystem:𝑓� = 𝜆𝑔� 𝑓¦ = 𝜆𝑔¦ 𝑓b = 𝜆𝑔b 𝑔 𝑥, 𝑦, 𝑧 = 𝑘Onceyoufindallpossiblevalues,thenyousimplyplugtheminto𝑓,andseewhichislargest/smallest.Thesearethenyourmax/min.


TwoConstraints

𝛻𝑓 𝑥, 𝑦, 𝑧 = 𝜆𝛻𝑔 𝑥, 𝑦, 𝑧 + 𝜇𝛻ℎ 𝑥, 𝑦, 𝑧 𝑓� = 𝜆𝑔� + 𝜇ℎ� 𝑓¦ = 𝜆𝑔¦ + 𝜇ℎ¦ 𝑓b = 𝜆𝑔b + 𝜇ℎb 𝑔 𝑥, 𝑦, 𝑧 = 𝑘r ℎ 𝑥, 𝑦, 𝑧 = 𝑘q

MultipleIntegralsDouble

𝑓 𝑥, 𝑦»

�

�

�𝑑𝑦𝑑𝑥 ≡ 𝑓 𝑥, 𝑦

ß𝑑𝐴, 𝑅 = 𝑥, 𝑦 𝑥 ∈ 𝑎, 𝑏 , 𝑦 ∈ 𝑐, 𝑑 ≡ 𝑅 = 𝑎, 𝑏 × 𝑐, 𝑑

Note1:

𝑓 𝑥, 𝑦»

�

�

�𝑑𝑦𝑑𝑥 = 𝑓 𝑥, 𝑦

�

�

»

�𝑑𝑥𝑑𝑦 ⇔ 𝑎 ≤ 𝑥 ≤ 𝑏 ∧ 𝑐 ≤ 𝑦 ≤ 𝑑

Note2:𝑓 𝑥, 𝑦 = 𝑓 𝑥 𝑔 𝑦

𝑓 𝑥, 𝑦»

�

�

�𝑑𝑦𝑑𝑥 = 𝑓 𝑥 𝑔 𝑦

»

�

�

�𝑑𝑦𝑑𝑥 = 𝑔 𝑦

»

�𝑑𝑦 𝑓 𝑥

�

�𝑑𝑥

AverageValue

1𝑑 − 𝑐

1𝑏 − 𝑎 𝑓 𝑥, 𝑦

»

�

�

�𝑑𝑦𝑑𝑥

TypeI

𝑓 𝑥, 𝑦h

𝑑𝐴 = 𝑓 𝑥, 𝑦È� �

È: �

�

�𝑑𝑦𝑑𝑥, 𝐷 = 𝑥, 𝑦 𝑥 ∈ 𝑎, 𝑏 , 𝑦 ∈ 𝑔r 𝑥 , 𝑔q 𝑥

TypeII

𝑓 𝑥, 𝑦h

𝑑𝐴 = 𝑓 𝑥, 𝑦ã� ¦

ã: ¦

»

�𝑑𝑥𝑑𝑦, 𝐷 = 𝑥, 𝑦 𝑥 ∈ ℎr 𝑦 , ℎq 𝑦 , 𝑦 ∈ 𝑐, 𝑑


Polar𝑟q = 𝑥q + 𝑦q 𝑥 = 𝑟 cos 𝜃 𝑦 = 𝑟 sin 𝜃

𝑓 𝑥, 𝑦ß

𝑑𝐴 = 𝑟𝑓 𝑟 cos 𝜃 , 𝑟 sin 𝜃Ô�

Ô:

<�

<:𝑑𝑟𝑑𝜃, 𝑅 = 𝑟, 𝜃 𝑟 ∈ 𝑟r, 𝑟q , 𝜃 ∈ 𝜃r, 𝜃q

Note:Donotforgettheextra𝑟multipliedby𝑓TypeIII𝑓iscontinuousonapolarregion

𝑓 𝑥, 𝑦ß

𝑑𝐴 = 𝑓 𝑟 cos 𝜃 , 𝑟 sin 𝜃 𝑟È� <

È: <

<�

<:𝑑𝑟𝑑𝜃,

𝑅 = 𝑟, 𝜃 𝑟 ∈ 𝑔r 𝜃 , 𝑔q 𝜃 , 𝜃 ∈ 𝜃r, 𝜃q

Moments&CenterofMassMoments𝑀� 𝑦𝜌 𝑥, 𝑦

h𝑑𝐴

𝑀¦ 𝑥𝜌 𝑥, 𝑦h

𝑑𝐴

Centerofmass

𝑥 =𝑀¦

𝑚 1𝑚 𝑦𝜌 𝑥, 𝑦

h𝑑𝐴, 𝑚 = 𝜌 𝑥, 𝑦

h𝑑𝐴

𝑦 =𝑀�

𝑚 1𝑚 𝑥𝜌 𝑥, 𝑦

h𝑑𝐴, 𝑚 = 𝜌 𝑥, 𝑦

h𝑑𝐴

MomentofInertia𝐼� 𝑦q𝜌 𝑥, 𝑦

h𝑑𝐴

𝐼¦ 𝑥q𝜌 𝑥, 𝑦h

𝑑𝐴

𝐼2(aboutorigin) 𝑥q + 𝑦q 𝜌 𝑥, 𝑦h

𝑑𝐴


SurfaceArea𝑧 = 𝑓 𝑥, 𝑦 , 𝑥, 𝑦 ∈ 𝐷,and𝑓�, 𝑓¦arecontinuous

𝐴Û = 1 +𝜕𝑧𝜕𝑥

q

+𝜕𝑧𝜕𝑦

q

h𝑑𝐴 = 1 + 𝑓� 𝑥, 𝑦

q + 𝑓¦ 𝑥, 𝑦q

ß𝑑𝐴

TripleIntegrals

𝑓 𝑥, 𝑦, 𝑧ß

𝑑𝑉, 𝑅 = 𝑥, 𝑦, 𝑧 𝑥r, 𝑥q × 𝑦r, 𝑦q × 𝑧r, 𝑧q ≡ 𝑓 𝑥, 𝑦, 𝑧��

�:

¦�

¦:

b�

b:𝑑𝑥𝑑𝑦𝑑𝑧

TypeIV:

𝑓 𝑥, 𝑦, 𝑧i

𝑑𝑉 = 𝑓 𝑥, 𝑦, 𝑧È� �,¦

È: �,¦𝑑𝑧

h𝑑𝐴

TypeV:


𝑑𝑉 = 𝑓 𝑥, 𝑦, 𝑧È� �,¦

È: �,¦𝑑𝑧

ã� �

ã: �𝑑𝑦

��

�:𝑑𝑥

TypeVI:


𝑑𝑉 = 𝑓 𝑥, 𝑦, 𝑧È� �,¦

È: �,¦𝑑𝑧

3� ¦

3: ¦𝑑𝑥

¦�

¦:𝑑𝑦

Moments&CenterofMassMoments𝑀�¦ 𝑧𝜌 𝑥, 𝑦, 𝑧

i𝑑𝑉

𝑀¦b 𝑥𝜌 𝑥, 𝑦, 𝑧i

𝑑𝑉

𝑀�b 𝑦𝜌 𝑥, 𝑦, 𝑧i

𝑑𝑉


CenterofMassThecentroidof𝐸isthecenterofmass(𝑥, 𝑦, 𝑧)forconstantdensity.

𝑚 = 𝜌 𝑥, 𝑦, 𝑧i

𝑑𝑉

𝑥 =𝑀¦b

𝑚 𝑦 =𝑀�b

𝑚 𝑧 =𝑀�¦

𝑚 MomentsofInertia

𝐼� = 𝑦q + 𝑧q 𝜌 𝑥, 𝑦, 𝑧i

𝑑𝑉 𝐼¦ = 𝑥q + 𝑧q 𝜌 𝑥, 𝑦, 𝑧i

𝑑𝑉 𝐼b = 𝑥q + 𝑦q 𝜌 𝑥, 𝑦, 𝑧i

𝑑𝑉

CylindricalCoordinates𝑟q = 𝑥q + 𝑦q tan 𝜃 =

𝑦𝑥

𝑥 = 𝑟 cos 𝜃 𝑦 = 𝑟 sin 𝜃 𝑧 = 𝑧


𝑑𝑉 = 𝑟𝑓 𝑟 cos 𝜃 , 𝑟 sin 𝜃 , 𝑧ã� Ô jk/ <,Ô /ó¯<

ã: Ô jk/ <,Ô /ó¯<

È� <

È: <

<�

<:𝑑𝑧𝑑𝑟𝑑𝜃

Note:Donotforgettheextra𝑟SphericalCoordinates𝑥 = 𝜌 sin𝜙 cos 𝜃 𝑦 = 𝜌 sin𝜙 sin 𝜃 𝑧 = 𝜌 cos𝜙 𝜌q = 𝑥q + 𝑦q + 𝑧q

𝐸 = 𝜌, 𝜃,𝜙 𝜌 ∈ 𝜌r,𝜌q , 𝜃 ∈ 𝜃r, 𝜃q ,𝜙 ∈ 𝜙r,𝜙q


𝑑𝑉 = 𝑓 𝜌 sin𝜙 cos 𝜃 ,𝜌 sin𝜙 sin 𝜃 ,𝜌 cos𝜙 𝜌q sin𝜙m�

m:

<�

<:

n�

n:𝑑𝜌𝑑𝜃𝑑𝜙

ChangeofVariables2DJacobian

𝜕 𝑥, 𝑦𝜕 𝑢, 𝑣 =

𝜕𝑥𝜕𝑢

𝜕𝑥𝜕𝑣

𝜕𝑦𝜕𝑢

𝜕𝑦𝜕𝑣

=𝜕𝑥𝜕𝑢

𝜕𝑦𝜕𝑣 −

𝜕𝑥𝜕𝑣𝜕𝑦𝜕𝑢 , 𝑥 = 𝑥 𝑢, 𝑣 ∧ 𝑦 = 𝑦 𝑢, 𝑣

𝑓 𝑥, 𝑦ß

𝑑𝐴 = 𝑓 𝑥 𝑢, 𝑣 , 𝑦 𝑢, 𝑣o

abs𝜕 𝑥, 𝑦𝜕 𝑢, 𝑣 𝑑𝑢𝑑𝑣


Note:Donotconfusethedeterminantwiththeabsolutevaluei.e.

𝜕𝑥𝜕𝑢

𝜕𝑥𝜕𝑣

𝜕𝑦𝜕𝑢

𝜕𝑦𝜕𝑣

≠𝜕 𝑥, 𝑦𝜕 𝑢, 𝑣 = abs

𝜕𝑥𝜕𝑢

𝜕𝑥𝜕𝑣

𝜕𝑦𝜕𝑢

𝜕𝑦𝜕𝑣

3DJacobian

𝐽 =

𝜕𝑥𝜕𝑢

𝜕𝑥𝜕𝑣

𝜕𝑥𝜕𝑤

𝜕𝑦𝜕𝑢

𝜕𝑦𝜕𝑣

𝜕𝑦𝜕𝑤

𝜕𝑧𝜕𝑢

𝜕𝑧𝜕𝑣

𝜕𝑧𝜕𝑤

, 𝑥 = 𝑥 𝑢, 𝑣,𝑤 ∧ 𝑦 = 𝑦 𝑢, 𝑣,𝑤 ∧ 𝑧 = 𝑧 𝑢, 𝑣,𝑤

𝑓 𝑥, 𝑦, 𝑧r

𝑑𝑉 ⇒ 𝑑𝑉 = 𝐽 𝑑𝑢𝑑𝑣𝑑𝑤, 𝐽 =𝜕 𝑥, 𝑦𝜕 𝑢, 𝑣

LineIntegralsGeneralSmooth

𝑓 𝑥, 𝑦á

𝑑𝑠 = 𝑓 𝑥 𝑡 , 𝑦 𝑡 𝑥Þ 𝑡 q + 𝑦Þ 𝑡 q©�

©:𝑑𝑡

NotSmooth

𝑓 𝑥, 𝑦á

𝑑𝑠 = 𝑓 𝑥, 𝑦á:

𝑑𝑠 + 𝑓 𝑥, 𝑦á�

𝑑𝑠 + ⋯ 𝑓 𝑥, 𝑦á�

𝑑𝑠

𝒙, 𝒚DerivativesRespectto𝑥 𝑓 𝑥, 𝑦

á𝑑𝑥 𝑓 𝑥 𝑡 , 𝑦 𝑡 𝑥Þ 𝑡

©�

©:𝑑𝑡

Respectto𝑦 𝑓 𝑥, 𝑦á

𝑑𝑦 𝑓 𝑥 𝑡 , 𝑦 𝑡 𝑦Þ 𝑡©�

©:𝑑𝑡

Note:Changingdirectionof𝑥, 𝑦 𝑓 𝑥, 𝑦

pá𝑑𝑠 = − 𝑓 𝑥, 𝑦

á𝑑𝑠

Arclength 𝑓 𝑥, 𝑦pá

𝑑𝑠 = 𝑓 𝑥, 𝑦á

𝑑𝑠


Vectorform

𝑓 𝑥, 𝑦, 𝑧á

𝑑𝑠 = 𝑓 𝑥 𝑡 , 𝑦 𝑡 , 𝑧 𝑡 𝑥Þ 𝑡 q + 𝑦Þ 𝑡 q + 𝑧Þ 𝑡 q©�

©:𝑑𝑡

∵ 𝐫 𝑡 = 𝑥 𝑡 , 𝑦 𝑡 , 𝑧 𝑡 ∧ 𝐫 𝑡 = 𝑥Þ 𝑡 q + 𝑦Þ 𝑡 q + 𝑧Þ 𝑡 q

∴ 𝑓 𝑥 𝑡 , 𝑦 𝑡 , 𝑧 𝑡 𝑥Þ 𝑡 q + 𝑦Þ 𝑡 q + 𝑧Þ 𝑡 q©�

©:𝑑𝑡 = 𝑓 𝐫 𝑡 𝐫 𝑡

©�

©:𝑑𝑡

Respectto𝒛

𝑓 𝑥 𝑡 , 𝑦 𝑡 , 𝑧 𝑡 𝑧Þ 𝑡©�

©:𝑑𝑡

MultipleFunctions𝑷,𝑸, 𝑹

𝑃 𝑥, 𝑦, 𝑧á

𝑑𝑥 + 𝑄 𝑥, 𝑦, 𝑧 𝑑𝑦 + 𝑅 𝑥, 𝑦, 𝑧 𝑑𝑧 = 𝑔r 𝑡 + 𝑔q 𝑡 + 𝑔o ©

©�

©:𝑑𝑡

WorkCaseI 𝑊 = 𝐅 𝑥, 𝑦, 𝑧 ⋅ 𝐓 𝑥, 𝑦, 𝑧

á𝑑𝑠

CaseII 𝑊 = 𝐅 ⋅ 𝐓

á𝑑𝑠

CaseIII

𝑊 = 𝐅 𝐫 𝑡 ⋅𝐫Þ 𝑡𝐫Þ 𝑡

©�

©:𝐫Þ 𝑡 𝑑𝑡

CaseIV

𝑊 = 𝐅 𝐫 𝑡 ⋅ 𝐫Þ 𝑡©�

©:𝑑𝑡

CaseV 𝑊 = 𝐅 𝐫 𝑡 ⋅ 𝑑𝐫

á

CaseVI 𝑊 = 𝑃𝑑𝑥 + 𝑄𝑑𝑦 + 𝑅𝑑𝑧

á, 𝐅 = 𝑃,𝑄, 𝑅


GradientLineIntegralCaseI:FundamentalTheorem 𝛻𝑓 ⋅ 𝑑𝐫

á= 𝑓 𝐫 𝑡q − 𝑓 𝐫 𝑡r

CaseII 𝛻𝑓 ⋅ 𝑑𝐫

á= 𝑓 xq, yq, 𝑧q − 𝑓 xr, 𝑦r, 𝑧r

CaseIII

𝛻𝑓 𝐫 𝑡 ⋅ 𝐫Þ 𝑡©�

©:𝑑𝑡 =

𝜕𝑓𝜕𝑥𝑑𝑥𝑑𝑡 +

𝜕𝑓𝜕𝑦𝑑𝑦𝑑𝑡 +

𝜕𝑓𝜕𝑧𝑑𝑧𝑑𝑡

©�

©:𝑑𝑡

CaseIV 𝑑

𝑑𝑡

©�

©:𝑓 𝐫 𝑡 𝑑𝑡 = 𝑓 𝐫 𝑡q − 𝑓 𝐫 𝑡r

ConservativeVectorField

𝛻𝑓 = 𝐅 x, y = 𝑃 x, y , 𝑄 x, y ∧ 𝜕𝑃𝜕𝑦 =

𝜕𝑄𝜕𝑥

Green’sTheorem

𝑃𝑑𝑥 + 𝑄𝑑𝑦á

=𝜕𝑄𝜕𝑥 −

𝜕𝑃𝜕𝑦h

𝑑𝐴

Curl𝜵Note:gradientof𝑓is𝛻𝑓,andcurl/divergenceof𝑓is𝛻×𝛻𝑓and𝛻 ⋅ 𝛻𝑓,where𝛻(nabla)isreferredtoasdel.

𝛻 =𝜕𝜕𝑥 𝐢 +

𝜕𝜕𝑦 𝐣+

𝜕𝜕𝑧 𝐤 ≡ 𝜕�,𝜕¦,𝜕b

𝛻×𝛻𝑓 = 𝛻×𝐅 =𝐢 𝐣 𝐤𝜕� 𝜕¦ 𝜕b𝜕�² 𝜕¦

² 𝜕b², 𝜕�

²,𝜕¦²,𝜕b

² ≡𝜕𝑓𝜕𝑥 𝐢 +

𝜕𝑓𝜕𝑦 𝐣+

𝜕𝑓𝜕 𝐤

Conservativeifcurl𝐅 = 0Divergence

𝛻 ⋅ 𝛻𝑓 = 𝛻 ⋅ 𝐅 = 𝜕�,𝜕¦,𝜕b ⋅ 𝜕�²,𝜕¦

²,𝜕b²

StokesTheorem


𝛻𝑓 ⋅ 𝑑𝐫á

= 𝐅 ⋅ 𝑑𝐫á

= ∇×𝐅o

⋅ 𝑑𝐒 = curl𝐅o

⋅ 𝑑𝐒

DivergenceTheorem

𝐅o⋅ 𝑑𝐒 = 𝛻 ⋅ 𝛻𝑓

i𝒅𝑽 = 𝛻 ⋅ 𝐅

i𝑑𝑉 = div𝐅

i𝑑𝑉

PreCalculusReviewArithmetic

𝑎𝑏 ± 𝑎𝑐 = 𝑎 𝑏 ± 𝑐 = 𝑏 ± 𝑐 𝑎𝑎𝑏𝑐 =

𝑎𝑏𝑐

𝑎𝑏 ±

𝑐𝑑 =


𝑎 − 𝑏𝑐 − 𝑑 =

𝑏 − 𝑎𝑑 − 𝑐

𝑎𝑏 + 𝑎𝑐𝑎 = 𝑏 + 𝑐, 𝑎 ≠ 0𝑎

𝑏𝑐 =

𝑎𝑏𝑐

𝑎𝑏𝑐=

𝑎1 ∙

𝑐𝑏 =

𝑎𝑐𝑏

𝑎 ± 𝑏𝑐 =

𝑎𝑐 ±

𝑏𝑐

𝑎𝑏𝑐𝑑

=𝑎𝑏 ∙𝑑𝑐 =

𝑎𝑑𝑏𝑐

Exponential

𝑎r = 𝑎𝑎2 = 1𝑎p0 =1𝑎0

1𝑎p0 = 𝑎0𝑎0𝑎� = 𝑎0��

𝑎0

𝑎� = 𝑎0p� 𝑎𝑏

0=𝑎0

𝑏0 𝑎𝑏

p0=𝑏0

𝑎0 𝑎0

r� = 𝑎

r�

0 𝑎0 � = 𝑎� 0

Radicals

𝑎��= 𝑎�� = 𝑎

r�0 𝑎0� = 𝑎, 𝑛𝑖𝑠𝑜𝑑𝑑 𝑎0� = 𝑎 , 𝑛𝑖𝑠𝑒𝑣𝑒𝑛

𝑎 = 𝑎� = 𝑎r� = 𝑎rq 𝑎�� = 𝑎

�0

𝑎𝑏

�=

𝑎�

𝑏� =𝑎r0

𝑏r0=

𝑎𝑏

r0

Fractions

𝑎𝑏 ±

𝑐𝑑 =


𝑔 𝑥𝑓 𝑥 ±

ℎ 𝑥𝑟 𝑥 =

𝑔 𝑥 𝑟 𝑥 ± 𝑓 𝑥 ℎ 𝑥𝑓 𝑥 𝑟 𝑥


Logarithmicln 𝑏ln 𝑎 = log� 𝑏 𝑦 = log� 𝑥 ⇔ 𝑥 = 𝑏¦𝑒 ≈ 2.72 log� 𝑎 = 1

log� 1 = 0 log� 𝑎3 = 𝑢 log¤ 𝑢 = ln 𝑢 log� 𝑢� = 𝑏 log� 𝑢

log� 𝑢𝑣 = log� 𝑢 + log� 𝑣 log�𝑢𝑣 = log� 𝑢 − log� 𝑣 log� 𝑏 =

ln 𝑏ln 𝑎

𝑣 = ln 𝑢 ⇒ 𝑢 = 𝑒4𝑣 = 𝑒3 ⇒ 𝑢 = ln 𝑣 𝑒 =1𝑛!

«

012

ln 𝑎 = undefined, 𝑎 ≤ 0 ln 1 = 0 ln 𝑒3 = 𝑢 ⇒ 𝑒®¯3 = 𝑢ln 𝑒r = 1 ⇒ 𝑒®¯ r = 1 ln 𝑢� = 𝑏 ln 𝑢 ln 𝑢𝑣 = ln 𝑢 + ln 𝑣 ln

𝑢𝑣 = ln 𝑢 − ln 𝑣

OtherFormulas/Equations

QuadraticFormula

𝑎𝑥q + 𝑏𝑥 + 𝑐 = 0 ⇒ 𝑥 =−𝑏 ± 𝑏q − 4𝑎𝑐

2𝑎

DiscriminantTwoRealSolutions𝑏q − 4𝑎𝑐 > 0RepeatedSolution𝑏q − 4𝑎𝑐 = 0ComplexSolution 𝑥 = 𝛼 ± 𝛽𝑖 if𝑏q − 4𝑎𝑐 < 0

CompletetheSquare

𝑦 = 𝑎𝑥q + 𝑏𝑥 + 𝑐 ⇒ 𝑦 = 𝑎 𝑥 +𝑏2𝑎

q

+ 𝑐 −𝑏q

4𝑎


OtherFormulasDistanceFormula

MidpointFormula

𝐷 = 𝑥 − 𝑥2 q + 𝑦 − 𝑦2 q

𝑀 =𝑥 + 𝑥22 ,

𝑦 + 𝑦22

EquationofaLine

𝑠𝑙𝑜𝑝𝑒 = 𝑚 =𝑦q − 𝑦r𝑥q − 𝑥r

𝑦 = 𝑚𝑥 + 𝑏

𝑦q − 𝑦r = 𝑚 𝑥q − 𝑥r

𝐴𝑥 + 𝐵𝑦 = 𝐶

EquationofParabolaVertex: ℎ, 𝑘


𝑦 = 𝑎 𝑥 − ℎ q + 𝑘

EquationofCircleCenter: ℎ, 𝑘 Radius:𝑟

𝑥 − ℎ q + 𝑦 − 𝑘 q = 𝑟q

EquationofEllipse

RightPoint: ℎ + 𝑎, 𝑘

LeftPoint: ℎ − 𝑎, 𝑘

TopPoint: ℎ, 𝑘 + 𝑏

BottomPoint: ℎ, 𝑘 − 𝑏

𝑥 − ℎ q

𝑎q +𝑦 − 𝑘 q

𝑏q = 1

EquationofHyperbolaCenter: ℎ, 𝑘 Slope:± �

�


Vertices: ℎ + 𝑎, 𝑘 , ℎ − 𝑎, 𝑘

𝑥 − ℎ q

𝑎q −𝑦 − 𝑘 q

𝑏q = 1

EquationofHyperbolaCenter: ℎ, 𝑘 Slope:± �

�


Vertices: ℎ, 𝑘 + 𝑏 , ℎ, 𝑘 − 𝑏

𝑦 − 𝑘 q

𝑎q −𝑥 − ℎ q

𝑏q = 1


AreasSquare:𝐴 = 𝐿q = 𝑊qRectangle:𝐴 = 𝐿 ∙ 𝑊Circle:𝐴 = 𝜋 ∙ 𝑟qEllipse:𝐴 = 𝜋 ∙ 𝑎𝑏Triangle:𝐴 = r

q𝑏 ∙ ℎTrapezoid:𝐴 = r

q𝑎 + 𝑏 ∙ ℎ

Parallelogram:𝑏 ∙ ℎRhombus:𝐴 = ÙÚ

q,𝑝and𝑞arethediagonals

SurfaceAreas

Cube:𝐴Û = 6𝐿q = 6𝑊qBox:𝐴Û = 2(𝐿𝑊 +𝑊𝐻 +𝐻𝐿)Sphere:𝐴Û = 4𝜋𝑟qCone:𝐴Û = 𝜋𝑟 𝑟 + ℎq + 𝑟q Cylinder:2𝜋𝑟ℎ + 2𝜋𝑟q

VolumesCube:𝑉 = 𝐿o = 𝑊oBox:𝑉 = 𝐿 ∙ 𝑊 ∙ 𝐻Sphere:𝑉 = Ü

o𝜋 ∙ 𝑟o

Cone:𝑉 = r

o𝜋 ∙ 𝑟qℎEllipsoid:𝑉 = Ü

o𝜋 ∙ 𝑎𝑏𝑐,𝑎, 𝑏, 𝑐aretheradii

DomainRestrictions

𝑦 =

𝑢𝑣 , 𝑣 ≠ 0𝑦 = 𝑢, 𝑢 ≥ 0𝑦 = ln 𝑢 , 𝑢 > 0

𝑦 = 𝑎3,none𝑦 = 𝑢� noneif𝑛isodd,𝑢 ≥ 0if𝑛iseven


RightTriangle

𝑥q + 𝑦q = 𝑟q ⇔ 𝑟 = 𝑥q + 𝑦q

cos 𝛼 =

𝑥𝑟 cos 𝛽 =

𝑦𝑟

tan 𝛼 =

𝑦𝑥 tan 𝛽 =

𝑥𝑦

sin 𝛼 =

𝑦𝑟 sin 𝛽 =

𝑥𝑟

𝑥 = 𝑟 cos 𝛼 𝑦 = 𝑟 cos 𝛽𝑦 = 𝑟 sin 𝛼 𝑥 = 𝑟 sin 𝛽

𝛼 = arctan𝑦𝑥 = tanpr

𝑦𝑥 𝛽 = arctan

𝑥𝑦 = tanpr

𝑥𝑦

ReciprocalIdentities

sin 𝜃 =1

csc 𝜃 csc 𝜃 =1

sin 𝜃 tan 𝜃 =1

cot 𝜃

csc 𝜃 =1

sec 𝜃 sec 𝜃 =1

cos 𝜃 cot 𝜃 =1

tan 𝜃

tan 𝜃 =sin 𝜃cos 𝜃 cot 𝜃 =

cos 𝜃sin 𝜃


DoubleAngleFormulassin 2𝜃 = 2 sin 𝜃 cos 𝜃cos 2𝜃 = 1 − 2 sinq 𝜃cos 2𝜃 = cosq 𝜃 − sinq 𝜃 cos 2𝜃 = 2 cosq 𝜃 − 1cos 2𝜃 = 1 − 2 sinq 𝜃 tan 2𝜃 = q -ò¯<

rp-ò¯� <Officia

HalfAngleFormulas

sinq 𝜃 =12 1 − cos 2𝜃 cosq 𝜃 =

12 1 + 𝑐𝑜𝑠 2𝜃 tanq 𝜃 =

1 − cos(2𝜃)1 + cos(2𝜃)

SumandDifferenceFormulas

sin 𝛼 ± 𝛽 = sin 𝛼 cos 𝛽 ± cos 𝛼 sin 𝛽

cos(𝛼 ± 𝛽) = cos 𝛼 cos 𝛽 ∓ sin 𝛼 sin 𝛽

tan 𝛼 ± 𝛽 =tan𝛼 ± tan𝛽1 ∓ tan𝛼 𝑡𝑎𝑛𝛽

ProducttoSumFormulas

sin 𝛼 sin 𝛽 =12 [cos 𝛼 − 𝛽 − cos(𝛼 + 𝛽)]

cos 𝛼 cos 𝛽 =

12 [cos 𝛼 − 𝛽 + cos(𝛼 + 𝛽)]

sin 𝛼 cos 𝛽 =12 [sin 𝛼 + 𝛽 + sin 𝛼 − 𝛽 ]

cos 𝛼 sin 𝛽 =

12 sin 𝛼 + 𝛽 − sin 𝛼 − 𝛽

SumtoProductFormulas

sin 𝛼 + sin 𝛽 = 2 sin𝛼 + 𝛽2 cos

𝛼 − 𝛽2

sin 𝛼 − sin 𝛽 = 2 cos

𝛼 + 𝛽2 sin

𝛼 − 𝛽2

cos 𝛼 + cos 𝛽 = 2 cos𝛼 + 𝛽2 cos

𝛼 − 𝛽2

cos 𝛼 − cos 𝛽 = −2 sin

𝛼 + 𝛽2 sin

𝛼 − 𝛽2


UnitCircle


Pre-CALCIIIReferenceDerivativeRules(primenotations)

DerivativeofaConstant

𝑐 Þ = 0

PowerRule 𝑥0 ′ = 𝑛𝑥0pr


𝑐𝑢 Þ = 𝑐𝑢′

ProductRule 𝑢𝑣 Þ = 𝑢𝑣Þ + 𝑣𝑢′

QuotientRule

𝑢𝑣

Þ=𝑣𝑢Þ − 𝑢𝑣′

𝑣q

ChainRule

[𝑢 𝑣 ]′ = 𝑢Þ 𝑣 ∙ 𝑣′

ExponentialandLogarithmic

Operator Primeexp{u} 𝑑

𝑑𝑥 𝑒² � = 𝑒² � ∙ 𝑓Þ 𝑥

𝑒3 Þ = 𝑒3 ⋅ 𝑢′

NaturalLog 𝑑𝑑𝑥 ln 𝑓 𝑥 =


ln 𝑢 Þ =𝑢Þ

𝑢

BaseLogNote:log� 𝑎 ≡

®¯ �®¯ �

𝑑𝑑𝑥 log� 𝑓 𝑥 =

1ln 𝑏 ⋅


log� 𝑢 Þ =1ln 𝑏 ⋅

𝑢Þ

𝑢


² � = 𝑎² � 𝑓Þ 𝑥 ln 𝑎

𝑎3 Þ = 𝑎3𝑢Þ ln 𝑎

InverseFunctionDerivative

𝑑𝑑𝑥 𝑓

pr 𝑥�=

1𝑓Þ 𝑓pr 𝑎

, 𝑓pr 𝑎 = 𝑏 ⇔ 𝑓 𝑏 = 𝑎


TrigDerivativesStandard

sin 𝑢 Þ = cos 𝑢 ∙ 𝑢Þ cos 𝑢 Þ = − sin 𝑢 ∙ 𝑢Þ tan 𝑢 Þ = secq 𝑢 ∙ 𝑢Þ

csc 𝑢 Þ = − csc 𝑢 cot 𝑢 ∙ 𝑢Þ sec 𝑢 Þ = sec 𝑢 tan 𝑢 ∙ 𝑢Þ cot 𝑢 Þ = − cscq 𝑢 ∙ 𝑢′

Inverse

sinpr 𝑢 Þ =𝑢′1 − 𝑢q

cospr 𝑢 Þ = −𝑢′1 − 𝑢q

tanpr 𝑢 Þ =𝑢′

1 + 𝑢q

cscpr 𝑢 Þ = −𝑢′

𝑢 𝑢q − 1 secpr 𝑢 Þ =

𝑢′𝑢 𝑢q − 1

cotpr 𝑢 Þ = −𝑢′

1 + 𝑢q

CommonDerivativesOperator𝑑𝑑𝑥 𝑦 =

𝑑𝑦𝑑𝑥

𝑑𝑑𝑥 𝑥

0 = 𝑛𝑥0pr𝑑𝑑𝑥 𝑦

0 = 𝑛𝑦0pr𝑑𝑦𝑑𝑥

𝑑𝑑𝑥 𝑒

� = 𝑒�𝑑𝑑𝑥 𝑒

² � = 𝑒² � 𝑓Þ 𝑥 𝑑𝑑𝑥 ln 𝑥 =

1𝑥

𝑑𝑑𝑥 ln 𝑓 𝑥 =


𝑑𝑑𝑥 𝑎

� = 𝑎� ln 𝑎𝑑𝑑𝑥 𝑎

² � = 𝑎² � 𝑓Þ 𝑥 ln 𝑎

𝑑𝑑𝑥 sin 𝑥 = cos 𝑥

𝑑𝑑𝑥 csc 𝑥 = −csc 𝑥 cot 𝑥

𝑑𝑑𝑥 cos 𝑥 = −sin 𝑥

𝑑𝑑𝑥 (sec 𝑥) = sec 𝑥 tan 𝑥

𝑑𝑑𝑥 tan 𝑥 = secq 𝑥

𝑑𝑑𝑥 cot 𝑥 = −cscq 𝑥

𝑑𝑑𝑥 sin

pr 𝑥 =1

1 − 𝑥q

𝑑𝑑𝑥 csc

pr 𝑥 =−1

𝑥 𝑥q − 1

𝑑𝑑𝑥 cos

pr 𝑥 =−11 − 𝑥q

𝑑𝑑𝑥 sec

pr 𝑥 =1

𝑥 𝑥q − 1

𝑑𝑑𝑥 tan

pr 𝑥 =1

1 + 𝑥q𝑑𝑑𝑥 cot

pr 𝑥 =−1

1 + 𝑥q

𝑑𝑑𝑥 sinh 𝑥 = cosh 𝑥

𝑑𝑑𝑥 csch 𝑥 = −csch 𝑥 coth 𝑥

𝑑𝑑𝑥 cosh 𝑥 = sinh 𝑥

𝑑𝑑𝑥 sech 𝑥 = −sech 𝑥 tanh 𝑥

𝑑𝑑𝑥 tanh 𝑥 = sechq 𝑥

𝑑𝑑𝑥 coth 𝑥 = −cschq 𝑥


Prime𝑒3 Þ = 𝑢Þ𝑒3

ln 𝑢 Þ =𝑢Þ

𝑢 𝑎3 Þ = 𝑢Þ𝑎3 ln 𝑎

sin 𝑢 Þ = 𝑢Þ cos 𝑢 cos 𝑢 Þ = −𝑢Þ sin 𝑢 tan 𝑢 Þ = 𝑢Þ secq 𝑢

csc 𝑢 Þ = −𝑢Þ csc 𝑢 cot 𝑢 sec 𝑢 Þ = 𝑢Þ sec 𝑢 tan 𝑢 cot 𝑢 Þ = −𝑢Þ cscq 𝑢

arcsin 𝑢 Þ =𝑢Þ

1 − 𝑢q arccos 𝑢 Þ =

−𝑢Þ

1 − 𝑢q arctan 𝑢 Þ =

𝑢Þ

1 + 𝑢q

arccsc 𝑢 Þ =−𝑢Þ

𝑢 𝑢q − 1 arcsec 𝑢 Þ =

𝑢Þ

𝑢 𝑢q − 1 arccot 𝑢 Þ =

−𝑢Þ

1 + 𝑢q

ImplicitDifferentiation𝑑𝑑 𝒙 𝒚

Alwayspayattentiontothevariables

𝑑𝑦𝑑𝑥 = 𝑦Þ

𝑑𝑑𝑥 𝑦

q 2 𝑦 qpr 𝑑𝑑𝑥 𝑦 = 2𝑦𝑦′

Chain/PowerRule 𝑑𝑑𝑥 𝑦

0 = 𝑛𝑦0pr𝑑𝑦𝑑𝑥 ≡ 𝑛𝑦0pr𝑦′

Chain/Product 𝑑𝑑𝑥 𝑥𝑦 = 𝑥

𝑑𝑦𝑑𝑥 + 𝑦

𝑑𝑥𝑑𝑥 ≡ 𝑥𝑦Þ + 𝑦

Chain/Quotient𝑑𝑑𝑥

𝑥𝑦 =

𝑦 𝑑𝑥𝑑𝑥 − 𝑥𝑑𝑦𝑑𝑥

𝑦q ≡𝑦 − 𝑥𝑦′𝑦q

Logarithmic 𝑑𝑑𝑥 ln 𝑦 =

𝑦Þ

𝑦


¦ = 𝑦Þ𝑎¦ ln 𝑎

Euler’sNumber 𝑑𝑑𝑥 𝑒

¦ = 𝑦Þ𝑒¦

Trigonometric 𝑑𝑑𝑥 sin 𝑦 = cos 𝑦 ⋅

𝑑𝑦𝑑𝑥 = cos 𝑦 ⋅ 𝑦′

TangentLine

𝑓 𝑥, 𝑦 = 0, 𝑃 𝑎, 𝑏 ⇒ 𝑦0 = 𝑓Þ 𝑎, 𝑏 𝑥 − 𝑎 + 𝑏


RelatedRatesTheideaforrelatedrates,ingeneral,istofindtheequationthatrelatesgeometricallytothequestion,implicitlydifferentiateit,andthenpluginthegivenvariablesandsolvefortheunknown.Hereareafewexamplesi.e.justusetheequation/formulathatmimicstheobjectinquestion.Righttriangle 𝑎q + 𝑏q = 𝑐q ⇒ 𝑎𝑎Þ 𝑡 + 𝑏𝑏Þ 𝑡 = 𝑐𝑐Þ 𝑡

Circle 𝐴 = 𝜋𝑟q ⇒ 𝑑𝐴𝑑𝑡 = 2𝜋𝑟𝑟Þ𝑟 𝑡

Sphere 𝑉 =43𝜋𝑟

o ⇒ 𝑉Þ 𝑡 = 4𝜋𝑟q𝑑𝑟𝑑𝑡

HyperbolicFunctions

Notation


2 csch 𝑥 =2

𝑒� + 𝑒p� tanh 𝑥 =𝑒� − 𝑒p�

𝑒� + 𝑒p�

sech 𝑥 =2

𝑒� + 𝑒p� cosh 𝑥 =𝑒� + 𝑒p�

2 coth 𝑥 =𝑒� + 𝑒p�

𝑒� − 𝑒p�

Identities





sinhpr 𝑥 = ln 𝑥 + 𝑥q + 1 , −∞ ≤ 𝑥 ≤∞


tanhpr 𝑥 =12 ln

1 + 𝑥1 − 𝑥 , −1 < 𝑥 < 1


DerivativesStandardsinh 𝑢 Þ = 𝑢′ cosh 𝑢 cosh 𝑢 Þ = 𝑢Þ sinh 𝑢 tanh 𝑢 Þ = 𝑢Þ sechq 𝑢

csch 𝑢 Þ = −𝑢Þ csch 𝑢 coth 𝑢 sech 𝑢 Þ = −𝑢Þ sech 𝑢 tanh 𝑢 coth 𝑢 Þ = −𝑢Þ cschq 𝑢

Inverse

sinhpr 𝑢 Þ =𝑢Þ

1 + 𝑢q coshpr 𝑢 Þ =

𝑢Þ

𝑢q − 1 tanhpr 𝑢 Þ =

𝑢Þ

1 − 𝑢q

cschpr 𝑢 Þ = −𝑢Þ

𝑢 1 + 𝑢q sechpr 𝑢 Þ = −

𝑢Þ

𝑢 1 − 𝑢q cothpr 𝑢 Þ =

𝑢Þ

1 − 𝑢q


1𝑛 + 1𝑥

0 + 𝐶 ⇔ 𝑛 ≠ −1


𝑎�

ln 𝑎 + 𝐶



𝑎𝑥 + 𝑏 ⇒ 𝑦 =1𝑎 ln 𝑎𝑥 + 𝑏 + 𝐶




�� + 𝐶


�� + 𝐶





𝐴 ≈ lim0→∞

𝑓 𝑥8∗0

81r

𝛥𝑥 , 𝛥𝑥 =𝑏 − 𝑎𝑛 , 𝑥8 = 𝑎 + 𝑖 ∙ 𝛥𝑥

𝑐0

81r

= 𝑐𝑛 𝑖0

81r

=𝑛 𝑛 + 1

2

𝑐𝑓 𝑥8

0

81r

= 𝑐 𝑓 𝑥8

0

81r

𝑖q0

81r

=𝑛 𝑛 + 1 2𝑛 + 1

6

𝑓 𝑥8 ± 𝑔 𝑥8

0

81r

= 𝑓 𝑥8

0

81r

± 𝑔 𝑥8

0

81r

𝑖o0

81r

=𝑛 𝑛 + 1

2

q


𝑓 𝑥�

�𝑑𝑥 ≈



𝑥q + 𝑥o2 +⋯

TrapezoidRule

𝑓 𝑥�

�𝑑𝑥 ≈




lim0→∞

𝑓(𝑥8∗)0

81r


�𝑑𝑥


𝑓 𝑥�


�

�𝑑𝑥 = 𝑐 𝑏 − 𝑎

𝑓 𝑥�


�


�

�𝑑𝑥

𝑓 𝑥�

p�𝑑𝑥 = 0




�


�

�𝑑𝑥

𝑓 𝑥�


�

2


𝑓 𝑥�


𝒌


�

𝒌𝑑𝑥

NOTE:


𝑓 𝑥�

�𝑑𝑥 = − 𝑓 𝑥

�

�𝑑𝑥

FundamentalTheorems



3⇒ 𝑦Þ = 𝑓 𝑣 ∙ 𝑣Þ − 𝑓 𝑢 ∙ 𝑢′








𝑓(𝑥8∗)0

81r



𝛥𝑥 =𝑏 − 𝑎𝑛 , 𝑥8 = 𝑎 + 𝑖 ∙ 𝛥𝑥




CommonIntegrals


q + 𝐶


o + 𝐶 𝑥0 𝑑𝑥 =1

𝑛 + 1𝑥0�r + 𝐶

⇔ 𝑛 ≠ −1

1𝑥 𝑑𝑥 = ln |𝑥| + 𝐶

𝑒� 𝑑𝑥 = 𝑒� + 𝐶 𝑒�� 𝑑𝑥 =1𝑎 𝑒

�� + 𝐶 𝑒�� 𝑑𝑥 =1𝑎 𝑒

�� + 𝐶

1𝑥 + 1𝑑𝑥 = ln 𝑥 + 1 + 𝐶





�= 𝐹 𝑏 − 𝐹 𝑎



𝑢Þ


−𝑢Þ


𝑢Þ



DefiniteIntegralRulesSubstitution

𝑓 𝑔 𝑥 𝑔Þ 𝑥�


È �

È �𝑑𝑢

IntegrationbyParts



� − 𝑔 𝑥 𝑓Þ 𝑥�

�𝑑𝑥


𝑢�


� − 𝑣�

�𝑑𝑢

TrigSubstitution𝑎q − 𝑥q 𝑎q + 𝑥q 𝑥q − 𝑎q

1 − sinq 𝜃 = cosq 𝜃 1 + tanq 𝜃 = secq 𝜃 secq 𝜃 − 1 = tanq 𝜃

𝑥 = 𝑎 sin 𝜃 𝑥 = 𝑎 tan 𝜃 𝑥 = 𝑎 sec 𝜃

𝜃 ∈ −𝜋2 ,𝜋2 𝜃 ∈ −

𝜋2 ,𝜋2 𝜃 ∈ 0,

𝜋2 ∨ 𝜃 ∈ 𝜋,

3𝜋2

TrigIdentity

tan 𝑥 𝑑𝑥 =sin 𝑥cos 𝑥 𝑑𝑥 = −

1cos 𝑥 ⋅ − sin 𝑥 𝑑𝑥,

𝑑𝑑𝑥 ln 𝑢 𝑥 =

1𝑢𝑑𝑢𝑑𝑥

= − ln cos 𝑥 + 𝐶 = ln1

cos 𝑥 + 𝐶 = ln sec 𝑥 + 𝐶

PartialFractions𝑝 𝑥

𝑥 𝑥 + 1 =𝐴𝑥 +

𝐵𝑥 + 1

𝑝 𝑥𝑥q 𝑥 + 1 =

𝐴𝑥 +

𝐵𝑥q +

𝐶𝑥 + 1

𝑝 𝑥𝑥 𝑥q + 1 =

𝐴𝑥 +

𝐵𝑥 + 𝐶𝑥q + 1

𝑝 𝑥𝑥 𝑥q + 1 q =

𝐴𝑥 +

𝐵𝑥 + 𝐶𝑥q + 1 +

𝐷𝑥 + 𝐸𝑥q + 1 q


PHYSICSINFOBasicsymbolsNote:Aboldletteri.e.𝒗isthesameassaying𝑣.Vector-hatsareusuallydonebyhand,andboldaregenerallyusedinprint(probablybecausethevectorhatwasnotonatypewriterinthepast).Bothwillbeusedthroughoutthistext.Time 𝑡Position

𝑟 𝑡 = 𝒓 𝑡 = 𝑟r 𝑡 , 𝑟q 𝑡 , 𝑟o 𝑡 𝑟 𝑡 = 𝑠

Velocity 𝑣 = 𝒗 =

𝑑𝑟𝑑𝑡 =

𝑑𝒓𝑑𝑡 = 𝑟′r 𝑡 , 𝑟′q 𝑡 , 𝑟′o 𝑡

𝑣 = 𝒗 =𝑑𝑠𝑑𝑡 = 𝑠Þ 𝑡 = 𝑠 = 𝑣 𝑡 = 𝑣

Acceleration 𝑎 = 𝒂 =

𝑑𝑣𝑑𝑡 =

𝑑𝒗𝑑𝑡 =

𝑑q𝑟𝑑𝑡q =

𝑑q𝒓𝑑𝑡q = 𝑟′′r 𝑡 , 𝑟′′q 𝑡 , 𝑟′′o 𝑡

𝑎 = 𝒂 = 𝒗′ =𝑑q𝑠𝑑𝑡q = 𝑠 =

𝑑𝑣𝑑𝑡 = 𝑣 = 𝑣Þ 𝑡 = 𝑎 𝑡 = 𝑎

DerivingformulasStartingwithconstantacceleration

𝑑𝑣𝑑𝑡 = 𝑎 ⇒ 𝑑𝑣 = 𝑎𝑑𝑡 ⇒ 𝑑𝑣 = 𝑎 𝑑𝑡 ⇒ 𝑣 = 𝑎𝑡 + 𝑣2

Nowvelocity

𝑑𝑠𝑑𝑡 = 𝑣 = 𝑎𝑡 + 𝑣2 ⇒ 𝑑𝑠 = 𝑎𝑡 + 𝑣2 𝑑𝑡 ⇒ 𝑠 =

12𝑎𝑡

q + 𝑣2𝑡 + 𝑠2

𝑠 =12𝑎𝑡

q + 𝑣2𝑡 + 𝑠2 =𝛥𝑥 = 𝑣2,�𝑡 +

12𝑎�𝑡

q

𝛥𝑦 = 𝑣2,¦𝑡 +12𝑎¦𝑡

q

Mostformulasarealreadyderivedinaphysicsbooksoeventhoughcalculusmaybeaprerequisite,itmaynotreallybeusedinproblemsolving.


UnitsSystemInternationalUnits(S.I.Units)Meters(m) Seconds(s) Kilograms(kg)Note:Unitsarenotitalicized,asvariablesaree.g.𝑚 = 16kgand𝛥𝑥 = 10m:𝑚isthevariableformass,wheremistheunitofmeasurementformeters.UnitconversionUnitconversionisprettystraightforward;let’slookatanexample.ExampleConvert100meterspersecondtoinchesperhoursFindtheappropriaterelationsandsolvefor1sothatwhentheproductcancelsouttheunits:

1m = 100cm ⇒ 1 =100cm1m 1in = 2.54cm ⇒ 1 =

1in2.54cm 3600s = 1hr ⇒ 1 =

3600s1hr

100ms100cm1m

1in2.54cm

3600s1hr =

100 ⋅ 100 ⋅ 36002.54

inhr =

100 ⋅ 100 ⋅ 36002.54

inhr = 1.42×10ë

inhr

Tosumupunitconversion,justmakesureyouhavetheappropriaterelations,andthensetthemequalto1(1multipliedbyanythingisstillthesamething)intheorderofcancelingunits.

VectorsNotation

𝑎 = 𝑎r, 𝑎q in2Dor𝑎 = 𝑎r, 𝑎q, 𝑎o in3D

Addition/Subtraction

𝑎 ± 𝑏 = 𝑎r, 𝑎q ± 𝑏r, 𝑏q = 𝑎r ± 𝑏r, 𝑎q ± 𝑏q

𝑎 ± 𝑏 = 𝑎r, 𝑎q, 𝑎o ± 𝑏r, 𝑏q, 𝑏o = 𝑎r ± 𝑏r, 𝑎q ± 𝑏q, 𝑎o ± 𝑏o Visually


DotProduct

𝑎 ⋅ 𝑏 = 𝑎r, 𝑎q ⋅ 𝑏r, 𝑏q = 𝑎r𝑏r + 𝑎q𝑏q

𝑎 ⋅ 𝑏 = 𝑎r, 𝑎q, 𝑎o ⋅ 𝑏r, 𝑏q, 𝑏o = 𝑎r𝑏r + 𝑎q𝑏q + 𝑎o𝑏oCrossProduct

𝑎×𝑏 = −𝑏×𝑎

𝑎×𝑏 =𝚤 𝚥 𝑘𝑎r 𝑎q 𝑎o𝑏r 𝑏q 𝑏o

= 𝚤

𝑎q 𝑎o𝑏q 𝑏o − 𝚥

𝑎r 𝑎o𝑏r 𝑏o + 𝑘

𝑎r 𝑎q𝑏r 𝑏q

= 𝚤 𝑎q 𝑏o − 𝑎o 𝑏q − 𝚥 𝑎r 𝑏o − 𝑎o 𝑏r + 𝑘 𝑎r 𝑏q − 𝑎q 𝑏r

𝚤, 𝚥,and𝑘arecalledunitvectors.Aunitvector,isavectoroflength1

𝚤 = 1, 0, 0 , 𝚥 = 0, 1, 0 , 𝑘 = 0, 0, 1

= 𝑎q 𝑏o − 𝑎o 𝑏q , 0, 0 − 0, 𝑎r 𝑏o − 𝑎o 𝑏r , 0 + 0, 0, 𝑎r 𝑏q − 𝑎q 𝑏r

= 𝑎q 𝑏o − 𝑎o 𝑏q , 𝑎r 𝑏o − 𝑎o 𝑏r , 𝑎r 𝑏q − 𝑎q 𝑏r


MagnitudeorLengthofavectorAboldletterisavectori.e.𝑎 = 𝒂 = 𝑎r, 𝑎q, 𝑎o

2𝐷, 𝒂 = 𝑎 = 𝑎 = 𝑎rq + 𝑎qq

3𝐷, 𝑎 = 𝒂 = 𝑎 = 𝑎rq + 𝑎qq + 𝑎oq

UnitizingavectorTomakethevectorbeoflength1butpreservethedirection.

2𝐷, 𝑎 =𝑎𝑎 =

𝑎r, 𝑎q𝑎rq + 𝑎qq

3𝐷, 𝑎 =𝑎𝑎 =

𝑎r, 𝑎q, 𝑎o𝑎rq + 𝑎qq + 𝑎oq

ResultantVector

𝑅 = 𝑎 + 𝑏Inphysicsyouwillbeusuallybegiventhevectore.g.(e.g.=forexample)𝑣(𝑣=velocity)Theresultantvector,𝑣wouldbeavectorthatcanbebrokenintoa𝑥and𝑦component.Anglewithrespecttox-axis Anglewithrespecttoy-axis

𝑣 = 𝑣 cos 𝜃 , 𝑣 sin 𝜃 𝑣 = 𝑣 sin𝜙 , 𝑣 cos𝜙


𝑣,𝑣� = 𝑣 cos 𝜃 , 𝑣� = 𝑣 cos 𝜃 , 0𝑣¦ = 𝑣 sin 𝜃 , 𝑣¦ = 0, 𝑣 sin 𝜃

𝑣,𝑣� = 𝑣 sin𝜙 , 𝑣� = 𝑣 sin𝜙 , 0𝑣¦ = 𝑣 cos𝜙 , 𝑣¦ = 0, 𝑣 cos𝜙

𝑅 = 𝑣 = 𝑣� + 𝑣¦ = 𝑣 cos 𝜃 , 0 + 0, 𝑣 sin 𝜃 = 𝑣 cos 𝜃 , 𝑣 sin 𝜃 𝑣 = 𝑣 cos 𝜃 q + 𝑣 sin 𝜃 q = 𝑣q cosq 𝜃 + 𝑣q sinq 𝜃 = 𝑣q cosq 𝜃 + sinq 𝜃 = 𝑣q 1 = 𝑣

Thismaybeslightlyconfusingwiththenotationbecauseofthevectorsbutinphysics,youwillbegivenanumberforthevectori.e.𝑣 = −25ÿ

/,𝜃 = 25°(avectorhasmagnitudeanddirection,

whichmeansitcanbe𝑣 = −25ÿ/cos 25° , −25ÿ

/sin 25° orformagnitude 𝑣 = 𝑣 = 25ÿ

/.

SummingitupSincewearealwaysworkingwithnumbersingeneral,notvectornotationi.e. 𝑎r, … ,andweareinaphysicscourse,wecansimplyrefertodistance,velocity,andaccelerationwithoutanyvectorhatsi.e.𝑣 …sovelocityis𝑣,accelerationis𝑎,anddistanceiseither𝑥or𝑦(incalculusdistanceis𝑠).Forthetimebeing.

𝑣,𝑣� = 𝑣 cos 𝜃𝑣¦ = 𝑣 sin 𝜃 , 𝑎,

𝑎� = 𝑎 cos 𝜃𝑎¦ = 𝑎 sin 𝜃 , 𝛥𝑥 = 𝑥 − 𝑥2, 𝛥𝑦 = 𝑦 − 𝑦2

Forsomevector𝐴,𝐴� = 𝐴 cos 𝜃𝐴¦ = 𝐴 sin 𝜃

Resultantvector:𝑅 = ∑𝐴� q + ∑𝐴¦q,𝐴isanyvector

Theexampleonthenextpageisaproblemusuallyfoundaroundchapters3-5.TheproblemdemonstrateshowtorelateafreebodydiagramtoaCartesiancoordinatesystem.Thisisextremelyimportanttounderstand.Moststudentsstrugglewiththis,somakesuretotakethetimetounderstanditfully.


FreeBodyDiagramA250-Nforceisdirectedhorizontallytopusha29-kgboxupaninclinedplaneataconstantspeed.Determinethemagnitudeofthenormalforce,FN,andthecoefficientofkineticfriction.Theangleoftheinclineis27degrees.Step1)Identifytheunknowns,andthegivens.Given Angle 𝜃 = 27°Force 𝐹 = 250N Gravity 𝑔 = 9.81

msq

Mass 𝑚 = 29kg Weight 𝑤 = 29 ∗ 9.81N

Step2)FreeBodyDiagram.

FreeBody GraphRelation

Step3)Locateallrelatedequations. 𝐹 = 0 ∧ 𝜇Ö =

²x0

𝒙-direction 𝒚-direction𝐹� = 𝐹 cos 𝜃r = 250 cos 333°N 𝐹¦ = 𝐹 sin 𝜃r = 250 sin 333°N

𝑤� = 𝑚𝑔 cos 𝜃q = 29 9.81 cos 243°N 𝑤¦ = 𝑚𝑔 sin 𝜃q = 29 9.81 sin 243°N

𝐹� = 𝑓Ö + 𝐹 cos 𝜃r + 𝑚𝑔 cos 𝜃q = 0

⇒ 𝑓Ö = − 250 cos 333° + 29 9.81 cos 243° N

⇒ 𝐹Ö ≈ −93.59587378N ⇒ 𝐹Ö = 𝑓Ö = 94N

𝐹¦ = 𝑛 + 𝐹 sin 𝜃r + 𝑚𝑔 sin 𝜃q = 0

⇒ 𝑛 = − 250 sin 333° + 29 9.81 sin 243° N

⇒ 𝑛 ≈ 366.980071 ⇒ 𝑛 = 367N

Step4)Pluginallvalues

∴ 𝑛 = 367N, 𝜇Ö =𝐹Ö𝑛 =

𝑓Ö𝑛 =

𝐹 cos 𝜃r + 𝑚𝑔 cos 𝜃q𝐹 sin 𝜃r + 𝑚𝑔 sin 𝜃q

=−93.59587378366.980071 ≈ 0.26


Averagevelocity(straight-line)

𝑣òy ≡ 𝑣 𝑣bar =𝛥𝑠𝛥𝑡 =

𝑠 − 𝑠2𝑡 − 𝑡2

=𝑠q − 𝑠r𝑡q − 𝑡r

Note:"𝑠"(italicized)isthestandardvariablefordistance/displacementincalculus,where"s"(notitalicized)istheunitfortime𝑡.Ofteninphysics,“𝑑”willbeusedfordistance/displacementbutincalculus,the𝑑isreservedforderivatives,whichiswhywewillstickwith𝑥and𝑦torepresentdistances.𝒙-direction y-direction

𝑣� ≡ 𝑣òyp� =𝑥 − 𝑥2𝑡 − 𝑡2

=𝛥𝑥𝛥𝑡 𝑣¦ ≡ 𝑣òyp¦ =

𝑦 − 𝑦2𝑡 − 𝑡2

=𝛥𝑦𝛥𝑡

Note:𝑥2maybereferredtoasx-initial,x-naughtorx-subzero.Instantaneousvelocity(Calculus)LimitDefinition 𝑣 = lim

â©→2

𝛥𝑠𝛥𝑡 , straight − line

𝑣 = limâ©→2

𝑠 𝑡 + 𝛥𝑡 − 𝑠 𝑡𝛥𝑡 , curve

𝒙-direction 𝑣� = limâ©→2

𝛥𝑥𝛥𝑡 , straight − line

= limâ©→2

𝑥 𝑡 + 𝛥𝑡 − 𝑥 𝑡𝛥𝑡

y-direction 𝑣¦ = limâ©→2

𝛥𝑦𝛥𝑡 , straight − line

= limâ©→2

𝑦 𝑡 + 𝛥𝑡 − 𝑦 𝑡𝛥𝑡 , curve

OperatorNotation 𝑣 =𝑑𝑑𝑡 𝑠 𝑡 =

𝑑𝑠𝑑𝑡 = 𝑠Þ 𝑡 ≡ 𝑠

𝒙-direction 𝑣� =𝑑𝑑𝑡 𝑥 𝑡 =

𝑑𝑥𝑑𝑡 = 𝑥Þ 𝑡 ≡ 𝑥

y-direction 𝑣¦ =𝑑𝑑𝑡 𝑦 𝑡 =

𝑑𝑦𝑑𝑡 = 𝑦Þ 𝑡 ≡ 𝑦


Note:Whenyouhaveafunction𝑦 = 𝑓 𝑥 ,itcanbeseparatedinto“parametricequations”𝑦 =𝑦 𝑡 and𝑥 = 𝑥 𝑡 ,whichgivestwonewgraphingsystems.ParametricEquationGraphingExample

𝑦 𝑥 = + 25 − 𝑥q

− 25 − 𝑥q

𝑥q + 𝑦q = 25

⇒𝑥5

q+

𝑦5

q= 1 = cosq 𝑡 + sinq 𝑡

⇒ 𝑥 𝑡 = 5 cos 𝑡 ∧ 𝑦 𝑡 = 5 sin 𝑡 , 𝑡 ∈ 0,2𝜋 AverageAcceleration(straight-line)

𝑎 ≡ 𝑎òy =𝑣 − 𝑣2𝑡 − 𝑡2

=𝛥𝑣𝛥𝑡

InstantaneousAcceleration(Calculus)Acceleration

𝑎 = limâ©→2

𝑣 𝛥𝑡 − 𝑡 − 𝑣 𝑡𝛥𝑡 =

𝑑𝑣𝑑𝑡 = 𝑣 =

𝑑𝑑𝑡

𝑑𝑠𝑑𝑡 = 𝑠

𝒙-direction 𝑎� =𝑑𝑣�𝑑𝑡 =

𝑑𝑑𝑡

𝑑𝑥𝑑𝑡

𝒚-direction𝑎¦ =

𝑑𝑣¦𝑑𝑡 =

𝑑𝑑𝑡

𝑑𝑦𝑑𝑡

Formulas(one-dimensional)Note:Thefollowingformulascanbeusedinthe𝑦-directionaswellbutbecarefulbecauseinthe𝑦-direction𝑎maybeequaltogravityandorthevelocitymaynotbeconstanti.e.dependingonhowthecoordinatesarechoseninreferencetotheobject(s)inquestion.


Velocity:Accelerationisconstantin𝑥-direction

𝑣� = 𝑣�,2 + 𝑎�𝑡

AverageVelocity:𝑎isconstantin𝑥-direction 𝑣òyp� = 𝑣�,2 +

12𝑎�𝑡

Distance:𝑎isconstantin𝑥-direction 𝛥𝑥 = 𝑥 − 𝑥2 = 𝑣�,2𝑡 +

12𝑎�𝑡

q

Velocity:𝑎isconstantin𝑥-direction

𝑣�q = 𝑣�,2q + 2𝑎�𝛥𝑥

𝑣²q = 𝑣8q + 2𝑎𝑑

Distance:𝑎isconstantin𝑥-direction 𝛥𝑥 = 𝑥 − 𝑥2 =

𝑣� + 𝑣�,22 𝑡

𝑑² − 𝑑8 =𝑣² + 𝑣82 𝑡

Note:𝑣�,2isreadastheinitialvelocityinthe𝑥-directionIntegrationDerivations(Calculus)Note:Aderivativeisthe“rate-of-change,”wereaderivationistoderivesomething.

𝑑𝑣𝑑𝑡 = 𝑎 ⇒ 𝑑𝑣 = 𝑎 𝑑𝑡 ⇒ 𝑣 = 𝑎𝑡 + 𝑣2

𝑑𝑠𝑑𝑡 = 𝑣 = 𝑎𝑡 + 𝑣2 ⇒ 𝑑𝑠 = 𝑎𝑡 + 𝑣2 𝑑𝑡 ⇒ 𝑠 − 𝑠2 =

𝑎𝑡q

2 + 𝑣2𝑡

VectorNotations

𝐫 = 𝑥 𝑡 𝐢 + 𝑦 𝑡 𝐣+ 𝑧 𝑡 𝐤 ≡ 𝑟 = 𝑥 𝑡 𝚤 + 𝑦 𝑡 𝚥 + 𝑧 𝑡 𝑘

𝑥 𝑡 𝚤 + 𝑦 𝑡 𝚥 + 𝑧 𝑡 𝑘 = 𝑥 𝑡 1, 0, 0 + 𝑦 𝑡 0, 1, 0 + 𝑧 𝑡 0, 0, 1 = 𝑥 𝑡 , 𝑦 𝑡 , 𝑧 𝑡 ∴ 𝑟 = 𝑥 𝑡 𝚤 + 𝑦 𝑡 𝚥 + 𝑧 𝑡 𝑘 = 𝑥 𝑡 , 𝑦 𝑡 , 𝑧 𝑡

VectorDerivatives

𝑣 =𝑑𝑟𝑑𝑡 =

𝑑𝑥𝑑𝑡 𝚤 +

𝑑𝑦𝑑𝑡 𝚥 +

𝑑𝑧𝑑𝑡 𝑘 = 𝑥Þ 𝑡 , 𝑦Þ 𝑡 , 𝑧Þ 𝑡 = 𝑥, 𝑦, 𝑧

𝑎 =𝑑𝑣𝑑𝑡 =

𝑑q𝑟𝑑𝑡q =

𝑑q𝑥𝑑𝑡q 𝚤 +

𝑑q𝑦𝑑𝑡q 𝚥 +

𝑑q𝑧𝑑𝑡q 𝑘 = 𝑥ÞÞ 𝑡 , 𝑦ÞÞ 𝑡 , 𝑧ÞÞ 𝑡 = 𝑥, 𝑦, 𝑧


Note:Sometimesyouwillseeaderivativesuchas𝑥Þ 𝑡 = 𝑣� =»�»©,butinthree-dimensional

calculus,𝑣� ={4{!,whichisthepartialderivativeof𝑣withrespectto𝑥.Justpayattentiontowhat

eachbookdefinesnotationsas,astheymaynotalwaysbeconsistent.

∴ 𝑣 =𝑑𝑟𝑑𝑡 =

𝑑𝑥𝑑𝑡 𝚤 +

𝑑𝑦𝑑𝑡 𝚥 +

𝑑𝑧𝑑𝑡 𝑘 = 𝑥Þ 𝑡 , 𝑦Þ 𝑡 , 𝑧Þ 𝑡 ≡ 𝑣�, 𝑣¦, 𝑣b

𝑎 =𝑑q𝑟𝑑𝑡q =

𝑑q𝑥𝑑𝑡q 𝚤 +

𝑑q𝑦𝑑𝑡q 𝚥 +

𝑑q𝑧𝑑𝑡q 𝑘 = 𝑎�, 𝑎¦, 𝑎b

Magnitudeofvector

v = 𝑣 = 𝑣 = 𝑥Þ 𝑡 q + 𝑦Þ 𝑡 q + 𝑧Þ 𝑡 q

ProjectileMotion𝑥 = 𝑣2 cos 𝜃2 𝑡 𝑦 = 𝑣2 sin 𝜃2 𝑡 −

12𝑔𝑡

q

𝑣� = 𝑣2 cos 𝜃2 𝑣¦ = 𝑣2 sin 𝜃2 − 𝑔𝑡

𝑦 = tan 𝜃2 𝑥 −𝑔𝑥q

2𝑣2q cosq 𝜃2 𝑡 =

𝑣2 sin 𝜃2 ± 𝑣2q sinq 𝜃2 − 2𝑔𝑦𝑔

CircularMotion𝑇istheperiod,𝑅istheradiusUniformcircularmotion

𝑎ñòî =𝑣q

𝑅

Uniformcircularmotion 𝑣 =2𝜋𝑅𝑇

Uniformcircularmotion𝑎ñòî =

4𝜋q𝑅𝑇q

Uniformcircularmotion𝑅 =

𝑣q

𝑎ñòî

Uniformcircularmotion𝑎-ò¯ =

𝑑 𝑣𝑑𝑡

Non-uniformcircularmotion𝑎-ò¯ =

𝑑𝑣𝑑𝑡


Uniformcircularmotion 𝑑𝑣𝑑𝑡 =

𝑎q

𝑟

Non-uniformcircularmotion 𝑑𝑣𝑑𝑡 = 𝑎ñòî q + 𝑎-ò¯ q

ForceResultantvector𝑹(thesumofallvectors)

𝑅� = 𝐹�,r + 𝐹�,q + 𝐹�,o + ⋯ = 𝐹� 𝑅¦ = 𝐹¦,r + 𝐹¦,q + 𝐹¦,o + ⋯ = 𝐹¦

𝑅 = 𝐹r + 𝐹q + 𝐹o + ⋯ = 𝐹 ⇒ 𝑅 = 𝐹 = 𝑅�q+ 𝑅¦

q= 𝐹�

q+ 𝐹¦

q

Newton’sFirstLawofMotionAbodyactedon,withzeronetforce,moveswithconstantvelocityandnoacceleration.Newtonunit𝐍

N = kg ⋅msq mass times acceleration

Newton’sSecondLawofMotionIfaforceisactedonabody,thenthebodyaccelerates,andthedirectionofaccelerationisthesameasthenetforcesdirection.

𝐹 = 𝑚𝑎

FormulasWeight 𝑤 = 𝑚𝑔,(mass)(gravity)

𝑤 = 𝑚𝑎ForceactingonanotherForce 𝐹r,q = −𝐹q,r

LINEARALGEBRA


Rankofmatrixandpivots

𝟏 1 , 𝑟𝑎𝑛𝑘 𝐴r = 1

𝟏 1 , 𝑟𝑎𝑛𝑘 𝐴ç = 1

𝟏1 , 𝑟𝑎𝑛𝑘 𝐴q = 1

𝟏0 , 𝑟𝑎𝑛𝑘 𝐴è = 1

𝟏 1 1 , 𝑟𝑎𝑛𝑘 𝐴o = 1

𝟏 1 1 , 𝑟𝑎𝑛𝑘 𝐴r2 = 1

𝟏11, 𝑟𝑎𝑛𝑘 𝐴Ü = 1

𝟏00, 𝑟𝑎𝑛𝑘 𝐴rr = 1

𝟏 00 𝟏 , 𝑟𝑎𝑛𝑘 𝐴é = 2

𝟏 1 11 1 11 1 1

, 𝑟𝑎𝑛𝑘 𝐴rq = 1

𝟏 0 00 𝟏 1 , 𝑟𝑎𝑛𝑘 𝐴ê = 2

𝟏 1 11 1 −𝟏1 1 1

, 𝑟𝑎𝑛𝑘 𝐴ro = 2

𝟏 00 00 𝟏

, 𝑟𝑎𝑛𝑘 𝐴ë = 2

𝟏 1 10 𝟏 10 0 𝟏

, 𝑟𝑎𝑛𝑘 𝐴rÜ = 3

Note:maxrankisthesmallerdimensionof𝑛×𝑚e.g.3×7meansthat3isthehighestpossiblerank.Itgoeswiththetransposeaswelli.e.7×3stillhasahighestrankof3.

𝐴 = 1 2−1 −2

1 11 1

1 11 1 𝑅1 + 𝑅2 ⇐ 𝑅2

~ 𝟏 20 0

1 1𝟐 2

1 12 2 ⇒ 𝑟𝑎𝑛𝑘 𝐴 = 2

𝐴𝑥 = 𝑏 ⇒3 2 31 3 33 2 1

131~𝟏 0 00 𝟏 00 0 𝟏

−37870

, 𝑟𝑎𝑛𝑘 𝐴 = 3𝑖. 𝑒.𝐴 = 𝑓𝑢𝑙𝑙𝑟𝑎𝑛𝑘

Lengthofavectorandtheunitvector


Givenavector𝒙 = 𝑥 = 𝑥r, 𝑥q, 𝑥o, … , 𝑥0 =

𝑥r𝑥q𝑥o⋮𝑥0

Thelengthofthevectoristhemagnitudeofthevector

𝒙 ≡ 𝑥 = 𝑥rq + 𝑥qq + 𝑥oq + ⋯+ 𝑥0q

Ex:Findthelengthof 1,2,3,4

1,2,3,4 =

1234

⇒ 1,2,3,4 = 1q + 2q + 3q + 4q = 1 + 4 + 9+ 16 = 30units

Ex:Fromthevectorabove,finditsunitvector.

𝑣𝑣 =

𝒗𝒗 ⇒

𝑣𝑣 =

𝒗𝒗 = 1units

𝒙𝒙 =

11 + 4 + 9+ 16

1234

=1,2,3,430

=130,230,330,430

𝑥𝑥 =

130

q

+230

q

+330

q

+430

q

=130 +

430 +

930 +

1630 =

3030 = 1units

SolutionsofAugmentedMatricesConsiderthebasicscenarioi.e.rememberfromalgebrawhenyouhave𝑎𝑥 + 𝑏𝑦 = 𝑐and𝑑𝑥 +𝑒𝑦 = 𝑓?Rememberthatthesetwolineseitherlyeoneachother,intersectornevertouch,andthismeanstheyhaveeitherauniquesolution,infinitesolutions,onnosolution.Thesamegoeswith𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑,exceptthisisaplane.Forℝo,considerthefollowingsystemanditsthreepossiblesolutionsafterreduction:CoefficientMatrix


𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑𝑒𝑥 + 𝑓𝑦 + 𝑔𝑦 = ℎ𝑖𝑥 + 𝑗𝑦 + 𝑘𝑧 = 𝑙

⇒ 𝑎 𝑏 𝑐𝑒 𝑓 𝑔𝑖 𝑗 𝑘

𝑥𝑦𝑧=

𝑑ℎ𝑙⇒

𝑎 𝑏 𝑐𝑒 𝑓 𝑔𝑖 𝑗 𝑘

𝑑ℎ𝑙

TheCoefficientMatrix=𝑎 𝑏 𝑐𝑒 𝑓 𝑔𝑖 𝑗 𝑘

UniqueSolution

~1 0 00 1 00 0 1

∗∗∗⇒

𝑥𝑦𝑧=

∗∗∗

In2𝐷/3𝐷hereisasinglepointofintersection

InfiniteSolution

~1 0 00 1 00 0 0

∗∗0⇒

𝑥𝑦𝑧=

∗∗0+ 𝑠

001

In3𝐷twoplaneslieontopofeachotherIn2𝐷twolineslieontopofeachother

NoSolution

~1 0 00 1 00 0 0

∗∗∗⇒

𝑥𝑦0=

∗∗∗

Twoplanes/linesnevertouch

SolvingSystemofEquations

𝑥q + 𝑥Ü = 5 ∧ 𝑥o − 4𝑥Ü = 4


Iliketosetupamatrixforthisproblem,andsolvethematrixi.e.

𝑥q + 𝑥Ü = 5 ⇒ 0𝑥r + 𝑥q + 0𝑥o + 𝑥Ü = 5

𝑥o − 4𝑥Ü = 4 ⇒ 0𝑥r + 0𝑥q + 𝑥o − 4𝑥Ü = 4

⇒ 0010011−4

54 ⇒

0 00 1

0 00 1

0 00 0

1 −40 0

0540

Thishelpstoseethepivots,andidentifythat𝑥r ∧ 𝑥Üarefreevariables.Whichmeansyoucansetthemequaltothemselves.

0 00 1

0 00 1

0 00 0

1 −40 0

0540

⇒ 𝒙 =

𝑥r𝑥q𝑥3𝑥4

=

𝑥r5 − 𝑥Ü4 + 4𝑥4𝑥4

=

0 + 𝑥r + 0𝑥Ü5 + 0𝑥r − 𝑥Ü4 + 0𝑥1 + 4𝑥40 + 0𝑥1 + 𝑥4

=

0540

+

1000

𝑥r +

0−141

𝑥Ü

Youcanchoose𝑥r ∧ 𝑥Ü = 𝑠 ∧ 𝑡sincetheyarefree

∴ 𝒙 =

0540

+ 𝑠

1000

+ 𝑡

0−141

= 𝑠, 5 − 𝑡, 4 + 4𝑡, 𝑡

GaussJordanAugmentedMatrix


2𝑥 − 𝑦 = 2𝑥 + 2𝑦 = 112𝑥 + 3𝑦 = 18

⇒2 −11 22 3

21118

𝑅2 ⇐12𝑅1 − 𝑅2 ∧ 𝑅3 ⇐ 𝑅1 − 𝑅3 ⇒

2 −10 −520 −4

2−10−16

𝑅2 ⇐ −25𝑅2 ∧ 𝑅3 ⇐ −

14𝑅3 ⇒

2 −10 10 1

244

𝑅3 ⇐ 𝑅2 − 𝑅3 ⇒2 −10 10 0

240

𝑅1 ⇐ 𝑅1 + 𝑅2 ⇒2 00 10 0

640

𝑅1 ⇐12𝑅1 ⇒

1 00 10 0

340⇒ 𝐼q𝒙 =

34 ⇒ 𝑥 = 3

𝑦 = 4

∴ 𝑥, 𝑦 = 3,4

RowOperationRulesandGuidelinesforSolveaSystemofMatrices

Solvethesystemofequationsrref[{-1/4,1,0,1},{1,0,1,2},{3,-1,1,2}]

−14𝑥 + 𝑦 = 1𝑥 + 𝑧 = 2

3𝑥 − 𝑦 + 𝑧 = 2

⇒−14 1 01 0 13 −1 1

122

Alwaysgointhefollowingorderunlessazeroalreadyexistsandorarowoperationmakesitexist.

1st

∗ ∗1 ∗

∗ ∗∗ ∗

2 43 5

∗ ∗6 ∗

∗∗∗∗

,2nd∗ 120 ∗

11 910 8

0 00 0

∗ 70 ∗

∗∗∗∗

,3rd1 00 1

0 00 0

0 00 0

1 00 1

∗∗∗∗

Generalallowedoperationswhensolveasystem(notthesameforamatrixA)


1. Rowswapping𝑅1 ⇔ 𝑅2(meansswaprow1withrow2)

2. Divide/multiplyaRowr

é𝑅2 ∧ −3𝑅4(meansdividerow2by5andmultiplyrow4by-3)

3. Adding/subtractingscaledrows5𝑅1 + 𝑅2 ⇐ 𝑅2(meansthenewrow2=5 row1 +

row2 )

−14 1 01 0 13 −1 1

122

Takealookatthematrix.Firstmultiplyrow1by−4andthenswaprow2androw3

−4𝑅1 = −4 −14 1 0 1 = 1 −4 0 −4 ⇐ 𝑅1

𝑅2 ⇔ 𝑅3

1 −4 03 −1 11 0 1

−422

Noweliminate3and1fromcolumn1

−3𝑅1 + 𝑅2 = −3 1 −4 0 −4 + 3 −1 1 2 = 0 11 1 14 ⇐ 𝑅2

−𝑅1 + 𝑅3 = (−1) 1 −4 0 −4 + 1 0 1 2 = 0 4 1 6 ⇐ 𝑅3

1 −4 00 11 10 4 1

−4146

Noweliminate4fromcolumn2

−411𝑅2 + 𝑅3 = −

411 0 11 1 14 + 0 4 1 6 = 0 0

711

1011

⇐ 𝑅3

1 −4 00 11 10 0 7

11

−4141011

Multiplyrow2by r

rrandrow3byrr

ë


111𝑅2 = 0 1

111

1411

117 𝑅3 = 0 0 1

107

1 −4 00 1 1

110 0 1

−41411107

⇒

𝑥 − 4𝑦 = −4 ⇒ 𝑥 = 487 − 4 =

47

𝑦 +111 𝑧 =

1411 ⇒ 𝑦 =

1411 −

111

107 =

87

𝑧 =107

∴ ! =

4787107

=17

4810

=27

245

EchelonForms:EF,REF,RREF

𝑥q − 𝑥r − 𝑥o = 22𝑥r − 𝑥q = 2

2𝑥r + 𝑥q + 𝑥o = 2⇒

−1 1 −12 −1 02 1 1

222

EchelonFormakaef−1 1 −12 −1 02 1 1

2222𝑅1 + 𝑅2 ⇐ 𝑅22𝑅1 + 𝑅3 ⇐ 𝑅3

~−1 1 −10 1 −20 3 −1

266−3𝑅2 + 𝑅3 ⇐ 𝑅3

~ −𝟏 1 −1𝟎 𝟏 −2𝟎 𝟎 𝟓

26−12

ReducedEchelonFormakaref

−1 1 −10 1 −20 0 5

26−12

−𝑅1 ⇐ 𝑅115𝑅3 ⇐ 𝑅3

~𝟏 −1 1𝟎 𝟏 −2𝟎 𝟎 𝟏

26

−125

ReducedRowEchelonFormakarref


1 −1 10 1 −20 0 1

−26

−1252𝑅3 + 𝑅2 ⇐ 𝑅2−𝑅3 + 𝑅1 ⇐ 𝑅1

~1 −1 00 1 00 0 1

2565

−125

𝑅2 + 𝑅1 ⇐ 𝑅1~

𝟏 𝟎 𝟎𝟎 𝟏 𝟎𝟎 𝟎 𝟏

8565

−125

[{-1,1,-1,2},{2,-1,0,2},{2,1,1,2}]

LinearDependence

LinearcombinationSayyouhaveℬ = 𝑢, 𝑣,𝑤 ,writeitasalinearcombination.Allthatmeansis𝑢𝑥r + 𝑣𝑥q + 𝑤𝑥oNowtoverifylinearindependence/dependencesetthecombinationequaltozero.Ifthereisauniquesolution,thevectorsarelinearlyindependentelsedependenti.e.if𝑢, 𝑣, 𝑜𝑟𝑤canbewrittenasalinearcombinationoftheotherse.g.𝑢 = 𝑣 − 𝑤or𝑣 = 𝑢 + 2𝑤thentheyaredependent.DifferentwaystoverifydependencyofvectorsEx1:Setu,v,wLinearlyDependentArethesetslineardependent/independent?(ℬdenotesbasis.𝒲isgenerallysubset/space)

𝒲r = 𝑢 − 2𝑣 + 𝑤,𝑤 + 𝑣 − 𝑢, 2𝑤 − 𝑣 (itcaneasilybeseenthat 𝑢 − 2𝑣 + 𝑤 + 𝑤 + 𝑣 − 𝑢 = 2𝑤 − 𝑣,whichmeansthesetisdependent)

𝒲r: 𝑢 − 2𝑣 + 𝑤 𝑥r + 𝑤 + 𝑣 − 𝑢 𝑥q + 2𝑤 − 𝑣 𝑥o

= 𝑢 − 2𝑣 + 𝑤 −𝑢 + 𝑣 + 𝑤 0𝑢 − 𝑣 + 2𝑤𝑥r𝑥q𝑥o

= 𝑢 𝑣 𝑤1−21

𝑢 𝑣 𝑤−111

𝑢 𝑣 𝑤0−12

𝑥r𝑥q𝑥o

= 𝑢 𝑣 𝑤1 −1 0−2 1 −11 1 2

𝑥r𝑥q𝑥o

rref1 −1 0−2 1 −11 1 2

=1 0 10 1 10 0 0

, notfullrank ∴ LD


Ex2:Setu,v,wLinearlyIndependent

𝒲q = 𝑢, 𝑣 + 2𝑢, 𝑢 − 2𝑣

𝑢𝑥r + 𝑣 + 2𝑢 𝑥q + 𝑢 − 2𝑣 + 𝑤 𝑥o = 0 ⇒

=

= 𝑢 𝑣 𝑤100

𝑢 𝑣 𝑤210

𝑢 𝑣 𝑤1−21

𝑥r𝑥q𝑥o

= 𝑢 𝑣 𝑤1 2 10 1 −20 0 1

𝑥r𝑥q𝑥o

rref1 2 10 1 −20 0 1

, fullrank ∴ LI

Ex3:VectorsLinearlyIndependentDetermineifthesetislinearlyindependentordependent

𝒲 =123

,321

,11−1

𝑢𝑥r + 𝑣𝑥q + 𝑤𝑥o =123

𝑥r +321

𝑥q +11−1

𝑥o = 0 𝐴𝒙 = 0

rref1 3 12 2 13 1 −1

000

=1 0 00 1 00 0 1

000

Thisisafullrankmatrixthereforeitisalinearlyindependentsetofvetors.Ex4:VectorsLinearlyDependent Determineifthesetislinearlyindependentordependent

𝒲 =123

,321

,−3−6−9


1 3 −32 2 −63 1 −9

000

𝐴𝒙 = 0 ⇒ 𝐴0𝐴pr𝐴𝒙 = 𝐴0𝐴pr0 ⇒ " # 𝐼 𝒙 = 0 ⇒ 𝐴0𝒙 = 0

∴1 2 33 2 1−3 −6 −9

000𝑅1 = −3𝑅3 ∴ LDi. e.notfullrank

3𝑅1 + 𝑅2 ⇐ 𝑅3 ⇒ 1 2 33 2 1−3 −6 −9

000~1 2 33 2 10 0 0

000

Ex5:Polynomials Determineifthesetislinearlyindependentordependent

𝑝r = 𝑥q + 𝑥, 𝑝q = 𝑥 − 𝑥q, 𝑝o = 1Don’tbescaredofthepolynomial,justfollowtherules,anditwillworkitselfout!(note:𝑝 𝑥 = 𝑎𝑥q + 𝑏𝑥 + 𝑐orhigherorderpolynomials)

𝑥q + 𝑥 𝑣r + 𝑥 − 𝑥q 𝑣q + 1 𝑣o = 0

𝑥q + 𝑥 𝑥 − 𝑥q 1𝑣r𝑣q𝑣o

=000⇒

𝑥q + 𝑥 + 0 −𝑥q + 𝑥 + 0 0𝑥q + 0𝑥 + 1𝑣r𝑣q𝑣o

=000⇒ 𝑥q 𝑥 1

1 −1 01 1 00 0 1

𝑣r𝑣q𝑣o

=000

−𝑅1 + 𝑅2 ⇒1 −1 00 2 00 0 1

Fullrankandlinearlyindependent

Ex6:(M_(2x2))

𝒲 = 1 01 1 , 1 1

0 1 , 1 00 1 , 1 0

1 0

1 01 1 𝑥r +

1 10 1 𝑥q +

1 00 1 𝑥o +

1 01 0 𝑥Ü = 0

𝑥r 0𝑥r 𝑥r

+𝑥q 𝑥q0 𝑥q + 𝑥o 0

0 𝑥o+ 𝑥Ü 0

𝑥Ü 0 = 0


𝑥r + 𝑥q + 𝑥o + 𝑥Ü 𝑥q𝑥r + 𝑥Ü 𝑥r + 𝑥q + 𝑥o

= 0 00 0

𝑥r + 𝑥q + 𝑥o + 𝑥Ü = 0

𝑥q = 0𝑥r + 𝑥Ü = 0

𝑥r + 𝑥q + 𝑥o = 0

⇒1 10 1

1 10 0

1 01 1

0 11 0

0000

~1 00 1

0 00 0

0 00 0

1 00 1

0000

Thisisafullrankmatrix,thereforeitisalinearlyindependentsetof2X2matrices

ColumnSpace-RowSpace-NullSpace-Kernel

𝐴 =−323

9−6−9

−24−2

−782

Step1rref(𝐴)rref[{-3,9,-2,-7},{2,-6,4,8},{3,-9,-2,2}]

−323

9−6−9

−24−2

−782

~𝟏00

−300

0𝟏0

32540

IdentifyRowSpace

𝟏𝟎0

−𝟑𝟎0

𝟎𝟏0

𝟑/𝟐𝟓/𝟒0

⇒ ℬßo =

1−303/2

,

0015/4

IdentifyColumnSpace

−𝟑𝟐𝟑

9−6−9

−𝟐𝟒−𝟐

−782

~𝟏00

−300

0𝟏0

3/25/40

⇒ ℬáo =−323

,−24−2

Checkyouworki.e.Note:CS*RS=A

−3 −22 43 −2

1 −3 0 3/20 0 1 5/4 =

−323

9−6−9

−24−2

−782

NullSpace(Kernel)


−323

9−6−9

−24−2

−782

~100

−𝟑00

010

𝟑/𝟐𝟓/𝟒0

⇒ ℬ�o =

3100

,

−60−54

Youcanextractthenullspacequicklybychangingthesignofthenon-pivotelementandaddingapivotwherethepivotwouldlineuptoanidentitymatrixbutthisishowtocomputeit:The‘NullSpace’isthesolutionto𝐴𝒙 = 𝟎

100

−300

010

3/25/40

𝑥r𝑥q𝑥o𝑥Ü

=

0000

⇒

𝑥r − 3𝑥q +32 𝑥Ü = 0

𝑥q = 𝑥q𝑓𝑟𝑒𝑒

𝑥o +54 𝑥Ü = 0

𝑥Ü = 𝑥Ü𝑓𝑟𝑒𝑒

⇒

𝑥r = 3𝑥q −32 𝑥Ü

𝑥q = 𝑥q + 0𝑥Ü

𝑥o = 0𝑥q −54 𝑥Ü

𝑥Ü = 0𝑥q + 𝑥Ü

⇒ 𝒙 =

3100

𝑥q +

−320

−541

𝑥Ü, 𝑥q = 1 ∧ 𝑥Ü = 4 ⇒ ℬ�o =

3100

,

−60−54

CheckyourworkA*NS=0:

−323

9−6−9

−24−2

−782

31

−6−5

00

04

=0 00 00 0

= 𝟎

LUDDecompositionandElementaryMatrices

𝐴 =1 2 13 2 11 2 4

, 𝐴 = 𝐿𝑈𝐷 =? note∗ 6𝑡ℎ 5𝑡ℎ1𝑠𝑡 ∗ 4𝑡ℎ2𝑛𝑑 3𝑟𝑑 ∗

∧∗ 12𝑡ℎ1𝑠𝑡 ∗

11𝑡ℎ 9𝑡ℎ10𝑡ℎ 8𝑡ℎ

2𝑛𝑑 4𝑡ℎ3𝑟𝑑 5𝑡ℎ

∗ 7𝑡ℎ6𝑡ℎ ∗

1 2 13 2 11 2 4

−3𝑅1 + 𝑅2 = −3 −6 −3 + 3 2 1 = 0 −4 −2 ⇐ 𝑅2 −1 𝑅1 + 𝑅3 = −1 −2 −1 + 1 2 4 = 0 0 3 ⇐ 𝑅3


⇒1 2 10 −4 −20 0 3

∨ $ 1𝐴 =1 0 0−3 1 0−1 0 1

1 2 13 2 11 2 4

=1 2 10 −4 −20 0 3

⇒1 2 10 −4 −20 0 3

23𝑅3 + 𝑅2 = 0 0 2 + 0 −4 −2 = 0 −4 0 ⇐ 𝑅2

−13𝑅3 + 𝑅1 = 0 0 −1 + 1 2 1 = 1 2 0 ⇐ 𝑅1

⇒1 2 00 −4 00 0 3

12𝑅2 + 𝑅1 = 0 −2 0 + 1 2 0 = 1 0 0 ⇐ 𝑅1

⇒1 0 00 −4 00 0 3

∨ $ 3𝐸q𝐸r𝐴 =1

12 0

0 1 00 0 1

1 0 −13

0 123

0 0 1

1 0 0−3 1 0−1 0 1

1 2 13 2 11 2 4

=1 0 00 −4 00 0 3

= 𝐷

𝐿 =1 0 0𝟑 1 0𝟏 0 1

, 𝑈 =1 −

𝟏𝟐

𝟏𝟑

0 1 −𝟐𝟑

0 0 1

, 𝐷 =1 0 00 −4 00 0 3

∴ 𝐴 = 𝐿𝑈𝐷 =1 0 0𝟑 1 0𝟏 0 1

1 −𝟏𝟐

𝟏𝟑

0 1 −𝟐𝟑

0 0 1

1 0 00 −4 00 0 3

=1 2 13 2 11 2 4

Transpose

𝑛×𝑚 0 = 𝑚×𝑛

Ex1:


12345

0

= 1 2 34 5

Ex2:

1 2 31 3 2

0=

1 12 33 2

Ex3:

1 2 34 5 67 8 9

0

=1 4 72 5 83 6 9

Ex4:

𝐴𝐵 0 = 𝐵0𝐴0

4×3 3×5 0 = 3×5 0 4×3 0 = 5×3 3×4 = 5×4

SymmetricmatrixforA=LDU=LDL^T

𝐴 =1 3 43 1 34 3 −1

𝐸r𝐴 =1 0 0−3 1 0−4 0 1

1 3 43 1 34 3 −1

=1 3 40 −8 −90 −9 −17


𝐸q 𝐸r𝐴 =

1 0 00 1 0

0 −98 1

1 3 40 −8 −90 −9 −17

=

1 3 40 −8 −9

0 0 −558

𝐴 = 𝐿𝐷𝐿0 = 𝐿𝐷𝑈 =

1 0 03 1 0

498 1

1 0 00 −8 0

0 0 −558

1 0 03 1 0

498 1

0

∴ 𝐴 =

1 0 03 1 0

498 1

1 0 00 −8 0

0 0 −558

1 3 4

0 198

0 0 1

Matrixadditionandsubtraction

𝑛×𝑚 ± 𝑛×𝑚 = 𝑛×𝑚 Ex1:

𝐴 = 1 2 11 1 1 ∧ 𝐵 = 1 1 1

1 2 1 , 𝐴 + 𝐵 =?

𝐴 + 𝐵 = 1 2 11 1 1 + 1 1 1

1 2 1 = 1 + 1 2 + 1 1 + 11 + 1 1 + 2 1 + 1 = 2 3 2

2 3 2 Ex2:

𝐶 =1 22 22 2

∧ 𝐷 =1 11 11 1

, 𝐶 − 𝐷 =?

𝐶 − 𝐷 =1 22 22 2

−1 11 11 1

=1 − 1 2 − 12 − 1 2 − 12 − 1 2 − 1

=0 11 11 1

Ex3:

𝐴 + 𝐵 0 − 5 𝐶 − 𝐷 =?

2 3 22 3 2

0− 5

0 11 11 1

=2 23 32 2

−0 55 55 5

=2 −3−2 −2−3 −3


Multiplythematrices(2x2)(2x3)

(𝑚×𝑛)(𝑛×𝑝) = (𝑚×𝑝)

(𝟐×2)(2×𝟑) = (2×3)

1 5−1 2

1 5 04 0 2 =

1 5 14 1 5 5

0 1 5 02

−1 2 14 −1 2 5

0 −1 2 02

= 1 1 + 5 4 1 5 + 5 0 1 0 + 5 2−1 1 + 2 4 −1 5 + 2 0 −1 0 + 2 2

= 21 5 10

7 −5 4

MatrixMultiplication(mxn)(nxp)

𝒎×𝒏 𝒏×𝒑 = 𝒎×𝒑 Matrixmultiplicationiskindoflikeagiant‘dotproduct’ThisistherowbycolumnmethodEx1:

2×2 2×1 = 2×1 : 1 21 1

31 =

1 2 31

1 1 31

= 1 3 + 2 11 3 + 1 1 = 5

4

Ex2:


2×3 3×1 = 2×1 : 1 2 22 1 2

131=

1 2 2131

2 1 2131

= 1 1 + 2 3 + 2 12 1 + 1 3 + 2 1 = 9

7

Ex3:

3×3 1𝑥3 = 𝐷𝑁𝐸:1 3 21 2 21 1 2

1 2 3

Ex4:

3×3 1𝑥3 0 = 3×3 3×1 = 3×1 :1 3 21 2 21 1 2

1 2 3 0 =1 3 21 2 21 1 2

123=

1 3 2123

1 2 2123

1 1 2123

=1 1 + 3 2 + 2 31 1 + 2 2 + 2 31 1 + 1 2 + 2 3

=13119

Ex5:

2×3 3×3 = 2×3 : 1 4 13 1 1

1 4 32 1 11 2 2

=1 4 1

121

1 4 1412

1 4 1312

3 1 1121

3 1 1412

3 1 1312

= 1 1 + 4 2 + 1 1 1 4 + 4 1 + 1 2 1 3 + 4 1 + 1 23 1 + 1 2 + 1 1 3 4 + 1 1 + 1 2 3 3 + 1 1 + 1 2

= 10 10 9

6 15 12 Ex6:


2×3 3×3 3×2 = 2×3 3×2 = 2×2 :

1 4 13 1 1

1 4 32 1 11 2 2

1 00 11 0

= 10 10 96 15 12

1 00 11 0

= 19 1018 15

Idempotentmatrix

𝐴𝐴 = 𝐴 ⇒ 𝐴0 = 𝐴Ex1:

𝐴 =2 −2 −4−1 3 41 −2 −3

⇒ 𝐴𝐴 =2 −2 −4−1 3 41 −2 −3

2 −2 −4−1 3 41 −2 −3

=2 −2 −4−1 3 41 −2 −3

⇒ 𝐴𝐴𝐴 =2 −2 −4−1 3 41 −2 −3

2 −2 −4−1 3 41 −2 −3

2 −2 −4−1 3 41 −2 −3

=2 −2 −4−1 3 41 −2 −3

⇒ 𝐴0 =2 −2 −4−1 3 41 −2 −3

0

=2 −2 −4−1 3 41 −2 −3

= 𝐴

Ex2:(2X2)DetermineifAisIdempotentwithoutmultiplication

𝐴 = 𝑎 𝑏𝑐 𝑑 ⇒ 𝐴Ö = 𝑎q + 𝑏𝑐 𝑎𝑏 + 𝑏𝑑

𝑐𝑎 + 𝑐𝑑 𝑏𝑐 + 𝑑qÖ= 𝑎q + 𝑏𝑐 𝑎𝑏 + 𝑏𝑑

𝑐𝑎 + 𝑐𝑑 𝑏𝑐 + 𝑑q

4 −112 −3

𝑎 = 𝑎q + 𝑏𝑐 ⇒ 4 = 4q + −1 12 = 16 − 12 = 4

𝑏 = 𝑎𝑏 + 𝑏𝑑 ⇒ −1 = 4 −1 + −1 −3 = −4 + 3 = −1

𝑐 = 𝑐𝑎 + 𝑐𝑑 ⇒ 12 = 12 4 + 12 −3 = 48 − 36 = 12

𝑑 = 𝑏𝑐 + 𝑑q ⇒ −3 = −1 12 + −3 q = −12 + 9 = −3

∴ 4 −112 −3

4 −112 −3 = 4 −1

12 −3 = 4 −112 −3

Ö,YesAisidempotent


RotationandTranslateEx.1Givethe4×4matrixthatrotatespointsinℝoaboutthe𝑧-axisthroughanangleof−30°,andthentranslatesby𝐩 = 5,−2, 1 Unitpointsinℝoandjust𝒗 = 𝒆r, 𝒆q, 𝒆o ,note:𝒆0 = 0, 0, … 1,… 0, 0 Aboutthe𝑧-axismeanstoCreatea3×3rotationmatrixfor−30°about𝑧-axis(1)Wewanttomoveapointonthe𝑥𝑦-plane30°towards– 𝑦-axis,whichiscos −30° , sin −30° , 0 = o

q, − r

q, 0 .Note:wearemoving𝑥inapositivedirection,and𝑦ina

negativedirectioni.e.choosetheanglewisely.(2)Nextwewanttomoveapointinthe𝑥𝑦-planetowardsthepositive𝑥-axis(twopositivecoordinates)Lookattheunitcircle,coordinate(0,1).Weanttomovethis−30°thecoordinateforthisis cos 60° , sin 60° = r

q, oqjustput

thisinto3×1columnvector rq, oq, 0

(3)Sincewearegoingaboutthe𝑧-axis,thethirdcoordinate 0,0,1 doesnotmove.Finally,wegetthefollowingmatrix𝐴

𝐴 =

32

12 0

−12

32 0

0 0 1

⇒

32

12

−12

32

0 00 0

0 00 0

1 00 1

Translatedvector:Mapping𝑥, 𝑦, 𝑧, 1 → 𝑥 + 5, 𝑦 − 2, 𝑧 + 1,1 givesthefollowingmatrix

𝑥 + 5𝑦 − 2𝑧 + 11

=

𝑥 + 0𝑦 + 0𝑧 + 50𝑥 − 0𝑦 + 0𝑧 − 20𝑥 + 0𝑦 + 𝑧 + 10𝑥 + 0𝑦 + 0𝑧 + 1

=1 00 1

0 50 −2

0 00 0

1 10 1

𝑥𝑦𝑧1


∴1 00 1

0 50 −2

0 00 0

1 10 1

32

12

−12

32

0 00 0

0 00 0

1 00 1

=

32

12

−12

32

0 50 −2

0 00 0

1 10 1

Ex.2TranslateTranslateby −2, 3 ,andthenscalethe𝑥-coordinateby0.8andthe𝑦-coordinateby1.2.Note:Wheneveryoutranslateavector,addanadditionaldimensionwithelementas#1i.e.

ℝ0×r ⟼ ℝ 0�r ×r ⇒ −23 →

−23𝟏

Translate:

𝑥, 𝑦, 1 → (𝑥 + 2, 𝑦 + 3, 1)Scale𝑥-coordinate:

𝟎.𝟖 1, 0, 0 = 𝟎.𝟖, 0, 0 Scale𝑦-coordinate:

𝟏.𝟐 0, 1, 0 = 0, 𝟏.𝟐, 0 Assemblethe3×3matrixwiththescaledpositions:

𝟎.𝟖 0 00 𝟏.𝟐 00 0 𝟏

Assemblethe3×3translatematrix

𝑥 + 2, 𝑦 + 3, 1 =𝑥 − 2𝑦 + 31

=𝑥 + 0𝑦 − 20𝑥 + 𝑦 + 30𝑥 + 0𝑦 + 1

=1 0 −20 1 30 0 1

𝑥𝑦1

∴0.8 0 00 1.2 00 0 1

1 0 −20 1 30 0 1

=0.8 0 −1.60 1.2 3.60 0 1


Rotateaboutapoint 𝒄, 𝒅 Note:𝐷=dilate,𝑅 =rotation,𝑇 =translate.Scale𝑥 ∧ 𝑦by𝛼,rotateby𝜃,andtranslate 𝑎, 𝑏

𝐷 =𝛼 0 00 𝛼 00 0 1

, 𝑅 =cos 𝜃 − sin 𝜃 0sin 𝜃 cos 𝜃 00 0 1

,𝑇 =1 0 𝑎0 1 𝑏0 0 1

1 0 𝑐0 1 𝑑0 0 1

cos 𝜃 − sin 𝜃 0sin 𝜃 cos 𝜃 00 0 1

1 0 −𝑐0 1 −𝑑0 0 1

Nilpotentmatrix(eigenvaluesarezero)

𝐴0�0Ö = 0, 𝑘 ≤ 𝑛Ex1:

𝐴 =0 1 10 0 10 0 0

𝐴q =0 1 10 0 10 0 0

0 1 10 0 10 0 0

=0 0 10 0 00 0 0

𝐴o =0 1 10 0 10 0 0

0 1 10 0 10 0 0

0 1 10 0 10 0 0

=0 0 00 0 00 0 0


Ex2:Find𝑘suchthat𝐴Ö = 0

10 −6 430 −18 1220 −12 8

10 −6 430 −18 1220 −12 8

=

10 −6 4103020

10 −6 4−6−18−12

10 −6 44128

30 −18 12103020

30 −18 12−6−18−12

30 −18 124128

20 −12 8103020

20 −12 8−6−18−12

20 −12 84128

Note: 𝑎 𝑏 𝑐𝑑𝑒𝑓= 𝑎 𝑑 + 𝑏 𝑒 + 𝑐 𝑓

=0 0 00 0 00 0 0

∴ 𝑘 = 2

Determinantrules

(1) det 𝐴0 = det 𝐴(2) det 𝐴pr = r

îï- �

(3) det 𝐴𝐵 = det 𝐴 det 𝐵 ⇔ 𝐴 = 𝑛𝑥𝑛 = 𝐵(4) det 𝑐𝐴 = 𝑐0 det 𝐴 𝑓𝑜𝑟𝑛𝑥𝑛

Givendet 𝐴pr = 5, det 𝐵0 = 6

Evaluatedet 𝐴𝐵 + det 5𝐴 𝐵0 ,𝐴 = 4𝑥4 = 𝐵

det 𝐴𝐵 + det 5𝐴 𝐵0 = det 𝐴 det 𝐵 + det 5𝐴 det 𝐵0 = det 𝐴 det 𝐵 + 5Ü det 𝐴 det 𝐵

∵ det 𝐴pr =1

det 𝐴 ∴ det 𝐴pr = 5 ⇒

1det 𝐴 = 5 ⇔ det 𝐴 =

15


=15 6 + 5Ü

15 6 =

65 + 5

o ⋅ 6 =66130

ProofsLet𝐴and𝑃besquarematrices,with𝑃invertible.Showthatdet 𝑃𝐴𝑃pr = det 𝐴.Usingdeterminantrule2i.e.det 𝐴pr = r

îï- �,wefind:

det 𝑃𝐴𝑃pr = det 𝑃 det 𝐴 det 𝑃pr =det 𝑃 det 𝐴det 𝑃 =

det 𝑃det 𝑃 det 𝐴 = det 𝐴

Findaformulafordet 𝑟𝐴 when𝐴isan𝑛×𝑛matrix.Longversion

𝐴 = 𝐿𝑈𝐷 ⇒𝑎rr ⋯ 𝑎r0⋮ ⋱ ⋮𝑎0r ⋯ 𝑎00

=1 0 0⋮ 1 0𝑙0r ⋯ 1

1 ⋯ 𝑢r00 1 ⋮0 0 1

𝑑rr ⋯ 0⋮ ⋱ ⋮0 ⋯ 𝑑00

1 0 0⋮ 1 0𝑙0r ⋯ 1

1 ⋯ 𝑢r00 1 ⋮0 0 1

𝑟𝑑rr ⋯ 0⋮ ⋱ ⋮0 ⋯ 𝑑00

= det1 0 0⋮ 1 0𝑙0r ⋯ 1

1 ⋯ 𝑢r00 1 ⋮0 0 1

𝑟𝑑rr ⋯ 0⋮ ⋱ ⋮0 ⋯ 𝑑00

det1 0 0⋮ 1 0𝑙0r ⋯ 1

det1 ⋯ 𝑢r00 1 ⋮0 0 1

det 𝑟𝑑rr ⋯ 0⋮ ⋱ ⋮0 ⋯ 𝑑00

= 1 1 det 𝑟𝑑rr ⋯ 0⋮ ⋱ ⋮0 ⋯ 𝑑00

= det𝑟𝑑rr ⋯ 0⋮ ⋱ ⋮0 ⋯ 𝑟𝑑00

= 𝑟𝑑rr𝑟𝑑qq … 𝑟𝑑00 = 𝑟0𝑑rr𝑑qq …𝑑00 = 𝑟0 det 𝐴

Shortversion

det 𝑟𝐴 = det 𝑟𝐿𝑈𝐷 = det 𝐿𝑈 𝑟𝐷 = det 𝐿𝑈 det 𝑟𝐷 = 1 𝑟𝑑r𝑟𝑑q𝑟𝑑o ⋯𝑟𝑑0

= 1 𝑟0 det 𝐷 = det 𝐿𝑈 𝑟 det𝐷 = 𝑟 det 𝐿𝑈 det𝐷 = 𝑟 det 𝐿𝑈𝐷 = 𝑟 det 𝐴


Determinate’sofa(2x2)matrixVariouswaystocheckdeterminant(2x2):

𝐴 = 𝑎 𝑏

𝑐 𝑑 ⇒ det 𝐴 = 𝐴 = 𝑎 𝑏𝑐 𝑑 = 𝑎 𝑑 − 𝑏 𝑐

𝐴 = 𝑎 𝑏

0 𝑐 ⇒ det 𝐴 = 𝐴 = 𝑎 𝑏0 𝑐 = 𝑎 𝑐

𝐴 = 𝑎 0

0 𝑏 ⇒ det 𝐴 = 𝐴 = 𝑎 00 𝑏 = 𝑎 𝑏

Formula:

1 −32 1 = 1 1 — 3 2 = 1 + 6 = 7

1 2−1 −2 = 1 −2 − 2 −1 = −2 + 2 = 0

1 30 −3 = 1 −3 = −3

1 00 −5 = 1 −5 = −5

Rowoperation:

1 34 5 ~ 1 3

0 −7 ⇒ 1 34 5 = 1 −7 = −7

−4𝑅1 + 𝑅2 = −4 −12 + 4 5 = 0 −7 ⇐ 𝑅2Noteforfuturereference:

𝐴 = 𝐸r𝑈 ⇒1 34 5 = 1 0

4 11 30 −7 ⇒ det 𝐴 = det𝐸r𝑈 = det𝐸r det 𝑈 =

1 04 1

1 30 −7

= 1 1 1 −7 = −7


Determinateofa(3x3)andhighermatricesCofactorExpansionNote:

𝑎 𝑏𝑐 𝑑 = 𝑎 𝑑 − 𝑏 𝑐

𝑎 𝑏 𝑐𝑑 𝑒 𝑓𝑔 ℎ 𝑖

= +𝑎 𝑒 𝑓ℎ 𝑖

− 𝑏 𝑑 𝑓𝑔 𝑖 + 𝑐 𝑑 𝑒

𝑔 ℎ

𝑎 𝑏𝑒 𝑓

𝑐 𝑑𝑔 ℎ

𝑖 𝑗𝑚 𝑛

𝑘 𝑙𝑜 𝑝

= +𝑎𝑓 𝑔 ℎ𝑗 𝑘 𝑙𝑛 𝑜 𝑝

− 𝑏𝑒 𝑔 ℎ𝑖 𝑘 𝑙𝑚 0 𝑝

+ 𝑐𝑒 𝑓 ℎ𝑖 𝑗 𝑙𝑚 𝑛 𝑝

− 𝑑𝑒 𝑓 𝑔𝑖 𝑗 𝑘𝑚 𝑛 𝑜

Cofactorexpansion:(bestfor3x3andhigherbutrowoperationscanbeeasierNote:Payattentionto0’s

𝑎 𝑏 𝑐𝑑 𝑒 𝑓0 ℎ 0

= +0 𝑏 𝑐𝑒 𝑓 − ℎ

𝑎 𝑐𝑑 𝑓 + 0 𝑎 𝑏

𝑑 𝑒 = −ℎ𝑎 𝑐𝑑 𝑓

Example1:

1 2 22 1 11 3 4

= + 1 1 13 4 − 2 2 1

1 4 + 2 2 11 3

= + 1 1 4 − 1 3 − 2 2 4 − 1 1 + 2 2 3 − 1 1 = −3

Example2:(lookforthezero’selseitwillbeapain!)

1 33 2

3 11 1

0 01 2

0 43 0

= +03 3 12 1 12 3 0

− 01 3 13 1 11 3 0

+ 01 3 13 2 11 2 0

− 41 3 33 2 11 2 3

= −41 3 33 2 11 2 3

= −4 −8 = 32


Example3:

1 03 2

0 11 1

0 01 2

0 43 0

= +00 0 12 1 12 3 0

− 01 0 13 1 11 3 0

+ 01 0 13 2 11 2 0

− 41 0 03 2 11 2 3

= −41 0 03 2 11 2 3

= −4 +1 2 12 3 − 0 3 1

1 3 + 0 3 21 2 = −4 2 3 − 1 2 = −4 4 = −16

RowOperations:Youmustfollowtheexactelementremovalforthistowork

∗ ∗ ∗1𝑠𝑡 ∗ ∗2𝑛𝑑 3𝑟𝑑 ∗

,

∗ ∗1𝑠𝑡 ∗

∗ ∗∗ ∗

2𝑛𝑑 4𝑡ℎ3𝑟𝑑 5𝑡ℎ

∗ ∗6𝑡ℎ ∗

𝐴 =1 2 22 1 11 3 4

⇒ det 𝐴 = −3

−2𝑅1 + 𝑅2 = −2 −4 −4 + 2 1 1 = 0 −3 −3 ⇐ 𝑅2

1 2 22 1 11 3 4

~1 2 20 −3 −31 3 4

−𝑅1 + 𝑅3 = −1 −2 −2 + 1 3 4 = 0 1 2 ⇐ 𝑅3

1 2 22 1 11 3 4

~1 2 20 −3 −31 3 4

~1 2 20 −3 −30 1 2

13𝑅2 + 𝑅3 = 0 −1 −1 + 0 1 2 = 0 0 1 ⇐ 𝑅3

∴1 2 22 1 11 3 4

~1 2 20 −3 −31 3 4

~1 2 20 −3 −30 1 2

~1 2 20 −3 −30 0 1

⇒ det 𝐴 = 1 −3 1 = −3

Thisworksbecausewedecomposed𝐴usingelementarymatrices.Youwillprobablytouchonthislaterbuthereitis.

𝐴 =1 2 22 1 11 3 4

=1 0 02 1 01 0 1

1 0 00 1 0

0 −13 1

1 2 20 −3 −30 0 1

=

1 0 02 1 0

1 −13 1

1 2 20 −3 −30 0 1


= 𝐸q𝐸r𝑈 = 𝐿𝑈 ⇒ det 𝐿𝑈 = det 𝐿 det 𝑈 = 1 −3 = −3

VectorSpace,SubspaceandSubsetAsetisvectorspaceif:

1. 𝒖+ 𝒗 ∈ 𝑉2. 𝒖+ 𝒗 = 𝒗 + 𝒖3. 𝒖+ 𝒗 +𝒘 = 𝒖+ 𝒗 +𝒘 4. 𝒖+ 𝟎 = 𝒖.5. 𝒖+ −𝒖 = 06. 𝑐𝒖 ∈ 𝑉.7. 𝑐 𝒖+ 𝒗 = 𝑐𝒖+ 𝑐𝒗8. 𝑐 + 𝑑 𝒖 = 𝑐𝒖+ 𝑑𝒖9. 𝑐 𝑑𝒖 = 𝑐𝑑 𝒖10. 1𝒖 = 𝒖

Asetisasubspaceif:

a. Thezerovectorof𝑉isin𝐻.b. 𝐻isclosedundervectoraddition.c. 𝐻isclosedunderscalarmultiplication.

Whenshowingthefollowingsetsaresubspaces,thezerovectorsisevaluatingallconstantsat0Determineifthesetisasubspaceofℙ0Allpolynomialsoftheform𝒑 𝑡 = 𝑎 + 𝑡q,where𝑎 ∈ ℝ1)𝑎 = 0 ⇒ 0 + 𝑡q = 𝑡q ∉ ℙo ∴notasubspaceNoAllpolynomialsinℙ0suchthat𝑃 0 = 0(1)

𝑃 0 = 0 ⇒ 𝑃 𝑡 = 𝑎r𝑡 + 𝑎q𝑡q + 𝑎o𝑡o + ⋯+ 𝑎0pr𝑡0pr ∴ 𝑎0pr = 𝟎 ⇒ 𝑃 0 = 0Case1:True

(2)

𝑃r + 𝑃q = 𝑎r𝑡 + 𝑎q𝑡q + 𝑎o𝑡o + ⋯+ 𝑎0pr𝑡0pr + 𝑏r𝑡 + 𝑏q𝑡q + 𝑏o𝑡o + ⋯+ 𝑏0pr𝑡0pr

= 𝑎 + 𝑏 r𝑡 + 𝑎 + 𝑏 q𝑡q + 𝑎 + 𝑏 o𝑡o + ⋯+ 𝑎 + 𝑏 0pr𝑡0pr ⇒ 𝑃r + 𝑃q 0 = 0Case2:True(3)


𝑐𝑃 = 𝑐 𝑎r𝑡 + 𝑎q𝑡q + 𝑎o𝑡o + ⋯+ 𝑎0pr𝑡0pr = 𝑐𝑎 r𝑡 + 𝑐𝑎 q𝑡q + 𝑐𝑎 o𝑡o + ⋯+ 𝑐𝑎 0pr𝑡0pr⇒ 𝑐𝑃 0 = 0

Case3:True-Yesitisasubspaceofℙ0

Cramer’srules

𝑥0 =det 𝐴0 𝒃det 𝐴

ThesolutioninawayyoumayalreadyknowisRref[{4,1,6},{3,2,7}]

4 13 2

67 ~ 𝐼q

12 ⇒ 𝒙 = 𝑥r = 1

𝑥q = 2

Letsfindthiswith“Cramer’sRule”

4 𝟏3 𝟐

𝟔𝟕 ⇒ 𝐴r 𝒃 = 𝟔 𝟏

𝟕 𝟐 ⇒ det 𝐴r 𝒃 = 2 6 − 1 7 = 12 − 7 = 5

∴ 𝑥r =det 𝐴r 𝒃det 𝐴 =

55 = 1

𝟒 1𝟑 2

𝟔𝟕 ⇒𝐴q 𝒃 = 𝟒 𝟔

𝟑 𝟕 ⇒ det 𝐴q 𝒃 = 4 7 − 6 3 = 10

∴ 𝑥q =det 𝐴q 𝒃det 𝐴 =

105 = 2

IMPORTANT:Noticethattheindexof𝐴isthelocationof𝒃i.e.ifyouhada4×4𝐴o 𝒃 = 𝒗r 𝒗q 𝒃 𝒗o

BasiscoordinatevectorGivenasetℬ,andvector𝑥putℬintoamatrixequation𝐴𝒖 = 𝑥 ⇒ 𝒖 = 𝑥 ℬ Ex.1

𝑏r =1−3 , 𝑏q =

−35 , 𝑥 = −7

5

1 −3−3 5 𝑥 ℬ =

−75 ⇒ 1 −3

−3 5−75 ~ 1 0

0 154 ⇒ 𝑥 ℬ = 5,4

∨∵ det 𝐴 ≠ 0, 𝑥 ℬ = 𝐴pr𝑥 =1

1 5 − −3 −35 33 1

−75 = −

14−20−16 = 5

4


∴ 𝑥 ℬ =

54

Ex.2

𝑏r =−31−4

, 𝑏q =75−6

, 𝑥 =1107

𝑥 ℬ =

𝑐r𝑐q⋮𝑐0

⇔ ℬ = 𝑏r, 𝑏q, … , 𝑏0 ⇒ 𝑏r𝑐r + 𝑏q𝑐q + ⋯+ 𝑏0𝑐0 = 𝒙

−31−4

𝑐r +75−6

𝑐q =1107

⇒−3 71 5−4 −6

1107

~1 00 10 0

−52120

∴ 𝑥 ℬ =12−52


Adjugateofamatrix

𝐴pr =ajd 𝐴det 𝐴

𝐴 = 𝑎 𝑏

𝑐 𝑑 ⇒ adj 𝐴q×q = +𝑑 −𝑏−𝑐 +𝑎

𝐴 =𝑎 𝑏 𝑐𝑑 𝑒 𝑓𝑔 ℎ 𝑖

⇒ adj 𝐴o×o =

+ 𝑒 𝑓ℎ 𝑖

− 𝑑 𝑓𝑔 𝑖 + 𝑑 𝑒

𝑔 ℎ

− 𝑏 𝑐ℎ 𝑖 +

𝑎 𝑐𝑔 𝑖 − 𝑎 𝑏

𝑔 ℎ

+ 𝑏 𝑐𝑒 𝑓 −

𝑎 𝑐𝑑 𝑓 + 𝑎 𝑏

𝑑 𝑒

0

𝐴 =

𝑎 𝑏𝑒 𝑓

𝑐 𝑑𝑔 ℎ

𝑖 𝑗𝑚 𝑛

𝑘 𝑙𝑜 𝑝

⇒ adj 𝐴Ü×Ü

=

+𝑓 𝑔 ℎ𝑗 𝑘 𝑙𝑛 𝑜 𝑝

−𝑒 𝑔 ℎ𝑖 𝑘 𝑙𝑚 𝑜 𝑝

−𝑏 𝑐 𝑑𝑗 𝑘 𝑙𝑛 𝑜 𝑝

+𝑎 𝑐 𝑑𝑖 𝑘 𝑙𝑚 𝑜 𝑝

+𝑒 𝑓 ℎ𝑖 𝑗 𝑙𝑚 𝑛 𝑝

−𝑒 𝑓 𝑔𝑖 𝑗 𝑘𝑚 𝑛 𝑜

−𝑎 𝑏 𝑑𝑖 𝑗 𝑙𝑚 𝑛 𝑝

+𝑎 𝑏 𝑐𝑖 𝑗 𝑘𝑚 𝑛 𝑜

+𝑏 𝑐 𝑑𝑓 𝑔 ℎ𝑛 𝑜 𝑝

−𝑎 𝑐 𝑑𝑒 𝑔 ℎ𝑚 𝑜 𝑝

−𝑏 𝑐 𝑑𝑓 𝑔 ℎ𝑗 𝑘 𝑙

+𝑎 𝑐 𝑑𝑒 𝑔 ℎ𝑖 𝑘 𝑙

+𝑎 𝑏 𝑑𝑖 𝑗 𝑙𝑚 𝑛 𝑝

−𝑎 𝑏 𝑐𝑖 𝑗 𝑘𝑚 𝑛 𝑜

−𝑎 𝑏 𝑑𝑒 𝑓 ℎ𝑒 𝑗 𝑙

+𝑎 𝑏 𝑐𝑒 𝑓 𝑔𝑖 𝑗 𝑘

0


ComputetheAdjugate

𝐴 =1 1 3−2 2 10 1 1

𝐵 =𝑎 𝑏 𝑐𝑑 𝒆 𝒇𝑔 𝒉 𝒊

⇒ adj 𝐵 =

+ 𝒆 𝒇𝒉 𝒊

− 𝑑 𝑓𝑔 𝑖 + 𝑑 𝑒

𝑔 ℎ

− 𝑏 𝑐ℎ 𝑖 +

𝑎 𝑐𝑔 𝑖 − 𝑎 𝑏

𝑔 ℎ

+ 𝑏 𝑐𝑒 𝑓 −

𝑎 𝑐𝑑 𝑓 + 𝑎 𝑏

𝑑 𝑒

0

adj A =

+ 2 11 1 − −2 1

0 1 + −2 20 1

− 1 31 1 + 1 3

0 1 − 1 10 1

+ 1 32 1 − 1 3

−2 1 + 1 1−2 2

0

=+ 2 1 − 1 1 − −2 1 − 1 0 + −2 1 − 2 0− 1 1 − 3 1 + 1 1 − 3 0 − 1 1 − 1 0+ 1 1 − 2 3 − 1 1 − 3 −2 + 1 2 − 1 −2

0

=𝟏 𝟐 −𝟐𝟐 𝟏 −𝟏−5 −7 4

0

=𝟏 𝟐 −5𝟐 𝟏 −7−𝟐 −𝟏 4

Inverseofa2x2Matrix

Inverseof 𝟐×𝟐 :

Option1)

𝐴 𝐼 ~ 𝐼 𝐴pr

−3 21 3

1 00 1 ~ 1 0

0 1− 311

211

111

311


Option2)

𝐴 = 𝑎 𝑏𝑐 𝑑 ⇒ 𝐴pr =

adj 𝐴det 𝐴 =

1𝑎𝑑 − 𝑏𝑐

𝑑 −𝑏−𝑐 𝑎 ∧ 𝐴𝑥 = 𝑏 ⇒ 𝑥 = 𝐴pr𝑏 ⇔ det 𝐴 ≠ 0

𝐴 = −3 21 3 ⇒ 𝐴pr =

1−3 3 − 2 1

3 −2−1 −3 = −

111

3 −2−1 −3 =

111

−3 21 3

SolveasystemofequationswithaninverseTheorem:

𝐴𝒙 = 𝑏 ⇒ 𝐴pr𝐴𝒙 = 𝐴pr𝑏 ⇒ 𝐼𝒙 = 𝐴pr𝑏 ⇒ 𝒙 = 𝐴pr𝑏 ⇔ det 𝐴 ≠ 0(Note:𝑥 = 𝒙 = 𝑥r, 𝑥q, … , 𝑥0 )

−3 21 3 𝒙 = 2

1 ⇒ 𝒙 = −111

3 −2−1 −3

21 = −

111

3 −2 21

−1 −3 21

= −111

3 2 + −2 1−1 2 + −3 1

= −111

6 − 2−2 − 3 = −

111

4−5 =

−411511

Similarlyyoucouldsolvewithrrefi.e.

−3 21 3 𝒙 = 2

1 ⇒ −3 21 3

21 ~ 1 0

0 1− 411511


Inverseof3x3

𝐴𝒙 = 𝑏 ⇒ 𝐴pr𝐴𝒙 = 𝐴pr𝑏 ⇒ 𝐼𝒙 = 𝐴pr𝑏 ⇒ 𝒙 = 𝐴pr𝑏 ⇔ det 𝐴 ≠ 0Option1)

𝐴 𝐼 ~ 𝐼 𝐴pr

𝐴 =1 1 21 2 12 1 1

(symmetricmatrix)

1 1 21 2 12 1 1

1 0 00 1 00 0 1

~1 0 00 1 00 0 1

−14 −1434

−1434 −14

34 −14 −14

⇒ 𝐴pr = −14

1 1 −31 −3 1−3 1 1

Option2)

𝐴pr =adj 𝐴det 𝐴

𝐴 =1 1 21 2 12 1 1

⇒ det 𝐴 = −4 ∧ adj 𝐴 = −

+ 2 11 1 − 1 1

2 1 + 1 22 1

1 21 1 + 1 2

2 1 − 1 12 1

+ 1 22 1 − 1 2

1 1 + 1 11 2

=+ 2 1 − 1 1 − 1 1 − 1 2 + 1 1 − 2 2− 1 1 − 1 2 + 1 1 − 2 2 − 1 1 − 1 2+ 1 1 − 2 2 − 1 1 − 1 2 + 1 2 − 1 1

=1 1 −31 −3 1−3 1 1

alsosymmetric

∴ 𝐴pr =adj 𝐴det 𝐴 = −

14

1 1 −31 −3 1−3 1 1

alsosymmetric


Trace

𝑡𝑟 𝐴 = 𝜆0

det 𝐴 − 𝐼𝜆 = 0Ex1:

𝐴 =1 20 2

2 22 2

0 00 0

1 10 −1

⇒ 𝑡𝑟 𝐴 = 1 + 2 + 1 − 1 = 3

Ex2:

𝐴 =1 2 22 2 11 2 2

⇒ det 𝐴 − 𝐼𝜆 =1 − 𝜆 2 22 2 − 𝜆 11 2 2 − 𝜆

= 0

⇒ 1 − 𝜆 2 − 𝜆 1

2 2 − 𝜆 − 2 2 11 2 − 𝜆 + 2 2 2 − 𝜆

1 2 = 0

⇒ 1 − 𝜆 2 − 𝜆 2 − 𝜆 − 2 1 − 2 2 2 − 𝜆 − 1 1 + (2) 2 2 − 1 2 − 𝜆

= 5 − 𝜆 𝜆q = 0 ⇒ 𝜆 = 0,5

∴ 𝑡𝑟 𝐴 = 0 + 5 = 5Ex3:Given𝑡𝑟 𝐴 = 5 ∧ det 𝐴 = 6,find𝐴

𝐴 = 𝑎 10 𝑑

det 𝐴 = 𝑎𝑑 = 6

𝑡𝑟 𝐴 = 𝑎 + 𝑑 = 5

∴ 𝑑 = 5 − 𝑎 ⇒ 𝑎 5 − 𝑎 = 6 ⇒ 5𝑎 − 𝑎q = 6 ⇒ 𝑎q − 5𝑎 + 6 = 𝑎 − 2 𝑎 − 3 = 0

⇔ 𝑎 = 2 ∨ 𝑎 = 3 ⇒ 𝑑 = 2 ∨ 𝑑 = 3

∴ 𝑎, 𝑑 = 2,3 ∨ 𝑎, 𝑑 = 3,2

⇒ 𝐴 = 2 1

0 3 ∨ 3 10 2


CholeskyDecomposition

𝐴 = 𝐿𝐷𝑈 = 𝐿𝐷𝐿0 = 𝐿𝐷rq𝐷

rq𝐿0 = 𝐿𝐷

rq 𝐷

rq𝐿0 = 𝐾∗𝐾

𝐴 =9 00 9

−27 18−9 −27

−27 −918 −27

99 −27−27 121

Useelementarymatricestofind𝐿(watchlessononElementaryMatricesandorLDUdecomposition)

𝐿 =1 00 1

0 00 0

−9 −36 −9

1 00 1

∧ 𝑈 = 𝐿0 =1 00 1

−9 6−3 −9

0 00 0

1 00 1

𝐷 =9 00 9

0 00 0

0 00 0

9 00 4

⇒ 𝐷rq =

3 00 3

0 00 0

0 00 0

3 00 2

∴ 𝐿𝐷rq𝐷

rq𝐿0 =

1 00 1

0 00 0

−9 −36 −9

1 00 1

3 00 3

0 00 0

0 00 0

3 00 2

3 00 3

0 00 0

0 00 0

3 00 2

1 00 1

−9 6−3 −9

0 00 0

1 00 1

=3 00 3

0 00 0

−9 −36 −9

3 00 2

3 00 3

−9 6−3 −9

0 00 0

3 00 2

= 𝐾∗𝐾


Eigenvalues

𝐴 =1 2 22 1 10 0 1

⇒ det 𝐴 − 𝐼𝜆 =1 − 𝜆 2 22 1 − 𝜆 10 0 1 − 𝜆

1 − 𝜆 2 22 1 − 𝜆 10 0 1 − 𝜆

= 0 ⋅ 2 21 − 𝜆 1 − 0 ⋅ 1 − 𝜆 2

2 1 + 1 − 𝜆 1 − 𝜆 22 1 − 𝜆

∴ 1 − 𝜆 1 − 𝜆 22 1 − 𝜆 = 1 − 𝜆 1 − 𝜆 q − 4 = 1 − 𝜆 −3 − 2𝜆 + 𝜆q

= 1 − 𝜆 𝜆 + 1 𝜆 − 3

CharacteristicPolynomial:𝑝 𝜆 = 1 − 𝜆 𝜆 + 1 𝜆 − 3 ⇒ 𝑝 𝜆 = 0 ⇔ 𝜆 = −1, 1, 3

𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒𝑠 𝐴 = −1, 1, 3

EigenvectorsPartII-EasiestwaytofindEigenVectors

𝐴 =1 2 22 1 10 0 1

⇒ det 𝐴 − 𝐼𝜆 =1 − 𝜆 2 22 1 − 𝜆 10 0 1 − 𝜆

= 0 ⇒ 𝜆 = −1, 1, 3

𝜆r ⇒1 − −1 2 2

2 1 − −1 10 0 1 − −1

=2 2 22 2 10 0 2

~1 1 00 0 10 0 0

⇒ 𝒙 = 𝑠r−110

⇒ 𝑣r =−110

𝜆q ⇒1 − 1 2 22 1 − 1 10 0 1 − 1

=0 2 22 0 10 0 0

~1 0

12

0 1 10 0 0

⇒ 𝒙 = % q−12

−11

⇒ 𝑣q =−1−22

𝜆o ⇒1 − 3 2 22 1 − 3 10 0 1 − 3

=−2 2 22 −2 10 0 −2

~1 −1 00 0 10 0 0

⇒ 𝒙 = 𝑠o110

⇒ 𝑣o =110


Note:𝑠isafreevariablei.e.𝑠r = 1, 𝑠q = 2, 𝑠o = 1

∴ 𝛬 =−110

,−1−22

,110

DiagonlizeaMatrixPartIII-Diagonlize𝐴 = 1,2,2 , 2,1,1 , 0,0,1 ,notedet 𝐴 ≠ 0 ∴ 𝐴 = 𝑆𝐷𝑆pr

𝐴 =1 2 22 1 10 0 1

⇒ det 𝐴 − 𝐼𝜆 =1 − 𝜆 2 22 1 − 𝜆 10 0 1 − 𝜆

= 0 ⇒ 𝜆 = −1, 1, 3

𝑣r, 𝑣q, 𝑣o =−110

,−1−22

,110

𝐴 = 𝑆𝐷𝑆pr = 𝑣r𝑣q𝑣o 𝜆r𝑒r𝜆q𝑒q𝜆o𝑒o 𝑣r𝑣q𝑣o pr

Note:𝑒0 = 0, 0,⋯ , 1,⋯ , 0

∴ 𝐴 = 𝑆𝐷𝑆pr =−1 −1 11 −2 10 2 0

−1 0 00 1 00 0 3

−1 −1 11 −2 10 2 0

pr

=−1 −1 11 −2 10 2 0

−1 0 00 1 00 0 3

−12

12

14

0 012

12

12

34


SingularValueDecomposition

𝐴 = 𝑈𝛴𝑉0 , 𝐴 =1 10 1−1 1

Identifytheunknowns

𝑉0 = 𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟𝑠 𝐴0𝐴 0 =𝑣r𝑣q

𝑈 =1𝜎r𝐴𝑣r

1𝜎q𝐴𝑣q

𝑁𝑆 𝐴0

𝑁𝑆 𝐴0, 𝛴 =

𝜎r 00 𝜎q0 0

⇔ 𝜎0 = 𝜆0

∴ 𝐴 =1𝜎r𝐴𝑣r

1𝜎q𝐴𝑣q

𝑁𝑆 𝐴0

𝑁𝑆 𝐴0𝜎r 00 𝜎q0 0

𝑣r 𝑣q 0

Findallvalues

𝐴 =1 10 1−1 1

⇒ 𝐴0𝐴 = 2 00 3 ⇒ 𝜆r, 𝜆q = 3,2 ∧ 𝑣r, 𝑣q = 0

1 , 10

∴ 𝑉0 = 0 1

1 0

𝑈 =13

1 10 1−1 1

01

12

1 10 1−1 1

10

𝑁𝑆 𝐴0

𝑁𝑆 𝐴0

𝑁𝑆 𝐴0 ⇒ 𝐴0𝑥 = 0 ⇒ 1 1 11 0 −1

00 ~ 1 0 −1

0 1 200 ⇒ 𝑥 = 𝑥o

1−21

, 𝑥o = 𝑓𝑟𝑒𝑒 = 1 ∴ 𝑢o

=1−21

𝑢o = 1 + 4 + 1 = 6


⇒ 𝑈 =

13

12

16

13

0 −26

13

−12

16

, ∴ 𝐴 =

13

12

16

13

0 −26

13

−12

16

3 00 20 0

0 11 0 =

1 10 1−1 1

SystemofdifferentialequationsNote: 𝑆𝐷𝑆pr Ö = 𝑆𝐷Ö𝑆pr(easytoprove,tryitoutwithk=1,2,3,4…hint𝑆𝑆pr = 𝐼)

𝑑𝑋𝑑𝑡 = 𝐴𝑋 ⇒ 𝑑𝑋 = 𝑋𝐴𝑑𝑡 ⇒

1𝑋 𝑑𝑥 = 𝐴𝑑𝑡 ⇒ ln 𝑋 = 𝐴𝑡 + 𝐶r ⇒ 𝑋 = 𝑒á:��© = 𝑒á:𝑒�© = 𝐶𝑒�©

∴ 𝑋 = 𝐶𝑒�© ⇒ 𝑋 = 𝐶𝐴Ö𝑡Ö

𝑘!

∞

Ö12

= 𝐶𝑆𝐷Ö𝑆pr𝑡Ö

𝑘!

∞

Ö12

= 𝐶𝑆𝐷Ö𝑡Ö

𝑘!

∞

Ö12

𝑆pr

𝑥rÞ = 3𝑥r + 𝑥q − 𝑥o𝑥qÞ = 𝑥r + 3𝑥q − 𝑥o𝑥oÞ = 3𝑥r + 3𝑥q − 𝑥o

⇒ 𝑋Þ =3 1 −11 3 −13 3 −1

𝑋 ⇒ 𝑋 = 𝐶3 1 −11 3 −13 3 −1

Ö𝑡Ö

𝑘!

∞

Ö12

Diagonlize𝐴,𝐴 =3 1 −11 3 −13 3 −1

⇒ " =1 1 −11 0 13 1 0

1 0 00 2 00 0 2

−1 −1 13 3 −21 2 −1

∴ 𝐶1 1 −11 0 13 1 0

1 0 00 2 00 0 2

−1 −1 13 3 −21 2 −1

Ö𝑡Ö

𝑘!

∞

Ö12

= 𝐶1 1 −11 0 13 1 0

1 0 00 2 00 0 2

Ö𝑡Ö

𝑘!

∞

Ö12

−1 −1 13 3 −21 2 −1

= 𝐶1 1 −11 0 13 1 0

1Ö𝑡Ö

𝑘!

∞

Ö12

0 0

02Ö𝑡Ö

𝑘!

∞

Ö12

0

0 02Ö𝑡Ö

𝑘!

∞

Ö12

−1 −1 13 3 −21 2 −1

= 𝑐r 𝑐q 𝑐o1 1 −11 0 13 1 0

𝑒 0 00 𝑒q 00 0 𝑒q

−1 −1 13 3 −21 2 −1


Oruseaformula(easiestwiththreeeigenvectors)

𝑋 = 𝑐r𝑣r𝑒�: + 𝑐q𝑣q𝑒�� + 𝑐o𝑣o𝑒�Ó = 𝑐r113

𝑒 + 𝑐q101

𝑒q + 𝑐o−110

𝑒q

LinearProgramming:SimplexMethodSolvethelinearprogrammingproblembythesimplexmethod.Maximize𝑃 = 5𝑥 + 4𝑦subjectto3𝑥 + 5𝑦 ≤ 145and4𝑥 + 𝑦 ≤ 104and𝑥 ≥ 0and𝑦 ≥ 0SimplexTableau:(note:𝑃 = 5𝑥 + 4𝑦 ⇒ 𝑃 − 5𝑥 − 4𝑦 = 0)

𝑥 𝑦 𝑢 𝑣 𝑃 Constant3 5 1 0 0 1454 1 0 1 0 104−5 −4 0 0 1 0

1stSince−4 > −5thesecondcolumnisthepivotcolumn2ndPerformr

é𝑅1 ∧ − r

Ü𝑅3tomakecolumn2have1’s

𝑥 𝑦 𝑢 𝑣 𝑃 Constant35

1 15

0 0 29

4 1 0 1 0 10454

1 0 0 14

0

3rdWewantcolumntwotobeaunitcolumni.e. 0,1,0 perform𝑅2 − 𝑅1 ∧ 𝑅2 − 𝑅3


0 −15

1 0 75

4 1 0 1 0 104114

0 0 1 −14

104

4thRepeatforcolumn1{1,0,0}−q2

rë𝑅1 + 𝑅2 ∧ − é

rërrÜ𝑅1 + 𝑅3


0 −15

1 0 75

0 1 417 −

317

0 26817


0 0 1168

1368 −

14

−294768

5th é

rë𝑅1 ∧ −4𝑅3

𝑥 𝑦 𝑢 𝑣 𝑃 Constant1 0 −

117

1 0 37517

0 1 417 −

317

0 26817

0 0 −1117 −

1317

1 294717

Maximize𝑃 = 5𝑥 + 4𝑦subjectto3𝑥 + 5𝑦 ≤ 145and4𝑥 + 𝑦 ≤ 104and𝑥 ≥ 0and𝑦 ≥ 0

max 𝑃 =294717 ⇔ 𝑥, 𝑦 =

117 375,268

DIFFERENTIALEQUATIONSIntrotothefirst-orderdifferentialequation

𝑦Þ = 𝑥 ⇒ 𝑑𝑦𝑑𝑥 = 𝑥

Type:First-order-nonhomogeneouslineardifferentialequationSolutionMethod:SeparablevariableAnswer:Explicit-generalsolution

𝑑𝑦 = 𝑥𝑑𝑥

⇒ 𝑑𝑦 = 𝑥 𝑑𝑥𝑦

⇒ 𝑦 + 𝑐r =12 𝑥

q + 𝑐q

⇒ 𝑦 =12 𝑥

q + 𝑐q − 𝑐r, 𝐧𝐨𝐭𝐞:𝑐q − 𝑐r = 𝑐o = 𝐶

∴ 𝑦 =12 𝑥

q + 𝐶


HomogeneousAdifferentialequationthathasafunctionofwhichdoescontainthevariablethatisbeingdifferentiated.

𝑦 0 +⋯𝑦0 +⋯ = 0

Example:

𝑦 é − 𝑦ÞÞ +𝑦Þ

𝑦 𝑥q − cos 𝑥𝑦 = 0

NonhomogeneousAdifferentialequationthathasafunctionofwhichdoesnotcontainthevariablethatisbeingdifferentiated.

𝑓��r𝑦 0 + ⋯𝑓� 𝑥 𝑦0 +⋯ = 𝑔 𝑥

Example:

𝑦 é − 𝑦ÞÞ +𝑦Þ

𝑦 𝑥q − cos 2𝑥 = 0

LinearAdifferentialequationthatcontainsonlyderivativesinthenumeratorstate,hasthehighestpowerofthevariablebeingdifferentiatedis1,andthedifferentiatedvariableisnotbeingoperatedon.

Example:

𝑦 0 + 𝑦 0pr + ⋯𝑦 = 0

𝑎0 𝑥 𝑦 0 + 𝑎0pr 𝑥 𝑦 0pr + ⋯ = 0

Non-linearAdifferentialequationwherethedifferentiatedvariableisalsobeingoperatedonbyfunctions.

Example:

𝑦ÞÞ +𝑦Þ

𝑦 𝑥q − cos 2𝑥 = 0,

1𝑦 = 𝑦pr

𝑦 é − 𝑦ÞÞ + 𝑦Þ𝑥q − cos 𝑥𝑦 = 0

𝑦𝑦Þ = 𝑥

Note:MakesuretounderstandhowthefollowingtermsrelatetoaDEi.e.thetypeofDEwilltellyouwhatmethodtouseinordertosolvetheDEType:Order–Linearity–HomogeneityProblem:

• Initial-ValueProblem(IVP)hasaParticularSolution• NON-IVPhasaGeneralSolution(Constant𝐶insolution)• ExplicitSolution:𝑦 = 𝑓 𝑥 • Implicitsolution:𝑦0 ⋯ = 𝑓 𝑥,…

Theorderofadifferentialequationisdependentuponthehighestderivativee.g.𝑦ÞÞÞ + 𝑦ÞÞ = 0isathird-orderdifferentialequation.Note:Donotconfuse𝑦0with𝑦(0)

• 𝑦(0)isthenthderivative• 𝑦0isthenthpower


e.g.𝑦Ü = 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦where𝑦(Ü) = »»�

»»�

»»�

»»�𝑦

Orderofderivativenotation:𝑌, 𝑦, 𝑦Þ, 𝑦ÞÞ, 𝑦ÞÞÞ, 𝑦 Ü , …𝑦 0 Respecttotime:𝑦, 𝑦, 𝑦 ≡ 𝑦Þ 𝑡 , 𝑦ÞÞ 𝑡 , 𝑦ÞÞÞ 𝑡 ≡ »¦

»©, »

�¦»©�

, »Ó¦»©Ó

Letslookatacoupleexamplesofequationsthatarelinearandnonlinear.

𝑦é + 𝑥𝑦ÞÞ − »�¦»��

= sin 𝑥𝑦 ,Sixth-Order-Nonlinearandhomogeneous𝑥𝑦ÞÞ − »�¦

»��= sin 𝑥 ,Sixth-Order-Linearandnonhomogeneous

𝑦′′+ 𝑦′+ 𝑦𝑥 = 0,Second-Order-Linearandhomogeneous𝑦ÞÞ + 𝑦𝑦Þ = ln 𝑥,Second-Order-NonlinearandnonhomogeneousNote:Althoughthepowerofyis1inthiscase,itisdependentupony’makingitnonlinear.𝑦′′′+ 𝑦q + 𝑥𝑒¦ = 0,Third-Order-Nonlinearandhomogeneous

1stOrderSolutionMethodsSeparableVariableScenarioTheseparablevariableequationisprettymuchjustanaverageintegrationproblemyoumayhaveencounteredincalculus.Theideaisthatyouhaveafirst-orderDEanditisintheformofafunctionof𝑥, 𝑦,and𝑓 𝑥, 𝑦 canbefoundinaDEe.g.𝑓 𝑥, 𝑦 𝑦Þ = 𝑝 𝑥 ,whichcanbeseparatedintotheform𝑔 𝑦 𝑑𝑦 = ℎ 𝑥 𝑑𝑥.Theseparablevariablecouldalsobeviewedas𝑦′ = 𝑓r 𝑦 𝑓q 𝑥 Ex.1(Explicitvs.Implicit)Given𝑓 𝑥, 𝑦 = 𝑥𝑦and𝑝 𝑥 = 𝑥qSolve𝑓 𝑥, 𝑦 𝑦Þ = 𝑝 𝑥 Type:First-Order-Nonlinear-Nonhomogeneous-ODEODE-OrdinaryDifferentialEquationPDE-PartialDifferentialEquationsSolutionMethod:SeparableVariable

𝑓 𝑥, 𝑦 𝑦Þ = 𝑝 𝑥

⇒ 𝑥𝑦𝑦Þ = 𝑥q

⇒ 𝑥𝑦𝑑𝑦𝑑𝑥 = 𝑥q

⇒ 𝑦𝑑𝑦 = 𝑥𝑑𝑥

⇒ 𝑦𝑑𝑦 = 𝑥𝑑𝑥


Answer:Implicit-GeneralSolution ⇒

𝑦q

2 + 𝑐r =𝑥q

2 + 𝑐q⇒ 𝑦q + 𝑐o = 𝑥q + 𝑐Ü

⇒ 𝑦q = 𝑥q + 𝑐Ü − 𝑐o = 𝑥q + 𝑐é

∴ 𝑦q = 𝑥q + 𝐶

Ex.2(SeparableVariable)SolvetheDE

𝑥𝑦Þ − 𝑥 = 2Type:first-order-linearnonhomogeneousdifferentialequationSolution:SeparableVariableAnswer:Explicit-GeneralSolution

𝑥𝑑𝑦𝑑𝑥 = 2 + 𝑥

⇒𝑑𝑦𝑑𝑥 =

2 + 𝑥𝑥

⇒ 𝑑𝑦 =2 + 𝑥𝑥 𝑑𝑥

⇒ 𝑑𝑦 =2 + 𝑥𝑥 𝑑𝑥

⇒ 𝑦 + 𝑐r = 2 ln 𝑥 + 𝑥 + 𝑐q

∴ 𝑦 = 2 ln |𝑥| + 𝑥 + 𝐶

Ex.3*(IVPProblem)GivetheimplicitsolutiontotheIVP:-𝑦𝑥pr = 𝑥Þ 𝑦 ; 𝑦 −3 = 4

−𝑦𝑥 =

𝑑𝑥𝑑𝑦 ⇒ 𝑦𝑑𝑦 = 𝑥𝑑𝑥 ⇒ 𝑦 𝑑𝑦 = − 𝑥 𝑑𝑥 ⇒

12 𝑦

q = −12𝑥

q + 𝑐q − 𝑐r

⇒ 𝑦q + 𝑥q = 2 𝑐q + 𝑐r = 𝐶

Solvingimplicitly

𝑦 = ± 𝐶 − 𝑥q

Letssolvethisimplicitlyfirst −3 q + 5 q = 25 = 𝐶so𝑥q + 𝑦q = 5qacirclecenteredattheoriginwithradius5.


Nowletstakealookatthisexplicitly

−3 =− 𝐶 − 4 q

+ 𝐶 − 5 q

Butwait!Because 𝑢 ≥ 0∀3∈ 𝑅 − 3 =1stOrderLinearNon-homogeneousi.e.y’+P(x)y=Q(x)ProcessGivenafirstorderlinearnon-homogeneousdifferentialequationoftheform𝑦Þ + 𝑃 𝑥 𝑦 = 𝑄 𝑥 thesolutionis:

𝑦 =1𝐼 𝑥 𝐼 𝑥 𝑄 𝑥 𝑑𝑥 + 𝐶 , 𝐼 𝑥 = 𝑒 Ù � »�

Ex.1

𝑑𝑦𝑑𝑥 +

9𝑥 𝑦 =

1𝑥ç

𝑃 𝑥 =9𝑥 , 𝑄 𝑥 =

1𝑥ç , 𝐼 𝑥 = 𝑒

è�»� = 𝑒è ®¯ � = 𝑒 ®¯ �� = 𝑥è

⇒ 𝑦 =1𝑥è 𝑥è ∙

1𝑥ç 𝑑𝑥 + 𝑐r =

1𝑥è 𝑥 𝑑𝑥 + 𝑐r =

1𝑥è

12 𝑥

q + 𝑐r =1𝑥è

𝑥q + 2 ∙ 𝑐r2

=𝑥q + 𝐶2𝑥è

Ex.2

𝑦 ∙ sin 𝑥 𝑑𝑦 − yq ∙ csc 𝑥 𝑑𝑥 = 𝑦 ∙ csc 𝑥 𝑑𝑥

TherearemanyformsDE’scantakeonandmanysolutionstothem,someareeasierthanotherssotheonlytruewaytounderstandwhentousewhatistoexposeyourselftomanysituations.

𝑦pr csc 𝑥1𝑑𝑥 𝑦 ∙ sin 𝑥 𝑑𝑦 − yq ∙ csc 𝑥 𝑑𝑥 = 𝑦 ∙ csc 𝑥 𝑑𝑥 ⇒

𝑑𝑦𝑑𝑥 − csc

q 𝑥 𝑦 = cscq 𝑥

∴ 𝑃 𝑥 = −cscq 𝑥 , 𝑄 𝑥 = sec 𝑥 , 𝐼 𝑥 = 𝑒 p j/j� � »� = 𝑒jk- �

𝑦 =1

𝑒jk- �𝑒jk- � ∙ cscq 𝑥 𝑑𝑥 + 𝐶 = 𝑒p jk- � −𝑒jk- � + 𝐶 = −1 + 𝐶𝑒p jk- �

∴ 𝑦 𝑥 = 𝐶𝑒p jk- � − 1


ExactDifferentialEquationTheexactequationisnotverycommonandstemsfrompartialderivatives.Thesolutionisgenerallyverysimple.Situation

𝑓� 𝑥, 𝑦 𝑑𝑥 + 𝑓¦ 𝑥, 𝑦 𝑑𝑦 = 0 ⇔ 𝜕𝜕𝑦 𝑓� =

𝜕𝜕𝑥 𝑓¦

Thisimpliesthatthereisacommonfunctionℎ 𝑥, 𝑦 ineachindividualantiderivativeEx.1

𝑥q − 𝑦q 𝑑𝑥 + 𝑦q − 2𝑥𝑦 𝑑𝑦 = 0

𝑓� = 𝑥q − 𝑦q ⇒ 𝜕𝜕𝑦 𝑓� = −2𝑦

𝑓¦ = 𝑦q − 2𝑥𝑦 ⇒ 𝜕𝜕𝑥 𝑓¦ = −2𝑦

∴ 𝜕𝜕𝑦 𝑓� = −2𝑦 =

𝜕𝜕𝑥 𝑓¦

SothisisanexactequationAllyouhavetodoisintegrateandfindthecommontermandrecallfromseveralvariablecalculusthatweareintegratingamultiplevariablefunctionthattheconstantaddedisafunctionofthevariablenotbeingintegratedi.e.

𝑥q − 𝑦q 𝑑𝑥 = 0 ⇒ 13 𝑥

o − 𝑥𝑦q + 𝑘 𝑦 = 𝑐r

𝑦q − 2𝑥𝑦 𝑑𝑦 = 0 ⇒ 13 𝑦

o − 𝑥𝑦q + 𝑙 𝑥 = 𝑐q

Fromthiswecaneasilyidentify𝑘 𝑦 and𝑙 𝑥 oritmayjustbeeasytoseethesimilarityi.e.thinkabouttakingpartialderivativesof𝑓 𝑥, 𝑦 = r

o𝑥o − 𝑥𝑦q + r

o𝑦o

𝑘 𝑦 =13𝑦

o, 𝑙 𝑥 =13 𝑥

o, 𝑐𝑜𝑚𝑚𝑜𝑛𝑡𝑒𝑟𝑚 = −𝑥𝑦q

∴ 𝑓 𝑥, 𝑦 = 13 𝑥

o − 𝑥𝑦q +13𝑦

o = 𝐶


Thereareotherapproachestothis;infactyoucouldsolvethisinonstraightshotbyjustintegratingthewholeequationandidentifyingthecommontermbutcheckwithyourteacherhowmuchdetailtheywouldprefer.Note:Thisproblemisnotlikelytoshowuponexams(maybeaquiz)becauseitissuchararecaseandisreallyverysimpletosolve.Yourexamswillmostlikelyhaveallsecondorderorhigherdifferentialequations.General,ParticularandSuperpositionSolutionsFordifferentialequationsofhigherorderthan1,therewillbemultiplesolutionsi.e.𝑦r, 𝑦q, … , 𝑦0whereeachindividualyandallthey’stogetheraresolutionstotheDE.GeneralSolutionThegeneralsolutioncontainsaconstante.g.𝑦 = 𝑥 + 𝐶𝑒/ó¯ � ParticularSolution Theparticularsolutioncontainsnoconstants,usuallyduetoaninitialvalueoraspartofanon-homogeneoussolution.SuperpositionSolutionForDE’swithmultiplesolutions,thesumofthesolutionsisalsoasolutioni.e.𝑦 = 𝑐r𝑦r + 𝑐q𝑦q +⋯+ 𝑐0𝑦0andinanon-homogeneoussituationthesolutionwillbethesumofthesolutiontothehomogenouspartoftheequationandthenon-homogenouspartgenerallynotedas𝑦 = 𝑦� + 𝑦Ù.Thegeneralsolutionisgenerallynotedas𝑦� andtheparticular𝑦Ùgivingthesolutiontobe𝑦 =𝑦� + 𝑦Ù

𝑦 = (𝑐r𝑦�,r + 𝑐q𝑦�,q + ⋯+ 𝑐0𝑦�,0) + (𝑦Ù,r + 𝑦Ù,q + ⋯+ 𝑦Ù,0)

LinearHomogenouswithConstantCoefficientsScenario

𝑎r𝑦 0 + ⋯+ 𝑎q𝑦 0pÖ + ⋯+ 𝑎�𝑦 = 0

AuxiliaryequationSubstitute𝑦 = 𝑒Ô©intotheequation,eliminate𝑒Ô©andsolveforr

𝑎r𝑟0 + ⋯+ 𝑎q𝑟 0pÖ + ⋯+ 𝑎� = 0

Solution(s)


If𝑟hasapairofsolutions

𝑦� = 𝑐r𝑒Ô:© + 𝑐q𝑒Ô�©

If𝑟hasnrepeatingsolutions

𝑦� = 𝑐r𝑒Ô© + 𝑐q𝑡𝑒Ô© + 𝑐o𝑡q𝑒Ô© + ⋯+ 𝑐0𝑡0pr𝑒Ô©

If𝑟hasapairofcomplexsolutions

𝑟 = 𝛼 ± 𝑖𝛽, 𝑦 = 𝑒�© cos 𝛽𝑡 + 𝑒�© sin 𝛽𝑡

GenerallySpeaking

𝑎𝑦ÞÞ + 𝑏𝑦 + 𝑐𝑦 = 0, 𝑦 = 𝑒��, 𝑦Þ = 𝑚𝑒��, 𝑦ÞÞ = 𝑚q𝑒��

⇒ 𝑎 𝑚q𝑒�� + 𝑏 𝑚𝑒�� + 𝑐 𝑒�� = 𝑒�� 𝑎𝑚q + 𝑏𝑚 + 𝑐 = 0

Identify𝑒�� > 0soithasnopurposeforoursolutionleaving𝑎𝑚q + 𝑏𝑚 + 𝑐 = 0,whichistheauxiliaryequationandthequadraticequationmaybeusedtosolveit.

𝑎𝑚q + 𝑏𝑚 + 𝑐 = 0, 𝑚 =−𝑏 ± 𝑏q − 4𝑎𝑐

2𝑎

Two-real𝑏q − 4𝑎𝑐 > 0

𝑦 = 𝑐r𝑒�:� + 𝑐q𝑒��

Repeated𝑏q − 4𝑎𝑐 = 0

𝑦 = 𝑐r𝑒�� + 𝑐q𝑥𝑒��

Complexi.e.𝑏q − 4𝑎𝑐 < 0 ⇒ 𝑥 = 𝛼 ± 𝑖𝛽

𝑦 = 𝑒�� 𝑐r cos 𝛽𝑥 + 𝑐q sin 𝛽𝑥

Ex.1

3𝑦ÞÞ + 4𝑦Þ + 5𝑦 = 0

Extractauxiliaryequationi.e.3𝑟q + 4𝑟 + 5 = 0,solvefor𝑟

𝑟 =−4 ± 16 − 4 ∙ 3 ∙ 5

2 ∙ 3 = −46 ±

−446 = −

23 ± 𝑖

26 11 = −

23 ± 𝑖

13 11

𝛼 = −23 , 𝛽 =

113


∴ 𝑦 = 𝑒 pqo © 𝑐r cos113 𝑡 + 𝑐q sin

113 𝑡

Theothertwoareeasytosolvewiththegivenformulas.WhatwearereallyinterestedinnowishowtousethegivenformulasforhigherorderDE’s.Ex.2

3𝑦ÞÞÞ + 4𝑦Þ = 0

Theauxiliaryequationis3𝑟o + 0 ∙ 𝑟q + 4𝑟 + 0 = 3𝑟o + 4𝑟 = 0

3𝑟o + 4𝑟 = 0 ⇒ 𝑟 3𝑟q + 4 = 0 ⇒ 𝑟 = 0&𝑟 = 0 ± 𝑖23

Wenowhave3solutionsi.e.𝑦 = 𝑐r𝑦r + 𝑐q𝑦q + 𝑐o𝑦oThecomplexscenarioshouldbeprettyobviousjustplugitintotheformula

𝑐r𝑦r + 𝑐q𝑦q = 𝑒2∙© 𝑐r cos23𝑡 + 𝑐q sin

23𝑡 = 𝑐r cos

23𝑡 + 𝑐q sin

23𝑡

Whataboutthesolution0?0isrepeatedoncehence

𝑐o𝑦o = 𝑐o𝑒2∙© = 𝑐o 1 = 𝑐o

∴ 𝑦 = 𝑐r cos23𝑡 + 𝑐q sin

23𝑡 + 𝑐o

Ex.3

𝑦ÞÞÞ + 8𝑦 = 0

𝑥o + 𝑎o = 𝑥 + 𝑎 𝑥q − 𝑎𝑥 + 𝑎q

𝑟o + 8 = 0 ⇒ 𝑟 + 2 𝑟q − 2𝑟 + 4 = 0 ⇒ 𝑟 = −2, 1 ± 𝑖 3

𝑦r,q = 𝑒© 𝑐r cos 3𝑡 + 𝑐q sin 3𝑡 , 𝑦o = 𝑐o𝑒pq©

∴ 𝑦 = 𝑒© 𝑐r cos 3𝑡 + 𝑐q sin 3𝑡 + 𝑐o𝑒pq©

Ex.4

𝑦 Ü + 8𝑦ÞÞÞ = 0 ⇒ 𝑟Ü + 8𝑟o = 𝑟o 𝑟 + 8 = 0 ⇒ 𝑟o = 0, 𝑟 = −8

Zeroisrepeatedthreetimeshere


𝑦r,q,o = 𝑐r𝑒2∙� + 𝑐q𝑥𝑒2∙� + 𝑐o𝑥q𝑒2∙� = 𝑐r + 𝑐q𝑥 + 𝑐o𝑥q, 𝑦Ü = 𝑐Ü𝑒pç�

∴ 𝑦 = 𝑐r + 𝑐q𝑥 + 𝑐o𝑥q + 𝑐Ü𝑒pç�

Ex.5IVPy(0)=1,y’(0)=2,y’’(0)=3,y’’’(0)=4Usingthesolutionfromexample4

𝑦� = 𝑐r + 𝑐q𝑥 + 𝑐o𝑥q + 𝑐Ü𝑒pç�

𝑦Þ = 𝑐q + 2𝑐o𝑥 − 8𝑐Ü𝑒pç�, 𝑦ÞÞ = 2𝑐o + 64𝑐Ü𝑒pç�, 𝑦ÞÞÞ = −512𝑐Ü𝑒pç�

𝑦 0 = 1 ⇒ 1 = 𝑐r + 𝑐q ∙ 0 + 𝑐o ∙ 0 + 𝑐Ü𝑒2 = 𝑐r + 𝑐Ü

𝑦Þ 0 = 2 ⇒ 2 = 𝑐q + 2𝑐o ∙ 0 − 8𝑐Ü𝑒2 = 𝑐q − 8𝑐Ü

𝑦ÞÞ 0 = 3 ⇒ 3 = 2𝑐o + 64𝑐Ü𝑒2 = 2𝑐o + 64𝑐Ü

𝑦ÞÞÞ 0 = 4 ⇒ 4 = −512𝑐Ü𝑒2 = −512𝑐Ü

Solvethesystem

𝑐r + 𝑐Ü = 1, 𝑐q − 8𝑐Ü = 2, 2𝑐o + 64𝑐Ü = 3, −512𝑐Ü = 4

𝑐r =129128 , 𝑐q =

3116 , 𝑐o =

74 , 𝑐Ü = −

1128

∴ 𝑦Ù =129128 +

3116 𝑥 +

74 𝑥

q −1128 𝑒

pç�

ReductionofOrderProcessGivenasecondorderlinearhomogeneousDEoftheform𝑦ÞÞ + 𝑃 𝑥 𝑦Þ + 𝑄 𝑥 = 0accompaniedwith𝑦r(𝑥)SolutionSincethefirstsolutionisgiven,youmustfindthesecondsolution,whichis:

𝑦q 𝑥 = 𝑦r 𝑥𝑒p à � »�

𝑦r 𝑥 q 𝑑𝑥, ∴ 𝑦 = 𝑐r𝑦r + 𝑐q 𝑦r 𝑥𝑒p à � »�

𝑦r 𝑥 q 𝑑𝑥


Ex.1

𝑥q𝑦ÞÞ + 2𝑥𝑦Þ − 6𝑦 = 0, 𝑦r = 𝑥q

Find𝑃 𝑥

1𝑥q 𝑥q𝑦ÞÞ + 2𝑥𝑦Þ − 6𝑦 = 0 ⇒ 𝑦ÞÞ +

2𝑥 𝑦

Þ −6𝑥q 𝑦 = 0 ⇒ 𝑃 𝑥 =

2𝑥

∴ 𝑦q = 𝑥q𝑒p

q�»�

𝑥q q 𝑑𝑥 = 𝑥q𝑒pq ®¯ �

𝑥Ü 𝑑𝑥 = 𝑥q𝑒®¯ �ü�

𝑥Ü 𝑑𝑥 = 𝑥q𝑥pq

𝑥Ü 𝑑𝑥 = 𝑥q 𝑥pê 𝑑𝑥

= 𝑥q1−5𝑥

pé = −15𝑥

po ⇒ 𝑦q =1𝑥o

Theconstantcanbeignoredbecauseaconstanttimesaconstantisaconstant

∴ 𝑦 = 𝑐r𝑥q +

𝑐q𝑥o

Atthispointitshouldbecomeobviousthat𝑐r + 𝑐q + ⋯+ 𝑐0 = 𝐶,thisisalsotruefornumbersi.e.𝑐r + 5 + 𝑒 + ln 10 + 𝑒�Ó + 6𝑐q = 𝐶.Inotherwords:aconstantwithaconstantisaconstant.SubstitutionGeneralSituationThemethodofsubstationworkswellwithDE’sthatlooksimilartoanexactequationi.e.

𝑥q + 4𝑥𝑦 𝑑𝑥 + 𝑦q − 4𝑥q 𝑑𝑦 = 0Wecancheckforanexactandseethat𝜕¦𝑓� = 4𝑥 ≠ −8𝑥 = 𝜕�𝑓¦soweknowwecannotusethatmethodbecausetheyarenotequal.SubstitutionSolutionMethodSet𝑦 = 𝑣 𝑥 ∙ 𝑥andtakethederivativei.e.𝑦Þ = 𝑣 𝑥 + 𝑥 ∙ 𝑣Þ 𝑥 andsolvefor𝑑𝑦.

𝑑𝑦𝑑𝑥 = 𝑣 + 𝑥 ∙

𝑑𝑣𝑑𝑥 ⇒ 𝑑𝑥


𝑑𝑣𝑑𝑥 ⇒ 𝑑𝑦 = 𝑣𝑑𝑥 + 𝑥𝑑𝑣

Ifyouusethismethodanditgetsreallysloppy,itprobablyisnotthebestchoicesotrysomethingelse;letsseehowthisDEplaysout.Substitute𝑦and𝑑𝑦andsimplifyi.e.

𝑥q + 4𝑥 𝑣𝑥 𝑑𝑥 + 𝑣𝑥 q − 4𝑥q 𝑣𝑑𝑥 + 𝑥𝑑𝑣 = 0


⇒ 𝑥q𝑑𝑥 + 4𝑥q𝑣𝑑𝑥 + 𝑥q𝑣o𝑑𝑥 + 𝑥o𝑣q𝑑𝑣 − 4𝑥q𝑣𝑑𝑥 − 4𝑥o𝑑𝑣 = 0

⇒ 𝑥q𝑑𝑥 + 𝑥q𝑣o𝑑𝑥 + 4𝑥q𝑣𝑑𝑥 − 4𝑥q𝑣𝑑𝑥 + 𝑥o𝑣q𝑑𝑣 − 4𝑥o𝑑𝑣 = 0

⇒1𝑥q 𝑥q𝑑𝑥 + 𝑥q𝑣o𝑑𝑥 + 𝑥o𝑣q𝑑𝑣 − 4𝑥o𝑑𝑣 = 0

⇒ 𝑑𝑥 + 𝑣o𝑑𝑥 + 𝑥𝑣q𝑑𝑣 − 4𝑥𝑑𝑣 = 0

⇒ 1 + 𝑣o 𝑑𝑥 + 𝑥 𝑣q − 4 𝑑𝑣 = 0 ⇒ 𝑥 𝑣q − 4 𝑑𝑣 = − 1 + 𝑣o 𝑑𝑥

⇒𝑣q − 41 + 𝑣o 𝑑𝑣 = −

1𝑥 𝑑𝑥

Nowintegratetheseparablevariabledifferentialequation.

𝑣q

1 + 𝑣o −4

1 + 𝑣o 𝑑𝑣 = −1𝑥 𝑑𝑥

Atthispoint,itisjustareallytediouscalculusproblem. 𝑣 = ¦

�Twooftheintegralsaresimplei.e.

13 ln 1 + 𝑣

o − 41

1 + 𝑣o 𝑑𝑣 = − ln 𝑥 + 𝐶 ⇒ 13 ln 1 +

𝑦𝑥

o− 4

11 + 𝑣o 𝑑𝑣 = − ln 𝑥 + 𝐶

Wejustneedtointegrate rr�4Ó

.Thisisaverycomplicatedintegraltodobyhand.IusedWolfram|Alphatocompletethis.

∴1

1 + 𝑣o =16 − ln 𝑣q − 𝑣 + 1 + 2 ln 𝑣 + 1 + 2 3 tanpr

2𝑣 − 13

⇒13 ln 1 +

𝑦𝑥

o− 4

16 − ln 𝑣q − 𝑣 + 1 + 2 ln 𝑣 + 1 + 2 3 tanpr

2𝑣 − 13

= − ln 𝑥 + 𝐶

⇒13 ln 1 +

𝑦𝑥

o− 4

16 − ln

𝑦𝑥

q−𝑦𝑥 + 1 + 2 ln

𝑦𝑥 + 1 + 2 3 tanpr

2 𝑦𝑥 − 1

3

= − ln 𝑥 + 𝐶Thiswasaveryloadedsituation;itishighlyunlikelytoseesomethinglikethisinanundergraduateDEcourse.Whentousethismethod?Ifyouhaveasituationwithafirst-orderdifferentialequationthatisnotlinearandtheexactequationmethoddoesnotworkoristwocomplicatedandviceversa.


IntegratingFactorsWhenyouhavethe“exactequation”lookingsituationbutitisnotanexactequationi.e.

𝑀𝑑𝑥 + 𝑁𝑑𝑦 = 0&𝜕𝑀𝜕𝑦 ≠

𝜕𝑁𝜕𝑥

Thenyoucanmultiplythewholeequationby𝜇 𝑥 or𝜇 𝑦 anditwillthenbecomeanexactequation.

𝜇 𝑥 = 𝑒� p�!

� »�, 𝜇 𝑦 = 𝑒�!p� � »¦

Ex.1

𝑥𝑦 𝑑𝑥 + 2𝑥q + 3𝑦q − 20 𝑑𝑦 = 0

𝑀 = 𝑥𝑦, 𝑁 = 2𝑥q + 3𝑦q − 20

𝜕𝑀𝜕𝑦 = 𝑥 ≠ 4𝑥 =

𝜕𝑁𝜕𝑥 ,

𝑀¦ − 𝑁�𝑁 =

−3𝑥2𝑥q + 3𝑦q − 20 ,

𝑁� −𝑀¦

𝑀 =3𝑥𝑥𝑦 =

3𝑦

Wearelookingfortheonethathasasinglevariableandalsoeasiesttointegrate.

∴ 𝜇 𝑦 = 𝑒o¦»¦ = 𝑒o ®¯ ¦ = 𝑒®¯ ¦Ó = 𝑦o

Nowmultiplytheoriginalequationby𝑦o

𝑦o ∙ 𝑥𝑦 𝑑𝑥 + 2𝑥q + 3𝑦q − 20 𝑑𝑦 = 0 ⇒ 𝑥𝑦Ü 𝑑𝑥 + 2𝑥q𝑦o + 3𝑦é − 20𝑦o 𝑑𝑦 = 0

∴ 𝜕𝜕𝑦 𝑥𝑦Ü = 4𝑥𝑦Üpr = 4𝑥𝑦o ⇒ 4𝑥𝑦o = 2 2𝑥𝑦o =

𝜕𝜕𝑥 2𝑥q𝑦o + 3𝑦é − 20𝑦o ⇔

𝜕𝑀𝜕𝑦

=𝜕𝑁𝜕𝑥

Thus,itisanexactequationnow.Integrateallthewaythroughandidentifytheequivalenttermi.e.

𝑥𝑦Ü 𝑑𝑥 = 𝑐r ⇒ 12 𝑥

q𝑦Ü + 𝑔 𝑦 = 𝑐r


2𝑥q𝑦o + 3𝑦é − 20𝑦o 𝑑𝑦 = 𝑐q ⇒ 12 𝑥

q𝑦Ü +12𝑦

ê − 5𝑦Ü + ℎ 𝑥 = 𝑐r

Settingthesetwoequationsequal(therearemanywaystofindthisbytheway,thisisjustonemethod,seeexactequations)finding𝑔 𝑦 andℎ 𝑥

12 𝑥

q𝑦Ü +12𝑦

ê − 5𝑦Ü + ℎ 𝑥 =12𝑥

q𝑦Ü + 𝑔 𝑦 + 0

⇒12 𝑥

q𝑦Ü =12 𝑥

q𝑦Ü = 𝑓 𝑥, 𝑦 , 𝑔 𝑦 =12𝑦

ê − 5𝑦Ü , ℎ 𝑥 = 0, 𝑐r + 𝑐q = 𝐶

𝑓 𝑥, 𝑦 + 𝑔 𝑦 + ℎ 𝑥 = 𝐶 ∴ 12 𝑥

q𝑦Ü +12𝑦

ê − 5𝑦Ü = 𝐶

SECONDORDERDIFFERENTIALEQUATIONSLinearHomogenouswithConstantCoefficientsScenario

𝑎r𝑦 0 + ⋯+ 𝑎q𝑦 0pÖ + ⋯+ 𝑎�𝑦 = 0AuxiliaryequationSubstitute𝑦 = 𝑒Ô©intotheequation,eliminate𝑒Ô©andsolveforr

𝑎r𝑟0 + ⋯+ 𝑎q𝑟 0pÖ + ⋯+ 𝑎� = 0Solution(s)If𝑟hasapairofsolutions

𝑦� = 𝑐r𝑒Ô:© + 𝑐q𝑒Ô�©If𝑟hasnrepeatingsolutions

𝑦� = 𝑐r𝑒Ô© + 𝑐q𝑡𝑒Ô© + 𝑐o𝑡q𝑒Ô© + ⋯+ 𝑐0𝑡0pr𝑒Ô©If𝑟hasapairofcomplexsolutions

𝑟 = 𝛼 ± 𝑖𝛽, 𝑦 = 𝑒�©𝑐r cos 𝛽𝑡 + 𝑒�©𝑐q sin 𝛽𝑡 GenerallySpeaking


𝑎𝑦ÞÞ + 𝑏𝑦 + 𝑐𝑦 = 0, 𝑦 = 𝑒��, 𝑦Þ = 𝑚𝑒��, 𝑦ÞÞ = 𝑚q𝑒��

⇒ 𝑎 𝑚q𝑒�� + 𝑏 𝑚𝑒�� + 𝑐 𝑒�� = 𝑒�� 𝑎𝑚q + 𝑏𝑚 + 𝑐 = 0

Identify𝑒�� > 0soithasnopurposeforoursolutionleaving𝑎𝑚q + 𝑏𝑚 + 𝑐 = 0,whichistheauxiliaryequationandthequadraticequationmaybeusedtosolveit.

𝑎𝑚q + 𝑏𝑚 + 𝑐 = 0, 𝑚 =−𝑏 ± 𝑏q − 4𝑎𝑐

2𝑎 Two-real𝑏q − 4𝑎𝑐 > 0

𝑦 = 𝑐r𝑒�:� + 𝑐q𝑒��Repeated𝑏q − 4𝑎𝑐 = 0

𝑦 = 𝑐r𝑒�� + 𝑐q𝑥𝑒��Complexi.e.𝑏q − 4𝑎𝑐 < 0 ⇒ 𝑥 = 𝛼 ± 𝑖𝛽

𝑦 = 𝑒�� 𝑐r cos 𝛽𝑥 + 𝑐q sin 𝛽𝑥 Ex.1

3𝑦ÞÞ + 4𝑦Þ + 5𝑦 = 0Extractauxiliaryequationi.e.3𝑟q + 4𝑟 + 5 = 0,solvefor𝑟

𝑟 =−4 ± 16 − 4 ∙ 3 ∙ 5

2 ∙ 3 = −46 ±

−446 = −

23 ± 𝑖

26 11 = −

23 ± 𝑖

13 11

𝛼 = −23 , 𝛽 =

113

∴ 𝑦 = 𝑒 pqo © 𝑐r cos113 𝑡 + 𝑐q sin

113 𝑡

Theothertwoareeasytosolvewiththegivenformulas.WhatwearereallyinterestedinnowishowtousethegivenformulasforhigherorderDE’s.Ex.2

3𝑦ÞÞÞ + 4𝑦Þ = 0Theauxiliaryequationis3𝑟o + 0 ∙ 𝑟q + 4𝑟 + 0 = 3𝑟o + 4𝑟 = 0


3𝑟o + 4𝑟 = 0 ⇒ 𝑟 3𝑟q + 4 = 0 ⇒ 𝑟 = 0&𝑟 = 0 ± 𝑖23

Wenowhave3solutionsi.e.𝑦 = 𝑐r𝑦r + 𝑐q𝑦q + 𝑐o𝑦oThecomplexscenarioshouldbeprettyobviousjustplugitintotheformula

𝑐r𝑦r + 𝑐q𝑦q = 𝑒2∙© 𝑐r cos23𝑡 + 𝑐q sin

23𝑡 = 𝑐r cos

23𝑡 + 𝑐q sin

23𝑡

Whataboutthesolution0?0isrepeatedoncehence

𝑐o𝑦o = 𝑐o𝑒2∙© = 𝑐o 1 = 𝑐o

∴ 𝑦 = 𝑐r cos23𝑡 + 𝑐q sin

23𝑡 + 𝑐o

Ex.3

𝑦ÞÞÞ + 8𝑦 = 0

𝑥o + 𝑎o = 𝑥 + 𝑎 𝑥q − 𝑎𝑥 + 𝑎q

𝑟o + 8 = 0 ⇒ 𝑟 + 2 𝑟q − 2𝑟 + 4 = 0 ⇒ 𝑟 = −2, 1 ± 𝑖 3

𝑦r,q = 𝑒© 𝑐r cos 3𝑡 + 𝑐q sin 3𝑡 , 𝑦o = 𝑐o𝑒pq©

∴ 𝑦 = 𝑒© 𝑐r cos 3𝑡 + 𝑐q sin 3𝑡 + 𝑐o𝑒pq©Ex.4

𝑦 Ü + 8𝑦ÞÞÞ = 0 ⇒ 𝑟Ü + 8𝑟o = 𝑟o 𝑟 + 8 = 0 ⇒ 𝑟o = 0, 𝑟 = −8Zeroisrepeatedthreetimeshere

𝑦r,q,o = 𝑐r𝑒2∙� + 𝑐q𝑥𝑒2∙� + 𝑐o𝑥q𝑒2∙� = 𝑐r + 𝑐q𝑥 + 𝑐o𝑥q, 𝑦Ü = 𝑐Ü𝑒pç�

∴ 𝑦 = 𝑐r + 𝑐q𝑥 + 𝑐o𝑥q + 𝑐Ü𝑒pç�Ex.5IVPy(0)=1,y’(0)=2,y’’(0)=3,y’’’(0)=4Usingthesolutionfromexample4


𝑦� = 𝑐r + 𝑐q𝑥 + 𝑐o𝑥q + 𝑐Ü𝑒pç�

𝑦Þ = 𝑐q + 2𝑐o𝑥 − 8𝑐Ü𝑒pç�, 𝑦ÞÞ = 2𝑐o + 64𝑐Ü𝑒pç�, 𝑦ÞÞÞ = −512𝑐Ü𝑒pç�

𝑦 0 = 1 ⇒ 1 = 𝑐r + 𝑐q ∙ 0 + 𝑐o ∙ 0 + 𝑐Ü𝑒2 = 𝑐r + 𝑐Ü

𝑦Þ 0 = 2 ⇒ 2 = 𝑐q + 2𝑐o ∙ 0 − 8𝑐Ü𝑒2 = 𝑐q − 8𝑐Ü

𝑦ÞÞ 0 = 3 ⇒ 3 = 2𝑐o + 64𝑐Ü𝑒2 = 2𝑐o + 64𝑐Ü

𝑦ÞÞÞ 0 = 4 ⇒ 4 = −512𝑐Ü𝑒2 = −512𝑐Ü

Solvethesystem

𝑐r + 𝑐Ü = 1, 𝑐q − 8𝑐Ü = 2, 2𝑐o + 64𝑐Ü = 3, −512𝑐Ü = 4

𝑐r =129128 , 𝑐q =

3116 , 𝑐o =

74 , 𝑐Ü = −

1128

∴ 𝑦Ù =129128 +

3116 𝑥 +

74 𝑥

q −1128 𝑒

pç�ReductionofOrderProcessGivenasecondorderlinearhomogeneousDEoftheform𝑦ÞÞ + 𝑃 𝑥 𝑦Þ + 𝑄 𝑥 = 0accompaniedwith𝑦r(𝑥)SolutionSincethefirstsolutionisgiven,youmustfindthesecondsolution,whichis:

𝑦q 𝑥 = 𝑦r 𝑥𝑒p à � »�

𝑦r 𝑥 q 𝑑𝑥, ∴ 𝑦 = 𝑐r𝑦r + 𝑐q 𝑦r 𝑥𝑒p à � »�

𝑦r 𝑥 q 𝑑𝑥

Ex.1

𝑥q𝑦ÞÞ + 2𝑥𝑦Þ − 6𝑦 = 0, 𝑦r = 𝑥qFind𝑃 𝑥

1𝑥q 𝑥q𝑦ÞÞ + 2𝑥𝑦Þ − 6𝑦 = 0 ⇒ 𝑦ÞÞ +

2𝑥 𝑦

Þ −6𝑥q 𝑦 = 0 ⇒ 𝑃 𝑥 =

2𝑥


∴ 𝑦q = 𝑥q𝑒p

q�»�

𝑥q q 𝑑𝑥 = 𝑥q𝑒pq ®¯ �

𝑥Ü 𝑑𝑥 = 𝑥q𝑒®¯ �ü�

𝑥Ü 𝑑𝑥 = 𝑥q𝑥pq

𝑥Ü 𝑑𝑥 = 𝑥q 𝑥pê 𝑑𝑥

= 𝑥q1−5𝑥

pé = −15𝑥

po ⇒ 𝑦q =1𝑥o

Theconstantcanbeignoredbecauseaconstanttimesaconstantisaconstant

∴ 𝑦 = 𝑐r𝑥q +𝑐q𝑥o

Atthispointitshouldbecomeobviousthat𝑐r + 𝑐q + ⋯+ 𝑐0 = 𝐶,thisisalsotruefornumbersi.e.𝑐r + 5 + 𝑒 + ln 10 + 𝑒�Ó + 6𝑐q = 𝐶.Inotherwords:aconstantwithaconstantisaconstant.SubstitutionGeneralSituationThemethodofsubstationworkswellwithDE’sthatlooksimilartoanexactequationi.e.

𝑥q + 4𝑥𝑦 𝑑𝑥 + 𝑦q − 4𝑥q 𝑑𝑦 = 0Wecancheckforanexactandseethat𝜕¦𝑓� = 4𝑥 ≠ −8𝑥 = 𝜕�𝑓¦soweknowwecannotusethatmethodbecausetheyarenotequal.SubstitutionSolutionMethodSet𝑦 = 𝑣 𝑥 ∙ 𝑥andtakethederivativei.e.𝑦Þ = 𝑣 𝑥 + 𝑥 ∙ 𝑣Þ 𝑥 andsolvefor𝑑𝑦.


𝑑𝑣𝑑𝑥 ⇒ 𝑑𝑥


𝑑𝑣𝑑𝑥 ⇒ 𝑑𝑦 = 𝑣𝑑𝑥 + 𝑥𝑑𝑣

Ifyouusethismethodanditgetsreallysloppy,itprobablyisnotthebestchoicesotrysomethingelse;letsseehowthisDEplaysout.Substitute𝑦and𝑑𝑦andsimplifyi.e.

𝑥q + 4𝑥 𝑣𝑥 𝑑𝑥 + 𝑣𝑥 q − 4𝑥q 𝑣𝑑𝑥 + 𝑥𝑑𝑣 = 0

⇒ 𝑥q𝑑𝑥 + 4𝑥q𝑣𝑑𝑥 + 𝑥q𝑣o𝑑𝑥 + 𝑥o𝑣q𝑑𝑣 − 4𝑥q𝑣𝑑𝑥 − 4𝑥o𝑑𝑣 = 0

⇒ 𝑥q𝑑𝑥 + 𝑥q𝑣o𝑑𝑥 + 4𝑥q𝑣𝑑𝑥 − 4𝑥q𝑣𝑑𝑥 + 𝑥o𝑣q𝑑𝑣 − 4𝑥o𝑑𝑣 = 0

⇒1𝑥q 𝑥q𝑑𝑥 + 𝑥q𝑣o𝑑𝑥 + 𝑥o𝑣q𝑑𝑣 − 4𝑥o𝑑𝑣 = 0

⇒ 𝑑𝑥 + 𝑣o𝑑𝑥 + 𝑥𝑣q𝑑𝑣 − 4𝑥𝑑𝑣 = 0

⇒ 1 + 𝑣o 𝑑𝑥 + 𝑥 𝑣q − 4 𝑑𝑣 = 0 ⇒ 𝑥 𝑣q − 4 𝑑𝑣 = − 1 + 𝑣o 𝑑𝑥


⇒𝑣q − 41 + 𝑣o 𝑑𝑣 = −

1𝑥 𝑑𝑥

Nowintegratetheseparablevariabledifferentialequation.

𝑣q

1 + 𝑣o −4

1 + 𝑣o 𝑑𝑣 = −1𝑥 𝑑𝑥

Atthispoint,itisjustareallytediouscalculusproblem. 𝑣 = ¦

�Twooftheintegralsaresimplei.e.

13 ln 1 + 𝑣

o − 41

1 + 𝑣o 𝑑𝑣 = − ln 𝑥 + 𝐶 ⇒ 13 ln 1 +

𝑦𝑥

o− 4

11 + 𝑣o 𝑑𝑣 = − ln 𝑥 + 𝐶

Wejustneedtointegrate rr�4Ó

.Thisisaverycomplicatedintegraltodobyhand.IusedWolfram|Alphatocompletethis.

∴1

1 + 𝑣o =16 − ln 𝑣q − 𝑣 + 1 + 2 ln 𝑣 + 1 + 2 3 tanpr

2𝑣 − 13

⇒13 ln 1 +

𝑦𝑥

o− 4

16 − ln 𝑣q − 𝑣 + 1 + 2 ln 𝑣 + 1 + 2 3 tanpr

2𝑣 − 13

= − ln 𝑥 + 𝐶

⇒13 ln 1 +

𝑦𝑥

o− 4

16 − ln

𝑦𝑥

q−𝑦𝑥 + 1 + 2 ln

𝑦𝑥 + 1 + 2 3 tanpr

2 𝑦𝑥 − 1

3

= − ln 𝑥 + 𝐶Thiswasaveryloadedsituation;itishighlyunlikelytoseesomethinglikethisinanundergraduateDEcourse.Whentousethismethod?Ifyouhaveasituationwithafirst-orderdifferentialequationthatisnotlinearandtheexactequationmethoddoesnotworkoristwocomplicatedandviceversa.Bessel’sEquationofOrder𝒗Form

𝑥q𝑦ÞÞ + 𝑥𝑦Þ + 𝑥q − 𝑣q 𝑦 = 0SolutiontoFirstKindBessel(𝒗 =fraction)

𝑦 = 𝑐r𝐽4 𝑥 + 𝑐q𝐽p4 𝑥


𝐽4 𝑥 =−1 0

𝑛!𝛤 1 + 𝑣 + 𝑛𝑥2

q0�4∞

012

, 𝐽p4 𝑥 =−1 0

𝑛!𝛤 1 − 𝑣 + 𝑛𝑥2

q0p4∞

012

16𝑥q𝑦ÞÞ + 16𝑥𝑦Þ + 16𝑥q − 1 𝑦 = 0

⇒ 𝑥q𝑦ÞÞ + 𝑥𝑦Þ + 𝑥q −14

q

𝑦 = 0

𝑣 =14

SolutiontoSecondKindBessel(𝒗 =integer)

𝑦 = 𝑐r𝐽4 𝑥 + 𝑐q𝑌4 𝑥 , 𝑌4 𝑥 =cos 𝑣𝜋 𝐽4 𝑥 − 𝑐q𝐽p4 𝑥

sin 𝑣𝜋

16𝑥q𝑦ÞÞ + 16𝑥𝑦Þ + 16𝑥q − 1 𝑦 = 0

⇒ 𝑥q𝑦ÞÞ + 𝑥𝑦Þ + 𝑥q − 9 𝑦 = 0

𝑣 = 3SolutiontoThirdKindBessel(𝜶𝒙 = 𝒕)

𝑥q𝑦ÞÞ + 𝑥𝑦Þ + 𝛼q𝑥q − 𝑣q 𝑦 = 0 ⇒ 𝑥q𝑦ÞÞ + 𝑥𝑦Þ + 𝑡q − 𝑣q 𝑦 = 0Solution

𝑦 = 𝑐r𝐽4 𝑡 + 𝑐q𝑌4 𝑡 = 𝑐r𝐽4 𝛼𝑥 + 𝑐q𝑌4 𝛼𝑥

16𝑥q𝑦ÞÞ + 16𝑥𝑦Þ + 16𝑥q − 1 𝑦 = 0

𝛼 = 4Variationofparameters

𝑦ÞÞ + 4𝑦′ = 3 sin 𝑥1stsolvehomogenoususingconstantcoefficients

𝑦ÞÞ + 4𝑦Þ = 0 ⇒ 𝑦ã = 𝑐r𝑒pÜ� + 𝑐q


2ndSolvetheparticularsolutionusingvariationsofparametersIdentify𝑦r, 𝑦q, 𝑔 𝑥

𝑦r = 𝑒pÜ�, 𝑦q = 1, 𝑔 𝑥 = 3 sin 𝑥 ComputeWronskian

𝑊 𝑥 =𝑦r 𝑦q𝑦rÞ 𝑦qÞ

⇒ 𝑒pÜ� 1−4𝑒pÜ� 0

= 𝑒pÜ� 0 − 1 −4𝑒pÜ� = 4𝑒pÜ�

𝑦Ù = 𝑢r𝑦r + 𝑢q𝑦q

𝑢r = −𝑦r𝑔𝑊 𝑑𝑥 = −

𝑒pÜ�3 sin 𝑥4𝑒pÜ� 𝑑𝑥 =

34 − sin 𝑥 𝑑𝑥 =

34 cos 𝑥

𝑢q =𝑦q𝑔𝑊 𝑑𝑥 =

3 sin 𝑥4𝑒pÜ� 𝑑𝑥 =

34 𝑒Ü� sin 𝑥 𝑑𝑥 = −

368 𝑒

Ü� 4 sin 𝑥 + cos 𝑥

∴ 𝑦 = 𝑦ã + 𝑦Ù

⇒

𝑦 = 𝑐r𝑒pÜ� + 𝑐q +34 cos 𝑥 𝑒pÜ� −

368 𝑒

Ü� 4 sin 𝑥 + cos 𝑥 Methodofundeterminedcoefficients

𝑦ÞÞ + 2𝑦Þ + 5𝑦 = 𝑥𝑒p�

𝑦 = 𝑦� + 𝑦Ùi)

𝑟q + 2𝑟 + 5 = 0 ⇔ 𝑟 = −1 ± 2𝑖 ⇒ 𝑦� = 𝑒p� 𝑐r cos 2𝑥 + 𝑐q sin 2𝑥 ii)

𝑦Ù = 𝐴𝑥𝑒p�, 𝑦Þ = 𝐴𝑒p� − 𝐴𝑥𝑒p�, 𝑦ÞÞ = −𝐴𝑒p� − 𝐴𝑒p� − 𝐴𝑥𝑒p� = −2𝐴𝑒p� + 𝐴𝑥𝑒p�

𝑦ÞÞ + 2𝑦Þ + 5𝑦 = 𝑥𝑒p�

⇒ 𝐴𝑥𝑒p� − 2𝐴𝑒p� + 2 𝐴𝑒p� − 𝐴𝑥𝑒p� + 5 𝐴𝑥𝑒p�, = 𝑥𝑒p�


⇒ 𝐴𝑥 − 2𝐴 + 2𝐴 − 2𝐴𝑥 + 5𝐴𝑥 = 𝑥

⇒ 4𝐴𝑥 = 𝑥 ⇒ 𝐴 =14

∴ 𝑦 = 𝑒p� 𝑐r cos 2𝑥 + 𝑐q sin 2𝑥 +𝑥4 𝑒

p�

SecondSolutionforReductionofOrderFindthegeneralsolutionofFrom𝑦 = 𝑦ã + 𝑦Ùtheℎimpliesthehomogeneoussolutionalsothegeneralsolution,the𝑝impliestheparticularsolution.Thisproblemwillnotbeeasytofindtheparticularsolution,henceitstates“findthegeneralsolution”.

𝑥q − 1 𝑦ÞÞ − 2𝑥𝑦Þ + 2𝑦 = 𝑥q + 1Weneedtomakeaguessonthesolutionfor𝑦r.Sincethecoefficientsarepolynomials,weshouldalsochooseapolynomialtobe𝑦r.Startwiththeeasiestoptioni.e.𝑦r = 𝑥.Whydidwechoosethis?Well,𝑦 = 𝑥 ⇒ 𝑦Þ = 1 ⇒ 𝑦ÞÞ = 0 ∴ 𝑥q − 1 0 − 2𝑥 1 + 2 𝑥 = −2𝑥 + 2𝑥 = 0.Wechoseitbecauseitzeroesthehomogeneoussolution.Formula:

𝑦ÞÞ + 𝑷 𝒙 𝑦Þ + 𝑄 𝑥 𝑦 = 0, 𝑦r = 𝒚𝟏 𝒙 , 𝑦q = 𝑣𝒚𝟏, 𝑣 =1𝒚𝟏 q 𝑒

p 𝑷 𝒙 »�𝑑𝑥

1

𝑥q − 1 𝑥q − 1 𝑦ÞÞ − 2𝑥𝑦Þ + 2𝑦 = 𝑥q + 1 ⇒ 𝑦ÞÞ + −𝟐𝒙

𝒙𝟐 − 𝟏 𝑦Þ +2

𝑥q − 1 𝑦 =𝑥q + 1𝑥q − 1

∴ 𝑷 𝒙 = −2𝑥

𝑥q − 1 ⇒ 𝑣 =1𝑥 q 𝑒

p p 𝟐𝒙𝒙𝟐p𝟏

»�𝑑𝑥 =1𝑥q 𝑒

»»� ��pr��pr »�𝑑𝑥 =

1𝑥q 𝑒

®¯ ��pr 𝑑𝑥

1𝑥q 𝑥q − 1 𝑑𝑥 = 1 − 𝑥pq 𝑑𝑥 = 𝑥 +

1𝑥

∴ 𝑦ã = 𝑐r𝑥 + 𝑥 𝑥 +1𝑥 𝑐q = 𝑐r𝑥 + 𝑥q + 1 𝑐q

Solvefortheparticularusingundeterminedcoefficients(Ifyoutry𝐴𝑥q + 𝐵𝑥 + 𝐶itwon’tworksomoveupthepolynomial)

𝑦Ù = 𝐴𝑥o + 𝐵𝑥q + 𝐶𝑥 + 𝐷 ⇒ 𝑦ÙÞ = 3𝐴𝑥q + 2𝐵𝑥 + 𝐶 ⇒ 𝑦ÙÞÞ = 6𝐴𝑥 + 2𝐵

𝑥q − 1 𝑦ÞÞ − 2𝑥𝑦Þ + 2𝑦 = 𝑥q + 1


⇒ 𝑥q − 1 6𝐴𝑥 + 2𝐵 − 2𝑥 3𝐴𝑥q + 2𝐵𝑥 + 𝐶 + 2 𝐴𝑥o + 𝐵𝑥q + 𝐶𝑥 + 𝐷 = 𝑥q + 1

⇒ 2𝐴𝑥o − 6𝐴𝑥 − 2𝐵 + 2𝐷 = 0 𝑥o + 1 𝑥q + 0 𝑥 + 1 𝑥2

⇒ 𝑛𝑜𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑎𝑔𝑎𝑖𝑛…

Tryvariationofparameters

𝑊 = 𝑥 𝑥q + 11 2𝑥

= 2𝑥q − 𝑥q + 1 = 𝑥q − 1

𝑢r =𝑦q𝑔𝑊 𝑑𝑥 =

𝑥q + 1 q

𝑥q − 1 𝑑𝑥

𝑢q =𝑦r𝑔𝑊 𝑑𝑥 =

𝑥 𝑥q + 1𝑥q − 1 𝑑𝑥

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