MathJax Help and Configuration

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and Print Page</a></li> </ul> </div> <nav id="menu" class="notes-nav"> <ul> <li><a href="/" class="mm-link">Home</a></li> <li><span>Classes</span></li> <li><a href="/Classes/Alg/Alg.aspx" class="mm-link">Algebra</a> <ul> <li><a href="/Classes/Alg/Preliminaries.aspx" class="mm-link">1. Preliminaries</a> <ul> <li><a href="/Classes/Alg/IntegerExponents.aspx" class="mm-link">1.1 Integer Exponents</a></li> <li><a href="/Classes/Alg/RationalExponents.aspx" class="mm-link">1.2 Rational Exponents</a></li> <li><a href="/Classes/Alg/Radicals.aspx" class="mm-link">1.3 Radicals</a></li> <li><a href="/Classes/Alg/Polynomials.aspx" class="mm-link">1.4 Polynomials</a></li> <li><a href="/Classes/Alg/Factoring.aspx" class="mm-link">1.5 Factoring Polynomials</a></li> <li><a href="/Classes/Alg/RationalExpressions.aspx" class="mm-link">1.6 Rational Expressions</a></li> <li><a href="/Classes/Alg/ComplexNumbers.aspx" class="mm-link">1.7 Complex Numbers</a></li> </ul> </li> <li><a href="/Classes/Alg/Solving.aspx" class="mm-link">2. Solving Equations and Inequalities</a> <ul> <li><a href="/Classes/Alg/SolutionSets.aspx" class="mm-link">2.1 Solutions and Solution Sets</a></li> <li><a href="/Classes/Alg/SolveLinearEqns.aspx" class="mm-link">2.2 Linear Equations</a></li> <li><a href="/Classes/Alg/LinearApps.aspx" class="mm-link">2.3 Applications of Linear Equations</a></li> <li><a href="/Classes/Alg/SolveMultiVariable.aspx" class="mm-link">2.4 Equations With More Than One Variable</a></li> <li><a href="/Classes/Alg/SolveQuadraticEqnsI.aspx" class="mm-link">2.5 Quadratic Equations - Part I</a></li> <li><a href="/Classes/Alg/SolveQuadraticEqnsII.aspx" class="mm-link">2.6 Quadratic Equations - Part II</a></li> <li><a href="/Classes/Alg/SolveQuadraticEqnSummary.aspx" class="mm-link">2.7 Quadratic Equations : A Summary</a></li> <li><a href="/Classes/Alg/QuadraticApps.aspx" class="mm-link">2.8 Applications of Quadratic Equations</a></li> <li><a href="/Classes/Alg/ReducibleToQuadratic.aspx" class="mm-link">2.9 Equations Reducible to Quadratic in Form</a></li> <li><a href="/Classes/Alg/SolveRadicalEqns.aspx" class="mm-link">2.10 Equations with Radicals</a></li> <li><a href="/Classes/Alg/SolveLinearInequalities.aspx" class="mm-link">2.11 Linear Inequalities</a></li> <li><a href="/Classes/Alg/SolvePolyInequalities.aspx" class="mm-link">2.12 Polynomial Inequalities</a></li> <li><a href="/Classes/Alg/SolveRationalInequalities.aspx" class="mm-link">2.13 Rational Inequalities</a></li> <li><a href="/Classes/Alg/SolveAbsValueEqns.aspx" class="mm-link">2.14 Absolute Value Equations</a></li> <li><a href="/Classes/Alg/SolveAbsValueIneq.aspx" class="mm-link">2.15 Absolute Value Inequalities</a></li> </ul> </li> <li><a href="/Classes/Alg/Graphing_Functions.aspx" class="mm-link">3. Graphing and Functions</a> <ul> <li><a href="/Classes/Alg/Graphing.aspx" class="mm-link">3.1 Graphing</a></li> <li><a href="/Classes/Alg/Lines.aspx" class="mm-link">3.2 Lines</a></li> <li><a href="/Classes/Alg/Circles.aspx" class="mm-link">3.3 Circles</a></li> <li><a href="/Classes/Alg/FunctionDefn.aspx" class="mm-link">3.4 The Definition of a Function</a></li> <li><a href="/Classes/Alg/GraphFunctions.aspx" class="mm-link">3.5 Graphing Functions</a></li> <li><a href="/Classes/Alg/CombineFunctions.aspx" class="mm-link">3.6 Combining Functions</a></li> <li><a href="/Classes/Alg/InverseFunctions.aspx" class="mm-link">3.7 Inverse Functions</a></li> </ul> </li> <li><a href="/Classes/Alg/CommonGraphs.aspx" class="mm-link">4. Common Graphs</a> <ul> <li><a href="/Classes/Alg/Lines_Circles_PWF.aspx" class="mm-link">4.1 Lines, Circles and Piecewise Functions</a></li> <li><a href="/Classes/Alg/Parabolas.aspx" class="mm-link">4.2 Parabolas</a></li> <li><a href="/Classes/Alg/Ellipses.aspx" class="mm-link">4.3 Ellipses</a></li> <li><a href="/Classes/Alg/Hyperbolas.aspx" class="mm-link">4.4 Hyperbolas</a></li> <li><a href="/Classes/Alg/MiscFunctions.aspx" class="mm-link">4.5 Miscellaneous Functions</a></li> <li><a href="/Classes/Alg/Transformations.aspx" class="mm-link">4.6 Transformations</a></li> <li><a href="/Classes/Alg/Symmetry.aspx" class="mm-link">4.7 Symmetry</a></li> <li><a href="/Classes/Alg/GraphRationalFcns.aspx" class="mm-link">4.8 Rational Functions</a></li> </ul> </li> <li><a href="/Classes/Alg/PolynomialFunctions.aspx" class="mm-link">5. Polynomial Functions</a> <ul> <li><a href="/Classes/Alg/DividingPolynomials.aspx" class="mm-link">5.1 Dividing Polynomials</a></li> <li><a href="/Classes/Alg/ZeroesOfPolynomials.aspx" class="mm-link">5.2 Zeroes/Roots of Polynomials</a></li> <li><a href="/Classes/Alg/GraphingPolynomials.aspx" class="mm-link">5.3 Graphing Polynomials</a></li> <li><a href="/Classes/Alg/FindingZeroesOfPolynomials.aspx" class="mm-link">5.4 Finding Zeroes of Polynomials</a></li> <li><a href="/Classes/Alg/PartialFractions.aspx" class="mm-link">5.5 Partial Fractions</a></li> </ul> </li> <li><a href="/Classes/Alg/ExpAndLog.aspx" class="mm-link">6. Exponential and Logarithm Functions</a> <ul> <li><a href="/Classes/Alg/ExpFunctions.aspx" class="mm-link">6.1 Exponential Functions</a></li> <li><a href="/Classes/Alg/LogFunctions.aspx" class="mm-link">6.2 Logarithm Functions</a></li> <li><a href="/Classes/Alg/SolveExpEqns.aspx" class="mm-link">6.3 Solving Exponential Equations</a></li> <li><a href="/Classes/Alg/SolveLogEqns.aspx" class="mm-link">6.4 Solving Logarithm Equations</a></li> <li><a href="/Classes/Alg/ExpLogApplications.aspx" class="mm-link">6.5 Applications</a></li> </ul> </li> <li><a href="/Classes/Alg/Systems.aspx" class="mm-link">7. Systems of Equations</a> <ul> <li><a href="/Classes/Alg/SystemsTwoVrble.aspx" class="mm-link">7.1 Linear Systems with Two Variables</a></li> <li><a href="/Classes/Alg/SystemsThreeVrble.aspx" class="mm-link">7.2 Linear Systems with Three Variables</a></li> <li><a href="/Classes/Alg/AugmentedMatrix.aspx" class="mm-link">7.3 Augmented Matrices</a></li> <li><a href="/Classes/Alg/AugmentedMatrixII.aspx" class="mm-link">7.4 More on the Augmented Matrix</a></li> <li><a href="/Classes/Alg/NonlinearSystems.aspx" class="mm-link">7.5 Nonlinear Systems</a></li> </ul> </li> </ul> </li> <li><a href="/Classes/CalcI/CalcI.aspx" class="mm-link">Calculus I</a> <ul> <li><a href="/Classes/CalcI/ReviewIntro.aspx" class="mm-link">1. Review</a> <ul> <li><a href="/Classes/CalcI/Functions.aspx" class="mm-link">1.1 Functions</a></li> <li><a href="/Classes/CalcI/InverseFunctions.aspx" class="mm-link">1.2 Inverse Functions</a></li> <li><a href="/Classes/CalcI/TrigFcns.aspx" class="mm-link">1.3 Trig Functions</a></li> <li><a href="/Classes/CalcI/TrigEquations.aspx" class="mm-link">1.4 Solving Trig Equations</a></li> <li><a href="/Classes/CalcI/TrigEquations_CalcI.aspx" class="mm-link">1.5 Trig Equations with Calculators, Part I</a></li> <li><a href="/Classes/CalcI/TrigEquations_CalcII.aspx" class="mm-link">1.6 Trig Equations with Calculators, Part II</a></li> <li><a href="/Classes/CalcI/ExpFunctions.aspx" class="mm-link">1.7 Exponential Functions</a></li> <li><a href="/Classes/CalcI/LogFcns.aspx" class="mm-link">1.8 Logarithm Functions</a></li> <li><a href="/Classes/CalcI/ExpLogEqns.aspx" class="mm-link">1.9 Exponential and Logarithm Equations</a></li> <li><a href="/Classes/CalcI/CommonGraphs.aspx" class="mm-link">1.10 Common Graphs</a></li> </ul> </li> <li><a href="/Classes/CalcI/limitsIntro.aspx" class="mm-link">2. Limits</a> <ul> <li><a href="/Classes/CalcI/Tangents_Rates.aspx" class="mm-link">2.1 Tangent Lines and Rates of Change</a></li> <li><a href="/Classes/CalcI/TheLimit.aspx" class="mm-link">2.2 The Limit</a></li> <li><a href="/Classes/CalcI/OneSidedLimits.aspx" class="mm-link">2.3 One-Sided Limits</a></li> <li><a href="/Classes/CalcI/LimitsProperties.aspx" class="mm-link">2.4 Limit Properties</a></li> <li><a href="/Classes/CalcI/ComputingLimits.aspx" class="mm-link">2.5 Computing Limits</a></li> <li><a href="/Classes/CalcI/InfiniteLimits.aspx" class="mm-link">2.6 Infinite Limits</a></li> <li><a href="/Classes/CalcI/LimitsAtInfinityI.aspx" class="mm-link">2.7 Limits At Infinity, Part I</a></li> <li><a href="/Classes/CalcI/LimitsAtInfinityII.aspx" class="mm-link">2.8 Limits At Infinity, Part II</a></li> <li><a href="/Classes/CalcI/Continuity.aspx" class="mm-link">2.9 Continuity</a></li> <li><a href="/Classes/CalcI/DefnOfLimit.aspx" class="mm-link">2.10 The Definition of the Limit</a></li> </ul> </li> <li><a href="/Classes/CalcI/DerivativeIntro.aspx" class="mm-link">3. Derivatives</a> <ul> <li><a href="/Classes/CalcI/DefnOfDerivative.aspx" class="mm-link">3.1 The Definition of the Derivative</a></li> <li><a href="/Classes/CalcI/DerivativeInterp.aspx" class="mm-link">3.2 Interpretation of the Derivative</a></li> <li><a href="/Classes/CalcI/DiffFormulas.aspx" class="mm-link">3.3 Differentiation Formulas</a></li> <li><a href="/Classes/CalcI/ProductQuotientRule.aspx" class="mm-link">3.4 Product and Quotient Rule</a></li> <li><a href="/Classes/CalcI/DiffTrigFcns.aspx" class="mm-link">3.5 Derivatives of Trig Functions</a></li> <li><a href="/Classes/CalcI/DiffExpLogFcns.aspx" class="mm-link">3.6 Derivatives of Exponential and Logarithm Functions</a></li> <li><a href="/Classes/CalcI/DiffInvTrigFcns.aspx" class="mm-link">3.7 Derivatives of Inverse Trig Functions</a></li> <li><a href="/Classes/CalcI/DiffHyperFcns.aspx" class="mm-link">3.8 Derivatives of Hyperbolic Functions</a></li> <li><a href="/Classes/CalcI/ChainRule.aspx" class="mm-link">3.9 Chain Rule</a></li> <li><a href="/Classes/CalcI/ImplicitDIff.aspx" class="mm-link">3.10 Implicit Differentiation</a></li> <li><a href="/Classes/CalcI/RelatedRates.aspx" class="mm-link">3.11 Related Rates</a></li> <li><a href="/Classes/CalcI/HigherOrderDerivatives.aspx" class="mm-link">3.12 Higher Order Derivatives</a></li> <li><a href="/Classes/CalcI/LogDiff.aspx" class="mm-link">3.13 Logarithmic Differentiation</a></li> </ul> </li> <li><a href="/Classes/CalcI/DerivAppsIntro.aspx" class="mm-link">4. Applications of Derivatives</a> <ul> <li><a href="/Classes/CalcI/RateOfChange.aspx" class="mm-link">4.1 Rates of Change</a></li> <li><a href="/Classes/CalcI/CriticalPoints.aspx" class="mm-link">4.2 Critical Points</a></li> <li><a href="/Classes/CalcI/MinMaxValues.aspx" class="mm-link">4.3 Minimum and Maximum Values</a></li> <li><a href="/Classes/CalcI/AbsExtrema.aspx" class="mm-link">4.4 Finding Absolute Extrema</a></li> <li><a href="/Classes/CalcI/ShapeofGraphPtI.aspx" class="mm-link">4.5 The Shape of a Graph, Part I</a></li> <li><a href="/Classes/CalcI/ShapeofGraphPtII.aspx" class="mm-link">4.6 The Shape of a Graph, Part II</a></li> <li><a href="/Classes/CalcI/MeanValueTheorem.aspx" class="mm-link">4.7 The Mean Value Theorem</a></li> <li><a href="/Classes/CalcI/Optimization.aspx" class="mm-link">4.8 Optimization</a></li> <li><a href="/Classes/CalcI/MoreOptimization.aspx" class="mm-link">4.9 More Optimization Problems</a></li> <li><a href="/Classes/CalcI/LHospitalsRule.aspx" class="mm-link">4.10 L'Hospital's Rule and Indeterminate Forms</a></li> <li><a href="/Classes/CalcI/LinearApproximations.aspx" class="mm-link">4.11 Linear Approximations</a></li> <li><a href="/Classes/CalcI/Differentials.aspx" class="mm-link">4.12 Differentials</a></li> <li><a href="/Classes/CalcI/NewtonsMethod.aspx" class="mm-link">4.13 Newton's Method</a></li> <li><a href="/Classes/CalcI/BusinessApps.aspx" class="mm-link">4.14 Business Applications</a></li> </ul> </li> <li><a href="/Classes/CalcI/IntegralsIntro.aspx" class="mm-link">5. Integrals</a> <ul> <li><a href="/Classes/CalcI/IndefiniteIntegrals.aspx" class="mm-link">5.1 Indefinite Integrals</a></li> <li><a href="/Classes/CalcI/ComputingIndefiniteIntegrals.aspx" class="mm-link">5.2 Computing Indefinite Integrals</a></li> <li><a href="/Classes/CalcI/SubstitutionRuleIndefinite.aspx" class="mm-link">5.3 Substitution Rule for Indefinite Integrals</a></li> <li><a href="/Classes/CalcI/SubstitutionRuleIndefinitePtII.aspx" class="mm-link">5.4 More Substitution Rule</a></li> <li><a href="/Classes/CalcI/AreaProblem.aspx" class="mm-link">5.5 Area Problem</a></li> <li><a href="/Classes/CalcI/DefnOfDefiniteIntegral.aspx" class="mm-link">5.6 Definition of the Definite Integral</a></li> <li><a href="/Classes/CalcI/ComputingDefiniteIntegrals.aspx" class="mm-link">5.7 Computing Definite Integrals</a></li> <li><a href="/Classes/CalcI/SubstitutionRuleDefinite.aspx" class="mm-link">5.8 Substitution Rule for Definite Integrals</a></li> </ul> </li> <li><a href="/Classes/CalcI/IntAppsIntro.aspx" class="mm-link">6. Applications of Integrals</a> <ul> <li><a href="/Classes/CalcI/AvgFcnValue.aspx" class="mm-link">6.1 Average Function Value</a></li> <li><a href="/Classes/CalcI/AreaBetweenCurves.aspx" class="mm-link">6.2 Area Between Curves</a></li> <li><a href="/Classes/CalcI/VolumeWithRings.aspx" class="mm-link">6.3 Volumes of Solids of Revolution / Method of Rings</a></li> <li><a href="/Classes/CalcI/VolumeWithCylinder.aspx" class="mm-link">6.4 Volumes of Solids of Revolution/Method of Cylinders</a></li> <li><a href="/Classes/CalcI/MoreVolume.aspx" class="mm-link">6.5 More Volume Problems</a></li> <li><a href="/Classes/CalcI/Work.aspx" class="mm-link">6.6 Work</a></li> </ul> </li> <li><a href="/Classes/CalcI/ExtrasIntro.aspx" class="mm-link">Appendix A. Extras</a> <ul> <li><a href="/Classes/CalcI/LimitProofs.aspx" class="mm-link">A.1 Proof of Various Limit Properties</a></li> <li><a href="/Classes/CalcI/DerivativeProofs.aspx" class="mm-link">A.2 Proof of Various Derivative Properties</a></li> <li><a href="/Classes/CalcI/ProofTrigDeriv.aspx" class="mm-link">A.3 Proof of Trig Limits</a></li> <li><a href="/Classes/CalcI/DerivativeAppsProofs.aspx" class="mm-link">A.4 Proofs of Derivative Applications Facts</a></li> <li><a href="/Classes/CalcI/ProofIntProp.aspx" class="mm-link">A.5 Proof of Various Integral Properties </a></li> <li><a href="/Classes/CalcI/Area_Volume_Formulas.aspx" class="mm-link">A.6 Area and Volume Formulas</a></li> <li><a href="/Classes/CalcI/TypesOfInfinity.aspx" class="mm-link">A.7 Types of Infinity</a></li> <li><a href="/Classes/CalcI/SummationNotation.aspx" class="mm-link">A.8 Summation Notation</a></li> <li><a href="/Classes/CalcI/ConstantofIntegration.aspx" class="mm-link">A.9 Constant of Integration</a></li> </ul> </li> </ul> </li> <li><a href="/Classes/CalcII/CalcII.aspx" class="mm-link">Calculus II</a> <ul> <li><a href="/Classes/CalcII/IntTechIntro.aspx" class="mm-link">7. Integration Techniques</a> <ul> <li><a href="/Classes/CalcII/IntegrationByParts.aspx" class="mm-link">7.1 Integration by Parts</a></li> <li><a href="/Classes/CalcII/IntegralsWithTrig.aspx" class="mm-link">7.2 Integrals Involving Trig Functions</a></li> <li><a href="/Classes/CalcII/TrigSubstitutions.aspx" class="mm-link">7.3 Trig Substitutions</a></li> <li><a href="/Classes/CalcII/PartialFractions.aspx" class="mm-link">7.4 Partial Fractions</a></li> <li><a href="/Classes/CalcII/IntegralsWithRoots.aspx" class="mm-link">7.5 Integrals Involving Roots</a></li> <li><a href="/Classes/CalcII/IntegralsWithQuadratics.aspx" class="mm-link">7.6 Integrals Involving Quadratics</a></li> <li><a href="/Classes/CalcII/IntegrationStrategy.aspx" class="mm-link">7.7 Integration Strategy</a></li> <li><a href="/Classes/CalcII/ImproperIntegrals.aspx" class="mm-link">7.8 Improper Integrals</a></li> <li><a href="/Classes/CalcII/ImproperIntegralsCompTest.aspx" class="mm-link">7.9 Comparison Test for Improper Integrals</a></li> <li><a href="/Classes/CalcII/ApproximatingDefIntegrals.aspx" class="mm-link">7.10 Approximating Definite Integrals</a></li> </ul> </li> <li><a href="/Classes/CalcII/IntAppsIntro.aspx" class="mm-link">8. Applications of Integrals</a> <ul> <li><a href="/Classes/CalcII/ArcLength.aspx" class="mm-link">8.1 Arc Length</a></li> <li><a href="/Classes/CalcII/SurfaceArea.aspx" class="mm-link">8.2 Surface Area</a></li> <li><a href="/Classes/CalcII/CenterOfMass.aspx" class="mm-link">8.3 Center of Mass</a></li> <li><a href="/Classes/CalcII/HydrostaticPressure.aspx" class="mm-link">8.4 Hydrostatic Pressure</a></li> <li><a href="/Classes/CalcII/Probability.aspx" class="mm-link">8.5 Probability</a></li> </ul> </li> <li><a href="/Classes/CalcII/ParametricIntro.aspx" class="mm-link">9. Parametric Equations and Polar Coordinates</a> <ul> <li><a href="/Classes/CalcII/ParametricEqn.aspx" class="mm-link">9.1 Parametric Equations and Curves</a></li> <li><a href="/Classes/CalcII/ParaTangent.aspx" class="mm-link">9.2 Tangents with Parametric Equations</a></li> <li><a href="/Classes/CalcII/ParaArea.aspx" class="mm-link">9.3 Area with Parametric Equations</a></li> <li><a href="/Classes/CalcII/ParaArcLength.aspx" class="mm-link">9.4 Arc Length with Parametric Equations</a></li> <li><a href="/Classes/CalcII/ParaSurfaceArea.aspx" class="mm-link">9.5 Surface Area with Parametric Equations</a></li> <li><a href="/Classes/CalcII/PolarCoordinates.aspx" class="mm-link">9.6 Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarTangents.aspx" class="mm-link">9.7 Tangents with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarArea.aspx" class="mm-link">9.8 Area with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarArcLength.aspx" class="mm-link">9.9 Arc Length with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarSurfaceArea.aspx" class="mm-link">9.10 Surface Area with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/ArcLength_SurfaceArea.aspx" class="mm-link">9.11 Arc Length and Surface Area Revisited</a></li> </ul> </li> <li><a href="/Classes/CalcII/SeriesIntro.aspx" class="mm-link">10. Series & Sequences</a> <ul> <li><a href="/Classes/CalcII/Sequences.aspx" class="mm-link">10.1 Sequences</a></li> <li><a href="/Classes/CalcII/MoreSequences.aspx" class="mm-link">10.2 More on Sequences</a></li> <li><a href="/Classes/CalcII/Series_Basics.aspx" class="mm-link">10.3 Series - The Basics</a></li> <li><a href="/Classes/CalcII/ConvergenceOfSeries.aspx" class="mm-link">10.4 Convergence/Divergence of Series</a></li> <li><a href="/Classes/CalcII/Series_Special.aspx" class="mm-link">10.5 Special Series</a></li> <li><a href="/Classes/CalcII/IntegralTest.aspx" class="mm-link">10.6 Integral Test</a></li> <li><a href="/Classes/CalcII/SeriesCompTest.aspx" class="mm-link">10.7 Comparison Test/Limit Comparison Test</a></li> <li><a href="/Classes/CalcII/AlternatingSeries.aspx" class="mm-link">10.8 Alternating Series Test</a></li> <li><a href="/Classes/CalcII/AbsoluteConvergence.aspx" class="mm-link">10.9 Absolute Convergence</a></li> <li><a href="/Classes/CalcII/RatioTest.aspx" class="mm-link">10.10 Ratio Test</a></li> <li><a href="/Classes/CalcII/RootTest.aspx" class="mm-link">10.11 Root Test</a></li> <li><a href="/Classes/CalcII/SeriesStrategy.aspx" class="mm-link">10.12 Strategy for Series</a></li> <li><a href="/Classes/CalcII/EstimatingSeries.aspx" class="mm-link">10.13 Estimating the Value of a Series</a></li> <li><a href="/Classes/CalcII/PowerSeries.aspx" class="mm-link">10.14 Power Series</a></li> <li><a href="/Classes/CalcII/PowerSeriesandFunctions.aspx" class="mm-link">10.15 Power Series and Functions</a></li> <li><a href="/Classes/CalcII/TaylorSeries.aspx" class="mm-link">10.16 Taylor Series</a></li> <li><a href="/Classes/CalcII/TaylorSeriesApps.aspx" class="mm-link">10.17 Applications of Series</a></li> <li><a href="/Classes/CalcII/BinomialSeries.aspx" class="mm-link">10.18 Binomial Series</a></li> </ul> </li> <li><a href="/Classes/CalcII/VectorsIntro.aspx" class="mm-link">11. Vectors</a> <ul> <li><a href="/Classes/CalcII/Vectors_Basics.aspx" class="mm-link">11.1 Vectors - The Basics</a></li> <li><a href="/Classes/CalcII/VectorArithmetic.aspx" class="mm-link">11.2 Vector Arithmetic</a></li> <li><a href="/Classes/CalcII/DotProduct.aspx" class="mm-link">11.3 Dot Product</a></li> <li><a href="/Classes/CalcII/CrossProduct.aspx" class="mm-link">11.4 Cross Product</a></li> </ul> </li> <li><a href="/Classes/CalcII/3DSpace.aspx" class="mm-link">12. 3-Dimensional Space</a> <ul> <li><a href="/Classes/CalcII/3DCoords.aspx" class="mm-link">12.1 The 3-D Coordinate System</a></li> <li><a href="/Classes/CalcII/EqnsOfLines.aspx" class="mm-link">12.2 Equations of Lines</a></li> <li><a href="/Classes/CalcII/EqnsOfPlanes.aspx" class="mm-link">12.3 Equations of Planes</a></li> <li><a href="/Classes/CalcII/QuadricSurfaces.aspx" class="mm-link">12.4 Quadric Surfaces</a></li> <li><a href="/Classes/CalcII/MultiVrbleFcns.aspx" class="mm-link">12.5 Functions of Several Variables</a></li> <li><a href="/Classes/CalcII/VectorFunctions.aspx" class="mm-link">12.6 Vector Functions</a></li> <li><a href="/Classes/CalcII/VectorFcnsCalculus.aspx" class="mm-link">12.7 Calculus with Vector Functions</a></li> <li><a href="/Classes/CalcII/TangentNormalVectors.aspx" class="mm-link">12.8 Tangent, Normal and Binormal Vectors</a></li> <li><a href="/Classes/CalcII/VectorArcLength.aspx" class="mm-link">12.9 Arc Length with Vector Functions</a></li> <li><a href="/Classes/CalcII/Curvature.aspx" class="mm-link">12.10 Curvature</a></li> <li><a href="/Classes/CalcII/Velocity_Acceleration.aspx" class="mm-link">12.11 Velocity and Acceleration</a></li> <li><a href="/Classes/CalcII/CylindricalCoords.aspx" class="mm-link">12.12 Cylindrical Coordinates</a></li> <li><a href="/Classes/CalcII/SphericalCoords.aspx" class="mm-link">12.13 Spherical Coordinates</a></li> </ul> </li> </ul> </li> <li><a href="/Classes/CalcIII/CalcIII.aspx" class="mm-link">Calculus III</a> <ul> <li><a href="/Classes/CalcIII/3DSpace.aspx" class="mm-link">12. 3-Dimensional Space</a> <ul> <li><a href="/Classes/CalcIII/3DCoords.aspx" class="mm-link">12.1 The 3-D Coordinate System</a></li> <li><a href="/Classes/CalcIII/EqnsOfLines.aspx" class="mm-link">12.2 Equations of Lines</a></li> <li><a href="/Classes/CalcIII/EqnsOfPlanes.aspx" class="mm-link">12.3 Equations of Planes</a></li> <li><a href="/Classes/CalcIII/QuadricSurfaces.aspx" class="mm-link">12.4 Quadric Surfaces</a></li> <li><a href="/Classes/CalcIII/MultiVrbleFcns.aspx" class="mm-link">12.5 Functions of Several Variables</a></li> <li><a href="/Classes/CalcIII/VectorFunctions.aspx" class="mm-link">12.6 Vector Functions</a></li> <li><a href="/Classes/CalcIII/VectorFcnsCalculus.aspx" class="mm-link">12.7 Calculus with Vector Functions</a></li> <li><a href="/Classes/CalcIII/TangentNormalVectors.aspx" class="mm-link">12.8 Tangent, Normal and Binormal Vectors</a></li> <li><a href="/Classes/CalcIII/VectorArcLength.aspx" class="mm-link">12.9 Arc Length with Vector Functions</a></li> <li><a href="/Classes/CalcIII/Curvature.aspx" class="mm-link">12.10 Curvature</a></li> <li><a href="/Classes/CalcIII/Velocity_Acceleration.aspx" class="mm-link">12.11 Velocity and Acceleration</a></li> <li><a href="/Classes/CalcIII/CylindricalCoords.aspx" class="mm-link">12.12 Cylindrical Coordinates</a></li> <li><a href="/Classes/CalcIII/SphericalCoords.aspx" class="mm-link">12.13 Spherical Coordinates</a></li> </ul> </li> <li><a href="/Classes/CalcIII/PartialDerivsIntro.aspx" class="mm-link">13. Partial Derivatives</a> <ul> <li><a href="/Classes/CalcIII/Limits.aspx" class="mm-link">13.1 Limits</a></li> <li><a href="/Classes/CalcIII/PartialDerivatives.aspx" class="mm-link">13.2 Partial Derivatives</a></li> <li><a href="/Classes/CalcIII/PartialDerivInterp.aspx" class="mm-link">13.3 Interpretations of Partial Derivatives</a></li> <li><a href="/Classes/CalcIII/HighOrderPartialDerivs.aspx" class="mm-link">13.4 Higher Order Partial Derivatives</a></li> <li><a href="/Classes/CalcIII/Differentials.aspx" class="mm-link">13.5 Differentials</a></li> <li><a href="/Classes/CalcIII/ChainRule.aspx" class="mm-link">13.6 Chain Rule</a></li> <li><a href="/Classes/CalcIII/DirectionalDeriv.aspx" class="mm-link">13.7 Directional Derivatives</a></li> </ul> </li> <li><a href="/Classes/CalcIII/PartialDerivAppsIntro.aspx" class="mm-link">14. Applications of Partial Derivatives</a> <ul> <li><a href="/Classes/CalcIII/TangentPlanes.aspx" class="mm-link">14.1 Tangent Planes and Linear Approximations</a></li> <li><a href="/Classes/CalcIII/GradientVectorTangentPlane.aspx" class="mm-link">14.2 Gradient Vector, Tangent Planes and Normal Lines</a></li> <li><a href="/Classes/CalcIII/RelativeExtrema.aspx" class="mm-link">14.3 Relative Minimums and Maximums</a></li> <li><a href="/Classes/CalcIII/AbsoluteExtrema.aspx" class="mm-link">14.4 Absolute Minimums and Maximums</a></li> <li><a href="/Classes/CalcIII/LagrangeMultipliers.aspx" class="mm-link">14.5 Lagrange Multipliers</a></li> </ul> </li> <li><a href="/Classes/CalcIII/MultipleIntegralsIntro.aspx" class="mm-link">15. Multiple Integrals</a> <ul> <li><a href="/Classes/CalcIII/DoubleIntegrals.aspx" class="mm-link">15.1 Double Integrals</a></li> <li><a href="/Classes/CalcIII/IteratedIntegrals.aspx" class="mm-link">15.2 Iterated Integrals</a></li> <li><a href="/Classes/CalcIII/DIGeneralRegion.aspx" class="mm-link">15.3 Double Integrals over General Regions</a></li> <li><a href="/Classes/CalcIII/DIPolarCoords.aspx" class="mm-link">15.4 Double Integrals in Polar Coordinates</a></li> <li><a href="/Classes/CalcIII/TripleIntegrals.aspx" class="mm-link">15.5 Triple Integrals</a></li> <li><a href="/Classes/CalcIII/TICylindricalCoords.aspx" class="mm-link">15.6 Triple Integrals in Cylindrical Coordinates</a></li> <li><a href="/Classes/CalcIII/TISphericalCoords.aspx" class="mm-link">15.7 Triple Integrals in Spherical Coordinates</a></li> <li><a href="/Classes/CalcIII/ChangeOfVariables.aspx" class="mm-link">15.8 Change of Variables</a></li> <li><a href="/Classes/CalcIII/SurfaceArea.aspx" class="mm-link">15.9 Surface Area</a></li> <li><a href="/Classes/CalcIII/Area_Volume.aspx" class="mm-link">15.10 Area and Volume Revisited</a></li> </ul> </li> <li><a href="/Classes/CalcIII/LineIntegralsIntro.aspx" class="mm-link">16. Line Integrals</a> <ul> <li><a href="/Classes/CalcIII/VectorFields.aspx" class="mm-link">16.1 Vector Fields</a></li> <li><a href="/Classes/CalcIII/LineIntegralsPtI.aspx" class="mm-link">16.2 Line Integrals - Part I</a></li> <li><a href="/Classes/CalcIII/LineIntegralsPtII.aspx" class="mm-link">16.3 Line Integrals - Part II</a></li> <li><a href="/Classes/CalcIII/LineIntegralsVectorFields.aspx" class="mm-link">16.4 Line Integrals of Vector Fields</a></li> <li><a href="/Classes/CalcIII/FundThmLineIntegrals.aspx" class="mm-link">16.5 Fundamental Theorem for Line Integrals</a></li> <li><a href="/Classes/CalcIII/ConservativeVectorField.aspx" class="mm-link">16.6 Conservative Vector Fields</a></li> <li><a href="/Classes/CalcIII/GreensTheorem.aspx" class="mm-link">16.7 Green's Theorem</a></li> </ul> </li> <li><a href="/Classes/CalcIII/SurfaceIntegralsIntro.aspx" class="mm-link">17.Surface Integrals</a> <ul> <li><a href="/Classes/CalcIII/CurlDivergence.aspx" class="mm-link">17.1 Curl and Divergence</a></li> <li><a href="/Classes/CalcIII/ParametricSurfaces.aspx" class="mm-link">17.2 Parametric Surfaces</a></li> <li><a href="/Classes/CalcIII/SurfaceIntegrals.aspx" class="mm-link">17.3 Surface Integrals</a></li> <li><a href="/Classes/CalcIII/SurfIntVectorField.aspx" class="mm-link">17.4 Surface Integrals of Vector Fields</a></li> <li><a href="/Classes/CalcIII/StokesTheorem.aspx" class="mm-link">17.5 Stokes' Theorem</a></li> <li><a href="/Classes/CalcIII/DivergenceTheorem.aspx" class="mm-link">17.6 Divergence Theorem</a></li> </ul> </li> </ul> </li> <li><a href="/Classes/DE/DE.aspx" class="mm-link">Differential Equations</a> <ul> <li><a href="/Classes/DE/IntroBasic.aspx" class="mm-link">1. Basic Concepts</a> <ul> <li><a href="/Classes/DE/Definitions.aspx" class="mm-link">1.1 Definitions</a></li> <li><a href="/Classes/DE/DirectionFields.aspx" class="mm-link">1.2 Direction Fields</a></li> <li><a href="/Classes/DE/FinalThoughts.aspx" class="mm-link">1.3 Final Thoughts</a></li> </ul> </li> <li><a href="/Classes/DE/IntroFirstOrder.aspx" class="mm-link">2. First Order DE's</a> <ul> <li><a href="/Classes/DE/Linear.aspx" class="mm-link">2.1 Linear Equations</a></li> <li><a href="/Classes/DE/Separable.aspx" class="mm-link">2.2 Separable Equations</a></li> <li><a href="/Classes/DE/Exact.aspx" class="mm-link">2.3 Exact Equations</a></li> <li><a href="/Classes/DE/Bernoulli.aspx" class="mm-link">2.4 Bernoulli Differential Equations</a></li> <li><a href="/Classes/DE/Substitutions.aspx" class="mm-link">2.5 Substitutions</a></li> <li><a href="/Classes/DE/IoV.aspx" class="mm-link">2.6 Intervals of Validity</a></li> <li><a href="/Classes/DE/Modeling.aspx" class="mm-link">2.7 Modeling with First Order DE's</a></li> <li><a href="/Classes/DE/EquilibriumSolutions.aspx" class="mm-link">2.8 Equilibrium Solutions</a></li> <li><a href="/Classes/DE/EulersMethod.aspx" class="mm-link">2.9 Euler's Method</a></li> </ul> </li> <li><a href="/Classes/DE/IntroSecondOrder.aspx" class="mm-link">3. Second Order DE's</a> <ul> <li><a href="/Classes/DE/SecondOrderConcepts.aspx" class="mm-link">3.1 Basic Concepts</a></li> <li><a href="/Classes/DE/RealRoots.aspx" class="mm-link">3.2 Real & Distinct Roots</a></li> <li><a href="/Classes/DE/ComplexRoots.aspx" class="mm-link">3.3 Complex Roots</a></li> <li><a href="/Classes/DE/RepeatedRoots.aspx" class="mm-link">3.4 Repeated Roots</a></li> <li><a href="/Classes/DE/ReductionofOrder.aspx" class="mm-link">3.5 Reduction of Order</a></li> <li><a href="/Classes/DE/FundamentalSetsofSolutions.aspx" class="mm-link">3.6 Fundamental Sets of Solutions</a></li> <li><a href="/Classes/DE/Wronskian.aspx" class="mm-link">3.7 More on the Wronskian</a></li> <li><a href="/Classes/DE/NonhomogeneousDE.aspx" class="mm-link">3.8 Nonhomogeneous Differential Equations</a></li> <li><a href="/Classes/DE/UndeterminedCoefficients.aspx" class="mm-link">3.9 Undetermined Coefficients</a></li> <li><a href="/Classes/DE/VariationofParameters.aspx" class="mm-link">3.10 Variation of Parameters</a></li> <li><a href="/Classes/DE/Vibrations.aspx" class="mm-link">3.11 Mechanical Vibrations</a></li> </ul> </li> <li><a href="/Classes/DE/LaplaceIntro.aspx" class="mm-link">4. Laplace Transforms</a> <ul> <li><a href="/Classes/DE/LaplaceDefinition.aspx" class="mm-link">4.1 The Definition</a></li> <li><a href="/Classes/DE/LaplaceTransforms.aspx" class="mm-link">4.2 Laplace Transforms</a></li> <li><a href="/Classes/DE/InverseTransforms.aspx" class="mm-link">4.3 Inverse Laplace Transforms</a></li> <li><a href="/Classes/DE/StepFunctions.aspx" class="mm-link">4.4 Step Functions</a></li> <li><a href="/Classes/DE/IVPWithLaplace.aspx" class="mm-link">4.5 Solving IVP's with Laplace Transforms</a></li> <li><a href="/Classes/DE/IVPWithNonConstantCoefficient.aspx" class="mm-link">4.6 Nonconstant Coefficient IVP's</a></li> <li><a href="/Classes/DE/IVPWithStepFunction.aspx" class="mm-link">4.7 IVP's With Step Functions</a></li> <li><a href="/Classes/DE/DiracDeltaFunction.aspx" class="mm-link">4.8 Dirac Delta Function</a></li> <li><a href="/Classes/DE/ConvolutionIntegrals.aspx" class="mm-link">4.9 Convolution Integrals</a></li> <li><a href="/Classes/DE/Laplace_Table.aspx" class="mm-link">4.10 Table Of Laplace Transforms</a></li> </ul> </li> <li><a href="/Classes/DE/SystemsIntro.aspx" class="mm-link">5. Systems of DE's</a> <ul> <li><a href="/Classes/DE/LA_Systems.aspx" class="mm-link">5.1 Review : Systems of Equations</a></li> <li><a href="/Classes/DE/LA_Matrix.aspx" class="mm-link">5.2 Review : Matrices & Vectors</a></li> <li><a href="/Classes/DE/LA_Eigen.aspx" class="mm-link">5.3 Review : Eigenvalues & Eigenvectors</a></li> <li><a href="/Classes/DE/SystemsDE.aspx" class="mm-link">5.4 Systems of Differential Equations</a></li> <li><a href="/Classes/DE/SolutionsToSystems.aspx" class="mm-link">5.5 Solutions to Systems</a></li> <li><a href="/Classes/DE/PhasePlane.aspx" class="mm-link">5.6 Phase Plane</a></li> <li><a href="/Classes/DE/RealEigenvalues.aspx" class="mm-link">5.7 Real Eigenvalues</a></li> <li><a href="/Classes/DE/ComplexEigenvalues.aspx" class="mm-link">5.8 Complex Eigenvalues</a></li> <li><a href="/Classes/DE/RepeatedEigenvalues.aspx" class="mm-link">5.9 Repeated Eigenvalues</a></li> <li><a href="/Classes/DE/NonhomogeneousSystems.aspx" class="mm-link">5.10 Nonhomogeneous Systems</a></li> <li><a href="/Classes/DE/SystemsLaplace.aspx" class="mm-link">5.11 Laplace Transforms</a></li> <li><a href="/Classes/DE/SystemsModeling.aspx" class="mm-link">5.12 Modeling</a></li> </ul> </li> <li><a href="/Classes/DE/SeriesIntro.aspx" class="mm-link">6. Series Solutions to DE's</a> <ul> <li><a href="/Classes/DE/PowerSeries.aspx" class="mm-link">6.1 Review : Power Series</a></li> <li><a href="/Classes/DE/TaylorSeries.aspx" class="mm-link">6.2 Review : Taylor Series</a></li> <li><a href="/Classes/DE/SeriesSolutions.aspx" class="mm-link">6.3 Series Solutions</a></li> <li><a href="/Classes/DE/EulerEquations.aspx" class="mm-link">6.4 Euler Equations</a></li> </ul> </li> <li><a href="/Classes/DE/IntroHigherOrder.aspx" class="mm-link">7. Higher Order Differential Equations</a> <ul> <li><a href="/Classes/DE/HOBasicConcepts.aspx" class="mm-link">7.1 Basic Concepts for <em>n</em><sup>th</sup> Order Linear Equations</a></li> <li><a href="/Classes/DE/HOHomogeneousDE.aspx" class="mm-link">7.2 Linear Homogeneous Differential Equations</a></li> <li><a href="/Classes/DE/HOUndeterminedCoeff.aspx" class="mm-link">7.3 Undetermined Coefficients</a></li> <li><a href="/Classes/DE/HOVariationOfParam.aspx" class="mm-link">7.4 Variation of Parameters</a></li> <li><a href="/Classes/DE/HOLaplaceTransforms.aspx" class="mm-link">7.5 Laplace Transforms</a></li> <li><a href="/Classes/DE/HOSystems.aspx" class="mm-link">7.6 Systems of Differential Equations</a></li> <li><a href="/Classes/DE/HOSeries.aspx" class="mm-link">7.7 Series Solutions</a></li> </ul> </li> <li><a href="/Classes/DE/IntroBVP.aspx" class="mm-link">8. Boundary Value Problems & Fourier Series</a> <ul> <li><a href="/Classes/DE/BoundaryValueProblem.aspx" class="mm-link">8.1 Boundary Value Problems</a></li> <li><a href="/Classes/DE/BVPEvals.aspx" class="mm-link">8.2 Eigenvalues and Eigenfunctions</a></li> <li><a href="/Classes/DE/PeriodicOrthogonal.aspx" class="mm-link">8.3 Periodic Functions & Orthogonal Functions</a></li> <li><a href="/Classes/DE/FourierSineSeries.aspx" class="mm-link">8.4 Fourier Sine Series</a></li> <li><a href="/Classes/DE/FourierCosineSeries.aspx" class="mm-link">8.5 Fourier Cosine Series</a></li> <li><a href="/Classes/DE/FourierSeries.aspx" class="mm-link">8.6 Fourier Series</a></li> <li><a href="/Classes/DE/ConvergenceFourierSeries.aspx" class="mm-link">8.7 Convergence of Fourier Series</a></li> </ul> </li> <li><a href="/Classes/DE/IntroPDE.aspx" class="mm-link">9. Partial Differential Equations </a> <ul> <li><a href="/Classes/DE/TheHeatEquation.aspx" class="mm-link">9.1 The Heat Equation</a></li> <li><a href="/Classes/DE/TheWaveEquation.aspx" class="mm-link">9.2 The Wave Equation</a></li> <li><a href="/Classes/DE/PDETerminology.aspx" class="mm-link">9.3 Terminology</a></li> <li><a href="/Classes/DE/SeparationofVariables.aspx" class="mm-link">9.4 Separation of Variables</a></li> <li><a href="/Classes/DE/SolvingHeatEquation.aspx" class="mm-link">9.5 Solving the Heat Equation</a></li> <li><a href="/Classes/DE/HeatEqnNonZero.aspx" class="mm-link">9.6 Heat Equation with Non-Zero Temperature Boundaries</a></li> <li><a href="/Classes/DE/LaplacesEqn.aspx" class="mm-link">9.7 Laplace's Equation</a></li> <li><a href="/Classes/DE/VibratingString.aspx" class="mm-link">9.8 Vibrating String</a></li> <li><a href="/Classes/DE/PDESummary.aspx" class="mm-link">9.9 Summary of Separation of Variables</a></li> </ul> </li> </ul> </li> <li><span>Extras</span></li> <li><a href="/Extras/AlgebraTrigReview/AlgebraTrig.aspx" class="mm-link">Algebra & Trig Review</a> <ul> <li><a href="/Extras/AlgebraTrigReview/AlgebraIntro.aspx" class="mm-link">1. Algebra</a> <ul> <li><a href="/Extras/AlgebraTrigReview/Exponents.aspx" class="mm-link">1.1 Exponents </a></li> <li><a href="/Extras/AlgebraTrigReview/AbsoluteValue.aspx" class="mm-link">1.2 Absolute Value</a></li> <li><a href="/Extras/AlgebraTrigReview/Radicals.aspx" class="mm-link">1.3 Radicals</a></li> <li><a href="/Extras/AlgebraTrigReview/Rationalizing.aspx" class="mm-link">1.4 Rationalizing </a></li> <li><a href="/Extras/AlgebraTrigReview/Functions.aspx" class="mm-link">1.5 Functions </a></li> <li><a href="/Extras/AlgebraTrigReview/MultPoly.aspx" class="mm-link">1.6 Multiplying Polynomials</a></li> <li><a href="/Extras/AlgebraTrigReview/Factoring.aspx" class="mm-link">1.7 Factoring</a></li> <li><a href="/Extras/AlgebraTrigReview/SimpRatExp.aspx" class="mm-link">1.8 Simplifying Rational Expressions</a></li> <li><a href="/Extras/AlgebraTrigReview/Graphing.aspx" class="mm-link">1.9 Graphing and Common Graphs</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveEqnPtI.aspx" class="mm-link">1.10 Solving Equations, Part I</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveEqnPtII.aspx" class="mm-link">1.11 Solving Equations, Part II</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveSystems.aspx" class="mm-link">1.12 Solving Systems of Equations</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveIneq.aspx" class="mm-link">1.13 Solving Inequalities</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveAbsValue.aspx" class="mm-link">1.14 Absolute Value Equations and Inequalities</a></li> </ul> </li> <li><a href="/Extras/AlgebraTrigReview/TrigIntro.aspx" class="mm-link">2. Trigonometry</a> <ul> <li><a href="/Extras/AlgebraTrigReview/TrigFunctions.aspx" class="mm-link">2.1 Trig Function Evaluation</a></li> <li><a href="/Extras/AlgebraTrigReview/TrigGraphs.aspx" class="mm-link">2.2 Graphs of Trig Functions</a></li> <li><a href="/Extras/AlgebraTrigReview/TrigFormulas.aspx" class="mm-link">2.3 Trig Formulas</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveTrigEqn.aspx" class="mm-link">2.4 Solving Trig Equations</a></li> <li><a href="/Extras/AlgebraTrigReview/InverseTrig.aspx" class="mm-link">2.5 Inverse Trig Functions</a></li> </ul> </li> <li><a href="/Extras/AlgebraTrigReview/ExpLogIntro.aspx" class="mm-link">3. Exponentials & Logarithms</a> <ul> <li><a href="/Extras/AlgebraTrigReview/ExponentialFcns.aspx" class="mm-link">3.1 Basic Exponential Functions</a></li> <li><a href="/Extras/AlgebraTrigReview/LogarithmFcns.aspx" class="mm-link">3.2 Basic Logarithm Functions</a></li> <li><a href="/Extras/AlgebraTrigReview/LogProperties.aspx" class="mm-link">3.3 Logarithm Properties</a></li> <li><a href="/Extras/AlgebraTrigReview/SimpLogs.aspx" class="mm-link">3.4 Simplifying Logarithms</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveExpEqn.aspx" class="mm-link">3.5 Solving Exponential Equations</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveLogEqn.aspx" class="mm-link">3.6 Solving Logarithm Equations</a></li> </ul> </li> </ul> </li> <li><a href="/Extras/CommonErrors/CommonMathErrors.aspx" class="mm-link">Common Math Errors</a> <ul> <li><a href="/Extras/CommonErrors/GeneralErrors.aspx" class="mm-link">1. General Errors</a> </li> <li><a href="/Extras/CommonErrors/AlgebraErrors.aspx" class="mm-link">2. Algebra Errors</a> </li> <li><a href="/Extras/CommonErrors/TrigErrors.aspx" class="mm-link">3. Trig Errors</a> </li> <li><a href="/Extras/CommonErrors/CommonErrors.aspx" class="mm-link">4. Common Errors</a> </li> <li><a href="/Extras/CommonErrors/CalculusErrors.aspx" class="mm-link">5. Calculus Errors</a> </li> </ul> </li> <li><a href="/Extras/ComplexPrimer/ComplexNumbers.aspx" class="mm-link">Complex Number Primer</a> <ul> <li><a href="/Extras/ComplexPrimer/Definition.aspx" class="mm-link">1. The Definition</a> </li> <li><a href="/Extras/ComplexPrimer/Arithmetic.aspx" class="mm-link">2. Arithmetic</a> </li> <li><a href="/Extras/ComplexPrimer/ConjugateModulus.aspx" class="mm-link">3. Conjugate and Modulus</a> </li> <li><a href="/Extras/ComplexPrimer/Forms.aspx" class="mm-link">4. Polar and Exponential Forms</a> </li> <li><a href="/Extras/ComplexPrimer/Roots.aspx" class="mm-link">5. Powers and Roots</a> </li> </ul> </li> <li><a href="/Extras/StudyMath/HowToStudyMath.aspx" class="mm-link">How To Study Math</a> <ul> <li><a href="/Extras/StudyMath/GeneralTips.aspx" class="mm-link">1. General Tips</a> </li> <li><a href="/Extras/StudyMath/TakingNotes.aspx" class="mm-link">2. Taking Notes</a> </li> <li><a href="/Extras/StudyMath/GettingHelp.aspx" class="mm-link">3. Getting Help</a> </li> <li><a href="/Extras/StudyMath/Homework.aspx" class="mm-link">4. Doing Homework</a> </li> <li><a href="/Extras/StudyMath/ProblemSolving.aspx" class="mm-link">5. Problem Solving</a> </li> <li><a href="/Extras/StudyMath/StudyForExam.aspx" class="mm-link">6. Studying For an Exam</a> </li> <li><a href="/Extras/StudyMath/TakingExam.aspx" class="mm-link">7. Taking an Exam</a> </li> <li><a href="/Extras/StudyMath/Errors.aspx" class="mm-link">8. Learn From Your Errors</a> </li> </ul> </li> <li><span>Misc Links</span></li> <li><a href="/contact.aspx" class="mm-link">Contact Me</a></li> <li><a href="/links.aspx" class="mm-link">Links</a></li> <li><a href="/mathjax.aspx" class="mm-link">MathJax Help and Configuration</a></li> <li><a href="/privacy.aspx" class="mm-link">Privacy Statement</a></li> <li><a href="/help.aspx" class="mm-link">Site Help & FAQ</a></li> <li><a href="/terms.aspx" class="mm-link">Terms of Use</a></li> </ul> </nav> <div class="top-info-bar"> <span id="mobile-title" class="mobile-header-title header-title">Paul's Online Notes</span> <br /> <span class="top-menu-breadcrumb"> <a href="/" id="breadcrumb_home_link" title="Go to Main Page">Home</a> <span id="breadcrumb_h_b_sep_span">/ </span> <span id="breadcrumb_book_span">MathJax Help and Configuration</span> </span> </div> <div class="header-divider"></div> <div class="content"> <span id="SHLink_NoteMobile" class="SH-Link content-note-link-mobile">Show Mobile Notice</span> <span id="SHImg_NoteMobile" class="fas fa-caret-right SH-padding content-note-link-mobile" aria-hidden="true"></span> <span id="SHALink_S_Note" class="SH-Link SH-Hide SH-Bracket">Show All Notes</span> <span id="SHALink_H_Note" class="SH-Link SH-Hide SH-Bracket">Hide All Notes</span> <div id="SHObj_NoteMobile" class="content-note-container content-note-container-mobile"> <div class="content-note-header">Mobile Notice</div> <div class="content-note">You appear to be on a device with a "narrow" screen width (<em>i.e.</em> you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.</div> </div> <p>In order to display the mathematics on this site I use <a href="https://www.mathjax.org/">MathJax</a>. The mathematics "source" is <a href="https://www.latex-project.org/">LaTeX</a> and the MathJax "processor" converts that source into "viewable" mathematics. <a href="#MoreInfo">Below</a> is a little more information about the decision to do this if you are curious as to why the change in how math is displayed on the site.</p> <p>For example, the following LaTeX code,</p> <div class="center-div"><pre>\[ a x^{2} + b x + c = 0 \hspace{0.5in} \Rightarrow \hspace{0.5in} x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a} \]</pre></div> <p>will produce the following (probably) very familiar mathematical formula.</p> \[a x^{2} + b x + c = 0 \hspace{0.5in} \Rightarrow \hspace{0.5in} x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a} \] <p>Quick note about the LaTeX in the pages... If you are thinking of using the "source" of the mathematics to learn LaTeX. Don't do it! The tool I used to convert the source documents into LaTeX generally did a good job but the result is often UGLY LaTeX "code" and not how I would have done it if I was writing it in LaTeX from the start. It put a lot of extra "junk" into the LaTeX that just isn't needed most of the time and if I had taken the time to remove that and generally clean it up it would have, quite likely, taken me closer to 3 YEARS (!! Yikes !!) to convert all the pages.</p> <p>Okay, on to some of the configuration options that MathJax provides. Please note as well that the options here are provided by the MathJax folks and not me and as such I can't provide tech support for something in these options that doesn't work. Sorry.</p> <p>If you right click on any equation (such as the one above) a menu will open that will allow you to "configure" a handful of MathJax options. Here is a listing of the menu items and descriptions of each of them (well, descriptions of those that I'm confident that I know what they are anyway...). MathJax will set a cookie on your computer to remember any changes you make to the default settings.</p> <ul class="notes-list"> <li><span class="bold">Show Math As</span> <ul> <li>MathML Code - This will open a window showing the MathML code required to display the same mathematics.</li> <li>TeX Commands - This will open a window showing the LaTeX code that I used to display the mathematics.</li> <li>Annotation - Not sure to be honest as I've never actually gotten this submenu to open. My best guess is that it is only active under specific circumstances and I don't know that those are...</li> <li>Show TeX hints in MathML - This will add in a few extra commands in the MathML Code above and it is my understanding it helps make the spacing in the mathematics look more like the LaTeX spacing.</li> <li>Add original form as annotation - This will add in a command in the MathML Code above that gives the original LaTeX command.</li> </ul> </li> <li><span class="bold">Math Settings</span> <ul class="notes-list"> <li><span class="bold">Zoom Trigger</span> - Options for expanding the mathematics. <ul> <li>Hover - The mathematics will expand whenever you hover your mouse over the mathematics. Note that there will be a small delay just to make sure that you aren't just moving the mouse across the screen and it just happened to go over a piece of mathematics.</li> <li>Click - Simply click the mathematics to get it to expand.</li> <li>Double-Click - Double click the mathematics to get it to expand.</li> <li>No Zoom <span class="bold">(default setting)</span> - The mathematics won't expand.</li> <li>Trigger Requires : Alt - The mathematics will only expand if you press the Alt key as you hover/click/double-click the mathematics depending on your choice of the actual trigger.</li> <li>Trigger Requires : Control - The mathematics will only expand if you press the Control key as you hover/click/double-click the mathematics depending on your choice of the actual trigger.</li> <li>Trigger Requires : Shift - The mathematics will only expand if you press the Shift key as you hover/click/double-click the mathematics depending on your choice of the actual trigger.</li> <li>Quick note on the "Trigger Requires". More than one requirement can be selected. So, for instance if you select both Alt and Shift then you will need to have both the Alt and Shift keys pressed to trigger the mathematics expanding. Also, from my experiments on this it seems like if you select Hover as the trigger it will always expand and ignore any choices you've made for requirements.</li> </ul> </li> <li><span class="bold">Zoom Factor</span> - This gives a variety of options from 125% to 400% on how much the equation should be expanded when triggered.</li> <li><span class="bold">Math Renderer</span> - This controls the method used by MathJax to actually render the mathematics. <ul> <li>HTML-CSS - This method should produce high quality output in pretty much all browsers. It's main advantage over the other methods is that it supports quite a few "legacy" browsers, such as Internet Explorer 6. However, the drawback of using this method is that it is probably the slowest processing method. It uses web based fonts downloaded from MathJax. It should only need to download the fonts once and then keep them in your browsers cache so that it won't delay the display of the mathematics.</li> <li>Common HTML <span class="bold">(default method)</span> - This method should produce high quality output for modern browsers and it should be consistent across browsers and operating systems. For Internet Explorer the method will work for IE9+ and will gracefully fall back to other methods on older versions. This method is faster than most of the other output processors. It uses web based fonts downloaded from MathJax. It should only need to download the fonts once and then keep them in your browsers cache so it won't delay the display of the mathematics.</li> <li>Preview HTML - This is a very fast processor but tends to produce low quality output. It does not use web based fonts but instead fonts installed on the local computer. Because of this the output may well vary based upon computer you are on. Note that, by default on this site, when a page first loads it first uses this method to quickly convert the LaTeX into readable mathematics before applying the actual processor.</li> <li>MathML - This method uses MathML to display the math. However, in order for it to actually display mathematics your browser must support MathML. Currently Firefox is the only browser with native support for MathML and even then does not include all "standard" features. It is not suggested that you use this method and the results are liable to be very sub par with some of the mathematics that simply won't render properly. If you use this method on a browser that doesn't support MathML it will simply change the equations to the source LaTeX code.</li> <li>SVG - This uses Scalable Vector Graphics to render the mathematics. Most major browsers support SVG but some older browsers don't so it should only be used on a "modern" browser. It should produce high quality output but has some issues with variable length tables in web pages (in particular they won't resize if the window size is changed).</li> <li>Plain Source - This will simply skip the rendering and only show the LaTeX source code. Note that, in some instances, on this site this may end up with the LaTeX code for equations that are side by side overlapping and becoming unreadable.</li> <li>Fast Preview - This option is selected by default and will first render the mathematics with Preview HTML before rendering it into the final form with the selected method</li> </ul> </li> <li><span class="bold">Scale All Math...</span> - This will open a dialog allowing you to scale all the mathematics on the page. Note that you can reduce or expand the mathematics.</li> </ul> </li> <li><span class="bold">Accessiblity</span> <ul class="notes-list"> <li><span class="bold">Assistive MathML</span> - I'm not quite sure what this option is to be honest. I believe it allows screen readers to read the mathematics using MathML and is enabled by default.</li> <li><span class="bold">Include in Tab Order</span> - This allows all the mathematics to be selected via the tab button and is enabled by default.</li> <li><span class="bold">Collapsible Math</span> - This enables users to be able to "collapse" certain portions of the mathematics and is not enabled by default.</li> <li><span class="bold">Auto Collapse</span> - This is supposed to auto collapse some of the mathematics so allow for easier fitting into smaller viewports and is not enabled by default.</li> <li><span class="bold">Explorer</span> - This can be activated (by clicking on the Active option at the top of the submenu) and enables a variety of accessibility options <ul> <li>Walker - Controls the style of exploration of expressions. It uses the following key bindings. <ul> <li>SHIFT+SPACE : Starts the application. This requires focus to be set on the equation (can be set with tab) and some screen readers might need to have 2024-11_MathWeb02reading mode disabled prior to starting the application.</li> <li>ESC : Stops the application. If the application is restarted it will pick up where it last left off.</li> <li>UP/DOWN : Moves up/down the expression tree. Moving down will start at the left most sub-expression of the level you are on. Reaching the bottom/top of the tree will trigger an aural indicator.</li> <li>LEFT/RIGHT : Move left/right through the expression tree.</li> <li>ENTER : Toggle the collapsed state of the current sub-expression. It will regenerate speech-text upon toggling.</li> <li>SPACE : Get positional information.</li> </ul> </li> <li>Highlight - Used to highlight certain complex expressions. Often used when collapsing expressions.</li> <li>Background and Foreground - Sets the appropriate colors when the walker is used for expression exploration.</li> <li>Speech Output - Selecting this will activates the generation of speech output and is enabled by default.</li> <li>Subtitles - Not enabled for this site as there are no videos....</li> <li>Generation - Determines how verbose the speech output should be.</li> <li>Mathspeak and ChromVox Rules - Determines how verbose these screen readers should be.</li> </ul> </li> </ul> </li> <li><span class="bold">Language</span> - This sets the language for the MathJax interface. Note that this does NOT set the language for the text in the site itself. Currently the site is only available in English.</li> <li><span class="bold">About MathJax</span> - Opens a dialog box with some basic version information.</li> <li><span class="bold">MathJax Help</span> - Opens a dialog box with some very basic help information.</li> </ul> <br /> <a name="MoreInfo"></a><br /> <p class="bold">More Info about the change</p> <p>If you've been using the site for a while you probably know that previously I used images to display the mathematics. When I first started doing this site that was the only really reliable method of displaying mathematics on the web as most browsers just had not reached the point where more sophisticated methods could be used. Things have now progressed to the point where images are not only not needed anymore but in fact produce unacceptable results on some platforms (most notably mobile platforms).</p> <p>However, because of the decision to use images none of the source material that is used to actually generate the pages on the site are in a form that can easily be used to take advantage of the new tools for displaying mathematics on the web. Because of this I have put off converting to one of the newer tools. Until now. While rewriting the back end of the site to make it more agile and easier to maintain I decided to figure out a way to use the new tools. The results are web pages that should look pretty much the same regardless of platform and because images are no longer being used to display the mathematics pages should load faster and look much better on mobile devices (or at least as well as long math equations are ever going to look on mobile devices anyway....).</p> <p>An added benefit is that pages should now be readable by anyone needing to use a screen reader. MathJax should be compatible with ChromeVox, Texthelp and JAWS and they are working to add other screen readers.</p> <p>The only drawback is that, as noted, the original source documents were not in a format that the new technologies can use. So, to generate the pages on the site I first need to use a tool to convert all the mathematics over to LaTeX and while the tool I used generally works fairly well I had to go in and hand edit many of the equations to be "readable" by MathJax. That means that I will, on occasion, miss something and/or errors will slip into the material. If you see an equation that is just gibberish and/or just doesn't look quite right feel free to contact me and I'll take a look at it and see if I can fix it up.</p> </div>  <div class="footer"> <div class="footer-links"> [<a href="/Contact.aspx">Contact Me</a>] [<a href="/Privacy.aspx">Privacy Statement</a>] [<a href="/Help.aspx">Site Help & FAQ</a>] [<a href="/Terms.aspx">Terms of Use</a>] </div> <div class="footer-dates"> <div class="footer-copyright"><span id="lblCopyRight">© 2003 - 2025 Paul Dawkins</span></div> <div class="footer-spacer"></div> <div class="footer-modified-date">Page Last Modified : <span id="lblModified">12/2/2024</span></div> </div> </div> </div>  </body> </html>

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MathJax Help and Configuration