Schaum's Outline of Calculus, Seventh Edition

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• PrefaceChapter 1 Linear Coordinate Systems. Absolute Value. Inequalities.Linear Coordinate SystemFinite IntervalsInfinite IntervalsInequalitiesSolved ProblemsSupplementary ProblemsChapter 2 Rectangular Coordinate SystemsCoordinate AxesCoordinatesQuadrantsThe Distance FormulaThe Midpoint FormulasProofs of Geometric TheoremsSolved ProblemsSupplementary ProblemsChapter 3 LinesThe Steepness of a LineThe Sign of the SlopeSlope and SteepnessEquations of LinesA Point–Slope EquationSlope–Intercept EquationParallel LinesPerpendicular LinesSolved ProblemsSupplementary ProblemsChapter 4 CirclesEquations of CirclesThe Standard Equation of a CircleSolved ProblemsSupplementary ProblemsChapter 5 Equations and Their GraphsThe Graph of an EquationParabolasEllipsesHyperbolasConic SectionsSolved ProblemsSupplementary ProblemsChapter 6 FunctionsSolved ProblemsSupplementary ProblemsChapter 7 LimitsLimit of a FunctionRight and Left LimitsTheorems on LimitsInfinitySolved ProblemsSupplementary ProblemsChapter 8 ContinuityContinuous FunctionSolved ProblemsSupplementary ProblemsChapter 9 The DerivativeDelta NotationThe DerivativeNotation for DerivativesDifferentiabilitySolved ProblemsSupplementary ProblemsChapter 10 Rules for Differentiating FunctionsDifferentiationComposite Functions. The Chain Rule.Chain RuleAlternative Formulation of the Chain RuleInverse FunctionsHigher DerivativesSolved ProblemsSupplementary ProblemsChapter 11 Implicit DifferentiationImplicit FunctionsDerivatives of Higher OrderSolved ProblemsSupplementary ProblemsChapter 12 Tangent and Normal LinesThe Angles of IntersectionSolved ProblemsSupplementary ProblemsChapter 13 Law of the Mean. Increasing and Decreasing Functions.Relative Maximum and MinimumIncreasing and Decreasing FunctionsSolved ProblemsSupplementary ProblemsChapter 14 Maximum and Minimum ValuesCritical NumbersSecond Derivative Test for Relative ExtremaFirst Derivative TestAbsolute Maximum and MinimumTabular Method for Finding the Absolute Maximum and MinimumSolved ProblemsSupplementary ProblemsChapter 15 Curve Sketching. Concavity. Symmetry.ConcavityPoints of InflectionVertical AsymptotesHorizontal AsymptotesSymmetryInverse Functions and SymmetryEven and Odd FunctionsHints for Sketching the Graph G of y = f (x)Solved ProblemsSupplementary ProblemsChapter 16 Review of TrigonometryAngle MeasureDirected AnglesSine and Cosine FunctionsSolved ProblemsSupplementary ProblemsChapter 17 Differentiation of Trigonometric FunctionsContinuity of cos x and sin xGraph of sin xGraph of cos xOther Trigonometric FunctionsDerivativesOther RelationshipsGraph of y = tan xGraph of y = sec xAngles Between CurvesSolved ProblemsSupplementary ProblemsChapter 18 Inverse Trigonometric FunctionsThe Derivative of sin−1 xThe Inverse Cosine FunctionThe Inverse Tangent FunctionSolved ProblemsSupplementary ProblemsChapter 19 Rectilinear and Circular MotionRectilinear MotionMotion Under the Influence of GravityCircular MotionSolved ProblemsSupplementary ProblemsChapter 20 Related RatesSolved ProblemsSupplementary ProblemsChapter 21 Differentials. Newton’s Method.The DifferentialDefinitionNewton’s MethodSolved ProblemsSupplementary ProblemsChapter 22 AntiderivativesLaws for AntiderivativesSolved ProblemsSupplementary ProblemsChapter 23 The Definite Integral. Area Under a Curve.Sigma NotationArea Under a CurveProperties of the Definite IntegralSolved ProblemsSupplementary ProblemsChapter 24 The Fundamental Theorem of CalculusMean Value Theorem for IntegralsAverage Value of a Function on a Closed IntervalFundamental Theorem of CalculusChange of Variable in a Definite IntegralSolved ProblemsSupplementary ProblemsChapter 25 The Natural LogarithmThe Natural LogarithmDefinitionProperties of the Natural LogarithmSolved ProblemsSupplementary ProblemsChapter 26 Exponential and Logarithmic FunctionsDefinitionProperties of exThe General Exponential FunctionGeneral Logarithmic FunctionsSolved ProblemsSupplementary ProblemsChapter 27 L’Hôpital’s RuleL’Hôpital’s RuleIndeterminate Type 0 · ∞Indeterminate Type ∞ − ∞Indeterminate Types 00, ∞0, and 1∞Solved ProblemsSupplementary ProblemsChapter 28 Exponential Growth and DecayHalf-LifeSolved ProblemsSupplementary ProblemsChapter 29 Applications of Integration I: Area and Arc LengthArea Between a Curve and the Y-AxisAreas Between CurvesArc LengthSolved ProblemsSupplementary ProblemsChapter 30 Applications of Integration II: VolumeDisk FormulaWasher MethodCylindrical Shell MethodDifference of Shells FormulaCross-Section Formula (Slicing Formula)Solved ProblemsSupplementary ProblemsChapter 31 Techniques of Integration I: Integration by PartsSolved ProblemsSupplementary ProblemsChapter 32 Techniques of Integration II: Trigonometric Integrands and Trigonometric SubstitutionsTrigonometric IntegrandsTrigonometric SubstitutionsSolved ProblemsSupplementary ProblemsChapter 33 Techniques of Integration III: Integration by Partial FractionsMethod of Partial FractionsSolved ProblemsSupplementary ProblemsChapter 34 Techniques of Integration IV: Miscellaneous SubstitutionsSolved ProblemsSupplementary ProblemsChapter 35 Improper IntegralsInfinite Limits of IntegrationDiscontinuities of the IntegrandSolved ProblemsSupplementary ProblemsChapter 36 Applications of Integration III: Area of a Surface of RevolutionSolved ProblemsSupplementary ProblemsChapter 37 Parametric Representation of CurvesParametric EquationsArc Length for a Parametric CurveSolved ProblemsSupplementary ProblemsChapter 38 CurvatureDerivative of Arc LengthCurvatureThe Radius of CurvatureThe Circle of CurvatureThe Center of CurvatureThe EvoluteSolved ProblemsSupplementary ProblemsChapter 39 Plane VectorsScalars and VectorsSum and Difference of Two VectorsComponents of a VectorScalar Product (or Dot Product)Scalar and Vector ProjectionsDifferentiation of Vector FunctionsSolved ProblemsSupplementary ProblemsChapter 40 Curvilinear MotionVelocity in Curvilinear MotionAcceleration in Curvilinear MotionTangential and Normal Components of AccelerationSolved ProblemsSupplementary ProblemsChapter 41 Polar CoordinatesPolar and Rectangular CoordinatesSome Typical Polar CurvesAngle of InclinationPoints of IntersectionAngle of IntersectionThe Derivative of the Arc LengthCurvatureSolved ProblemsSupplementary ProblemsChapter 42 Infinite SequencesInfinite SequencesLimit of a SequenceMonotonic SequencesSolved ProblemsSupplementary ProblemsChapter 43 Infinite SeriesGeometric SeriesSolved ProblemsSupplementary ProblemsChapter 44 Series with Positive Terms. The Integral Test. Comparison Tests.Series of Positive TermsSolved ProblemsSupplementary ProblemsChapter 45 Alternating Series. Absolute and Conditional Convergence. The Ratio Test.Alternating SeriesDefinitionSolved ProblemsSupplementary ProblemsChapter 46 Power SeriesPower SeriesUniform ConvergenceSolved ProblemsSupplementary ProblemsChapter 47 Taylor and Maclaurin Series. Taylor’s Formula with Remainder.Taylor and Maclaurin SeriesApplications of Taylor’s Formula with RemainderSolved ProblemsSupplementary ProblemsChapter 48 Partial DerivativesFunctions of Several VariablesLimitsContinuityPartial DerivativesPartial Derivatives of Higher OrderSolved ProblemsSupplementary ProblemsChapter 49 Total Differential. Differentiability. Chain Rules.Total DifferentialDifferentiabilityChain RulesChain Rule (2 → 1)Chain Rule (2 → 2)Implicit DifferentiationSolved ProblemsSupplementary ProblemsChapter 50 Space VectorsVectors in SpaceDirection Cosines of a VectorDeterminantsVector Perpendicular to Two VectorsVector Product of Two VectorsTriple Scalar ProductThe Straight LineThe PlaneSolved ProblemsSupplementary ProblemsChapter 51 Surfaces and Curves in SpacePlanesSpheresCylindrical SurfacesEllipsoidElliptic ParaboloidElliptic ConeHyperbolic ParaboloidHyperboloid of One SheetHyperboloid of Two SheetsTangent Line and Normal Plane to a Space CurveTangent Plane and Normal Line to a SurfaceSurface of RevolutionSolved ProblemsSupplementary ProblemsChapter 52 Directional Derivatives. Maximum and Minimum Values.Directional DerivativesRelative Maximum and Minimum ValuesAbsolute Maximum and Minimum ValuesSolved ProblemsSupplementary ProblemsChapter 53 Vector Differentiation and IntegrationVector DifferentiationSpace CurvesSurfacesThe Operation ∇Divergence and CurlIntegrationLine IntegralsSolved ProblemsSupplementary ProblemsChapter 54 Double and Iterated IntegralsThe Double IntegralThe Iterated IntegralSolved ProblemsSupplementary ProblemsChapter 55 Centroids and Moments of Inertia of Plane AreasPlane Area by Double IntegrationCentroidsMoments of InertiaSolved ProblemsSupplementary ProblemsChapter 56 Double Integration Applied to Volume Under a Surface and the Area of a Curved SurfaceSolved ProblemsSupplementary ProblemsChapter 57 Triple IntegralsCylindrical and Spherical CoordinatesThe Triple IntegralEvaluation of Triple IntegralsCentroids and Moments of InertiaSolved ProblemsSupplementary ProblemsChapter 58 Masses of Variable DensitySolved ProblemsSupplementary ProblemsChapter 59 Differential Equations of First and Second OrderSeparable Differential EquationsHomogeneous FunctionsIntegrating FactorsSecond-Order EquationsSolved ProblemsSupplementary ProblemsAppendix A Trigonometric FormulasAppendix B Geometric FormulasIndex