Calculus with Differential Equations

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Calculus with Differential Equations, 9/e ISBN: 0-13-230633-6Dale Varberg/Edwin J. Purcell/Steve E. Rigdon 0 PRELIMINARIES 0.1 Real Numbers, Estimation, and Logic 0.2 Inequalities and Absolute Values 0.3 The Rectangular Coordinate System 0.4 Graphs of Equations 0.5 Functions and Their Graphs 0.6 Operations on Functions 0.7 The Trigonometric Functions 0.8 Chapter Review 1 LIMITS 1.1 Introduction to Limits 1.2 Rigorous Study of Limits 1.3 Limit Theorems 1.4 Limits Involving Trigonometric Functions 1.5 Limits at Infinity; Infinite Limits 1.6 Continuity of Functions 1.7 Chapter Review 2 THE DERIVATIVE 2.1 Two Problems with One Theme 2.2 The Derivative 2.3 Rules for Finding Derivatives 2.4 Derivatives of Trigonometric Functions 2.5 The Chain Rule 2.6 Higher-Order Derivatives 2.7 Implicit Differentiation 2.8 Related Rates 2.9 Differentials and Approximations 2.10 Chapter Review 3 APPLICATIONS OF THE DERIVATIVE 3.1 Maxima and Minima 3.2 Monotonicity and Concavity 3.3 Local Extrema and Extrema on Open Intervals 3.4 Practical Problems 3.5 Graphing Functions Using Calculus 3.6 The Mean Value Theorem for Derivatives 3.7 Solving Equations Numerically 3.8 Antiderivatives 3.9 Introduction to Differential Equations 3.10 Chapter Review 4 THE DEFINITE INTEGRAL 4.1 Introduction to Area 4.2 The Definite Integral 4.3 The 1st Fundamental Theorem of Calculus 4.4 The 2nd Fundamental Theorem of Calculus and the Method of Substitution 4.5 The Mean Value Theorem for Integrals & the Use of Symmetry 4.6 Numerical Integration 4.7 Chapter Review 5 APPLICATIONS OF THE INTEGRAL 5.1 The Area of a Plane Region 5.2 Volumes of Solids: Slabs, Disks, Washers 5.3 Volumes of Solids of Revolution: Shells 5.4 Length of a Plane Curve 5.5 Work and Fluid Pressure 5.6 Moments, Center of Mass 5.7 Probability and Random Variables 5.8 Chapter Review 6 TRANSCENDENTAL FUNCTIONS 6.1 The Natural Logarithm Function 6.2 Inverse Functions and Their Derivatives 6.3 The Natural Exponential Function 6.4 General Exponential & Logarithmic Functions 6.5 Exponential Growth and Decay 6.6 First-Order Linear Differential Equations 6.7 Approximations for Differential Equations 6.8 The Inverse Trig Functions & Their Derivatives 6.9 The Hyperbolic Functions & Their Inverses 6.10 Chapter Review 7 TECHNIQUES OF INTEGRATION 7.1 Basic Integration Rules 7.2 Integration by Parts 7.3 Some Trigonometric Integrals 7.4 Rationalizing Substitutions 7.5 Integration of Rational Functions Using Partial Fractions 7.6 Strategies for Integration 7.7 Chapter Review 8 INDETERMINATE FORMS & IMPROPER INTEGRALS 8.1 Indeterminate Forms of Type 0/0 8.2 Other Indeterminate Forms 8.3 Improper Integrals: Infinite Limits of Integration 8.4 Improper Integrals: Infinite Integrands 8.5 Chapter Review 9 INFINITE SERIES 9.1 Infinite Sequences 9.2 Infinite Series 9.3 Positive Series: The Integral Test 9.4 Positive Series: Other Tests 9.5 Alternating Series, Absolute Convergence, and Conditional Convergence 9.6 Power Series 9.7 Operations on Power Series 9.8 Taylor and Maclaurin Series 9.9 The Taylor Approximation to a Function 9.10 Chapter Review 10 CONICS AND POLAR COORDINATES 10.1 The Parabola 10.2 Ellipses and Hyperbolas 10.3 Translation and Rotation of Axes 10.4 Parametric Representation of Curves in the Plane 10.5 The Polar Coordinate System 10.6 Graphs of Polar Equations 10.7 Calculus in Polar Coordinates 10.8 Chapter Review 11 GEOMETRY IN SPACE AND VECTORS 11.1 Cartesian Coordinates in Three-Space 11.2 Vectors 11.3 The Dot Product 11.4 The Cross Product 11.5 Vector Valued Functions & Curvilinear Motion 11.6 Lines in Three-Space 11.7 Curvature and Components of Acceleration 11.8 Surfaces in Three Space 11.9 Cylindrical and Spherical Coordinates 11.10 Chapter Review 12 DERIVATIVES OF FUNCTIONS OF TWO OR MORE VARIABLES 12.1 Functions of Two or More Variables 12.2 Partial Derivatives 12.3 Limits and Continuity 12.4 Differentiability 12.5 Directional Derivatives and Gradients 12.6 The Chain Rule 12.7 Tangent Planes and Approximations 12.8 Maxima and Minima 12.9 The Method of Lagrange Multipliers 12.10 Chapter Review 13 MULTIPLE INTEGRATION 13.1 Double Integrals over Rectangles 13.2 Iterated Integrals 13.3 Double Integrals over Nonrectangular Regions 13.4 Double Integrals in Polar Coordinates 13.5 Applications of Double Integrals 13.6 Surface Area 13.7 Triple Integrals in Cartesian Coordinates 13.8 Triple Integrals in Cylindrical & Spherical Coordinates 13.9 Change of Variables in Multiple Integrals 13.1 Chapter Review 14 VECTOR CALCULUS 14.1 Vector Fields 14.2 Line Integrals 14.3 Independence of Path 14.4 Green's Theorem in the Plane 14.5 Surface Integrals 14.6 Gauss's Divergence Theorem 14.7 Stokes's Theorem 14.8 Chapter Review 15 DIFFERENTIAL EQUATIONS 15.1 Linear Homogeneous Equations 15.2 Nonhomogeneous Equations 15.3 Applications of Second-Order Equations APPENDIX A.1 Mathematical Induction A.2 Proofs of Several Theorems