APEX Calculus LT Greg Hartman, VMI UND Math Dept Contents APEX Calculus Contributing Authors Preface Calculus I 1 Limits Chapter Introduction 1.0 Chapter Prerequisites 1.1 An Introduction To Limits 1.2 Epsilon-Delta Definition of a Limit 1.3 Finding Limits Analytically 1.4 One Sided Limits 1.5 Limits Involving Infinity 1.6 Continuity Chapter Summary 2 Derivatives Chapter Introduction 2.0 Chapter Prerequisites 2.1 Instantaneous Rates of Change: The Derivative 2.2 Interpretations of the Derivative 2.3 Basic Differentiation Rules 2.4 The Product and Quotient Rules 2.5 The Chain Rule 2.6 Implicit Differentiation 3 The Graphical Behavior of Functions Chapter Introduction 3.1 Extreme Values 3.2 The Mean Value Theorem 3.3 Increasing and Decreasing Functions 3.4 Concavity and the Second Derivative 3.5 Curve Sketching 4 Applications of the Derivative Chapter Introduction 4.1 Related Rates 4.2 Optimization 4.3 Differentials 4.4 Newton’s Method 5 Integration Chapter Introduction 5.1 Antiderivatives and Indefinite Integration 5.2 The Definite Integral 5.3 Riemann Sums 5.4 The Fundamental Theorem of Calculus 5.5 Substitution 6 Applications of Integration Chapter Introduction 6.1 Area Between Curves 6.2 Volume by Cross-Sectional Area; Disk and Washer Methods 6.3 The Shell Method 6.4 Work 6.5 Fluid Forces Calculus II 7 Inverse Functions and L’Hôpital’s Rule Chapter Introduction 7.1 Inverse Functions 7.2 Derivatives of Inverse Functions 7.3 Exponential and Logarithmic Functions 7.4 Hyperbolic Functions 7.5 L’Hôpital’s Rule 8 Techniques of Integration Chapter Introduction 8.1 Integration by Parts 8.2 Trigonometric Integrals 8.3 Trigonometric Substitution 8.4 Partial Fraction Decomposition 8.5 Integration Strategies 8.6 Improper Integration 8.7 Numerical Integration 9 Sequences and Series Chapter Introduction 9.1 Sequences 9.2 Infinite Series 9.3 The Integral Test 9.4 Comparison Tests 9.5 Alternating Series and Absolute Convergence 9.6 Ratio and Root Tests 9.7 Strategy for Testing Series 9.8 Power Series 9.9 Taylor Polynomials 9.10 Taylor Series 10 Curves in the Plane Chapter Introduction 10.0 Chapter Prerequisites — Conic Sections 10.1 Arc Length and Surface Area 10.2 Parametric Equations 10.3 Calculus and Parametric Equations 10.4 Introduction to Polar Coordinates 10.5 Calculus and Polar Functions Calculus III 11 Vectors Chapter Introduction 11.1 Introduction to Cartesian Coordinates in Space 11.2 An Introduction to Vectors 11.3 The Dot Product 11.4 The Cross Product 11.5 Lines 11.6 Planes 12 Vector Valued Functions Chapter Introduction 12.1 Vector-Valued Functions 12.2 Calculus and Vector-Valued Functions 12.3 The Calculus of Motion 12.4 Unit Tangent and Normal Vectors 12.5 The Arc Length Parameter and Curvature 13 Functions of Several Variables Chapter Introduction 13.1 Introduction to Multivariable Functions 13.2 Limits and Continuity of Multivariable Functions 13.3 Partial Derivatives 13.4 Differentiability and the Total Differential 13.5 The Multivariable Chain Rule 13.6 Directional Derivatives 13.7 Tangent Lines, Normal Lines, and Tangent Planes 13.8 Extreme Values 13.9 Lagrange Multipliers 14 Multiple Integration Chapter Introduction 14.1 Iterated Integrals and Area 14.2 Double Integration and Volume 14.3 Double Integration with Polar Coordinates 14.4 Center of Mass 14.5 Surface Area 14.6 Volume Between Surfaces and Triple Integration 14.7 Triple Integration with Cylindrical and Spherical Coordinates 15 Vector Analysis Chapter Introduction Integration Review 15.1 Introduction to Line Integrals 15.2 Vector Fields 15.3 Line Integrals over Vector Fields 15.4 Flow, Flux, Green’s Theorem and the Divergence Theorem 15.5 Parameterized Surfaces and Surface Area 15.6 Surface Integrals 15.7 The Divergence Theorem and Stokes’ Theorem Solutions To Selected Problems Index Important Formulas
