Apex Calculus

Description

A Calculus text covering limits, derivatives and the basics of integration. This book contains numerous examples and illustrations to help make concepts clear. The follow-up to this text is Calculus 2, which review the basic concepts of integration, then covers techniques and applications of integration, followed by sequences and series. Calculus 3 finishes this series by covering parametric equations, polar coordinates, vector valued functions, multivariable functions and vector analysis. A free .pdf version of all three can be obtained at apexcalculus.com. Table of Contents Preface 1 Limits 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 Continuity 1.6 Limits Involving Infinity 2 Derivatives 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 2.7 Derivatives of Inverse Functions 3 The Graphical Behavior of Functions 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 4.1 Newton's Method 4.2 Related Rates 4.3 Optimization 4.4 Differentials 5 Integration 5.1 Antiderivatives and Indefinite Integration 5.2 The Definite Integral 5.3 Riemann Sums 5.4 The Fundamental Theorem of Calculus 5.5 Numerical Integration 6 Techniques of Antidifferentiation 6.1 Substitution 6.2 Integration by Parts 6.3 Trigonometric Integrals 6.4 Trigonometric Substitution 6.5 Partial Fraction Decomposition 6.6 Hyperbolic Functions 6.7 L'Hôpital's Rule 6.8 Improper Integration 7 Applications of Integration 7.1 Area Between Curves 7.2 Volume by Cross-Sectional Area; Disk and Washer Methods 7.3 The Shell Method 7.4 Arc Length and Surface Area 7.5 Work 7.6 Fluid Forces 8 Sequences and Series 8.1 Sequences 8.2 Infinite Series 8.3 Integral and Comparison Tests 8.4 Ratio and Root Tests 8.5 Alternating Series and Absolute Convergence 8.6 Power Series 8.7 Taylor Polynomials 8.8 Taylor Series 9 Curves in the Plane 9.1 Conic Sections 9.2 Parametric Equations 9.3 Calculus and Parametric Equations 9.4 Introduction to Polar Coordinates 9.5 Calculus and Polar Functions 10 Vectors 10.1 Introduction to Cartesian Coordinates in Space 10.2 An Introduction to Vectors 10.3 The Dot Product 10.4 The Cross Product 10.5 Lines 10.6 Planes 11 Vector Valued Functions 11.1 Vector–Valued Functions 11.2 Calculus and Vector–Valued Functions 11.3 The Calculus of Motion 11.4 Unit Tangent and Normal Vectors 11.5 The Arc Length Parameter and Curvature 12 Functions of Several Variables 12.1 Introduction to Multivariable Functions 12.2 Limits and Continuity of Multivariable Functions 12.3 Partial Derivatives 12.4 Differentiability and the Total Differential 12.5 The Multivariable Chain Rule 12.6 Directional Derivatives 12.7 Tangent Lines, Normal Lines, and Tangent Planes 12.8 Extreme Values 13 Multiple Integration 13.1 Iterated Integrals and Area 13.2 Double Integration and Volume 13.3 Double Integration with Polar Coordinates 13.4 Center of Mass 13.5 Surface Area 13.6 Volume Between Surfaces and Triple Integration 13.7 Triple Integration with Cylindrical and Spherical Coordinates 14 Vector Analysis 14.1 Introduction to Line Integrals 14.2 Vector Fields 14.3 Line Integrals over Vector Fields 14.4 Flow, Flux, Green's Theorem and the Divergence Theorem 14.5 Parametrized Surfaces and Surface Area 14.6 Surface Integrals 14.7 The Divergence Theorem and Stokes' Theorem A Solutions To Selected Problems Index