Essential Calculus: Early Transcendental Functions
Note: Each chapter concludes with Review Exercises. 1. Limits and Their Properties 1.1 Linear Models and Rates of Change 1.2 Functions and Their Graphs 1.3 Inverse Functions 1.4 Exponential and Logarithmic Functions 1.5 Finding Limits Graphically and Numerically 1.6 Evaluating Limits Analytically 1.7 Continuity and One-Sided Limits 1.8 Infinite Limits 2. Differentiation 2.1 The Derivative and the Tangent Line Problem 2.2 Basic Differentiation Rules and Rates of Change 2.3 Product and Quotient Rules and Higher-Order Derivatives 2.4 The Chain Rule 2.5 Implicit Differentiation 2.6 Derivatives of Inverse Functions 2.7 Related Rates 2.8 Newton's Method 3. Applications of Differentiation 3.1 Extrema on an Interval 3.2 Rolle's Theorem and the Mean Value Theorem 3.3 Increasing and Decreasing Functions and the First Derivative Test 3.4 Concavity and the Second Derivative Test 3.5 Limits at Infinity 3.6 Optimization Problems 3.7 Differentials 4. Integration 4.1 Antiderivatives and Indefinite Integration 4.2 Area 4.3 Riemann Sums and Definite Integrals 4.4 The Fundamental Theorem of Calculus 4.5 Integration by Substitution 4.6 Numerical Integration 4.7 The Natural Logarithmic Function: Integration 4.8 Inverse Trigonometric Functions: Integration 4.9 Hyperbolic Functions 5. Applications of Integration 5.1 Area of a Region Between Two Curves 5.2 Volume: The Disk Method 5.3 Volume: The Shell Method 5.4 Arc Length and Surfaces of Revolution 5.5 Applications in Physics and Engineering 5.6 Differential Equations: Growth and Decay 6. Integration Techniques, L'Hopital's Rule, and Improper Integrals 6.1 Integration by Parts 6.2 Trigonometric Integrals 6.3 Trigonometric Substitution 6.4 Partial Fractions 6.5 Integration by Tables and Other Integration Techniques 6.6 Indeterminate Forms and L'Hopital's Rule 6.7 Improper Integrals 7. Infinite Series 7.1 Sequences 7.2 Series and Convergence 7.3 The Integral and Comparison Tests 7.4 Other Convergence Tests 7.5 Taylor Polynomials and Approximations 7.6 Power Series 7.7 Representation of Functions by Power Series 7.8 Taylor and Maclaurin Series 8. Conics, Parametric Equations, and Polar Coordinates 8.1 Plane Curves and Parametric Equations 8.2 Parametric Equations and Calculus 8.3 Polar Coordinates and Polar Graphs 8.4 Area and Arc Length in Polar Coordinates 8.5 Polar Equations and Conics and Kepler's Laws 9. Vectors and the Geometry of Space 9.1 Vectors in the Plane 9.2 Space Coordinates and Vectors in Space 9.3 The Dot Product of Two Vectors 9.4 The Cross Product of Two Vectors in Space 9.5 Lines and Planes in Space 9.6 Surfaces in Space 9.7 Cylindrical and Spherical Coordinates 10. Vector-Valued Functions 10.1 Vector-Valued Functions 10.2 Differentiation and Integration of Vector-Valued Functions 10.3 Velocity and Acceleration 10.4 Tangent Vectors and Normal Vectors 10.5 Arc Length and Curvature 11. Functions of Several Variables 11.1 Introduction to Functions of Several Variables 11.2 Limits and Continuity 11.3 Partial Derivatives 11.4 Differentials and the Chain Rule 11.5 Directional Derivatives and Gradients 11.6 Tangent Planes and Normal Lines 11.7 Extrema of Functions of Two Variables 11.8 Lagrange Multipliers 12. Multiple Integration 12.1 Iterated Integrals and Area in the Plane 12.2 Double Integrals and Volume 12.3 Change of Variables: Polar Coordinates 12.4 Center of Mass and Moments of Inertia 12.5 Surface Area 12.6 Triple Integrals and Applications 12.7 Triple Integrals in Cylindrical and Spherical Coordinates 12.8 Change of Variables: Jacobians 13. Vector Analysis 13.1 Vector Fields 13.2 Line Integrals 13.3 Conservative Vector Fields and Independence of Path 13.4 Green's Theorem 13.5 Parametric Surfaces 13.6 Surface Integrals 13.7 Divergence Theorem 13.8 Stokes's Theorem Appendix A Proofs of Selected Theorems Appendix B Integration Tables Appendix C Business and Economic Applications Answers to Odd-Numbered Exercises Index Additional Appendices Appendix D Precalculus Review D.1 Real Numbers and the Real Number Line D.2 The Cartesian Plane D.3 Review of Trigonometric Functions Appendix E Rotation and General Second-Degree Equation Appendix F Complex Numbers