Calculus

Chapter 1: Precalculus Review 1.1 Real Numbers, Functions, and Graphs 1.2 Linear and Quadratic Functions 1.3 The Basic Classes of Functions 1.4 Trigonometric Functions 1.5 Technology: Calculators and Computers Chapter Review Exercises Chapter 2: Limits 2.1 The Limit Idea: Instantaneous Velocity and Tangent Lines 2.2 Investigating Limits 2.3 Basic Limit Laws 2.4 Limits and Continuity 2.5 Indeterminate Forms 2.6 The Squeeze Theorem and Trigonometric Limits 2.7 Limits at Infinity 2.8 The Intermediate Value Theorem 2.9 The Formal Definition of a Limit Chapter Review Exercises Chapter 3: Differentiation 3.1 Definition of the Derivative 3.2 The Derivative as a Function 3.3 Product and Quotient Rules 3.4 Rates of Change 3.5 Higher Derivatives 3.6 Trigonometric Functions 3.7 The Chain Rule 3.8 Implicit Differentiation 3.9 Related Rates Chapter Review Exercises Chapter 4: Applications of the Derivative 4.1 Linear Approximation and Applications 4.2 Extreme Values 4.3 The Mean Value Theorem and Monotonicity 4.4 The Second Derivative and Concavity 4.5 Analyzing and Sketching Graphs of Functions 4.6 Applied Optimization 4.7 Newton’s Method Chapter Review Exercises Chapter 5: Integration 5.1 Approximating and Computing Area 5.2 The Definite Integral 5.3 The Indefinite Integral 5.4 The Fundamental Theorem of Calculus, Part I 5.5 The Fundamental Theorem of Calculus, Part II 5.6 Net Change as the Integral of a Rate of Change 5.7 The Substitution Method Chapter Review Exercises Chapter 6: Applications of the Integral 6.1 Area Between Two Curves 6.2 Setting Up Integrals: Volume, Density, Average Value 6.3 Volumes of Revolution: Disks and Washers 6.4 Volumes of Revolution: Cylindrical Shells 6.5 Work and Energy Chapter Review Exercises Chapter 7: Exponential and Logarithmic Functions 7.1 The Derivative of f (x) = bx and the Number e 7.2 Inverse Functions 7.3 Logarithmic Functions and Their Derivatives 7.4 Applications of Exponential and Logarithmic Functions 7.5 L’Hopital’s Rule 7.6 Inverse Trigonometric Functions 7.7 Hyperbolic Functions Chapter Review Exercises Chapter 8: Techniques of Integration 8.1 Integration by Parts 8.2 Trigonometric Integrals 8.3 Trigonometric Substitution 8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions 8.5 The Method of Partial Fractions 8.6 Strategies for Integration 8.7 Improper Integrals 8.8 Numerical Integration Chapter Review Exercises Chapter 9: Further Applications of the Integral 9.1 Probability and Integration 9.2 Arc Length and Surface Area 9.3 Fluid Pressure and Force 9.4 Center of Mass Chapter Review Exercises Chapter 10: Introduction to Differential Equations 10.1 Solving Differential Equations 10.2 Models Involving y=k(y-b) 10.3 Graphical and Numerical Methods 10.4 The Logistic Equation 10.5 First-Order Linear Equations Chapter Review Exercises Chapter 11: Infinite Series 11.1 Sequences 11.2 Summing an Infinite Series 11.3 Convergence of Series with Positive Terms 11.4 Absolute and Conditional Convergence 11.5 The Ratio and Root Tests and Strategies for Choosing Tests 11.6 Power Series 11.7 Taylor Polynomials 11.8 Taylor Series Chapter Review Exercises Chapter 12: Parametric Equations, Polar Coordinates, and Conic Sections 12.1 Parametric Equations 12.2 Arc Length and Speed 12.3 Polar Coordinates 12.4 Area and Arc Length in Polar Coordinates 12.5 Conic Sections Chapter Review Exercises Chapter 13: Vector Geometry 13.1 Vectors in the Plane 13.2 Three-Dimensional Space: Surfaces, Vectors, and Curves 13.3 Dot Product and the Angle Between Two Vectors 13.4 The Cross Product 13.5 Planes in 3-Space 13.6 A Survey of Quadric Surfaces 13.7 Cylindrical and Spherical Coordinates Chapter Review Exercises Chapter 14: Calculus of Vector-Valued Functions 14.1 Vector-Valued Functions 14.2 Calculus of Vector-Valued Functions 14.3 Arc Length and Speed 14.4 Curvature 14.5 Motion in 3-Space 14.6 Planetary Motion According to Kepler and Newton Chapter Review Exercises Chapter 15: Differentiation in Several Variables 15.1 Functions of Two or More Variables 15.2 Limits and Continuity in Several Variables 15.3 Partial Derivatives 15.4 Differentiability, Tangent Planes, and Linear Approximation 15.5 The Gradient and Directional Derivatives 15.6 Multivariable Calculus Chain Rules 15.7 Optimization in Several Variables 15.8 Lagrange Multipliers: Optimizing with a Constraint Chapter Review Exercises Chapter 16: Multiple Integration 16.1 Integration in Two Variables 16.2 Double Integrals over More General Regions 16.3 Triple Integrals 16.4 Integration in Polar, Cylindrical, and Spherical Coordinates 16.5 Applications of Multiple Integrals 16.6 Change of Variables Chapter Review Exercises Chapter 17: Line and Surface Integrals 17.1 Vector Fields 17.2 Line Integrals 17.3 Conservative Vector Fields 17.4 Parametrized Surfaces and Surface Integrals 17.5 Surface Integrals of Vector Fields Chapter Review Exercises Chapter 18: Fundamental Theorems of Vector Analysis 18.1 Green’s Theorem 18.2 Stokes’ Theorem 18.3 Divergence Theorem Chapter Review Exercises Appendices A. The Language of Mathematics B. Properties of Real Numbers C. Induction and the Binomial Theorem D. Additional Proofs ANSWERS TO ODD-NUMBERED EXERCISES REFERENCES INDEX Additional content can be accessed online at www.macmillanlearning.com/calculuset4e: Additional Proofs: L’Hôpital’s Rule Error Bounds for Numerical Integration Comparison Test for Improper Integrals Additional Content: Second-Order Differential Equations Complex Numbers