Synopses & Reviews
Written by acclaimed author and mathematician George Simmons, this revision is designed for the calculus course offered in two and four year colleges and universities. It takes an intuitive approach to calculus and focuses on the application of methods to real-world problems. Throughout the text, calculus is treated as a problem solving science of immense capability.
Table of Contents
CHAPTER 1: Numbers, Functions, and Graphs 1-1 Introduction 1-2 The Real Line and Coordinate Plane: Pythagoras 1-3 Slopes and Equations of Straight Lines 1-4 Circles and Parabolas: Descartes and Fermat 1-5 The Concept of a Function 1-6 Graphs of Functions 1-7 Introductory Trigonometry 1-8 The Functions Sin O and Cos O CHAPTER 2: The Derivative of a Function 2-0 What is Calculus ? 2-1 The Problems of Tangents 2-2 How to Calculate the Slope of the Tangent 2-3 The Definition of the Derivative 2-4 Velocity and Rates of Change: Newton and Leibriz 2-5 The Concept of a Limit: Two Trigonometric Limits 2-6 Continuous Functions: The Mean Value Theorem and Other Theorem CHAPTER 3: The Computation of Derivatives 3-1 Derivatives of Polynomials 3-2 The Product and Quotient Rules 3-3 Composite Functions and the Chain Rule 3-4 Some Trigonometric Derivatives 3-5 Implicit Functions and Fractional Exponents 3-6 Derivatives of Higher Order CHAPTER 4: Applications of Derivatives 4-1 Increasing and Decreasing Functions: Maxima and Minima 4-2 Concavity and Points of Inflection 4-3 Applied Maximum and Minimum Problems 4-4 More Maximum-Minimum Problems 4-5 Related Rates 4-6 Newtons Method for Solving Equations 4-7 Applications to Economics: Marginal Analysis CHAPTER 5: Indefinite Integrals and Differential Equations 5-1 Introduction 5-2 Differentials and Tangent Line Approximations 5-3 Indefinite Integrals: Integration by Substitution 5-4 Differential Equations: Separation of Variables 5-5 Motion Under Gravity: Escape Velocity and Black Holes CHAPTER 6: Definite Integrals 6-1 Introduction 6-2 The Problem of Areas 6-3 The Sigma Notation and Certain Special Sums 6-4 The Area Under a Curve: Definite Integrals 6-5 The Computation of Areas as Limits 6-6 The Fundamental Theorem of Calculus 6-7 Properties of Definite Integrals CHAPTER 7: Applications of Integration 7-1 Introduction: The Intuitive Meaning of Integration 7-2 The Area between Two Curves 7-3 Volumes: The Disk Method 7-4 Volumes: The Method of Cylindrical Shells 7-5 Arc Length 7-6 The Area of a Surface of Revolution 7-7 Work and Energy 7-8 Hydrostatic Force PART II CHAPTER 8: Exponential and Logarithm Functions 8-1 Introduction 8-2 Review of Exponents and Logarithms 8-3 The Number e and the Function y = e <^>x 8-4 The Natural Logarithm Function y = ln x 8-5 Applications Population Growth and Radioactive Decay 8-6 More Applications CHAPTER 9: Trigonometric Functions 9-1 Review of Trigonometry 9-2 The Derivatives of the Sine and Cosine 9-3 The Integrals of the Sine and Cosine 9-4 The Derivatives of the Other Four Functions 9-5 The Inverse Trigonometric Functions 9-6 Simple Harmonic Motion 9-7 Hyperbolic Functions CHAPTER 10 : Methods of Integration 10-1 Introduction 10-2 The Method of Substitution 10-3 Certain Trigonometric Integrals 10-4 Trigonometric Substitutions 10-5 Completing the Square 10-6 The Method of Partial Fractions 10-7 Integration by Parts 10-8 A Mixed Bag 10-9 Numerical Integration CHAPTER 11: Further Applications of Integration 11-1 The Center of Mass of a Discrete System 11-2 Centroids 11-3 The Theorems of Pappus 11-4 Moment of Inertia CHAPTER 12: Indeterminate Forms and Improper Integrals 12-1 Introduction. The Mean Value Theorem Revisited 12-2 The Interminate Form 0/0. L'Hospital's Rule 12-3 Other Interminate Forms 12-4 Improper Integrals 12-5 The Normal Distribution CHAPTER 13: Infinite Series of Constants 13-1 What is an Infinite Series ? 13-2 Convergent Sequences 13-3 Convergent and Divergent Series 13-4 General Properties of Convergent Series 13-5 Series on Non-negative Terms: Comparison Tests 13-6 The Integral Test 13-7 The Ratio Test and Root Test 13-8 The Alternating Series Test CHAPTER 14: Power Series 14-1 Introduction 14-2 The Interval of Convergence 14-3 Differentiation and Integration of Power Series 14-4 Taylor Series and Taylor's Formula 14-5 Computations Using Taylor's Formula 14-6 Applications to Differential Equations 14. 7 (optional) Operations on Power Series 14. 8 (optional) Complex Numbers and Euler's Formula PART III CHAPTER 15: Conic Sections 15-1 Introduction 15-2 Another Look at Circles and Parabolas 15-3 Ellipses 15-4 Hyperbolas 15-5 The Focus-Directrix-Eccentricity Definitions 15-6 (optional) Second Degree Equations CHAPTER 16: Polar Coordinates 16-1 The Polar Coordinate System 16-2 More Graphs of Polar Equations 16-3 Polar Equations of Circles, Conics, and Spirals 16-4 Arc Length and Tangent Lines 16-5 Areas in Polar Coordinates CHAPTER 17: Parametric Equations 17-1 Parametric Equations of Curves 17-2 The Cycloid and Other Similar Curves 17-3 Vector Algebra 17-4 Derivatives of Vector Function 17-5 Curvature and the Unit Normal Vector 17-6 Tangential and Normal Components of Acceleration 17-7 Kepler's Laws and Newton's Laws of Gravitation CHAPTER 18: Vectors in Three-Dimensional Space 18-1 Coordinates and Vectors in Three-Dimensional Space 18-2 The Dot Product of Two Vectors 18-3 The Cross Product of Two Vectors 18-4 Lines and Planes 18-5 Cylinders and Surfaces of Revolution 18-6 Quadric Surfaces 18-7 Cylindrical and Spherical Coordinates CHAPTER 19: Partial Derivatives 19-1 Functions of Several Variables 19-2 Partial Derivatives 19-3 The Tangent Plane to a Surface 19-4 Increments and Differentials 19-5 Directional Derivatives and the Gradient 19-6 The Chain Rule for Partial Derivatives 19-7 Maximum and Minimum Problems 19-8 Constrained Maxima and Minima 19-9 Laplace's Equation, the Heat Equation, and the Wave Equation 19-10 (optional) Implicit Functions CHAPTER 20: Multiple Integrals 20-1 Volumes as Iterated Integrals 20-2 Double Integrals and Iterated Integrals 20-3 Physical Applications of Double Integrals 20-4 Double Integrals in Polar Coordinates 20-5 Triple Integrals 20-6 Cylindrical Coordinates 20-7 Spherical Coordinates 20-8 Areas of curved Surfaces CHAPTER 21: Line and Surface Integrals 21-1 Green's Theorem, Gauss's Theorem, and Stokes' Theorem 21-2 Line Integrals in the Plane 21-3 Independence of Path 21-4 Green's Theorem 21-5 Surface Integrals and Gauss's Theorem 21-6 Maxwell's Equations : A Final Thought Appendices A: The Theory of Calculus A-1 The Real Number System A-2 Theorems About Limits A-3 Some Deeper Properties of Continuous Functions A-4 The Mean Value theorem A-5 The Integrability of Continuous Functions A-6 Another Proof of the Fundamental Theorem of Calculus A-7 Continuous Curves With No Length A-8 The Existence of e = lim h->0 (1 + h) <^>1/h A-9 Functions That Cannot Be Integrated A-10 The Validity of Integration by Inverse Substitution A-11 Proof of the Partial fractions Theorem A-12 The Extended Ratio Tests of Raabe and Gauss A-13 Absolute vs Conditional Convergence A-14 Dirichlet's Test A-15 Uniform Convergence for Power Series A-16 Division of Power Series A-17 The Equality of Mixed Partial Derivatives A-18 Differentiation Under the Integral Sign A-19 A Proof of the Fundamental Lemma A-20 A Proof of the Implicit Function Theorem A-21 Change of Variables in Multiple Integrals B: A Few Review Topics B-1 The Binomial Theorem B-2 Mathematical Induction