Calculus With Analytic Geometry 2nd edition

【实例简介】
MIT的单变量微积分课程的教材，强烈推荐，第二版。GEORGE F. SIMMONS Simmons, George F. Calculus with Analytic Geometry. 2nd ed. New York, NY: McGraw-Hill, October 1, 1995. ISBN: 0070576424. Readings in the textbook are listed by section numbers (e.g., § 2.1-2.4 means read sections 2.1 through 2.4.)
ABOUT THE AUTHOR George F. Simmons has the usual academic degrees(CalTech, Chicago, Yale) and taught at several colleges and universities before joining the faculty of Col orado College in 1962. He is also the author of Introduction to Topology and Modern Analysis (McGraw-Hill, 1963), Differential Equations with applications and Historical Notes(McGraw-Hill, 1972, 2nd edition 1991), Precalculus Math ematics in a Nutshell(Janson Publications, 1981), and Calculus Gems: Brief Lives and Memorable Mathematics(McGraw-Hill, 1992) When not working or talking or eating or drinking or cooking, Professor Sim- mons is likely to be traveling (Western and Southern Europe, Turkey, Israel Egypt, Russia, China, Southeast Asia), trout fishing (Rocky Mountain states) playing pocket billiards, or reading (literature, history, biography and autobiog- raphy, science, and enough thrillers to achieve enjoyment without guilt. One of his personal heroes is the older friend who once said to him, "I should probably spend about an hour a week revising my opinions For My Grandson Nicky- without young people to continue to wonder and care and study and learn, it's all over With all humility, I think "Whatsoever thy hand findeth to do, do it with thy might"infi nitely more important than the vain attempt to love one's neighbor as oneself. If you want to hit a bird on the wing, you must have all your will in a focus: you must not be think ing about yourself, and, equally, you must not be thinking about your neighbor: you must be living in your eye on that bird. Every achievement is a bird on the wing Oliver Wendell Holmes bring forth what is within you, what you do not bring forth will destroy you wou do not If you bring forth what is within you, what you bring forth will save you. If Jesus. The Gospel of ThoMas in the Nag Hammadi manuscripts The more I work and practice, the luckier I seem to get Gary player professional golfer) A witty chess master once said that the difference between a master and a beginning chess player is that the beginner has everything clearly fixed in mind, while to the master every thing is a myster N Ia. vilenkin Marshalls Generalized Iceberg Theorem: Seven-eighths of everything can't be seen Everything should be made as simple as possible, but not simpler. Albert first CONTENTS Preface to the Instructor To the student XLll PART I 11 Introduction NUMBERS, FUNCTIONS, 1. 2 The Real Line and Coordinate Plane. Pythagoras AND GRAPHS 1.3 Slopes and Equations of Straight Lines L 4 Circles and Parabolas. Descartes and fermat 1.5 The Concept of a Function 22 1. 6 Graphs of Functions 30 1. 7 Introductory Trigonometry. The Functions sin A and cos e 37 Review: Definitions, Concepts, Methods 46 Additional Problems 47 2 2.1 What is Calculus? The Problem of Tangents 51 THE DERIVATIVE OF A 2.2 How to Calculate the Slope of the Tangent 53 funcTIon 2 3 The Definition of the Derivative 58 2. 4 Velocity and Rates of Change. Newton and Leibniz 62 2.5 The Concept of a Limit. Two Trigonometric Limits 2.6 Continuous Functions. The Mean value 'Theorem and Other theorems 74 Review: Definitions, Concepts. Methods 81 Additional problems 3 3.1 Derivatives of polynomials "THE COMPUTATION OF 3.2 The Product and Quotient Rules 88 DERIVATIVES 3.3 Composite Functions and the Chain Rule 92 3.4 Some Trigonometric Derivatives 3.5 Implicit Functions and Fractional Exponents 102 3.6 Derivatives of Higher Order 107 Review: Concepts, Formulas, Methods 111 Additional Problems 111 4.1 Increasing and Decreasing Functions. Maxima and Minima 115 APPLICATIONS OF 4.2 Concavity and Points of Inflection 120 DERIVATIVES 4.3 Applied Maximum and Minimum Problems 123 4.4 More Maximum-Minimum Problems. Reflection and Refraction 13I 45 Related Rates CONTENTS 4.6 Newtons Method for Solving Equations 143 4.7(Optional) Applications to Economics. Marginal Analysis 46 Review: Concepts, Methods Additional Problens 156 D 5.1 Introduction 163 INDEFINITE INTEGRAL- 5.2 Differentials and Tangent Line Approximations 163 AND) DIFFERENTIAL. 5.3 Indefinite Integrals. Integration by Substitution 170 EQUATIONS 5.4 Differential Equations, Separation of variables 178 5.5 Motion under Gravity. Escape Velocity and Black Holes 181 Review: Concepts, Methods 88 Additional Problems 188 6.1 Introduction 190 DEFINITE INTEGRALS 6.2 The Problem of Areas 191 6. 3 The Sigma Notation and Certain Special Sums 194 6.4 The Area under a Curve, definite Integrals. riemann 197 6.5 The Computation of Areas as Limits 203 6. 6 The Fundamental Theorem of Calculus 206 6. 7 Properties of Definite Integrals 213 Review: Concepts, Methods. 217 Additional Problems 217 Appendix: The Lunes of Hippocrates 218 7.1 Introduction. The Intuitive Meaning of Integration 221 APPLICATIONS OF 7.2 The Area between Two Curves 222 INTEGRATION 7.3 Volumes: The Disk Method 225 7.4 Volumes: The Method of Cylindrical Shells 231 7.5 Are Length 7. 6 The Area of a Surface of Revolution 240 7. 7 Work and Energy 244 7.8 Hydrostatic Force Review: Concepts, Methods 254 Additional Probles 254 Appendix: Archimedes and the Volume of a Sphere 257 PART II 8 8.1 Introduction 260 EXPONENTIAL AND) 8. 2 Review of Exponents and Logarithms 261 LOGARITHM FUNCTIONS 8.3 The number e and the Function y= er 264 8.4 The Natural Logarithm Function y= In x. Euler 269 8.5 Applications. Population Growth and Radioactive Decay 8.6 More Applications. Inhibited Population Growth, etc 283 Review: Concepts, Formulas 287 Additional problems 288 9 9. Review of trigonometry 292 TRIGONOMETRIC. 9.2 The Derivatives of the Sine and Cosine 301 FUNCTIONS 9.3 The Integrals of the Sine and Cosine. The Needle Problem 306 CONTENTS IX 9. 4 The derivatives of the Other four functions 310 9.5 The Inverse ' Trigonometric Functions 313 9.6 Simple Harmonic Motion. The Pendulum 319 9.7 (Optional) Hyperbolic Functions 324 Review: Definitions, Formulas 330 Additional problems 10 10.1 Introduction. The Basic formulas METHOD)S OF 10.2 The Method of Substitution 337 INTEGRATION 10.3 Certain Trigonometric Integrals 340 10,4 Trigonometric substitutions 10.5 Completing the Square 348 10.6 The Method ot partial fractions 10.7 Integration by p 357 10.8 A Mixed Bag. Strategy for Dealing with Integrals of Miscellaneous Types 362 10.9 Numerical Integration, Simpsons Rule Review: Formulas. Methods 375 Additional Problerns 375 Appendix 1: The Catenary, or Curve of a Hanging Chain 378 Appendix2: Wallis 's product=}·3·3·t·3,号 380 Appendix 3: How Leibniz Discovered His Formula +3-}+ 382 11 11.1 The Center of Mass of a Discrete System 384 FURTHER APPLICATIONS OF 11.2 Centroids 386 INTEGRATION 11.3 The Theorems of Pappus 391 I 1.4 Moment of inertia 393 Review: Definitions, Concepts 396 Additional probles 396 12 12.1 Introduction. The Mean Value Theorem Revisited 398 INDETERMINATE FORMS 12.2 The Indeterminate Form O/0 L Hospital,s rule 400 AND) IMPROPER INTEGRALS 12. 3 Other Indeterminate Forms 404 12.4 Improper Integrals 409 12.5 The Normal Distribution. Gauss 414 Review: Definitions. Concepts 424 Additional problems 424 13 13. 1 What Is an Infinite Series? 427 INFINITE SERIES OF 13.2 Convergent Sequences 432 CONSTANTS 13.3 Convergent and Divergent Series 439 13.4 General Properties of Convergent Series 445 3.5 Series of Nonnegative Terms. Comparison Tests 13.6 The Integral Test. Euler's Constant 455 13.7 The Ratio Test and Root Test 46l 13.8 The Alternating Series Test. Absolute Convergence 465 Review: Definitions. Concepts, Tests 470 Additional problems 470 X CONTENTS Appendix 1: Euler's Discovery of the Formula 2i n2 6 476 Appendix 2: More about Irrational Numbers, T Is Irrationa! 78 Appendix 3: The Series 2l/pn of the Reciprocals of the Primes 480 14 14.1 Introduction 483 POWER SERIES 14.2 The Interval of Convergence 484 14.3 Differentiation and Integration of Power Series 14.4 Taylor Series and Taylors Formula 494 14.5 Computations Using Taylor's Formula 504 14.6 Applications to Differential Equations 509 14.7 (Optional) Operations on Power Series 514 14.8 (Optional) Complex Numbers and Euler's Formula 521 Review: Concepts, Formulas, Methods 523 Additional Problems 523 Appendix: The Bernoulli Numbers and Some Wonderful Discoveries of euler 525 PART III 15 15.1 Introduction, Sections of a Cone CONIC SECTIONS 15.2 Another Look at Circles and Parabolas 531 15.3 Ellipses 5.4 Hyperbolas 543 15.5 The Focus-Directrix- Eccentricity Definitions 550 15.6 (Optional) Second-Degree Equations. Rotation of Axes 552 Review: Definitions, Properties 557 Additional problems 558 16 16. The Polar Coordinate System 560 POLAR COORDINATES 16.2 More Graphs of Polar Equations 5 16.3 Polar Equations of Circles, Conics, and Spirals 16.4 Are Length and Tangent Lines 575 16.5 Areas in Polar Coordinates 580 Review: Concepts, Formulas 583 Additional problems 583 17.1 Parametric Equations of Curves 86 PARAMETRIC EQUATIONS. 17.2 The Cycloid and Other Similar Curves 592 VECTORS IN THE PLANE 17.3 Vector Algebra. The Unit Vectors i and j 17.4 Derivatives of vector Functions. velocity and Acceleration 605 17.5 Curvature and the Unit Normal vector 611 17.6 Tangential and Normal Components of Acceleration 615 17. 7 Kepler's Laws and Newton's Law of Gravitation 619 Review: Concepts, Formulas 627 Additional Problems 627 Appendix: Bernoulli's Solution of thte Brachistochrone Problen 629 CONTENTS XI 18 18.1 Coordinates and Vectors in Three-Dimensional Space 632 VECTORS IN THREE- 18.2 The Dot Product of two vector 636 DIMENSIONAL SPACE 8.3 The Cross product of Two vectors 640 SURFACES 18.4 Lines and planes 46 18.5 Cylinders and Surfaces of Revolution 653 18.6 Quadric Surfaces 656 18 7 Cylindrical and Spherical Coordinates 66l Review: Definitions, Equations 664 19.1 Functions of Several variables 665 PARTIAL DERIVATIVES 9.2 Partial derivatives 669 19.3 The Tangent Plane to a Surface 675 19.4 Increments and Differentials. The Fundamental Lemma 679 19.5 Directional Derivatives and the gradient 681 19.6 The Chain Rule for Partial Derivatives 686 19.7 Maximum and Minimum problems 692 19.8 Constrained Maxima and Minima. Lagrange Multipliers 696 19.9(Optional) Laplace's Equation, the Heat Equation, and the Wave Equation. Laplace and Fourier 702 19. 10 (Optional) Implicit Functions 708 Review: Definitions. Methods 713 20 20.1 Volumes as Iterated Integrals 714 MULTIPLE INTEGRALS 20.2 Double Integrals and Iterated Integrals 718 20.3 Physical Applications of Double Integrals 722 20. 4 Double Integrals in Polar Coordinates 726 20.5 Triple Integrals 73l 20.6 Cylindrical Coordinates 736 20.7 Spherical Coordinates. Gravitational Attraction 739 201.8 Areas of Curved Surfaces. Legendres Formula 744 Review: methods, formulas 748 Appendix: Euler's Formula 21= by Double Integration 748 21 21.1 Line Integrals in the Plane 751 LINE AND SURFACE 21.2 Independence of Path. Conservative Fields 758 INTEGRALS GREENS 213 Green,s Theorem 764 THEOREM, GAUSS'S 21.4 Surface Integrals and Gauss's Theorem 771 THEOREM. AND STOKES 21.5 Stokes Theorem 778 THEOREM 21.6 Maxwell's Equations A Final Thought 784 Review: Concepts, Theorems 786 APPENDICES A The Theory of Calculus 787 A. The Real Number System 787 A 2 Theorems about Limits 791 A.3 Some Deeper Properties of Continnous Functions 796 A.4 The mean value theore 800 A. 5 The Integrability of Continuous FunctiOns 804 A, 6 Another Proof of the Fundamental Theorem of Calculus 808 A7 Continuous Curves with No Length 808 A.8 The Existence of e=lim(1 +h)I/h 8l1 X11 CONTENTS A9 Functions That Cannot Be Integrated A10 The Validity of Integration by Inverse Substitution 817 A.1 Proof of the Partial Fractions Theorem 818 A12 The Extended Ratio Tests of Raabe and Gauss 821 A13 Absolute vs Conditional Convergence 825 A. 4 Dirichlet's Test. Dirichlet 830 A15 Uniform Convergence for Power Series 834 A16 Division of Power Series 836 A.7 The Equality of Mixed Partial Derivatives 837 AI8 Differentiation under the integral sign 838 A19 A Proof of the Fundamental Lemma 839 A20 A Proof of the Implicit Function Theorem 840 A21 Change of Variables in Multiple Integrals. Jacobians 84 B A Few Review Topics B. The binomial theore 845 B 2 Mathematical induction 849 Answers 856 Hde 877 【实例截图】
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