Calculus

Calculus is the basic mathematics course studied by all students of scienceas well as many other disciplines. The applications of calculus to solve practicaland theoretical problems started with Isaac Newton who first applied its greatcomputational power to describe the motion of the planets. Applications ofcalculus are found today in areas ranging from high energy physics to molecularbiology and from ecology and the environment to economics and finance; itis generally regarded as the introduction to the study of higher mathematics.With such a wide variety of interests in the students of today, it is impossible toproduce a text that caters to the special needs of every interest group. Instead thistext provides a solid foundation for understanding the subject, the broad rangeof its applicability and the techniques necessary to apply the subject to specificproblems. Front Cover Title Page Copyright Page Dedication Contents Preface 1 Preparations for Calculus 1.1 Real Numbers and Sets 1.2 Inequalities and Absolute Values 1.3 Coordinates 1.4 Lines and Slopes 1.5 Circles and Parabolas 1.6 The Ellipse and Hyperbola 1.7 Polar Coordinates 1.8 Language 2 Functions and Limits 2.1 Examples of Functions 2.2 Graphs of Functions 2.3 Composite Functions 2.4 Bounds and Limits 2.5 Limits of Functions 2.6 Limit Theorems 2.7 Limits at Infinity 2.8 pi and the Circle 2.9 Trigonometric Functions 3 Continuity and the Derivative 3.1 Continuous Functions 3.2 Definition of Derivative 3.3 Product and Quotient Rules 3.4 Derivatives of Trigonometric Functions 3.5 The Chain Rule 3.6 Derivatives of Implicit Functions 4 Applications of the Derivative 4.1 Extrema of Functions 4.2 The Mean Value Theorem 4.3 Applications of the Mean Value Theorem 4.4 Indeterminate Forms 4.5 Maximum and Minimum Function Values 4.6 More Maximum and Minimum Problems 4.7 The Second Derivative 4.8 Concavity 4.9 Velocity and Acceleration 4.10 Motion in the Plane 4.11 Related Rates 5 The Definite Integral 5.1 Summation Notation 5.2 Lower and Upper Sums 5.3 Definition of the Definite Integral 5.4 Proof of the Existence of the Definite Integral 5.5 Riemann Sums 5.6 Proof of the Fundamental Theorem of Calculus 5.7 Computation of Areas 5.8 Indefinite Integrals 5.9 Integration by Substitution 5.10 Areas in Polar Coordinates 6 Computations Using the Definite Integral 6.1 Volumes of Certain Solids 6.2 Work 6.3 Arc Length 6.4 Surface Area of Solids of Revolution 6.5 Moments and Center of Mass 7 Transcendental Functions 7.1 Inverse Functions 7.2 The Natural Logarithm Function 7.3 Derivatives and Integrals Involving ln(x) 7.4 The Exponential Function 7.5 The Derivative of the Exponential Function 7.6 Applications of the Exponential Function 7.7 Indeterminate Forms 7.8 Inverse Trigonometric Functions 7.9 Derivatives of the Inverse Trigonometric Functions 7.10 Improper Integrals 8 Methods of Integration 8.1 Integral Formulas and Integral Tables 8.2 Integration by Parts 8.3 Reduction Formulas 8.4 Partial Fraction Decomposition of Rational Functions 8.5 Integration of Rational Functions 8.6 Integrals of Algebraic Functions 8.7 Trigonometric and Other Substitutions 9 Taylor Polynomials and Sequences 9.1 Taylor Polynomials 9.2 Sequences 9.3 Limits of Sequences 10 Power Series 10.1 Taylor Series 10.2 Convergence of Infinite Series 10.3 The Interval of Convergence 10.4 Differentiation and Integration of Series 10.5 Computation of Taylor Series 10.6 Applications of Power Series 10.7 Additional Convergence Tests 10.8 Alternating Series and Conditional Convergence 10.9 The Hyperbolic and Binomial Series 11 Numerical Computations 11.1 Solution of Equations 11.2 Numerical Integration 11.3 Simpson's Rule 12 Vectors in Two and Three Dimensions 12.1 Coordinates in Three Dimensions 12.2 Vectors 12.3 Coordinates for Vectors in Three Dimensions 12.4 The Dot Product 12.5 The Cross Product 12.6 Equations of Lines 12.7 Equations of Planes 13 Vector Functions 13.1 Vector Functions 13.2 Integral of a Vector Function 13.3 Curves in Parametric and Vector Form 13.4 Tangents and Normals to Curves 13.5 Arc Length in Two and Three Dimensions 13.6 Curvature 13.7 Curves in Three Dimensions 13.8 Motion Along a Curve 13.9 Planetary Motion 14 Partial Derivatives 14.1 Surfaces in Three Dimensions 14.2 Quadric Surfaces 14.3 Functions of Several Variables 14.4 Partial Derivatives 14.5 Limits and Continuity 14.6 The Chain Rule 14.7 The Gradient 14.8 Directional Derivatives 14.9 Extrema of Functions of Several Variables 14.10 Extrema With Constraints 14.11 Properties of Continuous Functions 15 Integration in Higher Dimensions 15.1 Double Integrals 15.2 Iterated Integrals 15.3 More Volumes Using Double Integrals 15.4 Center of Mass and Moments of Inertia 15.5 Surface Area 15.6 Triple Integrals 15.7 Line Integrals 15.8 Path-Independent Line Integrals 15.9 Green's Theorem 15.10 Change of Variables 15.11 Triple Integrals by Spherical Coordinates 16 Two Theorems in Vector Calculus 16.1 Oriented Surfaces and Stokes' Theorem 16.2 The Curl and Divergence 16.3 The Divergence Theorem Appendix A: Answers to Selected Exercises Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Index Back Cover Discusses calculus, placing emphasis on proofs of many theories and on the interpretation of graphs rather than their production. The book contains a complete chapter on numerical computations. Numerical integration is also discussed with an emphasis on estimation of errors.