Synopses & Reviews
This modern calculus textbook places a strong emphasis on developing students' conceptual understanding and on building connections between key calculus topics and their relevance for the real world. It is written for the average student -- one who is mostly unfamiliar with the subject and who requires significant motivation. It follows a relatively standard order of presentation, with early coverage of transcendentals, and integrates thought-provoking applications, examples and exercises throughout. The text also provides balanced guidance on the appropriate role of technology in problem-solving, including its benefits and its potential pitfalls. Wherever practical, concepts are developed from graphical, numerical, algebraic and verbal perspectives (the "Rule of Four") to give students a complete understanding of calculus.
Table of Contents
0 Preliminaries0.1 Polynomial and Rational Functions 0.2 Graphing Calculators and Computer Algebra Systems0.3 Inverse Functions 0.4 Trigonometric and Inverse Trigonometric Functions0.5 Exponential and Logarithmic Functions0.6 Transformations of Functions0.7 Parametric Equations and Polar Coordinates1 Limits and Continuity1.1 A Brief Preview of Calculus1.2 The Concept of Limit1.3 Computation of Limits1.4 Continuity and its Consequences1.5 Limits Involving Infinity1.6 Limits and Loss-of-Significance Errors 2 Differentiation2.1 Tangent Lines and Velocity2.2 The Derivative2.3 Computation of Derivatives: The Power Rule2.4 The Product and Quotient Rules2.5 The Chain Rule2.6 Derivatives of Trigonometric and Inverse Trigonometric Functions2.7 Derivatives of Exponential and Logarithmic Functions 2.8 Implicit Differentiation2.9 The Mean Value Theorem3 Applications of Differentiation3.1 Linear Approximations and Newtons Method3.2 Indeterminate Forms and L'Hopital's Rule3.3 Maximum and Minimum Values3.4 Increasing and Decreasing Functions3.5 Concavity and Overview of Curve Sketching3.6 Optimization3.7 Rates of Change in Economics and the Sciences3.8 Related Rates and Parametric Equations4 Integration4.1 Area Under a Curve4.2 The Definite Integral4.3 Antiderivatives4.4 The Fundamental Theorem of Calculus4.5 Integration by Substitution4.6 Integration by Parts4.7 Other Techniques of Integration 4.8 Integration Tables and Computer Algebra Systems4.9 Numerical Integration4.10 Improper Integrals5 Applications of the Definite Integral5.1 Area Between Curves5.2 Volume5.3 Arc Length and Surface Area5.4 Projectile Motion5.5 Applications of Integration to Physics and Engineering5.6 Probability6 Differential Equations6.1 Growth and Decay Problems6.2 Separable Differential Equations6.3 Euler's Method6.4 Second Order Equations with Constant Coefficients6.5 Nonhomogeneous Equations: Undetermined Coefficients6.6 Applications of Differential Equations7 Infinite Series7.1 Sequences of Real Numbers7.2 Infinite Series7.3 The Integral Test and Comparison Tests7.4 Alternating Series7.5 Absolute Convergence and the Ratio Test7.6 Power Series7.7 Taylor Series7.8 Applications of Taylor Series7.9 Fourier Series7.10 Power Series Solutions of Differential Equations8 Vectors and the Geometry of Space8.1 Vectors in the Plane8.2 Vectors in Space8.3 The Dot Product8.4 The Cross Product8.5 Lines and Planes in Space8.6 Surfaces in Space9 Vector-Valued Functions9.1 Vector-Valued Functions9.2 Parametric Surfaces9.3 The Calculus of Vector-Valued Functions9.4 Motion in Space9.5 Curvature9.6 Tangent and Normal Vectors10 Functions of Several Variables and Differentiation10.1 Functions of Several Variables10.2 Limits and Continuity10.3 Partial Derivatives10.4 Tangent Planes and Linear Approximations10.5 The Chain Rule10.6 The Gradient and Directional Derivatives10.7 Extrema of Functions of Several Variables10.8 Constrained Optimization and Lagrange Multipliers11 Multiple Integrals11.1 Double Integrals11.2 Area, Volume and Center of Mass11.3 Double Integrals in Polar Coordinates11.4 Surface Area11.5 Triple Integrals11.6 Cylindrical Coordinates11.7 Spherical Coordinates11.8 Change of Variables in Multiple Integrals12 Vector Calculus12.1 Vector Fields12.2 Curl and Divergence 12.3 Line Integrals12.4 Independence of Path and Conservative Vector Fields12.5 Green's Theorem12.6 Surface Integrals12.7 The Divergence Theorem12.8 Stokes' Theorem12.9 Applications of Vector CalculusAppendix A Graphs of Additional Polar Equations Appendix B Formal Definition of LimitAppendix C Complete Derivation of Derivatives of sin x and cos xAppendix D Natural Logarithm Defined as an Integral; Exponential Defined as the Inverse of the Natural LogarithmAppendix E Conic Sections in Polar CoordinatesAppendix F Proofs of Selected Theorems