Calculus: concepts and connections with olc bi-card

The wide-ranging debate brought about by the calculus reform movement has had a significant impact on calculus textbooks. In response to many of the questions and concerns surrounding this debate, the authors have written a modern calculus textbook, intended for students majoring in mathematics, physics, chemistry, engineering and related fields. The text is written for the average student one who does not already know the subject, whose background is somewhat weak in spots, and who requires a significant motivation to study calculus. The authors follow a relatively standard order of presentation, while integrating technology and thought-provoking exercises throughout the text. Some minor changes have been made in the order of topics to reflect shifts in the importance of certain applications in engineering and science. Wherever practical, concepts are developed from graphical, numerical, and algebraic perspectives (the "Rule of Three") to give students a full understanding of calculus. This text places a significant emphasis on problem solving and presents realistic applications, as well as open-ended problems.

Chapter 0: Preliminaries 0.1 Polynomial and Rational Functions 0.2 Graphing Calculators and Computer Algebra Systems 0.3 Inverse Functions 0.4 Trigonometric and Inverse Trigonometric Functions 0.5 Exponential and Logarithmic Functions 0.6 Parametric Equations and Polar Coordinates Chapter 1: Limits and Continuity 1.1 Preview of Calculus 1.2 The Concept of Limit 1.3 Computation of Limits 1.4 Continuity and its Consequences Method of Bisections 1.5 Limits Involving Infinity 1.6 Limits and Loss-of-Significance Errors Chapter 2: Differentiation 2.1 Tangent Lines and Velocity 2.2 The Derivative 2.3 Computation of Derivatives: The Power Rule 2.4 The Product and Quotient Rules 2.5 The Chain Rule 2.6 Derivatives of Trigonometric and Inverse Trigonometric Functions 2.7 Derivatives of Exponential and Logarithmic Functions 2.8 Implicit Differentiation and Related Rates 2.9 The Mean Value Theorem Chapter 3: Applications of Differentiation 3.1 Linear Approximations and Newton's Method 3.2 Indeterminate Forms and L'Hopital's Rule 3.3 Maximum and Minimum Values 3.4 Increasing and Decreasing Functions 3.5 Concavity and Overview of Curve Sketching 3.6 Optimization 3.7 Rates of Change in Applications Chapter 4: Integration 4.1 Area under a Curve 4.2 The Definite Integral Average Value of a Function 4.3 Antiderivatives 4.4 The Fundamental Theorem of Calculus 4.5 Integration by Substitution Trigonometric Techniques of Integration 4.6 Integration by Parts 4.7 Other Techniques of Integration 4.8 Integration Tables and Computer Algebra Systems 4.9 Numerical Integration 4.10 Improper Integrals Comparison Test Chapter 5: Applications of the Definite Integral 5.1 Area Between Curves 5.2 Volume Slicing, Disks and Washers 5.3 Arc Length and Surface Area 5.4 Projectile Motion 5.5 Work, Moments, and Hydrostatic Force 5.6 Probability Chapter 6: Differential Equations 6.1 Growth and Decay Problems 6.2 Separable Differential Equations 6.3 Euler's Method 6.4 Second Order Equations with Constant Coefficients 6.5 Nonhomogeneous Equations: Undetermined Coefficients 6.6 Applications of Differential Equations Chapter 7: Infinite Series 7.1 Sequences of Real Numbers 7.2 Infinite Series 7.3 The Integral Test and Comparison Tests 7.4 Alternating Series 7.5 Absolute Convergence and the Ratio Test 7.6 Power Series 7.7 Taylor Series Taylor's Theorem 7.8 Applications of Taylor Series 7.9 Fourier Series 7.10 Power Series Solutions of Differential Equations Chapter 8: Vectors and the Geometry of Space 8.1 Vectors in the Plane 8.2 Vectors in Space 8.3 The Dot Product Components and Projections 8.4 The Cross Product 8.5 Lines and Planes in Space 8.6 Surfaces in Space Chapter 9: Vector-Valued Functions 9.1 Vector-Valued Functions 9.2 Parametric Surfaces 9.3 The Calculus of Vector-Valued Functions 9.4 Motion in Space 9.5 Curvature 9.6 Tangent and Normal Vectors Components of Acceleration, Kepler's Laws Chapter 10: Functions of Several Variables and Differentiation 10.1 Functions of Several Variables 10.2 Limits and Continuity 10.3 Partial Derivatives 10.4 Tangent Planes and Linear Approximations 10.5 The Chain Rule Implicit Differentiation 10.6 The Gradient and Directional Derivatives 10.7 Extrema of Functions of Several Variables 10.8 Constrained Optimization and Lagrange Multipliers Chapter 11: Multiple Integrals 11.1 Double Integrals 11.2 Area, Volume and Center of Mass 11.3 Double Integrals in Polar Coordinates 11.4 Surface Area 11.5 Triple Integrals 11.6 Cylindrical Coordinates 11.7 Spherical Coordinates 11.8 Change of Variables in Multiple Integrals Chapter 12: Vector Calculus 12.1 Vector Fields 12.2 Curl and Divergence 12.3 Line Integrals 12.4 Independence of Path and Conservative Vector Fields 12.5 Green's Theorem 12.6 Surface Integrals Parametric Representation of Surfaces 12.7 The Divergence Theorem 12.8 Stokes' Theorem 12.9 Applications of Vector Calculus Appendices A.1 Formal Definition of Limit A.2 Complete Derivation of Derivatives of sin x and cos x A.3 Natural Logarithm Defined as an Integral, Exponential Defined as the Inverse of the Natural Logarithm A.4 Hyperbolic Functions A.5 Conic Sections in Polar Coordinates A.6 Proofs of Selected Theorems