Calculus

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1 A LIBRARY OF FUNCTIONS 1.1 Functions and Change 1.2 Exponential Functions 1.3 New Functions from Old 1.4 Logarithmic Functions 1.5 Trigonometric Functions 1.6 Powers, Polynomials, and Rational Functions 1.7 Introduction to Continuity 1.8 Limits Review Problems Check Your Understanding Projects: Matching Functions to Data, Which Way Is the Wind Blowing? 2 KEY CONCEPT: THE DERIVATIVE 2.1 How Do We Measure Speed? 2.2 The Derivative at a Point 2.3 The Derivative Function 2.4 Interpretations of the Derivative 2.5 The Second Derivative 2.6 Differentiability Review Problems Check Your Understanding Projects: Hours of Daylight as a Function of Latitude, US Population 3 SHORT-CUTS TO DIFFERENTIATION 3.1 Powers and Polynomials 3.2 The Exponential Function 3.3 The Product and Quotient Rules 3.4 The Chain Rule 3.5 The Trigonometric Functions 3.6 The Chain Rule and Inverse Functions 3.7 Implicit Functions 3.8 Hyperbolic Functions 3.9 Linear Approximation and the Derivative 3.10 Theorems about Differentiable Functions Review Problems Check Your Understanding Projects: Rule of 70, Newtona s Method 4 USING THE DERIVATIVE 4.1 Using First and Second Derivatives 4.2 Optimization 4.3 Families of Functions 4.4 Optimization, Geometry, and Modeling 4.5 Applications to Marginality 4.6 Rates and Related Rates 4.7 La hopitala s Rule, Growth, and Dominance 4.8 Parametric Equations Review Problems Check Your Understanding Projects: Building a Greenhouse, Fitting a Line to Data, Firebreaks 5 KEY CONCEPT: THE DEFINITE INTEGRAL 5.1 How Do We Measure Distance Traveled? 5.2 The Definite Integral 5.3 The Fundamental Theorem and Interpretations 5.4 Theorems about Definite Integrals Review Problems Check Your Understanding Projects: The Car and the Truck, An Orbiting Satellite 6 CONSTRUCTING ANTIDERIVATIVES 6.1 Antiderivatives Graphically and Numerically 6.2 Constructing Antiderivatives Analytically 6.3 Differential Equations 6.4 Second Fundamental Theorem of Calculus 6.5 The Equations of Motion Review Problems Check Your Understanding Projects: Distribution of Resources, Yield from an Apple Orchard, Slope Fields 7 INTEGRATION 7.1 Integration by Substitution 7.2 Integration by Parts 7.3 Tables of Integrals 7.4 Algebraic Identities and Trigonometric Substitutions 7.5 Approximating Definite Integrals 7.6 Approximation Errors and Simpsona s Rule 7.7 Improper Integrals 7.8 Comparison of Improper Integrals Review Problems Check Your Understanding Projects: Taylor Polynomial Inequalities 8 USING THE DEFINITE INTEGRAL 8.1 Areas and Volumes 8.2 Applications to Geometry 8.3 Area and Arc Length in Polar Coordinates 8.4 Density and Center of Mass 8.5 Applications to Physics 8.6 Applications to Economics 8.7 Distribution Functions 8.8 Probability, Mean, and Median Review Problems Check Your Understanding Projects: Volume Enclosed by Two Cylinders, Length of a Hanging Cable, Surface Area of an Unpaintable Can of Paint, Maxwella s Distribution of Molecular Velocities 9 SEQUENCES AND SERIES 9.1 Sequences 9.2 Geometric Series 9.3 Convergence of Series 9.4 Tests for Convergence 9.5 Power Series and Interval of Convergence Review Problems Check Your Understanding Projects: A Definition of e, Probability of Winning in Sports, Prednisone 10 APPROXIMATING FUNCTIONS USING SERIES 10.1 Taylor Polynomials 10.2 Taylor Series 10.3 Finding and Using Taylor Series 10.4 The Error in Taylor Polynomial Approximations 10.5 Fourier Series Review Problems Check Your Understanding Projects: Shape of Planets, Machina s Formula and the Value of pi, Approximation the Derivative 11 DIFFERENTIAL EQUATIONS 11.1 What Is a Differential Equation? 11.2 Slope Fields 11.3 Eulera s Method 11.4 Separation of Variables 11.5 Growth and Decay 11.6 Applications and Modeling 11.7 The Logistic Model 11.8 Systems of Differential Equations 11.9 Analyzing the Phase Plane 11.10 Second-Order Differential Equations: Oscillations 11.11 Linear Second-Order Differential Equations Review Problems Check Your Understanding Projects: SARS Predictions for Hong Kong, A S-I-R Model for SARS, Paretoa s Law, Vibrations in a Molecule 12 FUNCTIONS OF SEVERAL VARIABLES 12.1 Functions of Two Variables 12.2 Graphs of Functions of Two Variables 12.3 Contour Diagrams 12.4 Linear Functions 12.5 Functions of Three Variables 12.6 Limits and Continuity Review Problems Check Your Understanding Projects: A Heater in a Room, Light in a Wave-Guide 13 A FUNDAMENTAL TOOL: VECTORS 13.1 Displacement Vectors 13.2 Vectors in General 13.3 The Dot Product 13.4 The Cross Product Review Problems Check Your Understanding Projects: Cross Product of Vectors in the Plane, The Dot Product in Genetics, A Warren Truss 14 DIFFERENTIATING FUNCTIONS OF SEVERAL VARIABLES 14.1 The Partial Derivative 14.2 Computing Partial Derivatives Algebraically 14.3 Local Linearity and the Differential 14.4 Gradients and Directional Derivatives in the Plane 14.5 Gradients and Directional Derivatives in Space 14.6 The Chain Rule 14.7 Second-Order Partial Derivatives 14.8 Differentiability Review Problems Check Your Understanding 15 OPTIMIZATION: LOCAL AND GLOBAL EXTREMA 15.1 Local Extrema 15.2 Optimization 15.3 Constrained Optimization: Lagrange Multipliers Review Problems Check Your Understanding Projects: Optimization in Manufacturing, Fitting a Line to Data Using Least Squares, Hockey and Entropy 16 INTEGRATING FUNCTIONS OF SEVERAL VARIABLES 16.1 The Definite Integral of a Function of Two Variables 16.2 Iterated Integrals 16.3 Triple Integrals 16.4 Double Integrals in Polar Coordinates 16.5 Integrals in Cylindrical and Spherical Coordinates 16.6 Applications of Integration to Probability 16.7 Change of Variables in a Multiple Integral Review Problems Check Your Understanding Projects: A Connection Between e and pi, Average Distance Walked to an Airport Gate 17 PARAMETERIZATION AND VECTOR FIELDS 17.1 Parameterized Curves 17.2 Motion, Velocity, and Acceleration 17.3 Vector Fields 17.4 The Flow of a Vector Field 17.5 Parameterized Surfaces Review Problems Check Your Understanding Projects: Shooting a Basketball, Keplera s Second Law, Flux Diagrams 18 LINE INTEGRALS 18.1 The Idea of a Line Integral 18.2 Computing Line Integrals Over Parameterized Curves 18.3 Gradient Fields and Path-Independent Fields 18.4 Path-Dependent Vector Fields and Greena s Theorem Review Problems Check Your Understanding Projects: Conservation of Energy, Planimeters, AmpA..rea s Law 19 FLUX INTEGRALS 19.1 The Idea of a Flux Integral 19.2 Flux Integrals for Graphs, Cylinders, and Spheres 19.3 Flux Integrals Over Parameterized Surfaces Review Problems Check Your Understanding Projects: Gaussa Law Applied to a Charged Wire and a Charged Sheet, Flux across a Cylinder Due to a Point Charge: Obtaining Gaussa Law from Coulomba s Law 20 CALCULUS OF VECTOR FIELDS 20.1 The Divergence of a Vector Field 20.2 The Divergence Theorem 20.3 The Curl of a Vector Field 20.4 Stokesa Theorem 20.5 The Three Fundamental Theorems Review Problems Check Your Understanding Projects: Divergence of Spherically Symmetric Vector Fields, Divergence of Cylindrically Symmetric Vector Fields Appendix A Roots, Accuracy, and Bounds B Complex Numbers C Newtona s Method D Vectors in the Plane E Determinants A

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