Calculus

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