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

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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

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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

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Projects: Rule of 70, Newtonâ ™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 Lâ ™hopitalâ ™s Rule, Growth, and Dominance

4.8 Parametric Equations

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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

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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

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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 Simpsonâ ™s Rule

7.7 Improper Integrals

7.8 Comparison of Improper Integrals

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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

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Projects: Volume Enclosed by Two Cylinders, Length of a Hanging Cable, Surface Area of an Unpaintable Can of Paint, Maxwellâ ™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

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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

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Projects: Shape of Planets, Machinâ ™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 Eulerâ ™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

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Projects: SARS Predictions for Hong Kong, A S-I-R Model for SARS, Paretoâ ™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

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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

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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

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15 OPTIMIZATION: LOCAL AND GLOBAL EXTREMA

15.1 Local Extrema

15.2 Optimization

15.3 Constrained Optimization: Lagrange Multipliers

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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

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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

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Projects: Shooting a Basketball, Keplerâ ™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 Greenâ ™s Theorem

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Projects: Conservation of Energy, Planimeters, Ampère⠙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

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Projects: Gaussâ ™ Law Applied to a Charged Wire and a Charged Sheet, Flux across a Cylinder Due to a Point Charge: Obtaining Gaussâ ™ Law from Coulombâ ™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 Stokesâ ™ Theorem

20.5 The Three Fundamental Theorems

Review Problems

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Projects: Divergence of Spherically Symmetric Vector Fields, Divergence of Cylindrically Symmetric Vector Fields

Appendix

A Roots, Accuracy, and Bounds

B Complex Numbers

C Newtonâ ™s Method

D Vectors in the Plane

E Determinants