Efnisyfirlit

• Cover
• Title
• Copyright
• Dedication
• About the Authors
• Brief Table of Contents
• Table of Contents
• Seeing the Beauty and Power of Mathematics
• Applications Index
• Chapter 0: Preliminaries
  • 0.1 Polynomials and Rational Functions
    • The Real Number System and Inequalities
    • Equations of Lines
    • Functions
  • 0.2 Graphing Calculators and Computer Algebra Systems
  • 0.3 Inverse Functions
  • 0.4 Trigonometric and Inverse Trigonometric Functions
    • The Inverse Trigonometric Functions
  • 0.5 Exponential and Logarithmic Functions
    • Hyperbolic Functions
    • Fitting a Curve to Data
  • 0.6 Transformations of Functions
• Chapter 1: Limits and Continuity
  • 1.1 A Brief Preview of Calculus: Tangent Lines and the Length of a Curve
  • 1.2 The Concept of Limit
  • 1.3 Computation of Limits
  • 1.4 Continuity and its Consequences
    • The Method of Bisections
  • 1.5 Limits Involving Infinity; Asymptotes
    • Limits at Infinity
  • 1.6 Formal Definition of the Limit
    • Exploring the Definition of Limit Graphically
    • Limits Involving Infinity
  • 1.7 Limits and Loss-of-Significance Errors
    • Computer Representation of Real Numbers
• Chapter 2: Differentiation
  • 2.1 Tangent Lines and Velocity
    • The General Case
    • Velocity
  • 2.2 The Derivative
    • Alternative Derivative Notations
    • Numerical Differentiation
  • 2.3 Computation of Derivatives: The Power Rule
    • The Power Rule
    • General Derivative Rules
    • Higher Order Derivatives
    • Acceleration
  • 2.4 The Product and Quotient Rules
    • Product Rule
    • Quotient Rule
    • Applications
  • 2.5 The Chain Rule
    • Concept Mapping
  • 2.6 Derivatives of Trigonometric Functions
    • Applications
  • 2.7 Derivatives of Exponential and Logarithmic Functions
    • Derivatives of the Exponential Functions
    • Derivative of the Natural Logarithm
    • Logarithmic Differentiation
  • 2.8 Implicit Differentiation and Inverse Trigonometric Functions
    • Derivatives of the Inverse Trigonometric Functions
  • 2.9 The Hyperbolic Functions
    • The Inverse Hyperbolic Functions
  • 2.10 The Mean Value Theorem
• Chapter 3: Applications of Differentiation
  • 3.1 Linear Approximations and Newton’s Method
    • Linear Approximations
    • Newton’s Method
  • 3.2 Indeterminate Forms and L’Hôpital’s Rule
    • Other Indeterminate Forms
  • 3.3 Maximum and Minimum Values
  • 3.4 Increasing and Decreasing Functions
    • What You See May Not Be What You Get
  • 3.5 Concavity and the Second Derivative Test
    • Concept Mapping
  • 3.6 Overview of Curve Sketching
  • 3.7 Optimization
  • 3.8 Related Rates
  • 3.9 Rates of Change in Economics and the Sciences
• Chapter 4: Integration
  • 4.1 Antiderivatives
  • 4.2 Sums and Sigma Notation
    • Principle of Mathematical Induction
  • 4.3 Area
  • 4.4 The Definite Integral
    • Average Value of a Function
  • 4.5 The Fundamental Theorem of Calculus
  • 4.6 Integration by Substitution
    • Substitution in Definite Integrals
  • 4.7 Numerical Integration
    • Simpson’s Rule
    • Error Bounds for Numerical Integration
  • 4.8 The Natural Logarithm as an Integral
    • The Exponential Function as the Inverse of the Natural Logarithm
• Chapter 5: Applications of the Definite Integral
  • 5.1 Area Between Curves
  • 5.2 Volume: Slicing, Disks and Washers
    • Volumes by Slicing
    • The Method of Disks
    • The Method of Washers
  • 5.3 Volumes by Cylindrical Shells
  • 5.4 Arc Length and Surface Area
    • Arc Length
    • Surface Area
  • 5.5 Projectile Motion
  • 5.6 Applications of Integration to Physics and Engineering
  • 5.7 Probability
• Chapter 6: Integration Techniques
  • 6.1 Review of Formulas and Techniques
    • Concept Mapping
  • 6.2 Integration by Parts
  • 6.3 Trigonometric Techniques of Integration
    • Integrals Involving Powers of Trigonometric Functions
    • Trigonometric Substitution
  • 6.4 Integration of Rational Functions Using Partial Fractions
    • Brief Summary of Integration Techniques
  • 6.5 Integration Tables and Computer Algebra Systems
    • Using Tables of Integrals
    • Integration Using a Computer Algebra System
  • 6.6 Improper Integrals
    • Improper Integrals with a Discontinuous Integrand
    • Improper Integrals with an Infinite Limit of Integration
    • A Comparison Test
• Chapter 7: First-Order Differential Equations
  • 7.1 Modeling with Differential Equations
    • Growth and Decay Problems
    • Compound Interest
  • 7.2 Separable Differential Equations
    • Logistic Growth
  • 7.3 First-Order Linear Differential Equations
    • General Solution of Linear Differential Equations of the First Order
  • 7.4 Direction Fields and Euler’s Method
  • 7.5 Systems of First-Order Differential Equations
    • Predator–Prey Systems
• Chapter 8: Infinite Series
  • 8.1 Sequences of Real Numbers
  • 8.2 Infinite Series
  • 8.3 The Integral Test and Comparison Tests
    • Comparison Tests
  • 8.4 Alternating Series
    • Estimating the Sum of an Alternating Series
  • 8.5 Absolute Convergence and the Ratio Test
    • The Ratio Test
    • The Root Test
    • Summary of Convergence Tests
    • Concept Mapping
  • 8.6 Power Series
  • 8.7 Taylor Series
    • Representation of Functions as Power Series
    • Proof of Taylor’s Theorem
  • 8.8 Applications of Taylor Series
    • The Binomial Series
  • 8.9 Fourier Series
    • Functions of Period Other Than 2π
    • Fourier Series and Music Synthesizers
• Chapter 9: Parametric Equations and Polar Coordinates
  • 9.1 Plane Curves and Parametric Equations
  • 9.2 Calculus and Parametric Equations
  • 9.3 Arc Length and Surface Area in Parametric Equations
  • 9.4 Polar Coordinates
  • 9.5 Calculus and Polar Coordinates
  • 9.6 Conic Sections
    • Parabolas
    • Ellipses
    • Hyperbolas
  • 9.7 Conic Sections in Polar Coordinates
• Chapter 10: Vectors and the Geometry of Space
  • 10.1 Vectors in the Plane
  • 10.2 Vectors in Space
    • Vectors in ℝ3
  • 10.3 The Dot Product
    • Components and Projections
  • 10.4 The Cross Product
  • 10.5 Lines and Planes in Space
    • Planes in ℝ3
  • 10.6 Surfaces in Space
    • Cylindrical Surfaces
    • Quadric Surfaces
    • An Application
• Chapter 11: Vector-Valued Functions
  • 11.1 Vector-Valued Functions
    • Arc Length in ℝ3
  • 11.2 The Calculus of Vector-Valued Functions
  • 11.3 Motion in Space
    • Equations of Motion
  • 11.4 Curvature
  • 11.5 Tangent and Normal Vectors
    • Tangential and Normal Components of Acceleration
    • Kepler’s Laws
  • 11.6 Parametric Surfaces
• Chapter 12: Functions of Several Variables and Partial Differentiation
  • 12.1 Functions of Several Variables
  • 12.2 Limits and Continuity
  • 12.3 Partial Derivatives
  • 12.4 Tangent Planes and Linear Approximations
    • Increments and Differentials
  • 12.5 The Chain Rule
    • Implicit Differentiation
  • 12.6 The Gradient and Directional Derivatives
  • 12.7 Extrema of Functions of Several Variables
    • Proof of the Second Derivatives Test
  • 12.8 Constrained Optimization and Lagrange Multipliers
• Chapter 13: Multiple Integrals
  • 13.1 Double Integrals
    • Double Integrals over a Rectangle
    • Double Integrals over General Regions
  • 13.2 Area, Volume and Center of Mass
    • Moments and Center of Mass
  • 13.3 Double Integrals in Polar Coordinates
  • 13.4 Surface Area
  • 13.5 Triple Integrals
    • Mass and Center of Mass
  • 13.6 Cylindrical Coordinates
  • 13.7 Spherical Coordinates
    • Triple Integrals in Spherical Coordinates
  • 13.8 Change of Variables in Multiple Integrals
• Chapter 14: Vector Calculus
  • 14.1 Vector Fields
  • 14.2 Line Integrals
  • 14.3 Independence of Path and Conservative Vector Fields
  • 14.4 Green’s Theorem
  • 14.5 Curl and Divergence
  • 14.6 Surface Integrals
    • Parametric Representation of Surfaces
  • 14.7 The Divergence Theorem
  • 14.8 Stokes’ Theorem
  • 14.9 Applications of Vector Calculus
• Chapter 15: Second-Order Differential Equations
  • 15.1 Second-Order Equations with Constant Coefficients
  • 15.2 Nonhomogeneous Equations: Undetermined Coefficients
  • 15.3 Applications of Second-Order Equations
  • 15.4 Power Series Solutions of Differential Equations
  • 15.5 Laplace Transforms
  • 15.6 Solving Differential Equations Using Laplace Transforms
• Appendix A: Proofs of Selected Theorems
• Credits
• Subject Index
• Derivative Formulas
• Table of Integrals

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