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Calculus

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Efnisyfirlit

  • Title Page
  • Copyright Page
  • Contents
  • Preface
  • To the Student
  • To the Instructor
  • Acknowledgments
  • What Is Calculus?
  • P. Preliminaries
    • P.1. Real Numbers and the Real Line
      • Intervals
      • The Absolute Value
      • Equations and Inequalities Involving Absolute Values
    • P.2. Cartesian Coordinates in the Plane
      • Axis Scales
      • Increments and Distances
      • Graphs
      • Straight Lines
      • Equations of Lines
    • P.3. Graphs of Quadratic Equations
      • Circles and Disks
      • Equations of Parabolas
      • Reflective Properties of Parabolas
      • Scaling a Graph
      • Shifting a Graph
      • Ellipses and Hyperbolas
    • P.4. Functions and Their Graphs
      • The Domain Convention
      • Graphs of Functions
      • Even and Odd Functions; Symmetry and Reflections
      • Reflections in Straight Lines
      • Defining and Graphing Functions with Maple
    • P.5. Combining Functions to Make New Functions
      • Sums, Differences, Products, Quotients, and Multiples
      • Composite Functions
      • Piecewise Defined Functions
    • P.6. Polynomials and Rational Functions
      • Roots, Zeros, and Factors
      • Roots and Factors of Quadratic Polynomials
      • Miscellaneous Factorings
    • P.7. The Trigonometric Functions
      • Some Useful Identities
      • Some Special Angles
      • The Addition Formulas
      • Other Trigonometric Functions
      • Maple Calculations
      • Trigonometry Review
  • 1. Limits and Continuity
    • 1.1. Examples of Velocity, Growth Rate, and Area
      • Average Velocity and Instantaneous Velocity
      • The Growth of an Algal Culture
      • The Area of a Circle
    • 1.2. Limits of Functions
      • One-Sided Limits
      • Rules for Calculating Limits
      • The Squeeze Theorem
    • 1.3. Limits at Infinity and Infinite Limits
      • Limits at Infinity
      • Limits at Infinity for Rational Functions
      • Infinite Limits
      • Using Maple to Calculate Limits
    • 1.4. Continuity
      • Continuity at a Point
      • Continuity on an Interval
      • There Are Lots of Continuous Functions
      • Continuous Extensions and Removable Discontinuities
      • Continuous Functions on Closed, Finite Intervals
      • Finding Roots of Equations
    • 1.5. The Formal Definition of Limit
      • Using the Definition of Limit to Prove Theorems
      • Other Kinds of Limits
    • Chapter Review
  • 2. Differentiation
    • 2.1. Tangent Lines and Their Slopes
      • Normals
    • 2.2. The Derivative
      • Some Important Derivatives
      • Leibniz Notation
      • Differentials
      • Derivatives Have the Intermediate-Value Property
    • 2.3. Differentiation Rules
      • Sums and Constant Multiples
      • The Product Rule
      • The Reciprocal Rule
      • The Quotient Rule
    • 2.4. The Chain Rule
      • Finding Derivatives with Maple
      • Building the Chain Rule into Differentiation Formulas
      • Proof of the Chain Rule (Theorem 6)
    • 2.5. Derivatives of Trigonometric Functions
      • Some Special Limits
      • The Derivatives of Sine and Cosine
      • The Derivatives of the Other Trigonometric Functions
    • 2.6. Higher-Order Derivatives
    • 2.7. Using Differentials and Derivatives
      • Approximating Small Changes
      • Average and Instantaneous Rates of Change
      • Sensitivity to Change
      • Derivatives in Economics
    • 2.8. The Mean-Value Theorem
      • Increasing and Decreasing Functions
      • Proof of the Mean-Value Theorem
    • 2.9. Implicit Differentiation
      • Higher-Order Derivatives
      • The General Power Rule
    • 2.10. Antiderivatives and Initial-Value Problems
      • Antiderivatives
      • The Indefinite Integral
      • Differential Equations and Initial-Value Problems
    • 2.11. Velocity and Acceleration
      • Velocity and Speed
      • Acceleration
      • Falling Under Gravity
    • Chapter Review
  • 3. Transcendental Functions
    • 3.1. Inverse Functions
      • Inverting Non–One-to-One Functions
      • Derivatives of Inverse Functions
    • 3.2. Exponential and Logarithmic Functions
      • Exponentials
      • Logarithms
    • 3.3. The Natural Logarithm and Exponential
      • The Natural Logarithm
      • The Exponential Function
      • General Exponentials and Logarithms
      • Logarithmic Differentiation
    • 3.4. Growth and Decay
      • The Growth of Exponentials and Logarithms
      • Exponential Growth and Decay Models
      • Interest on Investments
      • Logistic Growth
    • 3.5. The Inverse Trigonometric Functions
      • The Inverse Sine (or Arcsine) Function
      • The Inverse Tangent (or Arctangent) Function
      • Other Inverse Trigonometric Functions
    • 3.6. Hyperbolic Functions
      • Inverse Hyperbolic Functions
    • 3.7. Second-Order Linear DEs with Constant Coefficients
      • Recipe for Solving ay” + by’ + cy = 0
      • Simple Harmonic Motion
      • Damped Harmonic Motion
    • Chapter Review
  • 4. More Applications of Differentiation
    • 4.1. Related Rates
      • Procedures for Related-Rates Problems
    • 4.2. Finding Roots of Equations
      • Discrete Maps and Fixed-Point Iteration
      • Newton’s Method
      • “Solve” Routines
    • 4.3. Indeterminate Forms
      • l’H^opital’s Rules
    • 4.4. Extreme Values
      • Maximum and Minimum Values
      • Critical Points, Singular Points, and Endpoints
      • Finding Absolute Extreme Values
      • The First Derivative Test
      • Functions Not Defined on Closed, Finite Intervals
    • 4.5. Concavity and Inflections
      • The Second Derivative Test
    • 4.6. Sketching the Graph of a Function
      • Asymptotes
      • Examples of Formal Curve Sketching
    • 4.7. Graphing with Computers
      • Numerical Monsters and Computer Graphing
      • Floating-Point Representation of Numbers in Computers
      • Machine Epsilon and Its Effect on Figure 4.45
      • Determining Machine Epsilon
    • 4.8. Extreme-Value Problems
      • Procedure for Solving Extreme-Value Problems
    • 4.9. Linear Approximations
      • Approximating Values of Functions
      • Error Analysis
    • 4.10. Taylor Polynomials
      • Taylor’s Formula
      • Big-O Notation
      • Evaluating Limits of Indeterminate Forms
    • 4.11. Roundoff Error, Truncation Error, and Computers
      • Taylor Polynomials in Maple
      • Persistent Roundoff Error
      • Truncation, Roundoff, and Computer Algebra
    • Chapter Review
  • 5. Integration
    • 5.1. Sums and Sigma Notation
      • Evaluating Sums
    • 5.2. Areas as Limits of Sums
      • The Basic Area Problem
      • Some Area Calculations
    • 5.3. The Definite Integral
      • Partitions and Riemann Sums
      • The Definite Integral
      • General Riemann Sums
    • 5.4. Properties of the Definite Integral
      • A Mean-Value Theorem for Integrals
      • Definite Integrals of Piecewise Continuous Functions
    • 5.5. The Fundamental Theorem of Calculus
    • 5.6. The Method of Substitution
      • Trigonometric Integrals
    • 5.7. Areas of Plane Regions
      • Areas Between Two Curves
    • Chapter Review
  • 6. Techniques of Integration
    • 6.1. Integration by Parts
      • Reduction Formulas
    • 6.2. Integrals of Rational Functions
      • Linear and Quadratic Denominators
      • Partial Fractions
      • Completing the Square
      • Denominators with Repeated Factors
    • 6.3. Inverse Substitutions
      • The Inverse Trigonometric Substitutions
      • Inverse Hyperbolic Substitutions
      • Other Inverse Substitutions
      • The tan( /2) Substitution
    • 6.4. Other Methods for Evaluating Integrals
      • The Method of Undetermined Coefficients
      • Using Maple for Integration
      • Using Integral Tables
      • Special Functions Arising from Integrals
    • 6.5. Improper Integrals
      • Improper Integrals of Type I
      • Improper Integrals of Type II
      • Estimating Convergence and Divergence
    • 6.6. The Trapezoid and Midpoint Rules
      • The Trapezoid Rule
      • The Midpoint Rule
      • Error Estimates
    • 6.7. Simpson’s Rule
    • 6.8. Other Aspects of Approximate Integration
      • Approximating Improper Integrals
      • Using Taylor’s Formula
      • Romberg Integration
      • The Importance of Higher-Order Methods
      • Other Methods
    • Chapter Review
  • 7. Applications of Integration
    • 7.1. Volumes by Slicing—Solids of Revolution
      • Volumes by Slicing
      • Solids of Revolution
      • Cylindrical Shells
    • 7.2. More Volumes by Slicing
    • 7.3. Arc Length and Surface Area
      • Arc Length
      • The Arc Length of the Graph of a Function
      • Areas of Surfaces of Revolution
    • 7.4. Mass, Moments, and Centre of Mass
      • Mass and Density
      • Moments and Centres of Mass
      • Two- and Three-Dimensional Examples
    • 7.5. Centroids
      • Pappus’s Theorem
    • 7.6. Other Physical Applications
      • Hydrostatic Pressure
      • Work
      • Potential Energy and Kinetic Energy
    • 7.7. Applications in Business, Finance, and Ecology
      • The Present Value of a Stream of Payments
      • The Economics of Exploiting Renewable Resources
    • 7.8. Probability
      • Discrete Random Variables
      • Expectation, Mean, Variance, and Standard Deviation
      • Continuous Random Variables
      • The Normal Distribution
      • Heavy Tails
    • 7.9. First-Order Differential Equations
      • Separable Equations
      • First-Order Linear Equations
    • Chapter Review
  • 8. Conics, Parametric Curves, and Polar Curves
    • 8.1. Conics
      • Parabolas
      • The Focal Property of a Parabola
      • Ellipses
      • The Focal Property of an Ellipse
      • The Directrices of an Ellipse
      • Hyperbolas
      • The Focal Property of a Hyperbola
      • Classifying General Conics
    • 8.2. Parametric Curves
      • General Plane Curves and Parametrizations
      • Some Interesting Plane Curves
    • 8.3. Smooth Parametric Curves and Their Slopes
      • The Slope of a Parametric Curve
      • Sketching Parametric Curves
    • 8.4. Arc Lengths and Areas for Parametric Curves
      • Arc Lengths and Surface Areas
      • Areas Bounded by Parametric Curves
    • 8.5. Polar Coordinates and Polar Curves
      • Some Polar Curves
      • Intersections of Polar Curves
      • Polar Conics
    • 8.6. Slopes, Areas, and Arc Lengths for Polar Curves
      • Areas Bounded by Polar Curves
      • Arc Lengths for Polar Curves
    • Chapter Review
  • 9. Sequences, Series, and Power Series
    • 9.1. Sequences and Convergence
      • Convergence of Sequences
    • 9.2. Infinite Series
      • Geometric Series
      • Telescoping Series and Harmonic Series
      • Some Theorems About Series
    • 9.3. Convergence Tests for Positive Series
      • The Integral Test
      • Using Integral Bounds to Estimate the Sum of a Series
      • Comparison Tests
      • The Ratio and Root Tests
      • Using Geometric Bounds to Estimate the Sum of a Series
    • 9.4. Absolute and Conditional Convergence
      • The Alternating Series Test
      • Rearranging the Terms in a Series
    • 9.5. Power Series
      • Algebraic Operations on Power Series
      • Differentiation and Integration of Power Series
      • Maple Calculations
    • 9.6. Taylor and Maclaurin Series
      • Maclaurin Series for Some Elementary Functions
      • Other Maclaurin and Taylor Series
      • Taylor’s Formula Revisited
    • 9.7. Applications of Taylor and Maclaurin Series
      • Approximating the Values of Functions
      • Functions Defined by Integrals
      • Indeterminate Forms
    • 9.8. The Binomial Theorem and Binomial Series
      • The Binomial Series
      • The Multinomial Theorem
    • 9.9. Fourier Series
      • Periodic Functions
      • Fourier Series
      • Convergence of Fourier Series
      • Fourier Cosine and Sine Series
    • Chapter Review
  • 10. Vectors and Coordinate Geometry in 3-Space
    • 10.1. Analytic Geometry in Three Dimensions
      • Euclidean n-Space
      • Describing Sets in the Plane, 3-Space, and n-Space
    • 10.2. Vectors
      • Vectors in 3-Space
      • Hanging Cables and Chains
      • The Dot Product and Projections
      • Vectors in n-Space
    • 10.3. The Cross Product in 3-Space
      • Determinants
      • The Cross Product as a Determinant
      • Applications of Cross Products
    • 10.4. Planes and Lines
      • Planes in 3-Space
      • Lines in 3-Space
      • Distances
    • 10.5. Quadric Surfaces
    • 10.6. Cylindrical and Spherical Coordinates
      • Cylindrical Coordinates
      • Spherical Coordinates
    • 10.7. A Little Linear Algebra
      • Matrices
      • Determinants and Matrix Inverses
      • Linear Transformations
      • Linear Equations
      • Quadratic Forms, Eigenvalues, and Eigenvectors
    • 10.8. Using Maple for Vector and Matrix Calculations
      • Vectors
      • Matrices
      • Linear Equations
      • Eigenvalues and Eigenvectors
    • Chapter Review
  • 11. Vector Functions and Curves
    • 11.1. Vector Functions of One Variable
      • Differentiating Combinations of Vectors
    • 11.2. Some Applications of Vector Differentiation
      • Motion Involving Varying Mass
      • Circular Motion
      • Rotating Frames and the Coriolis Effect
    • 11.3. Curves and Parametrizations
      • Parametrizing the Curve of Intersection of Two Surfaces
      • Arc Length
      • Piecewise Smooth Curves
      • The Arc-Length Parametrization
    • 11.4. Curvature, Torsion, and the Frenet Frame
      • The Unit Tangent Vector
      • Curvature and the Unit Normal
      • Torsion and Binormal, the Frenet-Serret Formulas
    • 11.5. Curvature and Torsion for General Parametrizations
      • Tangential and Normal Acceleration
      • Evolutes
      • An Application to Track (or Road) Design
      • Maple Calculations
    • 11.6. Kepler’s Laws of Planetary Motion
      • Ellipses in Polar Coordinates
      • Polar Components of Velocity and Acceleration
      • Central Forces and Kepler’s Second Law
      • Derivation of Kepler’s First and Third Laws
      • Conservation of Energy
    • Chapter Review
  • 12. Partial Differentiation
    • 12.1. Functions of Several Variables
      • Graphs
      • Level Curves
      • Using Maple Graphics
    • 12.2. Limits and Continuity
    • 12.3. Partial Derivatives
      • Tangent Planes and Normal Lines
      • Distance from a Point to a Surface: A Geometric Example
    • 12.4. Higher-Order Derivatives
      • The Laplace and Wave Equations
    • 12.5. The Chain Rule
      • Homogeneous Functions
      • Higher-Order Derivatives
    • 12.6. Linear Approximations, Differentiability, and Differentials
      • Proof of the Chain Rule
      • Differentials
      • Functions from n-Space to m-Space
      • Differentials in Applications
      • Differentials and Legendre Transformations
    • 12.7. Gradients and Directional Derivatives
      • Directional Derivatives
      • Rates Perceived by a Moving Observer
      • The Gradient in Three and More Dimensions
    • 12.8. Implicit Functions
      • Systems of Equations
      • Choosing Dependent and Independent Variables
      • Jacobian Determinants
      • The Implicit Function Theorem
    • 12.9. Taylor’s Formula, Taylor Series, and Approximations
      • Approximating Implicit Functions
    • Chapter Review
  • 13. Applications of Partial Derivatives
    • 13.1. Extreme Values
      • Classifying Critical Points
    • 13.2. Extreme Values of Functions Defined on Restricted Domains
      • Linear Programming
    • 13.3. Lagrange Multipliers
      • The Method of Lagrange Multipliers
      • Problems with More than One Constraint
    • 13.4. Lagrange Multipliers in n-Space
      • Using Maple to Solve Constrained Extremal Problems
      • Significance of Lagrange Multiplier Values
      • Nonlinear Programming
    • 13.5. The Method of Least Squares
      • Linear Regression
      • Applications of the Least Squares Method to Integrals
    • 13.6. Parametric Problems
      • Differentiating Integrals with Parameters
      • Envelopes
      • Equations with Perturbations
    • 13.7. Newton’s Method
      • Implementing Newton’s Method Using a Spreadsheet
    • 13.8. Calculations with Maple
      • Solving Systems of Equations
      • Finding and Classifying Critical Points
    • 13.9. Entropy in Statistical Mechanics and Information Theory
      • Boltzmann Entropy
      • Shannon Entropy
      • Information Theory
    • Chapter Review
  • 14. Multiple Integration
    • 14.1. Double Integrals
      • Double Integrals over More General Domains
      • Properties of the Double Integral
      • Double Integrals by Inspection
    • 14.2. Iteration of Double Integrals in Cartesian Coordinates
    • 14.3. Improper Integrals and a Mean-Value Theorem
      • Improper Integrals of Positive Functions
      • A Mean-Value Theorem for Double Integrals
    • 14.4. Double Integrals in Polar Coordinates
      • Change of Variables in Double Integrals
    • 14.5. Triple Integrals
    • 14.6. Change of Variables in Triple Integrals
      • Cylindrical Coordinates
      • Spherical Coordinates
    • 14.7. Applications of Multiple Integrals
      • The Surface Area of a Graph
      • The Gravitational Attraction of a Disk
      • Moments and Centres of Mass
      • Moment of Inertia
    • Chapter Review
  • 15. Vector Fields
    • 15.1. Vector and Scalar Fields
      • Field Lines (Integral Curves, Trajectories, Streamlines)
      • Vector Fields in Polar Coordinates
      • Nonlinear Systems and Liapunov Functions
    • 15.2. Conservative Fields
      • Equipotential Surfaces and Curves
      • Sources, Sinks, and Dipoles
    • 15.3. Line Integrals
      • Evaluating Line Integrals
    • 15.4. Line Integrals of Vector Fields
      • Connected and Simply Connected Domains
      • Independence of Path
    • 15.5. Surfaces and Surface Integrals
      • Parametric Surfaces
      • Composite Surfaces
      • Surface Integrals
      • Smooth Surfaces, Normals, and Area Elements
      • Evaluating Surface Integrals
      • The Attraction of a Spherical Shell
    • 15.6. Oriented Surfaces and Flux Integrals
      • Oriented Surfaces
      • The Flux of a Vector Field Across a Surface
      • Calculating Flux Integrals
    • Chapter Review
  • 16. Vector Calculus
    • 16.1. Gradient, Divergence, and Curl
      • Interpretation of the Divergence
      • Distributions and Delta Functions
      • Interpretation of the Curl
    • 16.2. Some Identities Involving Grad, Div, and Curl
      • Scalar and Vector Potentials
      • Maple Calculations
    • 16.3. Green’s Theorem in the Plane
      • The Two-Dimensional Divergence Theorem
    • 16.4. The Divergence Theorem in 3-Space
      • Variants of the Divergence Theorem
    • 16.5. Stokes’s Theorem
    • 16.6. Some Physical Applications of Vector Calculus
      • Fluid Dynamics
      • Electromagnetism
      • Electrostatics
      • Magnetostatics
      • Maxwell’s Equations
    • 16.7. Orthogonal Curvilinear Coordinates
      • Coordinate Surfaces and Coordinate Curves
      • Scale Factors and Differential Elements
      • Grad, Div, and Curl in Orthogonal Curvilinear Coordinates
    • Chapter Review
  • 17. Differential Forms and Exterior Calculus
    • Differentials and Vectors
    • Derivatives versus Differentials
    • 17.1. k-Forms
      • Bilinear Forms and 2-Forms
      • k-Forms
      • Forms on a Vector Space
    • 17.2. Differential Forms and the Exterior Derivative
      • The Exterior Derivative
      • 1-Forms and Legendre Transformations
      • Maxwell’s Equations Revisited
      • Closed and Exact Forms
    • 17.3. Integration on Manifolds
      • Smooth Manifolds
      • Integration in n Dimensions
      • Sets of k-Volume Zero
      • Parametrizing and Integrating over a Smooth Manifold
    • 17.4. Orientations, Boundaries, and Integration of Forms
      • Oriented Manifolds
      • Pieces-with-Boundary of a Manifold
      • Integrating a Differential Form over a Manifold
    • 17.5. The Generalized Stokes Theorem
      • Proof of Theorem 4 for a k-Cube
      • Completing the Proof
      • The Classical Theorems of Vector Calculus
  • 18. Ordinary Differential Equations
    • 18.1. Classifying Differential Equations
    • 18.2. Solving First-Order Equations
      • Separable Equations
      • First-Order Linear Equations
      • First-Order Homogeneous Equations
      • Exact Equations
      • Integrating Factors
    • 18.3. Existence, Uniqueness, and Numerical Methods
      • Existence and Uniqueness of Solutions
      • Numerical Methods
    • 18.4. Differential Equations of Second Order
      • Equations Reducible to First Order
      • Second-Order Linear Equations
    • 18.5. Linear Differential Equations with Constant Coefficients
      • Constant-Coefficient Equations of Higher Order
      • Euler (Equidimensional) Equations
    • 18.6. Nonhomogeneous Linear Equations
      • Resonance
      • Variation of Parameters
      • Maple Calculations
    • 18.7. The Laplace Transform
      • Some Basic Laplace Transforms
      • More Properties of Laplace Transforms
      • The Heaviside Function and the Dirac Delta Function
    • 18.8. Series Solutions of Differential Equations
    • 18.9. Dynamical Systems, Phase Space, and the Phase Plane
      • A Differential Equation as a First-Order System
      • Existence, Uniqueness, and Autonomous Systems
      • Second-Order Autonomous Equations and the Phase Plane
      • Fixed Points
      • Linear Systems, Eigenvalues, and Fixed Points
      • Implications for Nonlinear Systems
      • Predator–Prey Models
    • Chapter Review
  • Appendices
    • Appendix I: Complex Numbers
      • Definition of Complex Numbers
      • Graphical Representation of Complex Numbers
      • Complex Arithmetic
      • Roots of Complex Numbers
    • Appendix II: Complex Functions
      • Limits and Continuity
      • The Complex Derivative
      • The Exponential Function
      • The Fundamental Theorem of Algebra
    • Appendix III: Continuous Functions
      • Limits of Functions
      • Continuous Functions
      • Completeness and Sequential Limits
      • Continuous Functions on a Closed, Finite Interval
    • Appendix IV: The Riemann Integral
      • Uniform Continuity
    • Appendix V: Doing Calculus with Maple
      • List of Maple Examples and Discussion
  • Answers to Odd-Numbered Exercises
  • Index
  • Back Cover

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Calculus

Vörumerki: Pearson
Vörunúmer: 9780134528656
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