# Calculus

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## Efnisyfirlit

• Title 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
• 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
• 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
• 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
• 10.6. 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
• Index
• Back Cover

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Vörumerki: Pearson
Vörunúmer: 9780134528656
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# Calculus

Vörumerki: Pearson
Vörunúmer: 9780134528656
Rafræn bók. Uppl. sendar á netfangið þitt eftir kaup

### Veldu vöru

4.990 kr.
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