Calculus
3rd Edition
Published on 10. May 2014
1364 pages
978-1-4832-6239-0 (ISBN)
Description
Calculus, Third Edition emphasizes the techniques and theorems of calculus, including many applied examples and exercises in both drill and applied-type problems.
This book discusses shifting the graphs of functions, derivative as a rate of change, derivative of a power function, and theory of maxima and minima. The area between two curves, differential equations of exponential growth and decay, inverse hyperbolic functions, and integration of rational functions are also elaborated. This text likewise covers the fluid pressure, ellipse and translation of axes, graphing in polar coordinates, proof of l'Hôpital's rule, and approximation using Taylor polynomials. Other topics include the rectangular coordinate system in space, higher-order partial derivatives, line integrals in space, and vibratory motion.
This publication is valuable to students taking calculus.
This book discusses shifting the graphs of functions, derivative as a rate of change, derivative of a power function, and theory of maxima and minima. The area between two curves, differential equations of exponential growth and decay, inverse hyperbolic functions, and integration of rational functions are also elaborated. This text likewise covers the fluid pressure, ellipse and translation of axes, graphing in polar coordinates, proof of l'Hôpital's rule, and approximation using Taylor polynomials. Other topics include the rectangular coordinate system in space, higher-order partial derivatives, line integrals in space, and vibratory motion.
This publication is valuable to students taking calculus.
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Content
¿Preface
To the Instructor
1 Preliminaries
1.1 Sets of Real Numbers
1.2 Absolute Value and Inequalities
1.3 The Cartesian Plane
1.4 Lines
1.5 Equations of a Straight Line
1.6 Functions
1.7 Operations with Functions
1.8 Shifting the Graphs of Functions (Optional)
Review Exercises for Chapter One
2 Limits And Derivatives
2.1 Introduction to the Derivative
2.2 The Calculation of Limits
2.3 The Limit Theorems
2.4 Infinite Limits and Limits at Infinity
2.5 Tangent Lines and Derivatives
2.6 The Derivative as a Rate of Change
2.7 Continuity
2.8 The Theory of Limits (Optional)
Review Exercises for Chapter Two
3 More About Derivatives
3.1 Some Differentiation Formulas
3.2 The Product and Quotient Rules
3.3 The Derivative of Composite Functions: The Chain Rule
3.4 The Derivative of a Power Function
3.5 The Derivatives of the Trigonometric Functions
3.6 Implicit Differentiation
3.7 Higher-Order Derivatives
3.8 Approximation and Differentials
Review Exercises for Chapter Three
4 Applications Of The Derivative
4.1 Related Rates of Change
4.2 The Mean Value Theorem
4.3 Elementary Curve Sketching I: Increasing and Decreasing Functions and the First Derivative Test
4.4 Elementary Curve Sketching II: Concavity and the Second Derivative Test
4.5 The Theory of Maxima and Minima
4.6 Maxima and Minima: Applications
4.7 Some Applications in Economics (Optional)
4.8 Newton's Method for Solving Equations
Review Exercises for Chapter Four
5 The Integral
5.1 Introduction
5.2 Antiderivatives
5.3 The S Notation
5.4 Approximations to Area
5.5 The Definite Integral
5.6 The Fundamental Theorem of Calculus
5.7 Integration by Substitution
5.8 The Area Between Two Curves
5.9 Work, Power, and Energy (Optional)
5.10 Additional Integration Theory (Optional)
Review Exercises for Chapter Five
6 Exponentials And Logarithms
6.1 Inverse Functions
6.2 The Exponential and Logarithmic Functions I
6.3 The Derivatives and Integrals of logax and ax
6.4 The Exponential and Logarithmic Functions II
6.5 Differentiation and Integration of More General Exponential and Logarithmic Functions
6.6 Differential Equations of Exponential Growth and Decay
6.7 Applications in Economics (Optional)
6.8 A Model for Epidemics (Optional)
Review Exercises for Chapter Six
7 More On Trigonometric Functions And The Hyperbolic Functions
7.1 Integration of Trigonometric Functions
7.2 The Inverse Trigonometric Functions
7.3 Periodic Motion (Optional)
7.4 The Hyperbolic Functions
7.5 The Inverse Hyperbolic Functions (Optional)
Review Exercises for Chapter Seven
8 Techniques Of Integration
8.1 Review of the Basic Formulas of Integration
8.2 Integration by Parts
8.3 Integrals of Certain Trigonometric Functions
8.4 The Idea Behind Integration by Substitution
8.5 Integrals Involving va2 - x2, va2 + x2, and vx2 - a2: Trigonometric Substitutions
8.6 The Integration of Rational Functions I: Linear and Quadratic Denominators
8.7 The Integration of Rational Functions II: The Method of Partial Fractions
8.8 Other Substitutions
8.9 Using the Integral Tables
8.10 Numerical Integration
Review Exercises for Chapter Eight
9 Further Applications Of The Definite Integral
9.1 Volumes
9.2 Arc Length
9.3 Surface Area
9.4 Center of Mass and the First Moment
9.5 The Centroid of a Plane Region
9.6 Moments of Intertia and Kinetic Energy (Optional)
9.7 Fluid Pressure (Optional)
Review Exercises for Chapter Nine
10 Topics In Analytic Geometry
10.1 The Ellipse and Translation of Axes
10.2 The Parabola
10.3 The Hyperbola
10.4 Second-Degree Equations and Rotation of Axes
Review Exercises for Chapter Ten
11 Polar Coordinates
11.1 The Polar Coordinate System
11.2 Graphing in Polar Coordinates
11.3 Points of Intersection of Graphs of Polar Equations
11.4 Derivatives and Tangent Lines
11.5 Areas in Polar Coordinates
Review Exercises for Chapter Eleven
12 Indeterminate Forms And Improper Integrals
12.1 The Indeterminate Form 0/0 and L'Hôpital's Rule
12.2 Proof of L'Hôpital's Rule (Optional)
12.3 Other Indeterminate Forms
12.4 Improper Integrals
Review Exercises for Chapter Twelve
13 Taylor Polynomials And Approximation
13.1 Taylor's Theorem and Taylor Polynomials
13.2 A Proof of Taylor's Theorem, Estimates on the Remainder Term, and a Uniqueness Theorem (Optional)
13.3 Approximation Using Taylor Polynomials
Review Exercises for Chapter Thirteen
14 Sequences And Series
14.1 Sequences of Real Numbers
14.2 Bounded and Monotonic Sequences
14.3 Geometric Series
14.4 Infinite Series
14.5 Series with Nonnegative Terms I: Two Comparison Tests and the Integral Test
14.6 Series with Nonnegative Terms II: The Ratio and Root Tests
14.7 Absolute and Conditional Convergence: Alternating Series
14.8 Power Series
14.9 Differentiation and Integration of Power Series
14.10 Taylor and Maclaurin Series
Review Exercises for Chapter Fourteen
15 Vectors In The Plane
15.1 Vectors and Vector Operations
15.2 The Dot Product
15.3 Some Applications of Vectors (Optional)
Review Exercises for Chapter Fifteen
16 Vector Functions, Vector Differentiation, And Parametric Equations
16.1 Vector Functions and Parametric Equations
16.2 The Equation of the Tangent Line to a Parametric Curve
16.3 The Differentiation and Integration of a Vector Function
16.4 Some Differentiation Formulas
16.5 Arc Length Revisited
16.6 Arc Length as a Parameter
16.7 Velocity, Acceleration, Force, and Momentum (Optional)
16.8 Curvature and the Acceleration Vector (Optional)
Review Exercises for Chapter Sixteen
17 Vectors In Space
17.1 The Rectangular Coordinate System in Space
17.2 Vectors in R3
17.3 Lines in R3
17.4 The Cross Product of Two Vectors
17.5 Planes
17.6 Quadric Surfaces
17.7 Vector Functions and Parametric Equations in R3
17.8 Cylindrical and Spherical Coordinates
Review Exercises for Chapter Seventeen
18 Differentiation Of Functions Of Two And Three Variables
18.1 Functions of Two and Three Variables
18.2 Limits and Continuity
18.3 Partial Derivatives
18.4 Higher-Order Partial Derivatives
18.5 Differentiability and the Gradient
18.6 The Chain Rule
18.7 Tangent Planes, Normal Lines, and Gradients
18.8 Directional Derivatives and the Gradient
18.9 Conservative Vector Fields and the Gradient (Optional)
18.10 The Total Differential and Approximation
18.11 Exact Vector Fields or How to Obtain a Function from Its Gradient
18.12 Maxima and Minima for a Function of Two Variables
18.13 Constrained Maxima and Minima-Lagrange Multipliers
Review Exercises for Chapter Eighteen
19 Multiple Integration
19.1 Volume Under a Surface and the Double Integral
19.2 The Calculation of Double Integrals
19.3 Density, Mass, and Center of Mass (Optional)
19.4 Double Integrals in Polar Coordinates
19.5 Surface Area
19.6 The Triple Integral
19.7 The Triple Integral in Cylindrical and Spherical Coordinates
Review Exercises for Chapter Nineteen
20 Introduction To Vector Analysis
20.1 Vector Fields
20.2 Work, Line Integrals in the Plane, and Independence of Path
20.3 Green's Theorem in the Plane
20.4 Line Integrals in Space
20.5 Surface Integrals
20.6 Divergence and Curl of a Vector Field in R3
20.7 Stokes's Theorem
20.8 The Divergence Theorem
20.9 Changing Variables in Multiple Integrals and the Jacobian
Review Exercises for Chapter Twenty
21 Ordinary Differential Equations
21.1 Introduction
21.2 First-Order Equations-Separation of Variables
21.3 First-Order Linear Equations
21.4 Second-Order Linear, Homogeneous Equations with Constant Coefficients
21.5 Second-Order Nonhomogeneous Equations with Constant Coefficients: The Method of Undetermined Coefficients
21.6 Vibratory Motion (Optional)
Review Exercises for Chapter Twenty-One
Appendix 1 Review Of Trigonometry
1.1 Angles and Radian Measure
1.2 The Trigonometric Functions and Basic Identities
1.3 Other Trigonometric Functions
1.4 Triangles
Appendix 2 Mathematical Induction
Appendix 3 Determinants
Appendix 4 The Binomial Theorem
Appendix 5 The Proofs Of Some Theorems On Limits, Continuity, And Differentiation
Appendix 6 Complex Numbers
Tables
A.1 Exponential Functions
A.2 Natural Logarithms
A.3 Hyperbolic Functions
A.4 Integrals
Answers to Odd-Numbered Problems and Review Exercises
Index
To the Instructor
1 Preliminaries
1.1 Sets of Real Numbers
1.2 Absolute Value and Inequalities
1.3 The Cartesian Plane
1.4 Lines
1.5 Equations of a Straight Line
1.6 Functions
1.7 Operations with Functions
1.8 Shifting the Graphs of Functions (Optional)
Review Exercises for Chapter One
2 Limits And Derivatives
2.1 Introduction to the Derivative
2.2 The Calculation of Limits
2.3 The Limit Theorems
2.4 Infinite Limits and Limits at Infinity
2.5 Tangent Lines and Derivatives
2.6 The Derivative as a Rate of Change
2.7 Continuity
2.8 The Theory of Limits (Optional)
Review Exercises for Chapter Two
3 More About Derivatives
3.1 Some Differentiation Formulas
3.2 The Product and Quotient Rules
3.3 The Derivative of Composite Functions: The Chain Rule
3.4 The Derivative of a Power Function
3.5 The Derivatives of the Trigonometric Functions
3.6 Implicit Differentiation
3.7 Higher-Order Derivatives
3.8 Approximation and Differentials
Review Exercises for Chapter Three
4 Applications Of The Derivative
4.1 Related Rates of Change
4.2 The Mean Value Theorem
4.3 Elementary Curve Sketching I: Increasing and Decreasing Functions and the First Derivative Test
4.4 Elementary Curve Sketching II: Concavity and the Second Derivative Test
4.5 The Theory of Maxima and Minima
4.6 Maxima and Minima: Applications
4.7 Some Applications in Economics (Optional)
4.8 Newton's Method for Solving Equations
Review Exercises for Chapter Four
5 The Integral
5.1 Introduction
5.2 Antiderivatives
5.3 The S Notation
5.4 Approximations to Area
5.5 The Definite Integral
5.6 The Fundamental Theorem of Calculus
5.7 Integration by Substitution
5.8 The Area Between Two Curves
5.9 Work, Power, and Energy (Optional)
5.10 Additional Integration Theory (Optional)
Review Exercises for Chapter Five
6 Exponentials And Logarithms
6.1 Inverse Functions
6.2 The Exponential and Logarithmic Functions I
6.3 The Derivatives and Integrals of logax and ax
6.4 The Exponential and Logarithmic Functions II
6.5 Differentiation and Integration of More General Exponential and Logarithmic Functions
6.6 Differential Equations of Exponential Growth and Decay
6.7 Applications in Economics (Optional)
6.8 A Model for Epidemics (Optional)
Review Exercises for Chapter Six
7 More On Trigonometric Functions And The Hyperbolic Functions
7.1 Integration of Trigonometric Functions
7.2 The Inverse Trigonometric Functions
7.3 Periodic Motion (Optional)
7.4 The Hyperbolic Functions
7.5 The Inverse Hyperbolic Functions (Optional)
Review Exercises for Chapter Seven
8 Techniques Of Integration
8.1 Review of the Basic Formulas of Integration
8.2 Integration by Parts
8.3 Integrals of Certain Trigonometric Functions
8.4 The Idea Behind Integration by Substitution
8.5 Integrals Involving va2 - x2, va2 + x2, and vx2 - a2: Trigonometric Substitutions
8.6 The Integration of Rational Functions I: Linear and Quadratic Denominators
8.7 The Integration of Rational Functions II: The Method of Partial Fractions
8.8 Other Substitutions
8.9 Using the Integral Tables
8.10 Numerical Integration
Review Exercises for Chapter Eight
9 Further Applications Of The Definite Integral
9.1 Volumes
9.2 Arc Length
9.3 Surface Area
9.4 Center of Mass and the First Moment
9.5 The Centroid of a Plane Region
9.6 Moments of Intertia and Kinetic Energy (Optional)
9.7 Fluid Pressure (Optional)
Review Exercises for Chapter Nine
10 Topics In Analytic Geometry
10.1 The Ellipse and Translation of Axes
10.2 The Parabola
10.3 The Hyperbola
10.4 Second-Degree Equations and Rotation of Axes
Review Exercises for Chapter Ten
11 Polar Coordinates
11.1 The Polar Coordinate System
11.2 Graphing in Polar Coordinates
11.3 Points of Intersection of Graphs of Polar Equations
11.4 Derivatives and Tangent Lines
11.5 Areas in Polar Coordinates
Review Exercises for Chapter Eleven
12 Indeterminate Forms And Improper Integrals
12.1 The Indeterminate Form 0/0 and L'Hôpital's Rule
12.2 Proof of L'Hôpital's Rule (Optional)
12.3 Other Indeterminate Forms
12.4 Improper Integrals
Review Exercises for Chapter Twelve
13 Taylor Polynomials And Approximation
13.1 Taylor's Theorem and Taylor Polynomials
13.2 A Proof of Taylor's Theorem, Estimates on the Remainder Term, and a Uniqueness Theorem (Optional)
13.3 Approximation Using Taylor Polynomials
Review Exercises for Chapter Thirteen
14 Sequences And Series
14.1 Sequences of Real Numbers
14.2 Bounded and Monotonic Sequences
14.3 Geometric Series
14.4 Infinite Series
14.5 Series with Nonnegative Terms I: Two Comparison Tests and the Integral Test
14.6 Series with Nonnegative Terms II: The Ratio and Root Tests
14.7 Absolute and Conditional Convergence: Alternating Series
14.8 Power Series
14.9 Differentiation and Integration of Power Series
14.10 Taylor and Maclaurin Series
Review Exercises for Chapter Fourteen
15 Vectors In The Plane
15.1 Vectors and Vector Operations
15.2 The Dot Product
15.3 Some Applications of Vectors (Optional)
Review Exercises for Chapter Fifteen
16 Vector Functions, Vector Differentiation, And Parametric Equations
16.1 Vector Functions and Parametric Equations
16.2 The Equation of the Tangent Line to a Parametric Curve
16.3 The Differentiation and Integration of a Vector Function
16.4 Some Differentiation Formulas
16.5 Arc Length Revisited
16.6 Arc Length as a Parameter
16.7 Velocity, Acceleration, Force, and Momentum (Optional)
16.8 Curvature and the Acceleration Vector (Optional)
Review Exercises for Chapter Sixteen
17 Vectors In Space
17.1 The Rectangular Coordinate System in Space
17.2 Vectors in R3
17.3 Lines in R3
17.4 The Cross Product of Two Vectors
17.5 Planes
17.6 Quadric Surfaces
17.7 Vector Functions and Parametric Equations in R3
17.8 Cylindrical and Spherical Coordinates
Review Exercises for Chapter Seventeen
18 Differentiation Of Functions Of Two And Three Variables
18.1 Functions of Two and Three Variables
18.2 Limits and Continuity
18.3 Partial Derivatives
18.4 Higher-Order Partial Derivatives
18.5 Differentiability and the Gradient
18.6 The Chain Rule
18.7 Tangent Planes, Normal Lines, and Gradients
18.8 Directional Derivatives and the Gradient
18.9 Conservative Vector Fields and the Gradient (Optional)
18.10 The Total Differential and Approximation
18.11 Exact Vector Fields or How to Obtain a Function from Its Gradient
18.12 Maxima and Minima for a Function of Two Variables
18.13 Constrained Maxima and Minima-Lagrange Multipliers
Review Exercises for Chapter Eighteen
19 Multiple Integration
19.1 Volume Under a Surface and the Double Integral
19.2 The Calculation of Double Integrals
19.3 Density, Mass, and Center of Mass (Optional)
19.4 Double Integrals in Polar Coordinates
19.5 Surface Area
19.6 The Triple Integral
19.7 The Triple Integral in Cylindrical and Spherical Coordinates
Review Exercises for Chapter Nineteen
20 Introduction To Vector Analysis
20.1 Vector Fields
20.2 Work, Line Integrals in the Plane, and Independence of Path
20.3 Green's Theorem in the Plane
20.4 Line Integrals in Space
20.5 Surface Integrals
20.6 Divergence and Curl of a Vector Field in R3
20.7 Stokes's Theorem
20.8 The Divergence Theorem
20.9 Changing Variables in Multiple Integrals and the Jacobian
Review Exercises for Chapter Twenty
21 Ordinary Differential Equations
21.1 Introduction
21.2 First-Order Equations-Separation of Variables
21.3 First-Order Linear Equations
21.4 Second-Order Linear, Homogeneous Equations with Constant Coefficients
21.5 Second-Order Nonhomogeneous Equations with Constant Coefficients: The Method of Undetermined Coefficients
21.6 Vibratory Motion (Optional)
Review Exercises for Chapter Twenty-One
Appendix 1 Review Of Trigonometry
1.1 Angles and Radian Measure
1.2 The Trigonometric Functions and Basic Identities
1.3 Other Trigonometric Functions
1.4 Triangles
Appendix 2 Mathematical Induction
Appendix 3 Determinants
Appendix 4 The Binomial Theorem
Appendix 5 The Proofs Of Some Theorems On Limits, Continuity, And Differentiation
Appendix 6 Complex Numbers
Tables
A.1 Exponential Functions
A.2 Natural Logarithms
A.3 Hyperbolic Functions
A.4 Integrals
Answers to Odd-Numbered Problems and Review Exercises
Index
