Calculus
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Calculus, Second Edition discusses the techniques and theorems of calculus. This edition introduces the sine and cosine functions, distributes ?-? material over several chapters, and includes a detailed account of analytic geometry and vector analysis. This book also discusses the equation of a straight line, trigonometric limit, derivative of a power function, mean value theorem, and fundamental theorems of calculus. The exponential and logarithmic functions, inverse trigonometric functions, linear and quadratic denominators, and centroid of a plane region are likewise elaborated. Other topics include the sequences of real numbers, dot product, arc length as a parameter, quadric surfaces, higher-order partial derivatives, and Green's theorem in the plane. This publication is a good source for students learning calculus.
Table of Contents
Preface
To the Instructor
One Preliminaries
1.1 Sets of Real Numbers
1.2 Absolute Value and Inequalities
1.3 The Cartesian Plane
1.4 Lines
1.5 The Equation of a Straight Line
1.6 Circles
1.7 Functions
1.8 Operations with Functions
1.9 Shifting the Graphs of Functions
1.10 Second-Degree Equations
Review Exercises for Chapter One
Two Limits And Derivatives
2.1 Introduction to Limits
2.2 The Calculation of Limits
2.3 The Limit Theorems
2.4 Infinite Limits and Limits at Infinity
2.5 One-Sided Limits
2.6 A Trigonometric Limit
2.7 Tangent Lines and Derivatives
2.8 Tangent Lines and Derivatives (Continued)
2.9 The Derivative as a Rate of Change
2.10 Continuity
2.11 The Theory of Limits (Optional)
Review Exercises for Chapter Two
Three 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 Sines and Cosines
3.6 Implicit Differentiation
3.7 Higher-Order Derivatives
3.8 Approximation and Differentials
Review Exercises for Chapter Three
Four 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
4.8 Newton's Method for Solving Equations
Review Exercises for Chapter Four
Five The Integral
5.1 The Area Problem
5.2 The Σ Notation
5.3 Approximations to Area
5.4 The Definite Integral
5.5 The Antiderivative
5.6 The Fundamental Theorems 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
Six Exponentials And Logarithms
6.1 The Exponential and Logarithmic Functions
6.2 The Derivatives and Integrals of logax and ax
6.3 The Exponential and Logarithmic Functions II
6.4 Differentiation and Integration of More General Exponential and Logarithmic Functions
6.5 Differential Equations of Exponential Growth and Decay
6.6 Applications in Economics (Optional)
6.7 Inverse Functions
Review Exercises for Chapter Six
Seven The Trigonometric And Hyperbolic Functions
7.1 Differentiation of Trigonometric Functions
7.2 Integration of Trigonometric Functions
7.3 The Inverse Trigonometric Functions
7.4 Periodic Motion (Optional)
7.5 The Hyperbolic Functions
7.6 The Inverse Hyperbolic Functions (Optional)
Review Exercises for Chapter Seven
Eight 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 √a2 — x2, √a2 + x2, and √x2 — 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
Nine 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 Inertia and Kinetic Energy (Optional)
9.7 Fluid Pressure (Optional)
Review Exercises for Chapter Nine
Ten 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
Eleven 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
Twelve 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
Thirteen Taylor Polynomials And Approximation
13.1 Taylor's Theorem and Taylor Polynomials
13.2 A Proof of Taylor's Theorem and Estimates on the Remainder Term (Optional)
13.3 Approximation Using Taylor Polynomials
Review Exercises for Chapter Thirteen
Fourteen 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
Fifteen 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
Sixteen 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
Seventeen 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
Eighteen 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 Differentials 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
Nineteen Multiple Integrations
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
Twenty 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' Theorem
20.8 The Divergence Theorem
20.9 Changing Variables in Multiple Integrals and the Jacobian
Review Exercises for Chapter Twenty
Twenty-One 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 Common Logarithms
A.4 Trigonometric Functions
A.5 Hyperbolic Functions
A.6 Integrals
Answers to Odd-Numbered Problems and Review Exercises
Index
Product details
- No. of pages: 1174
- Language: English
- Copyright: © Academic Press 1981
- Published: January 1, 1981
- Imprint: Academic Press
- eBook ISBN: 9781483262437