Table of Contents

1 Introduction

Part 1 Historical Introduction
 1.1 The Two Basic Concepts of Calculus
 1.2 Historical Background
 1.3 The Method of Exhaustion for the Area of a Parabolic Segment
 1.4 Exercises
 1.5 A Critical Analysis of Archimedes’ Method
 1.6 The Approach to Calculus to Be Used in This Book

Part 2 Some Basic Concepts of the Theory of Sets
 2.1 Introduction to Set Theory
 2.2 Notations for Designating Sets
 2.3 Subsets
 2.4 Unions, Intersections, Complements
 2.5 Exercises

Part 3 A Set of Axioms for the RealNumber System
 3.1 Introduction
 3.2 The Field Axioms
 3.3 Exercises
 3.4 The Order Axioms
 3.5 Exercises
 3.6 Integers and Rational Numbers
 3.7 Geometric Interpretation of Real Numbers as Points on a Line
 3.8 Upper Bound of a Set, Maximum Element, Least Upper Bound (Supremum)
 3.9 The LeastUpperBound Axiom (Completeness Axiom)
 3.10 The Archimedean Property of the RealNumber System
 3.11 Fundamental Properties of the Supremum and Infimum
 3.12 Exercises
 3.13 Existence of Square Roots of Nonnegative Real Numbers
 3.14 Roots of Higher Order. Rational Powers
 3.15 Representation of Real Numbers by Decimals

Part 4 Mathematical Induction, Summation Notation, and Related Topics
 4.1 An Example of a Proof by Mathematical Induction

4.2 The Principle of Mathematical Induction
 4.3 The WellOrdering Principle
 4.4 Exercises
 4.5 Proof of the WellOrdering Principle
 4.6 The Summation Notation
 4.7 Exercises

4.8 Absolute Values and the Triangle Inequality
 4.9 Exercises
 4.10 Miscellaneous Exercises Involving Induction

Part 1 Historical Introduction

1 The Concepts of Integral Calculus
 1.1 The Basic Ideas of Cartesian Geometry
 1.2 Functions. Informal Description and Examples
 1.3 Functions. Formal Definition as a Set of Ordered Pairs
 1.4 More Examples of Real Functions
 1.5 Exercises
 1.6 The Concept of Area as a Set Function
 1.7 Exercises
 1.8 Intervals and Ordinate Sets
 1.9 Partitions and Step Functions
 1.10 Sum and Product of Step Functions
 1.11 Exercises
 1.12 The Definition of the Integral for Step Functions
 1.13 Properties of the Integral of a Step Function
 1.14 Other Notations for Integrals
 1.15 Exercises
 1.16 The Integral of More General Functions
 1.17 Upper and Lower Integrals
 1.18 The Area of an Ordinate Set Expressed as an Integral
 1.19 Informal Remarks on the Theory and Technique of Integration
 1.20 Monotonic and Piecewise Monotonic Functions. Definitions and Examples
 1.21 Integrability of Bounded Monotonic Functions
 1.22 Calculation of the Integral of a Bounded Monotonic Function
 1.23 Calculation of the Integral ∫^𝑏_0 𝑥^𝑝 𝑑𝑥 When 𝑝 Is a Positive Integer
 1.24 The Basic Properties of the Integral
 1.25 Integration of Polynomials
 1.26 Exercises
 1.27 Proofs of the Basic Properties of the Integral

2 Some Applications of Integration
 2.1 Introduction
 2.2 The Area of a Region between Two Graphs Expressed as an Integral
 2.3 Worked Examples
 2.4 Exercises
 2.5 The Trigonometric Functions
 2.6 Integration Formulas for the Sine and Cosine
 2.7 A Geometric Description of the Sine and Cosine Functions
 2.8 Exercises
 2.9 Polar Coordinates
 2.10 The Integral for Area in Polar Coordinates
 2.11 Exercises

2.12 Application of Integration to the Calculation of Volume
 Disc Method for Rotating around a Horizontal
 Disc Method for Rotating around a Vertical
 Washer Method for Rotating around a Vertical
 Washer Method for Rotating around a Horizontal
 Shell Method for Rotating around a Horizontal
 Shell Method for Rotating around a Vertical
 Volumes of Solids of Revolution
 2.13 Exercises
 2.14 Application of Integration to the Concept of Work
 2.15 Exercises
 2.16 Average Value of a Function
 2.17 Exercises
 2.18 The Integral as a Function of the Upper Limit. Indefinite Integrals
 2.19 Exercises

3 Continuous Functions
 3.1 Informal Description of Continuity
 3.2 The Definition of the Limit of a Function
 3.3 The Definition of Continuity of a Function
 3.4 The Basic Limit Theorems. More Examples of Continuous Functions
 3.5 Proofs of the Basic Limit Theorems
 3.6 Exercises
 3.7 Composite Functions and Continuity
 3.8 Exercises
 3.9 Bolzano’s Theorem for Continuous Functions
 3.10 The IntermediateValue Theorem for Continuous Functions
 3.11 Exercises
 3.12 The Process of Inversion
 3.13 Properties of Functions Preserved by Inversion
 3.14 Inverses of Piecewise Monotonic Functions
 3.15 Exercises
 3.16 The ExtremeValue Theorem for Continuous Functions
 3.17 The SmallSpan Theorem for Continuous Functions (Uniform Continuity)
 3.18 The Integrability Theorem for Continuous Functions
 3.19 MeanValue Theorems for Integrals of Continuous Functions
 3.20 Exercises

4 Differential Calculus
 4.1 Historical Introduction
 4.2 A Problem Involving Velocity
 4.3 The Derivative of a Function
 4.4 Examples of Derivatives
 4.5 The Algebra of Derivatives
 4.6 Exercises
 4.7 Geometric Interpretation of the Derivative as a Slope
 4.8 Other Notations for Derivatives
 4.9 Exercises
 4.10 The Chain Rule for Differentiating Composite Functions
 4.11 Applications of the Chain Rule. Related Rates and Implicit Differentiation
 4.12 Exercises
 4.13 Applications of Differentiation to Extreme Values of Functions
 4.14 The MeanValue Theorem for Derivatives
 4.15 Exercises
 4.16 Applications of the MeanValue Theorem to Geometric Properties of Functions
 4.17 SecondDerivative Test for Extrema
 4.18 Curve Sketching
 4.19 Exercises
 4.20 Worked Examples of Extremum Problems
 4.21 Exercises
 4.22 Partial Derivatives
 4.23 Exercises

5 The Relation between Integration and Differentiation
 5.1 The Derivative of an Indefinite Integral. The First Fundamental Theorem of Calculus
 5.2 The ZeroDerivative Theorem
 5.3 Primitive Functions and the Second Fundamental Theorem of Calculus
 5.4 Properties of a Function Deduced from Properties of Its Derivative
 5.5 Exercises
 5.6 The Leibniz Notation for Primitives
 5.7 Integration by Substitution
 5.8 Exercises
 5.9 Integration by Parts
 5.10 Exercises
 5.11 Miscellaneous Review Exercises

6 The Logarithm, the Exponential, and the Inverse Trigonometric Functions
 6.1 Introduction
 6.2 Motivation for the Definition of the Natural Logarithm as an Integral
 6.3 The Definition of the Logarithm. Basic Properties
 6.4 The Graph of the Natural Logarithm
 6.5 Consequences of the Functional Equation 𝐿(𝑎𝑏) = 𝐿(𝑎) + 𝐿(𝑏)
 6.6 Logarithms Referred to Any Positive Base 𝑏 ≠ 1
 6.7 Differentiation and Integration Formulas Involving Logarithms
 6.8 Logarithmic Differentiation
 6.9 Exercises
 6.10 Polynomial Approximations to the Logarithm
 6.11 Exercises
 6.12 The Exponential Function
 6.13 Exponentials Expressed as Powers of 𝑒
 6.14 The Definition of 𝑒^𝑥 for Arbitrary Real 𝑥
 6.15 The Definition of 𝑎^𝑥 for 𝑎 > 0 and 𝑥 Real
 6.16 Differentiation and Integration Formulas Involving Exponentials
 6.17 Exercises
 6.18 The Hyperbolic Functions
 6.19 Exercises

6.20 Derivatives of Inverse Functions
 6.21 Inverses of the Trigonometric Functions
 6.22 Exercises
 6.23 Integration by Partial Fractions
 6.24 Integrals Which Can Be Transformed into Integrals of Rational Functions
 6.25 Exercises
 6.26 Miscellaneous Review Exercises

7 Polynomial Approximations to Functions

7.1 Introduction

7.2 The Taylor Polynomials Generated by a Function
 7.3 Calculus of Taylor Polynomials
 7.4 Exercises
 7.5 Taylor’s Formula with Remainder
 7.6 Estimates for the Error in Taylor’s Formula
 7.7 Other Forms of the Remainder in Taylor’s Formula
 7.8 Exercises
 7.9 Further Remarks on the Error in Taylor’s Formula. The oNotation
 7.10 Applications to Indeterminate Forms
 7.11 Exercises
 7.12 L’Hôpital’s Rule for the Indeterminate Form 0/0
 7.13 Exercises
 7.14 The Symbols +∞ and −∞. Extension of L’Hôpital’s Rule

7.15 Infinite Limits
 7.16 The Behavior of Log 𝑥 and 𝑒^𝑥 for Large 𝑥
 7.17 Exercises

7.1 Introduction

8 Introduction to Differential Equations
 8.1 Introduction
 8.2 Terminology and Notation
 8.3 A FirstOrder Differential Equation for the Exponential Function
 8.4 FirstOrder Linear Differential Equations
 8.5 Exercises
 8.6 Some Physical Problems Leading to FirstOrder Linear Differential Equations
 8.7 Exercises
 8.8 Linear Equations of Second Order with Constant Coefficients
 8.9 Existence of Solutions of the Equation 𝑦′′ + 𝑏𝑦 = 0
 8.10 Reduction of the General Equation to the Special Case 𝑦′′ + 𝑏𝑦 = 0
 8.11 Uniqueness Theorem for the Equation 𝑦′′ + 𝑏𝑦 = 0
 8.12 Complete Solution of the Equation 𝑦′′ + 𝑏𝑦 = 0
 8.13 Complete Solution of the Equation 𝑦′′ + 𝑎𝑦′ + 𝑏𝑦 = 0
 8.14 Exercises
 8.15 Nonhomogeneous Linear Equations of Second Order with Constant Coefficients
 8.16 Special Methods for Determining a Particular Solution of the Nonhomogeneous Equation 𝑦′′ + 𝑎𝑦′ + 𝑏𝑦 = 𝑅
 8.17 Exercises
 8.18 Examples of Physical Problems Leading to Linear SecondOrder Equations with Constant Coefficients
 8.19 Exercises
 8.20 Remarks Concerning Nonlinear Differential Equations
 8.21 Integral Curves and Direction Fields
 8.22 Exercises
 8.23 FirstOrder Separable Equations
 8.24 Exercises
 8.25 Homogeneous FirstOrder Equations
 8.26 Exercises
 8.27 Some Geometrical and Physical Problems Leading to FirstOrder Equations
 8.28 Miscellaneous Review Exercises

9 Complex Numbers
 9.1 Historical Introduction
 9.2 Definitions and Field Properties
 9.3 The Complex Numbers as an Extension of the Real Numbers
 9.4 The Imaginary Unit 𝑖
 9.5 Geometric Interpretation. Modulus and Argument
 9.6 Exercises
 9.7 Complex Exponentials
 9.8 ComplexValued Functions
 9.9 Examples of Differentiation and Integration Formulas
 9.10 Exercises

10 Sequences, Infinite Series, Improper Integrals
 10.1 Zeno’s Paradox
 10.2 Sequences
 10.3 Monotonic Sequences of Real Numbers
 10.4 Exercises
 10.5 Infinite Series
 10.6 The Linearity Property of Convergent Series
 10.7 Telescoping Series
 10.8 The Geometric Series
 10.9 Exercises
 10.10 Exercises on Decimal Expansions
 10.11 Tests for Convergence
 10.12 Comparison Tests for Series of Nonnegative Terms
 10.13 The Integral Test
 10.14 Exercises
 10.15 The Root Test and the Ratio Test for Series of Nonnegative Terms
 10.16 Exercises
 10.17 Alternating Series
 10.18 Conditional and Absolute Convergence
 10.19 The Convergence Tests of Dirichlet and Abel
 10.20 Exercises
 10.21 Rearrangements of Series
 10.22 Miscellaneous Review Exercises
 10.23 Improper Integrals
 10.24 Exercises

11 Sequences and Series of Functions
 11.1 Pointwise Convergence of Sequences of Functions
 11.2 Uniform Convergence of Sequences of Functions
 11.3 Uniform Convergence and Continuity
 11.4 Uniform Convergence and Integration
 11.5 A Sufficient Condition for Uniform Convergence

11.6 Power Series. Circle of Convergence
 11.7 Exercises
 11.8 Properties of Functions Represented by Real Power Series
 11.9 The Taylor’s Series Generated by a Function
 11.10 A Sufficient Condition for Convergence of a Taylor’s Series
 11.11 PowerSeries Expansions for the Exponential and Trigonometric Functions
 11.12 Bernstein’s Theorem
 11.13 Exercises
 11.14 Power Series and Differential Equations
 11.15 The Binomial Series
 11.16 Exercises

12 Vector Algebra
 12.1 Historical Introduction
 12.2 The Vector Space of 𝑛Tuples of Real Numbers
 12.3 Geometric Interpretation for 𝑛 ≤ 3
 12.4 Exercises
 12.5 The Dot Product
 12.6 Length or Norm of a Vector
 12.7 Orthogonality of Vectors
 12.8 Exercises
 12.9 Projections. Angle between Vectors in 𝑛Space
 12.10 The Unit Coordinate Vectors
 12.11 Exercises
 12.12 The Linear Span of a Finite Set of Vectors
 12.13 Linear Independence
 12.14 Bases
 12.15 Exercises
 12.16 The Vector Space 𝑉_𝑛(𝐶) of 𝑛Tuples of Complex Numbers
 12.17 Exercises

13 Applications of Vector Algebra to Analytic Geometry
 13.1 Introduction
 13.2 Lines in nSpace
 13.3 Some Simple Properties of Straight Lines
 13.4 Lines and VectorValued Functions
 13.5 Exercises
 13.6 Planes in Euclidean nSpace
 13.7 Planes and VectorValued Functions
 13.8 Exercises
 13.9 The Cross Product
 13.10 The Cross Product Expressed as a Determinant
 13.11 Exercises
 13.12 The Scalar Triple Product
 13.13 Cramer’s Rule for Solving a System of Three Linear Equations
 13.14 Exercises
 13.15 Normal Vectors to Planes
 13.16 Linear Cartesian Equations for Planes
 13.17 Exercises
 13.18 The Conic Sections
 13.19 Eccentricity of Conic Sections
 13.20 Polar Equations for Conic Sections
 13.21 Exercises
 13.22 Conic Sections Symmetric about the Origin
 13.23 Cartesian Equations for the Conic Sections
 13.24 Exercises
 13.25 Miscellaneous Exercises on Conic Sections

14 Calculus of VectorValued Functions
 14.1 VectorValued Functions of a Real Variable
 14.2 Algebraic Operations. Components
 14.3 Limits, Derivatives, and Integrals
 14.4 Exercises
 14.5 Applications to Curves. Tangency
 14.6 Applications to Curvilinear Motion. Velocity, Speed, and Acceleration
 14.7 Exercises
 14.8 The Unit Tangent, the Principal Normal, and the Osculating Plane of a Curve
 14.9 Exercises
 14.10 The Definition of Arc Length
 14.11 Additivity of Arc Length
 14.12 The ArcLength Function
 14.13 Exercises
 14.14 Curvature of a Curve
 14.15 Exercises
 14.16 Velocity and Acceleration in Polar Coordinates
 14.17 Plane Motion with Radial Acceleration
 14.18 Cylindrical Coordinates
 14.19 Exercises
 14.20 Applications to Planetary Motion
 14.21 Miscellaneous Review Exercises

15 Linear Spaces
 15.1 Introduction
 15.2 The Definition of a Linear Space
 15.3 Examples of Linear Spaces
 15.4 Elementary Consequences the Axioms
 15.5 Exercises
 15.6 Subspaces of a Linear Space
 15.7 Dependent and Independent Sets in a Linear Space
 15.8 Bases and Dimension
 15.9 Exercises
 15.10 Inner Products, Euclidian Spaces, Norms
 15.11 Orthogonality in a Euclidean Space
 15.12 Exercises
 15.13 Construction of Orthogonal Sets. The GramSchmidt Process
 15.14 Orthogonal Complements. Projections
 15.15 Best Approximation of Elements in a Euclidean Space by Elements in a FiniteDimensional Subspace
 15.16 Exercises

16 Linear Transformations and Matrices
 16.1 Linear Transformations
 16.2 Nul1 Space and Range
 16.3 Nullity and Rank
 16.4 Exercises
 16.5 Algebraic Operations on Linear Transformations
 16.6 Inverses
 16.7 OnetoOne Linear Transformations
 16.8 Exercises
 16.9 Linear Transformations with Prescribed Values
 16.10 Matrix Representations of Linear Transformations
 16.11 Construction of a Matrix Representation in Diagonal Form
 16.12 Exercises
 16.13 Linear Spaces of Matrices
 16.14 Isomorphism between Linear Transformations and Matrices
 16.15 Multiplication of Matrices
 16.16 Exercises
 16.17 Systems of Linear Equations
 16.18 Computation Techniques
 16.19 Inverses of Square Matrices
 16.20 Exercises
 16.21 Miscellaneous Exercises on Matrices