# Alignment: Calculus • Volume 1 • One-Variable Calculus, with an Introduction to Linear Algebra • Second Edition

Use Nagwa in conjunction with your preferred textbook. The recommended lessons from Nagwa for each section of this textbook are provided below. This alignment is not affiliated with, sponsored by, or endorsed by the publisher of the referenced textbook. Nagwa is a registered trademark of Nagwa Limited. All other trademarks and registered trademarks are the property of their respective owners.

• 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 Real-Number 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 Least-Upper-Bound Axiom (Completeness Axiom)
• 3.10 The Archimedean Property of the Real-Number 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 Well-Ordering Principle
• 4.4 Exercises
• 4.5 Proof of the Well-Ordering 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
• 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
• 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 Intermediate-Value 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 Extreme-Value Theorem for Continuous Functions
• 3.17 The Small-Span Theorem for Continuous Functions (Uniform Continuity)
• 3.18 The Integrability Theorem for Continuous Functions
• 3.19 Mean-Value 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 Mean-Value Theorem for Derivatives
• 4.15 Exercises
• 4.16 Applications of the Mean-Value Theorem to Geometric Properties of Functions
• 4.17 Second-Derivative 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 Zero-Derivative 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 o-Notation
• 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
• 8 Introduction to Differential Equations
• 8.1 Introduction
• 8.2 Terminology and Notation
• 8.3 A First-Order Differential Equation for the Exponential Function
• 8.4 First-Order Linear Differential Equations
• 8.5 Exercises
• 8.6 Some Physical Problems Leading to First-Order 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 Second-Order 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 First-Order Separable Equations
• 8.24 Exercises
• 8.25 Homogeneous First-Order Equations
• 8.26 Exercises
• 8.27 Some Geometrical and Physical Problems Leading to First-Order 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 Complex-Valued Functions
• 9.9 Examples of Differentiation and Integration Formulas
• 9.10 Exercises
• 10 Sequences, Infinite Series, Improper Integrals
• 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 Power-Series 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 𝑛-Tuples 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 𝑛-Space
• 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 𝑛-Tuples of Complex Numbers
• 12.17 Exercises
• 13 Applications of Vector Algebra to Analytic Geometry
• 13.1 Introduction
• 13.2 Lines in n-Space
• 13.3 Some Simple Properties of Straight Lines
• 13.4 Lines and Vector-Valued Functions
• 13.5 Exercises
• 13.6 Planes in Euclidean n-Space
• 13.7 Planes and Vector-Valued 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 Vector-Valued Functions
• 14.1 Vector-Valued 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 Arc-Length 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 Gram-Schmidt Process
• 15.14 Orthogonal Complements. Projections
• 15.15 Best Approximation of Elements in a Euclidean Space by Elements in a Finite-Dimensional 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 One-to-One 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