Calculus with Precalculus, 3rd edition

Original Calculus Content

• 1: Functions
  • Introduction
  • 1.1 Review of Functions
  • 1.2 Representing Functions
  • 1.3 Inverse, Exponential, and Logarithmic Functions
  • 1.4 Trigonometric Functions and Their Inverses
  • Review Exercises
  • Guided Projects
• 2: Limits
  • Introduction
  • 2.1 The Idea of Limits
  • 2.2 Definitions of Limits
  • 2.3 Techniques for Computing Limits
  • 2.4 Infinite Limits
  • 2.5 Limits at Infinity
  • 2.6 Continuity
  • 2.7 Precise Definitions of Limits
  • Review Exercises
  • Guided Projects
• 3: Exponentials and Logarithms
  • Introduction
  • 3.1 Introducing the Derivative
  • 3.2 The Derivative as a Function
  • 3.3 Rules of Differentiation
  • 3.4 The Product and Quotient Rules
  • 3.5 Derivatives of Trigonometric Functions
  • 3.6 Derivatives as Rates of Change
  • 3.7 The Chain Rule
  • 3.8 Implicit Differentiation
  • 3.9 Derivatives of Logarithmic and Exponential Functions
  • 3.10 Derivatives of Inverse Trigonometric Functions
  • 3.11 Related Rates
  • Review Exercises
  • Guided Projects
• 4: Applications of the Derivative
  • Introduction
  • 4.1 Maxima and Minima
  • 4.2 Mean Value Theorem
  • 4.3 What Derivatives Tell Us
  • 4.4 Graphing Functions
  • 4.5 Optimization Problems
  • 4.6 Linear Approximation and Differentials
  • 4.7 L’Hôpital’s Rule
  • 4.8 Newton’s Method
  • 4.9 Antiderivatives
  • Review Exercises
  • Guided Projects
• 5: Integration
  • Introduction
  • 5.1 Approximating Areas under Curves
  • 5.2 Definite Integrals
  • 5.3 Fundamental Theorem of Calculus
  • 5.4 Working with Integrals
  • 5.5 Substitution Rule
  • Review Exercises
  • Guided Projects
• 6: Applications of Integration
  • Introduction
  • 6.1 Velocity and Net Change
  • 6.2 Regions Between Curves
  • 6.3 Volume by Slicing
  • 6.4 Volume by Shells
  • 6.5 Length of Curves
  • 6.6 Surface Area
  • 6.7 Physical Applications
  • Review Exercises
  • Guided Projects
• 7: Logarithmic, Exponential, and Hyperbolic Functions
  • Introduction
  • 7.1 Logarithmic and Exponential Functions Revisited
  • 7.2 Exponential Models
  • 7.3 Hyperbolic Functions
  • Review Exercises
  • Guided Projects
• 8: Integration Techniques
  • Introduction
  • 8.1 Basic Approaches
  • 8.2 Integration by Parts
  • 8.3 Trigonometric Integrals
  • 8.4 Trigonometric Substitutions
  • 8.5 Partial Fractions
  • 8.6 Integration Strategies
  • 8.7 Other Methods of Integration
  • 8.8 Numerical Integration
  • 8.9 Improper Integrals
  • Review Exercises
  • Guided Projects
• 9: Vectors
  • Introduction
  • 9.1 Basic Ideas
  • 9.2 Direction Fields and Euler’s Method
  • 9.3 Separable Differential Equations
  • 9.4 Special First-Order Linear Differential Equations
  • 9.5 Modeling with Differential Equations
  • Review Exercises
  • Guided Projects
• 10: Sequences and Infinite Series
  • Introduction
  • 10.1 An Overview
  • 10.2 Sequences
  • 10.3 Infinite Series
  • 10.4 The Divergence and Integral Tests
  • 10.5 Comparison Tests
  • 10.6 Alternating Series
  • 10.7 The Ratio and Root Tests
  • 10.8 Choosing a Convergence Test
  • Review Exercises
  • Guided Projects
• 11: Power Series
  • Introduction
  • 11.1 Approximating Functions with Polynomials
  • 11.2 Properties of Power Series
  • 11.3 Taylor Series
  • 11.4 Working with Taylor Series
  • Review Exercises
  • Guided Projects
• 12: Parametric and Polar Curves
  • Introduction
  • 12.1 Parametric Equations
  • 12.2 Polar Coordinates
  • 12.3 Calculus in Polar Coordinates
  • 12.4 Conic Sections
  • Review Exercises
  • Guided Projects
• 13: Vectors and the Geometry of Space
  • Introduction
  • 13.1 Vectors in the Plane
  • 13.2 Vectors in Three Dimensions
  • 13.3 Dot Products
  • 13.4 Cross Products
  • 13.5 Lines and Planes in Space
  • 13.6 Cylinders and Quadric Surfaces
  • Review Exercises
  • Guided Projects
• 14: Vector-Valued Functions
  • Introduction
  • 14.1 Vector-Valued Functions
  • 14.2 Calculus of Vector-Valued Functions
  • 14.3 Motion in Space
  • 14.4 Length of Curves
  • 14.5 Curvature and Normal Vectors
  • Review Exercises
  • Guided Projects
• 15: Functions of Several Variables
  • Introduction
  • 15.1 Graphs and Level Curves
  • 15.2 Limits and Continuity
  • 15.3 Partial Derivatives
  • 15.4 The Chain Rule
  • 15.5 Directional Derivatives and the Gradient
  • 15.6 Tangent Planes and Linear Approximation
  • 15.7 Maximum/Minimum Problems
  • 15.8 Lagrange Multipliers
  • Review Exercises
  • Guided Projects
• 16: Multiple Integration
  • Introduction
  • 16.1 Double Integrals over Rectangular Regions
  • 16.2 Double Integrals over General Regions
  • 16.3 Double Integrals in Polar Coordinates
  • 16.4 Triple Integrals
  • 16.5 Triple Integrals in Cylindrical and Spherical Coordinates
  • 16.6 Integrals for Mass Calculations
  • 16.7 Change of Variables in Multiple Integrals
  • Review Exercises
  • Guided Projects
• 17: Vector Calculus
  • Introduction
  • 17.1 Vector Fields
  • 17.2 Line Integrals
  • 17.3 Conservative Vector Fields
  • 17.4 Green’s Theorem
  • 17.5 Divergence and Curl
  • 17.6 Surface Integrals
  • 17.7 Stokes’ Theorem
  • 17.8 Divergence Theorem
  • Review Exercises
  • Guided Projects
• Chapter D2: Second-order Differential Equations
  • D2 Introduction
  • D2.1 Basic Ideas
  • D2.2 Linear Homogenous Equations
  • D2.3 Linear Nonhomogenous Equations
  • D2.4 Applications
  • D2.5 Complex Forcing Functions
• Appendices
  • A: Proof of Selected Theorems
  • B: Algebra; Set of Real Numbers; Absolute Value; Cartesian Coordinate System; Equation of Lines; Answers
  • C: Complex Numbers; Complex Arithmetic; Answers

Answers

Index

Index of Applications

Table of Integrals

Useful References

Precalculus Content

• 1: Functions
  • 1.1 Introduction to Functions
  • 1.2 Graphs
  • 1.3 Linear Functions
  • 1.4 Combination of Functions
  • 1.5 Families of Functions
  • Summary and Review Exercises
• 2: Polynomial and Rational Functions
  • 2.1 Quadratic Functions
  • 2.2 Polynomial Functions
  • 2.3 Real Roots and Factors of Polynomial Functions
  • 2.4 Complex Numbers
  • 2.5 Complex Roots of Polynomial Functions
  • 2.6 Rational Functions
  • 2.7 Inequalities
  • Summary and Review Exercises
• 3: Exponentials and Logarithms
  • 3.1 Exponential Functions
  • 3.2 Inverse Functions
  • 3.3 Logarithmic Functions
  • 3.4 Logarithmic Identities
  • 3.5 Solving Exponential and Logarithmic Equations
  • Summary and Review Exercises
• 4: Unit Circle Trigonometry
  • 4.1 Angles and their measures
  • 4.2 Unit Circle Definitions of Sine, Cosine, and Tangent
  • 4.3 Sine, Cosine, and Tangent Functions
  • 4.4 Secant, Cosecant, and Cotangent Functions
  • 4.5 Inverse Trigonometric Functions
  • Summary and Review Exercises
• 5: Triangle Trigonometry
  • 5.1 Right Triangle Trigonometry
  • 5.2 Right Triangles and the Unit Circle
  • 5.3 Law of Sines
  • 5.4 Law of Cosines
  • 5.5 Application of Triangles
  • Summary and Review Exercises
• 6: Trigonometric Identities
  • 6.1 Fundamental Identities
  • 6.2 Sum, Difference, and Double-Angle Identities
  • 6.3 Power-Reducing, Half-Angle, and Product-Sum Identities
  • 6.4 Solving Trigonometric Equations
  • Summary and Review Exercises
• 7: Parametric and Polar Graphs
  • 7.1 Parametric Equations
  • 7.2 Polar Coordinates
  • 7.3 Polar Graphs
  • 7.4 Polar Form of Complex Numbers; DeMoivre's Theorem
  • Summary and Review Exercises
• 8: Conic Sections
  • 8.1 Parabolas
  • 8.2 Ellipses and Circles
  • 8.3 Hyperbolas
  • 8.4 Eccentricity and Polar Equations of Conic Sections
  • Summary and Review Exercises
• 9: Vectors
  • 9.1 Vectors in Two Dimensions
  • 9.2 Vectors in Three Dimensions
  • 9.3 Dot Product
  • 9.4 Cross Product
  • Summary and Review Exercises
• 10: Systems of Equations and Matrices
  • 10.1 Systems of Equations and Inequalities
  • 10.2 Matrices
  • 10.3 Method of Partial Fractions
  • Summary and Review Exercises<
• Appendices
  • A: Looking Forward to Calculus: Limits
  • B: Looking Forward to Calculus: Sequence and Series

Answers to Odd Exercises

Index

Application Index