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by Ron Larson

Edition: 4TH 07Copyright: 2007

Publisher: Houghton Mifflin Harcourt

Published: 2007

International: No

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Designed for the three-semester engineering calculus course, Calculus: Early Transcendental Functions, 4/e, continues to offer instructors and students innovative teaching and learning resources. Two primary objectives guided the authors in the revision of this book: to develop precise, readable materials for students that clearly define and demonstrate concepts and rules of calculus; and to design comprehensive teaching resources for instructors that employ proven pedagogical techniques and save time. The Larson/Hostetler/Edwards Calculus program offers a solution to address the needs of any calculus course and any level of calculus student.

Every edition from the first to the fourth of Calculus: Early Transcendental Functions, 4/e has made the mastery of traditional calculus skills a priority, while embracing the best features of new technology and, when appropriate, calculus reform ideas. Now, the Fourth Edition is part of the first calculus program to offer algorithmic homework and testing created in Maple so that answers can be evaluated with complete mathematical accuracy.

- Exercise sets have been carefully examined and revised to ensure they cover all calculus topics appropriately. Many new exercises have been added.
- A variety of exercise types are included in each exercise set. Questions involving skills, writing, critical thinking, problem-solving, applications, and real-data applications are included throughout the text. Exercises are presented in a variety of question formats, including matching, free response, true/false, modeling, and fill-in the blank.
- Putnam Exam Questions--taken from the William Lowell Putnam Mathematical Competition--offer challenging problems that often require students to look for creative solutions; Graphical Analysis exercises offer the opportunity to analyze graphs; Think About It exercises require students to use critical reasoning skills to explore the intricacies of calculus.
- Explanations, theorems, and definitions in the text have been thoroughly reviewed to ensure the text is mathematically precise and easily comprehensible.
- Clear, multi-step examples with worked-out solutions help students learn difficult mathematical concepts. Examples correspond to the exercises, serving as a supportive reference for students. This is the only text on the market where every example, proof, and explanation begins and ends on the same page.
- Explorations help students develop their intuitive understanding of calculus concepts. These optional activities are short enough to integrate into class, but they can also be omitted without loss of continuity.
- Theorem boxes clearly explain important mathematical concepts.
- The Integrated Learning System resources are available in print, CD-ROM, and online formats.
- Eduspace, powered by Blackboard, Houghton Mifflin's online learning tool, offers your students quality online homework, tutorials, multimedia, and testing that correspond to the Calculus: Early Transcendental Functions text. This content is paired with the course management tools of Blackboard. In addition, eSolutions, the complete solutions to the odd-numbered text exercises, provides students with a convenient and comprehensive way to do homework and view the course materials.
- SMARTHINKING online tutoring brings students real-time, online tutorial support when they need it most.

**1. Preparation for Calculus**

1.1 Graphs and Models

1.2 Linear Models and Rates of Change

1.3 Functions and Their Graphs

1.4 Fitting Models to Data

1.5 Inverse Functions

1.6 Exponential and Logarithmic Functions

**2. Limits and Their Properties**

2.1 A Preview of Calculus

2.2 Finding Limits Graphically and Numerically

2.3 Evaluating Limits Analytically

2.4 Continuity and One-Sided Limits

2.5 Infinite Limits

Section Project: Graphs and Limits of Trig Functions

**3. Differentiation**

3.1 The Derivative and the Tangent Line Problem

3.2 Basic Differentiation Rules and Rates of Change

3.3 The Product and Quotient Rules and Higher-Order Derivatives

3.4 The Chain Rule

3.5 Implicit Differentiation

Section Project: Optical Illusions

3.6 Derivatives of Inverse Functions

3.7 Related Rates

3.8 Newton's Method

**4. Applications of Differentiation**

4.1 Extrema on an Interval

4.2 Rolle's Theorem and the Mean Value Theorem

4.3 Increasing and Decreasing Functions and the First Derivative Test

Section Project: Rainbows

4.4 Concavity and the Second Derivative Test

4.5 Limits at Infinity

4.6 A Summary of Curve Sketching

4.7 Optimization Problems

Section Project: Connecticut River

4.8 Differentials

**5. Integration**

5.1 Antiderivatives and Indefinite Integration

5.2 Area

5.3 Riemann Sums and Definite Integrals

5.4 The Fundamental Theorem of Calculus

Section Project: Demonstrating the Fundamental Theorem

5.5 Integration by Substitution

5.6 Numerical Integration

5.7 The Natural Logarithmic Function: Integration

5.8 Inverse Trigonometric Functions: Integration

5.9 Hyperbolic Functions

Section Project: St. Louis Arch

**6. Differential Equations**

6.1 Slope Fields and Euler's Method

6.4 Differential Equations: Growth and Decay

6.5 Differential Equations: Separation of Variables

6.4 The Logistic Equation

6.5 First-Order Linear Differential Equations

Section Project: Weight Loss

6.6 Predator-Prey Differential Equations

**7. Applications of Integration**

7.1 Area of a Region Between Two Curves

7.2 Volume: The Disk Method

7.3 Volume: The Shell Method

7.4 Arc Length and Surfaces of Revolution

7.5 Work

7.6 Moments, Centers of Mass, and Centroids

7.7 Fluid Pressure and Fluid Force

**8. Integration Techniques, L'Hôpital's Rule, and Improper Integrals**

8.1 Basic Integration Rules

8.2 Integration by Parts

8.3 Trigonometric Integrals

8.4 Trigonometric Substitution

8.5 Partial Fractions

8.6 Integration by Tables and Other Integration Techniques

8.7 Indeterminate Forms and L'Hôpital's Rule

8.8 Improper Integrals

**9. Infinite Series**

9.1 Sequences

9.2 Series and Convergence

9.3 The Integral Test and p-Series

9.4 Comparisons of Series

9.5 Alternating Series

9.6 The Ratio and Root Tests

9.7 Taylor Polynomials and Approximations

9.8 Power Series

9.9 Representation of Functions by Power Series

9.10 Taylor and Maclaurin Series

**10. Conics, Parametric Equations, and Polar Coordinates**

10.1 Conics and Calculus

10.2 Plane Curves and Parametric Equations

10.3 Parametric Equations and Calculus

10.4 Polar Coordinates and Polar Graphs

10.5 Area and Arc Length in Polar Coordinates

10.6 Polar Equations of Conics and Kepler's Laws

**11. Vectors and the Geometry of Space**

11.1 Vectors in the Plane

11.2 Space Coordinates and Vectors in Space

11.3 The Dot Product of Two Vectors

11.4 The Cross Product of Two Vectors in Space

11.5 Lines and Planes in Space

11.6 Surfaces in Space

11.7 Cylindrical and Spherical Coordinates

**12. Vector-Valued Functions**

12.1 Vector-Valued Functions

12.2 Differentiation and Integration of Vector-Valued Functions

12.3 Velocity and Acceleration

12.4 Tangent Vectors and Normal Vectors

12.5 Arc Length and Curvature

**13. Functions of Several Variables**

13.1 Introduction to Functions of Several Variables

13.2 Limits and Continuity

13.3 Partial Derivatives

13.4 Differentials

13.5 Chain Rules for Functions of Several Variables

13.6 Directional Derivatives and Gradients

13.7 Tangent Planes and Normal Lines

13.8 Extrema of Functions of Two Variables

13.9 Applications of Extrema of Functions of Two Variables

13.10 Lagrange Multipliers

**14. Multiple Integration**

14.1 Iterated Integrals and Area in the Plane

14.2 Double Integrals and Volume

14.3 Change of Variables: Polar Coordinates

14.4 Center of Mass and Moments of Inertia

14.5 Surface Area

14.6 Triple Integrals and Applications

14.7 Triple Integrals in Cylindrical and Spherical Coordinates

14.8 Change of Variables: Jacobians

**15. Vector Analysis**

15.1 Vector Fields

15.2 Line Integrals

15.3 Conservative Vector Fields and Independence of Path

15.4 Green's Theorem

15.5 Parametric Surfaces

15.6 Surface Integrals

15.7 Divergence Theorem

15.8 Stoke's Theorem

**Appendices**

Appendix A Proofs of Selected Theorems

Appendix B Integration Tables

Appendix C Business and Economic Applications

Additional Appendices

The following appendices are available at the textbook website, on the HM mathSpace Student CD-ROM, and the HM ClassPrep with HM Testing CD-ROM:

Appendix D Precalculus Review

Appendix E Rotation and the General Second-Degree Equation

Appendix F Complex Numbers

Summary

Designed for the three-semester engineering calculus course, Calculus: Early Transcendental Functions, 4/e, continues to offer instructors and students innovative teaching and learning resources. Two primary objectives guided the authors in the revision of this book: to develop precise, readable materials for students that clearly define and demonstrate concepts and rules of calculus; and to design comprehensive teaching resources for instructors that employ proven pedagogical techniques and save time. The Larson/Hostetler/Edwards Calculus program offers a solution to address the needs of any calculus course and any level of calculus student.

Every edition from the first to the fourth of Calculus: Early Transcendental Functions, 4/e has made the mastery of traditional calculus skills a priority, while embracing the best features of new technology and, when appropriate, calculus reform ideas. Now, the Fourth Edition is part of the first calculus program to offer algorithmic homework and testing created in Maple so that answers can be evaluated with complete mathematical accuracy.

- Exercise sets have been carefully examined and revised to ensure they cover all calculus topics appropriately. Many new exercises have been added.
- A variety of exercise types are included in each exercise set. Questions involving skills, writing, critical thinking, problem-solving, applications, and real-data applications are included throughout the text. Exercises are presented in a variety of question formats, including matching, free response, true/false, modeling, and fill-in the blank.
- Putnam Exam Questions--taken from the William Lowell Putnam Mathematical Competition--offer challenging problems that often require students to look for creative solutions; Graphical Analysis exercises offer the opportunity to analyze graphs; Think About It exercises require students to use critical reasoning skills to explore the intricacies of calculus.
- Explanations, theorems, and definitions in the text have been thoroughly reviewed to ensure the text is mathematically precise and easily comprehensible.
- Clear, multi-step examples with worked-out solutions help students learn difficult mathematical concepts. Examples correspond to the exercises, serving as a supportive reference for students. This is the only text on the market where every example, proof, and explanation begins and ends on the same page.
- Explorations help students develop their intuitive understanding of calculus concepts. These optional activities are short enough to integrate into class, but they can also be omitted without loss of continuity.
- Theorem boxes clearly explain important mathematical concepts.
- The Integrated Learning System resources are available in print, CD-ROM, and online formats.
- Eduspace, powered by Blackboard, Houghton Mifflin's online learning tool, offers your students quality online homework, tutorials, multimedia, and testing that correspond to the Calculus: Early Transcendental Functions text. This content is paired with the course management tools of Blackboard. In addition, eSolutions, the complete solutions to the odd-numbered text exercises, provides students with a convenient and comprehensive way to do homework and view the course materials.
- SMARTHINKING online tutoring brings students real-time, online tutorial support when they need it most.

Table of Contents

**1. Preparation for Calculus**

1.1 Graphs and Models

1.2 Linear Models and Rates of Change

1.3 Functions and Their Graphs

1.4 Fitting Models to Data

1.5 Inverse Functions

1.6 Exponential and Logarithmic Functions

**2. Limits and Their Properties**

2.1 A Preview of Calculus

2.2 Finding Limits Graphically and Numerically

2.3 Evaluating Limits Analytically

2.4 Continuity and One-Sided Limits

2.5 Infinite Limits

Section Project: Graphs and Limits of Trig Functions

**3. Differentiation**

3.1 The Derivative and the Tangent Line Problem

3.2 Basic Differentiation Rules and Rates of Change

3.3 The Product and Quotient Rules and Higher-Order Derivatives

3.4 The Chain Rule

3.5 Implicit Differentiation

Section Project: Optical Illusions

3.6 Derivatives of Inverse Functions

3.7 Related Rates

3.8 Newton's Method

**4. Applications of Differentiation**

4.1 Extrema on an Interval

4.2 Rolle's Theorem and the Mean Value Theorem

4.3 Increasing and Decreasing Functions and the First Derivative Test

Section Project: Rainbows

4.4 Concavity and the Second Derivative Test

4.5 Limits at Infinity

4.6 A Summary of Curve Sketching

4.7 Optimization Problems

Section Project: Connecticut River

4.8 Differentials

**5. Integration**

5.1 Antiderivatives and Indefinite Integration

5.2 Area

5.3 Riemann Sums and Definite Integrals

5.4 The Fundamental Theorem of Calculus

Section Project: Demonstrating the Fundamental Theorem

5.5 Integration by Substitution

5.6 Numerical Integration

5.7 The Natural Logarithmic Function: Integration

5.8 Inverse Trigonometric Functions: Integration

5.9 Hyperbolic Functions

Section Project: St. Louis Arch

**6. Differential Equations**

6.1 Slope Fields and Euler's Method

6.4 Differential Equations: Growth and Decay

6.5 Differential Equations: Separation of Variables

6.4 The Logistic Equation

6.5 First-Order Linear Differential Equations

Section Project: Weight Loss

6.6 Predator-Prey Differential Equations

**7. Applications of Integration**

7.1 Area of a Region Between Two Curves

7.2 Volume: The Disk Method

7.3 Volume: The Shell Method

7.4 Arc Length and Surfaces of Revolution

7.5 Work

7.6 Moments, Centers of Mass, and Centroids

7.7 Fluid Pressure and Fluid Force

**8. Integration Techniques, L'Hôpital's Rule, and Improper Integrals**

8.1 Basic Integration Rules

8.2 Integration by Parts

8.3 Trigonometric Integrals

8.4 Trigonometric Substitution

8.5 Partial Fractions

8.6 Integration by Tables and Other Integration Techniques

8.7 Indeterminate Forms and L'Hôpital's Rule

8.8 Improper Integrals

**9. Infinite Series**

9.1 Sequences

9.2 Series and Convergence

9.3 The Integral Test and p-Series

9.4 Comparisons of Series

9.5 Alternating Series

9.6 The Ratio and Root Tests

9.7 Taylor Polynomials and Approximations

9.8 Power Series

9.9 Representation of Functions by Power Series

9.10 Taylor and Maclaurin Series

**10. Conics, Parametric Equations, and Polar Coordinates**

10.1 Conics and Calculus

10.2 Plane Curves and Parametric Equations

10.3 Parametric Equations and Calculus

10.4 Polar Coordinates and Polar Graphs

10.5 Area and Arc Length in Polar Coordinates

10.6 Polar Equations of Conics and Kepler's Laws

**11. Vectors and the Geometry of Space**

11.1 Vectors in the Plane

11.2 Space Coordinates and Vectors in Space

11.3 The Dot Product of Two Vectors

11.4 The Cross Product of Two Vectors in Space

11.5 Lines and Planes in Space

11.6 Surfaces in Space

11.7 Cylindrical and Spherical Coordinates

**12. Vector-Valued Functions**

12.1 Vector-Valued Functions

12.2 Differentiation and Integration of Vector-Valued Functions

12.3 Velocity and Acceleration

12.4 Tangent Vectors and Normal Vectors

12.5 Arc Length and Curvature

**13. Functions of Several Variables**

13.1 Introduction to Functions of Several Variables

13.2 Limits and Continuity

13.3 Partial Derivatives

13.4 Differentials

13.5 Chain Rules for Functions of Several Variables

13.6 Directional Derivatives and Gradients

13.7 Tangent Planes and Normal Lines

13.8 Extrema of Functions of Two Variables

13.9 Applications of Extrema of Functions of Two Variables

13.10 Lagrange Multipliers

**14. Multiple Integration**

14.1 Iterated Integrals and Area in the Plane

14.2 Double Integrals and Volume

14.3 Change of Variables: Polar Coordinates

14.4 Center of Mass and Moments of Inertia

14.5 Surface Area

14.6 Triple Integrals and Applications

14.7 Triple Integrals in Cylindrical and Spherical Coordinates

14.8 Change of Variables: Jacobians

**15. Vector Analysis**

15.1 Vector Fields

15.2 Line Integrals

15.3 Conservative Vector Fields and Independence of Path

15.4 Green's Theorem

15.5 Parametric Surfaces

15.6 Surface Integrals

15.7 Divergence Theorem

15.8 Stoke's Theorem

**Appendices**

Appendix A Proofs of Selected Theorems

Appendix B Integration Tables

Appendix C Business and Economic Applications

Additional Appendices

The following appendices are available at the textbook website, on the HM mathSpace Student CD-ROM, and the HM ClassPrep with HM Testing CD-ROM:

Appendix D Precalculus Review

Appendix E Rotation and the General Second-Degree Equation

Appendix F Complex Numbers

Publisher Info

Publisher: Houghton Mifflin Harcourt

Published: 2007

International: No

Published: 2007

International: No

Designed for the three-semester engineering calculus course, Calculus: Early Transcendental Functions, 4/e, continues to offer instructors and students innovative teaching and learning resources. Two primary objectives guided the authors in the revision of this book: to develop precise, readable materials for students that clearly define and demonstrate concepts and rules of calculus; and to design comprehensive teaching resources for instructors that employ proven pedagogical techniques and save time. The Larson/Hostetler/Edwards Calculus program offers a solution to address the needs of any calculus course and any level of calculus student.

Every edition from the first to the fourth of Calculus: Early Transcendental Functions, 4/e has made the mastery of traditional calculus skills a priority, while embracing the best features of new technology and, when appropriate, calculus reform ideas. Now, the Fourth Edition is part of the first calculus program to offer algorithmic homework and testing created in Maple so that answers can be evaluated with complete mathematical accuracy.

- Exercise sets have been carefully examined and revised to ensure they cover all calculus topics appropriately. Many new exercises have been added.
- A variety of exercise types are included in each exercise set. Questions involving skills, writing, critical thinking, problem-solving, applications, and real-data applications are included throughout the text. Exercises are presented in a variety of question formats, including matching, free response, true/false, modeling, and fill-in the blank.
- Putnam Exam Questions--taken from the William Lowell Putnam Mathematical Competition--offer challenging problems that often require students to look for creative solutions; Graphical Analysis exercises offer the opportunity to analyze graphs; Think About It exercises require students to use critical reasoning skills to explore the intricacies of calculus.
- Explanations, theorems, and definitions in the text have been thoroughly reviewed to ensure the text is mathematically precise and easily comprehensible.
- Clear, multi-step examples with worked-out solutions help students learn difficult mathematical concepts. Examples correspond to the exercises, serving as a supportive reference for students. This is the only text on the market where every example, proof, and explanation begins and ends on the same page.
- Explorations help students develop their intuitive understanding of calculus concepts. These optional activities are short enough to integrate into class, but they can also be omitted without loss of continuity.
- Theorem boxes clearly explain important mathematical concepts.
- The Integrated Learning System resources are available in print, CD-ROM, and online formats.
- Eduspace, powered by Blackboard, Houghton Mifflin's online learning tool, offers your students quality online homework, tutorials, multimedia, and testing that correspond to the Calculus: Early Transcendental Functions text. This content is paired with the course management tools of Blackboard. In addition, eSolutions, the complete solutions to the odd-numbered text exercises, provides students with a convenient and comprehensive way to do homework and view the course materials.
- SMARTHINKING online tutoring brings students real-time, online tutorial support when they need it most.

**1. Preparation for Calculus**

1.1 Graphs and Models

1.2 Linear Models and Rates of Change

1.3 Functions and Their Graphs

1.4 Fitting Models to Data

1.5 Inverse Functions

1.6 Exponential and Logarithmic Functions

**2. Limits and Their Properties**

2.1 A Preview of Calculus

2.2 Finding Limits Graphically and Numerically

2.3 Evaluating Limits Analytically

2.4 Continuity and One-Sided Limits

2.5 Infinite Limits

Section Project: Graphs and Limits of Trig Functions

**3. Differentiation**

3.1 The Derivative and the Tangent Line Problem

3.2 Basic Differentiation Rules and Rates of Change

3.3 The Product and Quotient Rules and Higher-Order Derivatives

3.4 The Chain Rule

3.5 Implicit Differentiation

Section Project: Optical Illusions

3.6 Derivatives of Inverse Functions

3.7 Related Rates

3.8 Newton's Method

**4. Applications of Differentiation**

4.1 Extrema on an Interval

4.2 Rolle's Theorem and the Mean Value Theorem

4.3 Increasing and Decreasing Functions and the First Derivative Test

Section Project: Rainbows

4.4 Concavity and the Second Derivative Test

4.5 Limits at Infinity

4.6 A Summary of Curve Sketching

4.7 Optimization Problems

Section Project: Connecticut River

4.8 Differentials

**5. Integration**

5.1 Antiderivatives and Indefinite Integration

5.2 Area

5.3 Riemann Sums and Definite Integrals

5.4 The Fundamental Theorem of Calculus

Section Project: Demonstrating the Fundamental Theorem

5.5 Integration by Substitution

5.6 Numerical Integration

5.7 The Natural Logarithmic Function: Integration

5.8 Inverse Trigonometric Functions: Integration

5.9 Hyperbolic Functions

Section Project: St. Louis Arch

**6. Differential Equations**

6.1 Slope Fields and Euler's Method

6.4 Differential Equations: Growth and Decay

6.5 Differential Equations: Separation of Variables

6.4 The Logistic Equation

6.5 First-Order Linear Differential Equations

Section Project: Weight Loss

6.6 Predator-Prey Differential Equations

**7. Applications of Integration**

7.1 Area of a Region Between Two Curves

7.2 Volume: The Disk Method

7.3 Volume: The Shell Method

7.4 Arc Length and Surfaces of Revolution

7.5 Work

7.6 Moments, Centers of Mass, and Centroids

7.7 Fluid Pressure and Fluid Force

**8. Integration Techniques, L'Hôpital's Rule, and Improper Integrals**

8.1 Basic Integration Rules

8.2 Integration by Parts

8.3 Trigonometric Integrals

8.4 Trigonometric Substitution

8.5 Partial Fractions

8.6 Integration by Tables and Other Integration Techniques

8.7 Indeterminate Forms and L'Hôpital's Rule

8.8 Improper Integrals

**9. Infinite Series**

9.1 Sequences

9.2 Series and Convergence

9.3 The Integral Test and p-Series

9.4 Comparisons of Series

9.5 Alternating Series

9.6 The Ratio and Root Tests

9.7 Taylor Polynomials and Approximations

9.8 Power Series

9.9 Representation of Functions by Power Series

9.10 Taylor and Maclaurin Series

**10. Conics, Parametric Equations, and Polar Coordinates**

10.1 Conics and Calculus

10.2 Plane Curves and Parametric Equations

10.3 Parametric Equations and Calculus

10.4 Polar Coordinates and Polar Graphs

10.5 Area and Arc Length in Polar Coordinates

10.6 Polar Equations of Conics and Kepler's Laws

**11. Vectors and the Geometry of Space**

11.1 Vectors in the Plane

11.2 Space Coordinates and Vectors in Space

11.3 The Dot Product of Two Vectors

11.4 The Cross Product of Two Vectors in Space

11.5 Lines and Planes in Space

11.6 Surfaces in Space

11.7 Cylindrical and Spherical Coordinates

**12. Vector-Valued Functions**

12.1 Vector-Valued Functions

12.2 Differentiation and Integration of Vector-Valued Functions

12.3 Velocity and Acceleration

12.4 Tangent Vectors and Normal Vectors

12.5 Arc Length and Curvature

**13. Functions of Several Variables**

13.1 Introduction to Functions of Several Variables

13.2 Limits and Continuity

13.3 Partial Derivatives

13.4 Differentials

13.5 Chain Rules for Functions of Several Variables

13.6 Directional Derivatives and Gradients

13.7 Tangent Planes and Normal Lines

13.8 Extrema of Functions of Two Variables

13.9 Applications of Extrema of Functions of Two Variables

13.10 Lagrange Multipliers

**14. Multiple Integration**

14.1 Iterated Integrals and Area in the Plane

14.2 Double Integrals and Volume

14.3 Change of Variables: Polar Coordinates

14.4 Center of Mass and Moments of Inertia

14.5 Surface Area

14.6 Triple Integrals and Applications

14.7 Triple Integrals in Cylindrical and Spherical Coordinates

14.8 Change of Variables: Jacobians

**15. Vector Analysis**

15.1 Vector Fields

15.2 Line Integrals

15.3 Conservative Vector Fields and Independence of Path

15.4 Green's Theorem

15.5 Parametric Surfaces

15.6 Surface Integrals

15.7 Divergence Theorem

15.8 Stoke's Theorem

**Appendices**

Appendix A Proofs of Selected Theorems

Appendix B Integration Tables

Appendix C Business and Economic Applications

Additional Appendices

The following appendices are available at the textbook website, on the HM mathSpace Student CD-ROM, and the HM ClassPrep with HM Testing CD-ROM:

Appendix D Precalculus Review

Appendix E Rotation and the General Second-Degree Equation

Appendix F Complex Numbers