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# Calculus I with Integrated Precalculus

## First Edition| ©2014 Laura Taalman

Taalman’s *Calculus I with Integrated Precalculus* helps students with weak mathematical backgrounds be successful in the calculus sequence, without retaking a precalculus course.

Taalman’s innovative text is the only book to interweave calculus with precalculus and algebra in

Taalman’s *Calculus I with Integrated Precalculus* helps students with weak mathematical backgrounds be successful in the calculus sequence, without retaking a precalculus course.

Taalman’s innovative text is the only book to interweave calculus with precalculus and algebra in a manner suitable for math and science majors— not a rehashing or just-in-time review of precalculus and algebra, but rather a new approach that uses a calculus-level toolbox to examine the structure and behavior of algebraic and transcendental functions.

This book was written specifically to tie in with the material covered in Taalman/Kohn *Calculus.* Students who begin their calculus sequence with *Calculus I with Integrated Precalculus* can easily continue on to Calculus II using the Taalman/Kohn text.

**Maximize Teaching and Learning with WebAssign Premium**

Macmillan Learning and WebAssign have partnered to deliver WebAssign *Premium* – a comprehensive and flexible suite of resources for your calculus course. Combining the most widely used online homework platform with authoritative textbook content and Macmillan’s esteemed Calctools, WebAssign *Premium* extends and enhances the classroom experience for instructors and students. Preview course content and sample assignments at www.webassign.net/whfreeman.

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Taalman’s *Calculus I with Integrated Precalculus* helps students with weak mathematical backgrounds be successful in the calculus sequence, without retaking a precalculus course.

Taalman’s innovative text is the only book to interweave calculus with precalculus and algebra in a manner suitable for math and science majors— not a rehashing or just-in-time review of precalculus and algebra, but rather a new approach that uses a calculus-level toolbox to examine the structure and behavior of algebraic and transcendental functions.

This book was written specifically to tie in with the material covered in Taalman/Kohn *Calculus.* Students who begin their calculus sequence with *Calculus I with Integrated Precalculus* can easily continue on to Calculus II using the Taalman/Kohn text.

**Maximize Teaching and Learning with WebAssign Premium**

Macmillan Learning and WebAssign have partnered to deliver WebAssign *Premium* – a comprehensive and flexible suite of resources for your calculus course. Combining the most widely used online homework platform with authoritative textbook content and Macmillan’s esteemed Calctools, WebAssign *Premium* extends and enhances the classroom experience for instructors and students. Preview course content and sample assignments at www.webassign.net/whfreeman.

Features

**A Calculus 1 Course with Precalculus and Algebra**

This one-year alternative to Calculus I covers everything from factoring, functions, and graphs to delta/epsilon limit statements, derivatives, Riemann sums, and integration techniques. Coverage is sufficient for many students to complete their calculus studies, while paving the way for others to continue to Calculus II and III using any other textbook.

**Calculus First, Complex Computation Later**

Part I explores the basic concepts, definitions, and theorems of functions, limits, and differential calculus with examples based on simple calculations. In Part II, students study different classes of functions (general algebraic, exponential and logarithmic, and trigonometric functions). Increasing computational complexity over time allows students to gain skills and confidence. The third part of the book covers integration.

**Compact, Structured Exposition**

*Calculus I with Integrated Precalculus*offers students clear and compact instruction in a linear, straightforward, and predictable format. The book’s patient, uncluttered approach is structured to maximize student comprehension. Each section begins with uncluttered exposition that introduces and explains the concepts at hand and includes definitions, short examples and figures, and key theorems with proofs. The section is completed by longer, more complex examples that prepare students for the exercise sets.

**Categorized Exercises**

Exercises reflect concepts and skills introduced in the section examples; but also test student understanding of the development of the material and the reading itself and. Each exercise set includes the following categories of problems: *Thinking Back*, *Concepts,* *Skills*, *Applications*, Proofs, and *Thinking Forward.* Each chapter concludes with a set of *Chapter Review* exercises.

New to This Edition

**
Calculus I with Integrated Precalculus**

First Edition| ©2014

Laura Taalman

# Digital Options

**Calculus I with Integrated Precalculus**

First Edition| 2014

Laura Taalman

## Table of Contents

**I Differential Calculus**

**0 Functions and Precalculus**

0.1 Numbers and Sets

0.2 Equations

0.3 Inequalities

0.4 Functions and Graphs

0.5 A Basic Library of Functions

0.6 Operations, Transformations, and Inverses

0.7 Logic and Mathematical Thinking**Chapter Review, Self-Test, and Capstones*

**1 Limits**

1.1 An Intuitive Introduction to Limits

1.2 Formal Definition of a Limit

1.3 Delta-Epsilon Proofs*

1.4 Continuity and Its Consequences

1.5 Limit Rules and Calculating Basic Limits

1.6 Infinite Limits and Indeterminate Forms*Chapter Review, Self-Test, and Capstones*

**2 Derivatives**

2.1 An Intuitive Introduction to Derivatives

2.2 Formal Definition of the Derivative

2.3 Rules for Calculating Basic Derivatives

2.4 The Chain Rule and Implicit Differentiation*Chapter Review, Self-Test, and Capstones*

**3 Applications of the Derivative**

3.1 The Mean Value Theorem

3.2 The First Derivative and Curve Sketching

3.3 The Second Derivative and Curve Sketching

3.4 Optimization

3.5 Related Rates*Chapter Review, Self-Test, and Capstones*

**II Exploring Functions with Calculus**

**4 Calculus with Power, Polynomial, and Rational Functions**4.1 Advanced Algebraic Techniques

4.2 Power Functions

4.3 Polynomial Functions

4.4 Rational Functions

*Chapter Review, Self-Test, and Capstones*

**5 Calculus with Exponential and Logarithmic Functions**

5.1 Defining Exponential and Logarithmic Functions

5.2 Limits of Exponential and Logarithmic Functions

5.3 Derivatives of Exponential and Logarithmic Functions

5.4 Applications of Exponential Functions

5.5 L’Hopital’s Rule*Chapter Review, Self-Test, and Capstones*

**6 Calculus with Trignometric and Inverse Trigonometric Functions**

6.1 Defining the Trigonometric Functions

6.2 Trigonometric Identities

6.3 Limits and Derivatives of Trigonometric Functions

6.4 Inverse Trigonometric Functions*Chapter Review, Self-Test, and Capstones*

**III Integral Calculus**

**7 Definite Integrals**

7.1 Addition and Accumulation

7.2 Riemann Sums

7.3 Definite Integrals

7.4 Indefinite Integrals

7.5 The Fundamental Theorem of Calculus

7.6 Areas and Average Values

7.7 Functions Defined by Integrals*Chapter Review, Self-Test, and Capstones*

**8 Techniques of Integration**

8.1 Integration by Substitution

8.2 Integration by Parts

8.3 Partial Fractions and Other Algebraic Techniques

8.4 Trigonometric Integrals

8.5 Trigonometric Substitution

8.6 Improper Integrals

8.7 Numerical Integration**Chapter Review, Self-Test, and Capstones*

**9 Applications of Integration**

9.1 Volumes by Slicing

9.2 Volumes by Shells

9.3 Arc Length and Surface Area

9.4 Real-World Applications of Integration

9.5 Differential Equations**Chapter Review, Self-Test, and Capstones*

Answers to Odd Problems

Index

## Authors

### Laura Taalman

**Laura Taalman**and

**Peter Kohn**are professors of mathematics at James Madison University, where they have taught calculus for a combined total of over 30 years.

**Laura Taalman**received her undergraduate degree from the University of Chicago

and master’s and Ph.D. degrees in mathematics from Duke University. Her research includes singular algebraic geometry, knot theory, and the mathematics of games and puzzles. She is a recipient of both the Alder Award and the Trevor Evans award from the Mathematical Association of America, and the author of five books on Sudoku and the mathematics of Sudoku. In her spare time, she enjoys being a geek.

**Peter Kohn**received his undergraduate degree from Antioch College, a master’s

degree from San Francisco State University, and a Ph.D. in mathematics from the University of Texas at Austin. His main areas of research are low-dimensional topology and knot theory. He has been a national judge for MathCounts since 2001. In his spare time, he enjoys hiking and riding his bicycle in the beautiful Shenandoah Valley.

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**Calculus I with Integrated Precalculus**

First Edition| 2014

Laura Taalman

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