Topics in Calculus

COURSE MATERIALS (Algebra I and II, Geometry, Precalculus, Calculus)

CALCULUS INDEX CARDS: hold the entire course in the palm of your hand

CALCULUS INDEX CARDS: hold the entire course in the palm of your hand

A COMPLETE TEXTBOOK (pdf files):

by Dr. Carol JVF Burns

I wrote a Calculus book (which incorporates math language ideas, of course!) years ago.
My wonderful husband was able to recover the TeX files from my old floppy disks.
I got TeXWorks to process them.
I have to scan my hand-drawn images and insert them in the pdf files, which takes
some time. But—soon—I'll have my entire Calculus course online!

(All the solutions will be posted, too.

There are selected solutions at the end of the main text,

and then a supplemental complete Solution Manual.)

TABLE OF CONTENTS (i–ii)

PREFACE (iii–v)

ACKNOWLEDGEMENTS (vi)

TABLE OF SYMBOLS (viii–ix)

1.1 The Language of Mathematics—Expressions versus Sentences (1–11)

1.2 The Role of Variables (12–21)

1.3 Sets and Set Notation (22–28)

1.4 Mathematical Equivalence (29–38)

1.5 Graphs (39–53)

2.1 Functions and Function Notation (54–68)

2.2 Graphs of Functions (69–81)

2.3 Composite Functions (82–91)

2.4 One-to-One Functions and Inverse Functions (92–103)

SAMPLE TEST, Chapters 1 and 2 (104–107)

3.1 Limits—The Idea (108–119)

3.2 Limits—Making it Precise (120–132)

3.3 Properties of Limits (133–144)

3.4 Continuity (145–153)

3.5 Indeterminate Forms (154–159)

3.6 The Intermediate Value Theorem (160–170)

3.7 The Max-Min Theorem (171–178)

SAMPLE TEST, Chapter 3 (179–181)

4.1 Tangent Lines

4.2 The Derivative

4.3 Some Very Basic Differentiation Formulas

4.4 Instantaneous Rates of Change

4.5 The Chain Rule (Differentiating Composite Functions)

4.6 Differentiating Products and Quotients

4.7 Higher Order Derivatives

4.8 Implicit Differentiation (optional)

4.9 The Mean Value Theorem

SAMPLE TEST, Chapter 4

5.1 Increasing and Decreasing Functions

5.2 Local Maxima and Minima—Critical Points

5.3 The Second Derivative—Inflection Points

5.4 Graphing Functions—Putting it All Together

5.5 More Graphing Techniques

5.6 Asymptotes—Checking Behavior at Infinity

SAMPLE TEST, Chapter 5

6.1 Antiderivatives

6.2 Some Basic Antidifferentiation Formulas

6.3 Analyzing a Falling Object (optional)

6.4 The Substitution Technique for Antidifferentiation

6.5 More on Substitution

6.6 Integration By Parts

SAMPLE TEST, Chapter 6

7.1 Using Antiderivatives to Find Area

7.2 The Definite Integral

7.3 The Definite Integral as the Limit of Riemann Sums

7.4 The Substitution Technique applied to Definite Integrals

7.5 The Area Between Two Curves

7.6 Finding the Volume of a Solid of Revolution—Disks

7.7 Finding the Volume of a Solid of Revolution—Shells

SAMPLE TEST, Chapter 7

SELECTED SOLUTIONS

INDEX