The Text A Companion to Calculus


After instituting the Calculus I with Review course in 1988, the Moravian College faculty found that the greatest difficulty in teaching the integrated course was the lack of published material to provide the needed background and review, in a calculus setting. In 1991, with FIPSE funding, a first draft of the text A Companion to Calculus was developed and class-tested at Moravian by a team of authors, all involved in teaching the integrated course: Doris Schattschneider, Alicia Sevilla, Kay Somers, and Dennis Ebersole. In 1992 these materials were revised and class-tested at several additional institutions.  In 1994, Brooks/Cole published a preliminary version, and after further class-testing and revision, the first edition was published in 1995. A second edition was published in 2005. 

A Companion to Calculus facilitates the integration of precalculus concepts with calculus concepts in a ‘just in time’ manner.  It is designed to be used with any calculus text ; chapters are keyed to calculus topics. The Companion uses the same notation as a calculus text, so there is no confusion about how to use the concepts and techniques to solve calculus problems.  Information is presented in four different modes: in words, in figures (graphs and diagrams), with symbols (algebraic formulations), and as numerical data. Students learn how to translate information from one mode to the other. The text contains many examples and exercises, and encourages the use of technology, but does not depend on it. The Companion can be used in several ways:

      In a Calculus with Review course that integrates, just in time, the review of precalculus topics

     As a supplement for a first calculus course with either scheduled or informal “help” sessions

     As a supplement for individual study when taking a first calculus course
Key features of the text:
      Each chapter links to a central topic in Calculus I
      Extensive instruction on how to set up calculus problems
      Warnings about common student errors—avoid these pitfalls!
      Strong emphasis on visualization to aid in understanding key concepts
      Carefully written examples give detailed solutions to typical calculus problems
      Exercises designed to develop both conceptual understanding and important algebra skills
 

The following parallel listing outlines how chapters in A Companion to Calculus can be integrated with chapters in a calculus text.
Section in Companion
Section in Calculus text
Introduction
Symbols and Notation
Modes of Communication
Cartesian Coordinates
The Cartesian Coordinate Plane
Graphs
Lines and Their Equations
Parallel and Intersecting Lines
Distance between Two Points
The Circle		 Coordinate Geometry and Lines
Functions
Function Notation
Domain and Range of a Function
Different Ways to Represent Functions
The Graph of a Function
Special Classes of Functions
Transformations of Graphs		 Functions and Graphs
Companion to Limits
Combinations of Functions
Algebraic Simplification of Functions
Inequalities:  Linear inequalities;
  Absolute Value; Equations and Inequalities
If-then Statements
 Limit of a Function
Calculating Limits Using Limit Laws
 Definition of Limit
Companion to Continuous Functions
Polynomials
Zeros of a Function:  Finding Zeros of a Polynomial;
  Approximation of Zeros of Continuous Functions
More on Domains of Functions:  Composite Functions
 Continuity
The Role of Infinity
Graphical Interpretation of Asymptotes
Algebraic Manipulations: Finding Asymptotes		 Limits at Infinity, Horizontal Asymptotes
Infinite Limits, Vertical Asymptotes
Problem-Solving and Rates of Change
Problem-Solving
Applications:  Average Rates of Change
Secant and Tangent Lines
 Tangents, Velocity, and Other Rates of Change
Companion to Rules of Differentiation
Negative and Rational Exponents
Decomposition of Functions
Simplifying Derivatives
 Derivatives
Differentiation Formulas;
Chain Rule
Review of Trigonometric Functions
Angle measures
Definition and Evaluation of the Trigonometric Functions
Properties and Identities for the Trigonometric Functions
Domain, Range, and Graphs of the Trigonometric Functions
Combining Functions with the Trigonometric Functions
 Review of Trigonometry
 Derivatives of the Trigonometric Functions
Companion to Implicit Differentiation
Implicitly Defined Functions
Solving Equations That Contain dy/dx		 Implicit Differentiation
Companion to Repeated Differentiation
Iteration and Patterns in Higher Derivatives
Rate of Change of Rate of Change		 Higher Derivatives
Companion to Related Rates
Setting up Equations for Related Rates Problems
Problem-Solving Strategies for Related Rates Problems		 Related Rates
Linear Approximations and Differentials
Tangent Line Approximation
The Differential		 The Differential and Tangent Approximation
Companion to Exponential Functions
Rules of Exponents
The Natural Exponential Function		 Exponential Functions
Derivatives of Exponential Functions
Companion to Inverse Functions
One-to-One Functions
Properties of a Function and its Inverse:
   Domain, Range, Graph
Finding the Inverse of a Function:
   When the Function is One-to one
   When the Function is Not One-to-one
 Inverse Functions
Companion to Logarithmic Functions
Definition and Properties of Logarithmic Functions
Graphs of Logarithmic Functions
Solving Equations with Logarithmic and Exponential Functions
 Logarithmic Functions
Derivatives of Logarithmic Functions
 Exponential Growth and Decay
Companion to Extreme Values of a Function
Extreme Values and Critical Values
Setting Up Functions to Solve Extreme Value Problems
 Maximum and Minimum Values
Applied Maximum and Minimum Problems
Companion to Curve Sketching
Solving Inequalities
Graphical Interpretation
Putting It All Together
 First Derivative Test
Concavity and Points of Inflection
Curve Sketching
Companion to Antidifferentiation
Antidifferentiation as the Inverse of Differentiation
Recognizing Antiderivatives
Substitution for Antiderivatives		 Antiderivatives
Companion to Area and Riemann Sums
Exact Areas as Sums of Basic Geometric Shapes
Approximation of Areas
Riemann Sums and Their Interpretations		 Area
Companion to the Definite Integral
Area Under a Curve as a Definite Integral
Other Interpretations of the Definite Integral
The Fundamental Theorem of Calculus
Change of Variable in Definite Integrals
 The Definite Integral
Properties of the Definite Integral
 The Fundamental Theorem of Calculus
The Substitution Rule
Appendix (review sections as needed)

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