Calculus Problems for a New Century ( MAA Notes Series #28: Resources for Calculus )

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Overview

Even in our age of calculators and computers, we still need problems that will help students develop fundamental skills and give them a sense of progress in their study. These problems must be phrased differently, however, than the traditional lists of the past. At the very least, they cannot be rendered trivial by available electronic aids; at best they should make use of such aids to lead the student to greater understanding.

This volume contains problems written with these objectives in mind. The authors have tried to emphasize conceptual understanding over rote drill. Although many of the problems require the use of a calculator or computer algebra system, most do not. A deliberate effort has been made to stress graphs and tables, rather than rules to define function, in the belief that "real world" data generally come that way.

The problems are organized in groups that parallel traditional grouping of ideas, making it possible to use them as supplements to most texts. All of the problems are given with commentaries that frequently give a bit of history about the problem, as well as show how the question can be extended and viewed in a different context. Our aim is to provide teachers with problems and exercises to challenge the current calculus student.

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Editorial Reviews

Connie Buller
"This book should be a boon to new teachers of calculus who don't already have a stockpile of great questions and hate to use only the examples presented in their particular textbook."
The Mathematics Teacher
S. M. Nugent
"All the problems are carefully worded to lead the student through not so copiously supplied with hints that they become trivial...students should, as soon as possible, be introduced to problems of the kind posed in this book, and indeed in this Collection.... Whereas some students see the power and beauty of calculus at once, all will benefit from the life and inspiration injected into the subject by problems such as these."
Mathematical Gazette
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Product Details

  • ISBN-13: 9780883850848
  • Publisher: Mathematical Association of America
  • Publication date: 1/28/1993
  • Series: MAA Notes Series , #28
  • Edition description: New Edition
  • Pages: 448
  • Product dimensions: 8.58 (w) x 11.02 (h) x 0.90 (d)

Read an Excerpt

This volume of problems has been produced under the auspices of a calculus reform project funded by the National Science Foundation. Its aim is to address the need, expressed by many mathematicians, for a fresh approach to the study of elementary calculus at the college level. In particular, we seek to provide a collection of textbook problems which stimulates students to understand the power and the beauty of calculus rather than to regard the subject as a hodgepodge of unrelated techniques.
 The current calculus reform movement has developed a number of themes and ideas which this collection attempts to incorporate. It is now apparent that the existence of computer algebra systems (CAS) transforms the study of calculus by providing students with a powerful resource for computations, graphical representation, and symbolic manipulation. In this environment, many of the problems calculus students traditionally see become trivialities, and we have, therefore, avoided such problems. Our exercises stress conceptual understanding over rote drill. Occasionally this is done by asking students to use the computer's output to make a conjecture. Sometimes we ask students to formulate the answer to a problem and to leave the calculation to a computer package or a calculator. In this way, interesting problems that were once through too hard for students because of their computational difficulties are now accessible. More often, however, our exercises neither require the use of a CAS nor are they trivialized by one. An effort has been made to convey functions by graphs and tables, rather than by rules in the belief that "real world" data generally come that way.
 Our exercises are designed to help students achieve a better understanding of this rich and useful subject. We have tried to frame questions that require students to grapple with ideas rather than techniques, and we have shunned problems involving tricky computations. We have written, in fact, some questions that may seem, at first, too easy, but not all students will find them trivial. These questions may be especially good for in-class discussions.
 The writers of this volume are: Deborah Hart of Knox College, Eugene Herman of Grinnel College, James Stein of California State University of Long Beach, Lyle Welch of Monmouth College, and its editor, Robert Fraga of Ripon College. Besides creating out own problems, we have drawn on a wealth of existing material, a code for which is given on the page following this foreword. In some instances, problems have been adapted or revised to accord with the author's goals.
 A word about the source of material is in order. Indication of source in this volume means that the writes used a form of the problem available in the source cited in square brackets at the end of the problem commentary. It has been our goal to assemble a set of problems, both old and new, which we feel will be useful to teachers of calculus. Sometimes this has meant rescuing some problems which have been jettisoned during the transition form one edition to another of a standard textbook. Such classics are designated by [CP]. In other cases, the problems are new insofar as they reflect the resources now available to calculus students. Problems which require computer or calculator assistance are marked by the symbol [C] at the beginning of the statement of the problem.
 The problem sets are structured around syllabi for a two-semester course in elementary calculus worked out by colleagues in the project. The commentaries which accompany the problems frequently contain more than answers to the questions posed. Often there is an indication about a way in which a question can be extended, or how it can be construed in a different (frequently physical) context, or something about its history as well as the reason why the problem was selected. This volume gives teachers a variety of exercises suitable for calculus students in the 1990's.
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Table of Contents

Introduction: Resources for Calculus Collection
The Five Volumes of the Resources for Calculus Collection
Acknowledgements
Foreword
Acknowledgements for Volume II
Suggestions to the Student
Suggestions to the Instructor

I. Functions and Graphs
1. Domain and Range. Elementary Functions
2. Trigonometric Functions
3. Exponential and Logarithmic Functions
4. Composite Functions
5. Functions Described by Tables or Graphs
6. Parametric Equations
7. Polar Coordinates

II. The Derivative
1. Average Rates of change
2. Introduction to the Derivative
3. Graphical Differential Problems
4. Limits
5. Continuity
6. Power, Sum, and Product Rules
7. The Chain Rule
8. Implicit Differentiation and Derivatives of Inverse
9. Derivatives of Trigonometric, Log, and Exponential Functions
10. Root Finding Methods
11. Related Rates

III. Extreme Values
1. Increasing and Decreasing Functions and Relative Extrema
2. Concavity and the Second Derivative
3. Man-Min Story Problems

IV. Antiderivatives and Differential Equations
1. Antiderivatives
2. Introduction to Differential Equations

V. The Definite Integral
1. Riemann Sums
2. Properties of Integrals
3. Geometric Integrals
4. The Fundamental Theorem of Calculus
5. Functions Defined by Integrals

VI. The Definite Integral Revisited
1. Exact Values from the Fundamental Theorem of Calculus
2. Techniques of Integration
3. Approximation Techniques and Error Analysis

VII. Sequences and Series of Numbers
1. Sequences of Numbers
2. Series of Numbers. Geometric Series
3. Convergence Tests: Positive Series
4. Newton's Method
5. Improper Integrals

VIII. Sequences and Series of Functions
1. Sequences of Functions. Taylor Polynomials
2. Series of Functions. Taylor Series
3. Power Series

IX. The Integral of R Squared and R Cubed
1. Real-valued Functions of Two and Three Variables
2. Definition of Double and Triple Integrals
3. Evaluation of Double Integrals

X. Vectors and Vector Geometry
1. Vectors
2. Velocity and Acceleration
3. Arc Length

XI. The Derivative in Two and Three Variables
1. Partial Derivatives
2. Gradient and Directional Derivatives
3. Equation of the Tangent Plane
4. Optimization

XII. Line Integrals
1. Line Integrals
2. Conservative Vector Fields and Green's Theorem

 

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