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Problems for Student Investigation

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Overview

The authors of this volume have assembled a collection of projects students will find lively and stimulating. They can be used by the average calculus student, and are solvable with guidance and instruction from the teacher.

Some of the projects cover a variety of calculus topics for the first year of a typical single-variable calculus program, while others are applicable to multivariable calculus. The subject matter is as diverse as the prerequisites. Some of the material involves concepts you would expect to find in any calculus course, while other material will lead the student to examine an interesting application or theory that is tangential to the core material. Several projects involve maxima and minima applications, others grapple with concepts such as surfaces and Riemann sums, and still others encourage expansions on the work of Newton and Archimedes.

Students will learn how to use calculus to solve real problems. How to use the library to ding mathematical sources, how to read and write mathematical material, and how to cooperate with their peers in the solution of a difficult problem. Learning that they can solve what at first seems an inscrutable mathematical problem can only increase their mathematical confidence.

Each project is self-contained, including a brief statement of the problem for the students and more thorough information for the teacher. The detailed information provided by the authors will lessen the amount of time such a project might require of the teacher.

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What People Are Saying

Harry Sedinger
"This book is interesting and fun to read. It is "meant to be in the public domain" and, like the other four volumes, should be available wherever calculus is being studied."
The Mathematics Teacher
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Product Details

  • ISBN-13: 9780883850862
  • Publisher: Mathematical Association of America
  • Publication date: 1/28/1993
  • Series: Resources for Calculus Ser. , #30
  • Edition description: New Edition
  • Pages: 224
  • Product dimensions: 8.30 (w) x 10.80 (h) x 0.70 (d)

Table of Contents

Introduction: Resources for Calculus Collection
The Five Volumes of the Resources for Calculus Collection
Acknowledgements
Table of Contents
Preface
Suggestions for Using This Volume
Not to Students
Syllabus for Calculus I
I. Derivatives
Optimal Design of a Steel Drum: John Ramsey, College of Wooster
Finding the Most Economical Speed for Trucks: John Ramsey, College or Wooster
Designer Polynomials: Charles Jones, Grinnel College
Cruise Control: Eric Robinson, John Maceli, Diane Schwartz, Stan Seltzer, and Steve Hilber, Ithaca College
Security System Design: Steve Hilbert, John Maceli, Eric Robinson, Diane Schwartz, and Stan Seltzer, Ithaca College
Designing a Pipeline With Minimum Cost: John Ramsey, College of Wooster
Crankshaft Design: Steve Boyce, Berea College
Valve Cover Design: Steve Boyce, Berea College
The Tape Deck Problem: Matt Richey, St. Olaf College
II. Antiderivatives and Definite Integrals (Pre-Fundamental Theorem)
Population Growth, Wayne Roberts, Macalester College
Drug Dosage: Diane Schwartz, John Maceli, Eric Robinson, Stan Seltzer, and Steve Hilbert, Ithaca College
Logarithms: You Figure it Out: Matt Richer, St. Olaf College
Numerical Integration and Error Estimation: Steve Boyce, Berea College
An Integral Existence Theorem: Steve Boyce, Berea College
A Fundamental Project: Charles Jones, Grinnel College
III. Applications of Integrationl
Inventory Decisions: Steve Boyce, Berea College
Tile Design: John Ramsey, College of Wooster
Minimizing the Area Between a Graph and Its Tangent Lines: Steve Boyce, Berea College (problem suggested by R.C. Buck)
Riemann Sums, Integrals, and Average Values, Eugene Herman, and Charles Jones, Grinnell College
The Ice Cream Cone Problem: Matt Richey, St. Olaf College
III. Multivariate Calculus
Waste Container Construction: John Ramsey, College of Wooster
Own your Own Function of Two Variables: Eugene Herman and Anita Solow, Grinnell College
Three Cylinder Intersection Problem: John Ramsey, College of Wooster
Gradient Method Optimization: John Ramsey, College of Wooster
IV. Historic Projects
Archimedes' Determination of the Area of a Circle: Mic Jackson and Sarah Angley, Earlham College
Archimedes' Approximation of Pi: Mic Jackson and David May, Earlham College
Zeno's Paradoxes: Charles Jones, Grinnell College, Mic Jackson and Will Carter, Earlham College
Archimedes' Determination of the Surface Area of a Sphere: Mic Jackson and Krista Briese, Eaerlham College
Newton's Investigation of Cubic Curves, Jeffrey Nunemacher, Ohio Wesleyan University
Cavalieri's Integration Method, Mic Jackson, Earlham College
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Preface

Preface
In the interest of making calculus more lively, this volume of projects can be used by an instructor to give her students an opportunity to work with a mathematical problem that can be posed easily, but which is impossible for most students to solve as part of an overnight homework assignment. The projects are not intended to be for honor students, but are problems that a small group of typical calculus students can solve given a reasonable amount of time and effort, with some timely guidance from the instructor. Experience indicated that through applying themselves to projects of this kind students develop a better notion of ways in which calculus can be used to solve realistic problems, have the opportunity to look more closely at some of the important concepts of calculus, and gain a sense of personal ownership of some piece of calculus. Some learn how to use the library effectively to find mathematical sources, and all improve their ability to read and write mathematical material and to cooperate with peers in the solution of a difficult problem. Finally, the experience of developing solutions to problems which on the first reading seem inscrutable, increases the confidence of students.
 The contents of the projects are distributed over the first year of a typical single-variable calculus program, with some projects applicable to multivariable calculus. Some projects involve the application or extension of a mathematical concept that is part of the content of the usual course, while others give students the opportunity to examine an interesting application to theory somewhat tangential to the core material. Each project is self-contained, including a brief statement of the problem for the students and more through information from the instructor. The first item in the information for the instructor is an abstract of the project in which we explain what makes this problem interesting, what we expect students to learn from doing the problem, and some rationale for why we pose the problem as we do. A description of prerequisite knowledge and skills for each project will help instructors determine where that project could fit into a particular course. To further assist in determining proper placement of the projects in your course, a recommended first-semester calculus syllabus is given in outline form with projects of the first three sections of this volume included according to their prerequisite.  (An annotated syllabus for Calculus I and II worked out by colleges in the project can be found in the appendix of Volume I of the Resources in Calculus set.) Each project description also identifies essential or recommended library or computing resources. The bulk of information for the instructor is a section containing one or more sample solutions. The solutions presented have been chosen to represent approaches students may take on the particular project. They are not necessarily the most concise, most elegant, or even the most intuitive for the instructor.
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