Volume 3 Why Slopes A Calculus Intro Etc
Foreword
Fall 1983 Calculus Appetizer
Chapter 1.Introduction
Chapter 2. Slopes and Ski Trails
Chapter 3. Slope Sign Analysis
Chapter 4. More Slope Sign Analysis
Chapter 5. Slope Sign Tests
Chapter 6. Slopes and Vertical Shifts
Chapter 7 Slopes and Velocity
Chapter 8. Slope Interpretation
Chapter 9 About First Courses in Calculus
Chapter 10 Slopes and Units
Chapter 11. Graphing Slope versus Position
Chapter 12. Units and Slopes
Chapter 13. Acceleration
Chapter 14 Limits and Continuity with and sans Decimals
Chapter 15. Algebraic Evaluation of Limits
Chapter 15. Slope Approximation
Chapter 16. Velocity Approximation
Chapter 17. Area Approximation
Chapter 18. Slopes Areas Integration
Chapter 19. Exponentials and Natural Logarithms
Chapter 20 Vectors and Complex Numbers
Chapter 21 Arrow Addition
Chapter 22 Complex Numbers
Chapter 23 Links To Trigonometry
Chapter 24 Logarithms Powers and Exponentials
Postscript Pythagorean Theorem yet another proof
Foreword
The physicist Richard Feynman (1918-1988) gave three public lectures at
McGill University in 1976 1979. His work
on physics has been followed by many scientists and students.
In the lectures, partly tongue-in-cheek, he suggested that physics was
based on two easily described operations, namely the addition and
multiplication of arrows in the plane. His description of arrow addition
and multiplication for a general, non-mathematical audience was a model
for the informal, very visual, most adequate, presentation of
mathematical ideas. But he gave it under the guise of describing physics.
And he avoided panic among the mathematically shy by not saying that the
arrows, with their addition and multiplication, represent what pure and
applied mathematicians (since Gauss) regard as the complex numbers.
No mastery of the algebraic way of writing and thinking was required to
understand his live description of addition and multiplication.
When I attended Feynman‘s lectures, I thought his description of arrows
in the plane could be an excellent way to introduce complex numbers. The
chapters on complex numbers elaborate on Feynman’s live presentation,
although their on-paper presentation employs the algebraic way of writing
and reasoning.
With Feynman's energetic presentation as a model, I looked for and found
in 1983, a preview and simple tour of calculus (slope-related
calculations) which likewise required a minimal knowledge of algebra.
Just the definition of a slope to a straight line needs to be understood
to follow it.
The why slopes chapters extend this tour and provide a geometric
motivation for calculus, easy to describe and to repeat without a great
dependence on algebra and without requiring a mastery of the rules of
differentiation, that is slope calculation, for nonlinear functions.
This book is one of three volumes on understanding and explaining
reasoning skills and mathematics. The objective of this volume is to
complement other texts in algebra, trigonometry and calculus. Students
may be able to read the first part of this book during their high school
days and keep the rest of this work for consultation during their college
studies.
The first why slopes chapters gradually illustrate the algebraic or
symbolic way of writing and thinking. The later is employed more deeply
in some later chapters and at full strength in proper calculus courses.
The aim of the first chapters is to provide a simple image-based preview
or review of calculus. In it, dependence on symbols or algebra is kept to
a minimum. The images may help readers to see and physically grasp the
simplest slope-related ideas in calculus. The remaining chapters cover
more topics – see the table of contents. Appendices present the most
advanced topics. Theorems in first courses on calculus are often stated
without proof. The appendices state the theorems and give or indicate the
proofs. This should provide a context for the decimal-free approach
favored in advance calculus or modern mathematical analysis.
This is a book which a student could begin reading in high school and
continuing reading through further college math courses. Material
elementary to advanced is covered.
Alan Selby
Montreal
March 1996
Copyright © 1995, 1996 by A. M. Selby
Canadian Cataloguing in Publication Data
Selby, Alan M,
Understanding and Explaining reason and math
Contents: v. 1. Elements of Reason - v. 2. Three Skills
for algebra - v.3. Why Slopes and more math.
ISBN 0-9697564-4-5 (set) -
ISBN 0-9697564-1-0 (v. 1) -
ISBN 0-9697564-2-9 (v. 2) -
ISBN 0-9697564-3-7 (v. 3) -
1. Mathematics–Philosophy. 2. Reason.
3. Algebra. 4. Calculus. I. Title. II. Title: Elements of reason.
III.Three Skills for algebra. IV. Title: Why Slopes and more math.
QA8.4.S44 1995 510’.1 C95-900945-0
Reprinting may lead to new ISBN numbers
In fall 1983, I gave three lessons to extend or complete the skills of
students starting calculus - recent high school graduates.
-
The first lesson three skills for algebra
gave a remedy for olde gaps in the high school introduction of
mathematics Exercise for students: Find the fourth skill
for algebra.
-
The second lesson two logic puzzles fostered
precision reading and writings skills, and hinted at the role of
logic in maths. Exercise for math and English teachers:
Present this puzzle in senior high school classes.
-
The third lesson why slopes - a geometric calculus
appetizer gave a starter lesson for calculus. It explains why
slopes may be met in high school maths, and non-algebraically informs
students where calculus will head after a coming review of high
school maths and a discussion of limits and continuity.
Chapters 2 to 14 in the 1996 site Volumes 2, Three Skills for Algebra ,
and chapters 2 to 6 plus 14 in the 1996 Volume 3, Why Slopes and More
Maths , present these three lessons and add to them. In doing so,
they provide words and stories to introduce logic and provide a clearer
oral and geometric paths for introduction of algebra in calculus and
earlier high school maths. Newer site area on Solving Linear
Equations may offers a geometric introduction for algebra at the the
junior high school level.
|