|
| |
1 Differential Calculus Preview and Starter Lessons
The first preview is geometric. The second preview in chapters
2 to 6 are algebraic. Flash videos are included in some. Decide for yourself
whether or not seeing the videos is better than reading the text.
The live presentation of these previews or starter lessons is
recommended to teachers and tutors as motivation for precalculus and as starter
lessons for calculus. The presentation of these preview may not be as
exciting for self-study. That being said, these previews or starter
lessons give a context for the earlier study of slopes and factored polynomials,
and they illustrate or develop algebraic reasoning that will be very much needed
in calculus. Good luck.
Saying how to approximate a number better and better may not only give the
number in the limit, but may also be the way to define it - a first twist
Read Later: Slopes of a curve y =
f(x) (derivatives of f(x))
are approximated by slopes of straight line segments (secants) and defined as the
limit of these approximations. That provides the first view of
what is a slope or derivative. BUT properties of limits imply
rules for obtaining derivatives which depend on the algebraic form
of a function f(x), rules in which limits are not seen, albeit
limits are beneath the surface in that they implied the rules:
Those rules include sum, difference, product, quotient and chain
rules plus all the rules for differentiating basic functions:
trig, polynomial, exponential, logarithmic.
Read Later: Derivatives are formally introduced as limits of slopes to secants - the slope
of a ski viewpoint in the starter lessons are too informal. But in a
second twist, rules
for evaluation these limits (rules based on the properties of limits, etc)
appear to give formulas for the calculation of derivatives without mention of
derivatives.
2. Integral Calculus Preview or Starter Lessons
17 What
is Area
18
Integration
18
Area Calculation
|
Chapter 17 and 18 offer a context for the
discussion of areas under curves plus statements of the
fundamental theorem of calculus. All this is done without
reference to summation notation for Riemann Sums
This summation notation free approach provides tutors
and teachers a simpler route for defining the definite integral as limit
of Riemann sums approximations to what area should be. |
3. Limits Revisited
- Limits
via Algebra may develop the algebraic way of writing and
thinking
with limits and point the way to a better understand of
differentiation. Think of this as a partial remedy for the
incomplete development of algebra skills in calculus.
- Chapters 15 to 18 develop further the view that
taking the limit of a sequence of
approximations to geometric or physical quantity - what they might
be - may be taken as a
definition
of the quantity in the case of convergence. The common theme is as
follows: Saying how to calculate a number or amount directly or
indirectly (for example as a limit) defines it.
- Chapter 14 takes an pre-modern approach to limits, continuity
and convergence. It frames these concepts in terms of decimals, or
the decimal representation of numbers, and from numerical
analysis, the practical or in-principle question of how to control
errors in the decimal calculation of functions and limits.
Earlier zeal for modern mathematics lead to an axiomatic approach
in high school and college in which theory made no mention of
decimals, but decimal themselves were employed in calculations and
in the discussion of approximations. Here is a correction
that is sufficient by itself for most students which may also
serve as a base for making the modern decimal free theory more
accessible - two birds with one stone.
Duplicate Material: Chapter 14 in Volume 3
with its discussion of error control in calculation of y = f(x) at or
near a point x = a, in intervals centered at x = a to be almost
precise, provides a simpler, old-fashion way to think about limits.
That old view is enough (sufficient) for students who will not be
specializing in undergraduate mathematics. But that old view also
makes the the epsilon-delta e-d view more
accessible. As a student I met the the epsilon-delta e-d
view and struggled to understand it, and did so temporarily
from time to time. The decimal perspective of error control and
convergence however makes the decimal free view redundant and for
those who must have, makes the latter more accessible.
- chapter 19 describes the area under a curve
definition of the natural logarithm ln(x), derives its properties
and defines the exponential as the inverse function to it. The
treatment here is simple and geometrically oriented.
4. And the More Mathematics
Chapters 21 to 24 introduce vectors (optional reading, an alternative
exposition will be provided sooner or later) and give
an introduction to complex numbers, the first site effort in that
direction. Writing is an iterative affair. The latest site development appears in the site Euclidean-Geometry
To Complex Numbers. The briefest introduction is in the webpage www.whyslopes.com/complex.html.
I am planning to present alternate
Origin of the Complex Number Chapters: I saw the late Richard
Feynmann in 20 minutes of three evenings of 1979 public lectures at
McGill, describe his discipline as a the multiplication of arrows in the plane
using a rule add the angles, multiple the lengths, all without saying the
two words: complex numbers. Since then I
have wondered how to include a development of complex numbers into the exposition of
high school mathematics and calculus. Chapters 22 and 23 do so with the help of
assumptions above rigid body motions - a concept from Euclidean
Geometry.
| |
|
Why Slopes
and
More Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
|
|
Read slowly, Volumes 2 & 3 may ease or avoid
calculus difficulties. Take the risk.
|
Chapters and Appendices
Content Guide
Foreword
2nd Content Guide
1. Introduction
Geometric Calculus Preview (1983)
2. Algebraic Preview Begins
2 Skier in Motion (V)
2 The Skier (V)
2. Position Dependent (V)
3 Slope & Extrema (V)
4 Single Factor Analysis (V)
4 Two Factor (V)
4 More Factors (V)
4 With Divisors (V)
5 Maxima & Minima Tests
6 Jumps & Discontinuities
8 Review (optional)
9 On Calculus Studies
11 Slope of Slope
13 Acceleration
14 Limits & Error Control (V)
14 Limit of a Fn.
14. Limited Error Control
14 Signif. Digits
14 Cauchy Limits
14 Sequence Limits
14 Decimal Arith.
15 What is Slope (V)
15 Slope Calculation (V)
15 Slope, a Limit
15 Tangent Lines
15 Linear Approx.,
15 Limits via Algebra (V)
15 Recap.
PS.Chain Rule for Polys
PS Chain Rule- General (V) -
PS More Chain Rule (V)
PS - Sign Analysis (V)
16 What is Velocity
17 What is Area
18 Integration
18 Area Calculation
18 Fn DefN, 6 Ways
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc
Units in Calculations:
7 Velocity
7 Varying Velocity Example
7. Velocity Calculation
7 Changing Units
7 Same Velocity Motions
10 Slopes without Units.
10 Units & Slopes
10 Units in Cost vs. Quantity
10 How Units Appear
10 Unit Elimination
10 Partial Elimination
10 Interest & Units
12 More on Units
Appendices:
Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
Theorem
Integration
& Lipschitz
Continuity
These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
Range Theorem is a postscript,
not in printed version.
What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent
or Independent
Variable, a Matter of Choice
|
|
|
For Parents & Teachers: Speaking
Skills, Reading
& Writing,
Preparing for Science, ends,
values and methods for work and study, parent- friendly mathematics booklets for ages 4-14.
-
Math
Education
Essays (opinions,
possibilities, references)
- POMME, a two
level program for future skill development in
schools and colleges worldwide. Address content &
motivation gaps with ends, values & methods for skill
development to say which way to go, how and why. -
Present Day Curriculum:
(A) Secondary
I Mathematics
consolidate fractions and measurement, skills and
sense consolidation,
(B)
Secondary II Mathematics
year of algebra and proportionality
(C) See too the following:
- Arithmetic
& Number Theory Practices (horribly put, but
useful)
- Algebra and
Logic SubProgram
(well put, extremely useful)
For Avid Readers in School & Out -
Online Books
1. Elements of
Reason. 1996
1A. Pattern
Based Reason 1995
1B. Math
Curriculum Notes 1996
2. Three
Skills for Algebra 1995
3.Why
Slopes & More.Math
1995
Tour their forewords.
Calculus Prep or Help: See Volumes 2 & 3,
and this bigger
Calculus
Guide.
|
|
|
|
Senior
High School &
Calculus Students
|
|
|
|
Free Live Lesson
- Operations with Decimals - Comparison, Subtraction and Long Division
- Click here
to attend.
|
|
|
|
For Senior
High School Mathematics & Calculus
Intro to Solving
Linear Equations
- a different paths for junior and even senior high
school students.
5
wordy Logic
Chapters
4 curious Algebra
Chapters
Words before & besides symbols. A Key Algebra
forward & backwards Chapter
First Calculus
Preview (1st intro)
Four Calculus
Chapters
(2nd intro)
Intro to Complex
Numbers (long)
Intro to Mathematical
Induction (romantic & wordy at first)
|
|
|
|
Many More Topics
1. Decimal
Arithmetic Reference!
2. Integers
- Intro to Signed No.s
3. Fractions
- fully explained.
4. Fractions
with Units
5. Number
Theory,
6. Solving
Linear Equations
7 Formulas
Use Forward & backwards -
8. Proportionality,
Back- & For-wards.
9. Logic
Chapters:
10. Euclidean-Geometry
11. Slopes
& Equations of Straight Lines. (Take
I. See take II below)
12. Why
Study Slopes.
13. Maps,
Plans, Similarity & Trig,
(Take II included here)
14. Quadratics:
Starter lessons
15. Polynomials:
Starter lessons
16 Why
Factor Polynomials:
17 Functions
- Forwards & Backwards.
18. Exponents,
Radicals & logs.
19. Complex
Numbers before trig (new advance/ starter lesson)
20. DC
Electric
Circuits Etc
21. Real
Analysis
22. The
Olde Complex No, Trig
& Vector Section.
23. More
Calculus Stuff
- written after Volumes 2 and 3.
More For Instructors
-
Education
Essays
(opinions,
possibilities, references)
- POMME, a two
level program for instruction K1-14
- Math & Logic How-TOs
1. Arithmetic
2. Algebra
3. More Algebra
4. Beginner Geometry
5. More Geometry
6. Calculus
7. Show Work or Logic
These may be too dense for students.
Level I Material: New Stuff
Time and Date Matters
Level I Arithmetic.
Money Matters
Measurement Matters
Matters of Chance (Risk Control)
Logic
Chapters
(leave what's not clear in Level I to Level II)
Using/Making Maps and Plans.
(A variant of
Maps,
Plans, Similarity & Trig, to
appear here).
|
|
|
|
Skill Development Tips
For All
Standards: (A) Take
care to avoid the domino effect of errors & approximations; (B) Do and
record steps in an manner that allows skill mastery to be seen or
corrected. Anything represent substandard work.
Key Numerical Methods
- To multiply signed numbers, prefix the product of their signs to the product
of their lengths or unsigned parts. The product is negative if the no of
negative sign in it is odd.
- To add signed numbers with like signs, prefix the common sign to the sum of the
lengths.
- To add signed numbers with opposite signs, prefix the sign of the longest to
the difference: length of longest minus length of shortest.
- Should we study roots and powers of real numbers with formulas involving exponential and log.
- How does adding and multiplying points in the plane and rotating the midpoint
of a line segment lead to mastery of complex numbers and the thought-based
development of their properties, all before trig?
- New Axioms for High School Mathematics: In accounting, totals of assets
and debts may be calculated by dividing the assets and debts into
non-overlapping (disjoint) groups and then adding subtotals. In general, sums
(and products) of counts and numbers, positive and negative numbers
included, can be obtained by adding subtotals (and multiplying
subproducts, respectively). These practices may be cast as axioms in
secondary mathematics. Then operations on polynomials are easily implied
justified by these "axioms" and the geometric introduction of column
methods for expanding a products of two sums. While set theory in pure
mathematics may imply the above axioms in university mathematics programs
instruction, an earlier and more accessible explanation based on easily accepted
and understood geometric and counting practices derivation of the above
axioms is possible at the high school for students heading for college programs
in science.
In Volume 2: Prep for Calculus
- What is the difference between saying A if B and saying A if and
only if B. Being aware of the difference will sharpen ye wits.
- What is a chain of reason?
-Are your arithmetic skills OK?
-Have words been missing in the introduction of algebra?
- Can ye talk about numbers & quantities varying apart from or before the
use of letters & functions?
- Do ye know about the forward & backward use of formulas?
-Contrapositive: is that backward use of A if B?
-What is a variable x? Answer before speaking of function f(x) = x.
-What a twist! There are no rules of algebra for subtraction and division. But
if you replace them by addition of -x and multiplication by 1/x, rules of
algebra (properties of arithmetic) can be used.
In Volume 3: Calculus Slowly?
-Why are slopes studied and polynomials factored in high school?
- Volume 3 suggests how to ease or delay algebra shock in
calculus *& beyond. In Calculus, derivatives and integrals introduced and defined by limits, but calculated
without when possible by using differentiation rules forwards and backwards. The
second site calculus section may help in differential calculus.
|
|
|
|