Appetizers and Lessons for Mathematics and Reason (www.whyslopes.com)
||Définition d'une variable || Algèbre || Arithmetique || Logique ||La raison basée sur les règles et modelés||

Online Volumes
1,  Elements of Reason.
1A. Pattern Based Reason 
1B. Math Curriculum Notes
2. Three Skills for Algebra
3. Why Slopes & More Math

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More Site Areas 
1. Help Your Child or Teen Learn 
2. Solving Linear Equations
3. Fractions Ratios Rates Proportions & Units
4. Euclidean Geometry
5. Analytic Geometry/Functions 
6. Number Theory
7. More Calculus
More Site Areas 
8. Complex Numbers 
9. Qc Maths  Education  
10. Secondary IV(?) maths
11. Real  Analysis 
12. LaTeX2HotEqn:
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14.  Français
15. Algebra, Odds & Ends, Etc
More Site Areas 
16. Math Education Essays
17. Telling & Working with Time
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19. Quantitative Skills for  home, shopping and work 
20. Statistics Useful, or Not.
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YOU are better than YOU think. Show yourself  how:  

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Read  logic chapters 1 to 5  in online volume Three Skills for Algebra  for greater skills & confidence in  work 
and study

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 Logic chapters 1 to 5  re- appear not in sequence, as is or longer,  in  Volume 1A,  Pattern Based Reason, Bon Appetite.

Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and
writing

Logic mastery makes the hard, easier. Logic mastery  leads to better, stronger and richer comprehension.  Logic mastery  improves reading and writing.  Logic mastery ease learning difficulties.  Logic mastery gives a headstart.  In sum, logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.


After logic  (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;

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Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought

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What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.


Try the Twiddla Whiteboard. In principle, it  allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio.  Visit www.twiddla.com to set up whiteboards to work with the webpage of your choice.

For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

Lipschitz Continuity and Riemann Sums

Theorem H.3 [Riemann Sums & Lipschitz Functions] Suppose f(x) is defined on an interval [a,b] and differentiable when a < x < b. Further suppose for some K > 0 that
|f(x2)-f(x1)| £ K·|x2-x1|
whenever x1 and x2 are both in the interval [a,b]. Suppose

whenever x1 and x2 are both in the interval  [a,b] Suppose
x0 = a < x1 < x2 < xj < xn < xn+1 = b.

Further suppose xj < wn < xj+1 . Then  all sums of the form
n
å
j = 0 
f(wj)·(xj+1-xj)
where 0 < xj+1-xj  < d  differ by at most e = K(b-a)d   Note if e = (1/2)10-k   given first, put d  = [e/((b-a)K)].

Proof of Theorem. Let K > 0 be a Lipschitz constant for f(x) on the interval [a,b]. Let e = [1/4][1/(10k)][1/(|b-a|)]. Put d = [(e)/((b-a)K)]. Then for every pair of numbers u and v in the interval [a,b], the inequality |u-v| £ d implies |f(u)-f(v)| £ . The rest of the proof is exactly the same as the previous one.

Remark. Most piecewise continuous functions met in practice, that is, in calculus courses, are Lipschitz continuous on each interval [a,b] which does not include a singularity - a point where division by zero occurs. But in principle, the exceptions are more frequent. This is analogous to the situation with real numbers, where, in every-day practice and computation, most people and computing machines handle fractions and finite decimal or binary expansions, but where, in principle, there are more irrationals than rationals among the real numbers. In any event, Lipschitz continuous functions and criteria which identify them provide further examples of continuous functions and another link to error control or error bounding in computations. This author is undecided as to whether or not Lipschitz continuity should be emphasized in first courses on calculus.

 

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Real Analysis - Decimal View


Here are the Appendices from  Volume 3, Why Slopes and More Math,  Chapters 14 to 19 in Vol 3 are related. Here is a  reference for college or university mathematics, electrical engineering and physics.

A. What's Next
B. Pigeon Hole Principle
B. Bolzano-Weierstrass
C1. Triangle Inequality
C2. Triangle Inequality
C. More T.Inequality
D. Sets & Sequences
D. Monotone Sequences
E. Limits,  Properties
E Limits & Error Control
F. Continuous Functions
F. Closed Range Thm
F. Intermediate Val. Thm
F. Compactness Thm
F. Equicontinuity Thm
F Extreme Value Thm
G. Rolle's Theorem etc
G. Mean Val. Thm.
G. Constant Difference Thm
G. Lipschitz Continuity I
PS: One Sided Range Theorems
G. Velocity Revisited
G. Sufficient Conditions
H. Riemann Sums Conv
H. Lipschitz Continuity II

The site area More Calculus contains a one-sided theorem with proof that should be of interest too.

Vol 1A Logic Postscripts
online only:-

Proof by Absurdity alias proof by contradiction

How the demand for consistency supports the law of the excluded middle

Reality versus or with the aid of Imagination

Science, Engineering & Math Students: Have you seen a simpler  geometric introduction to complex numbers? ( java applet included) . Can you explain what is a variable without using a symbol? Can you derive trig expression for dot & cross cosine law from complex number properties? For truth tables and indirect methods of reason, see  chapters 19-24 & postscripts in  Pattern Based Reason  and visit Volume 1A, Pattern Based Reason, striving for objectivity, the empirical challenge & limits.  

Vol 1A Postscripts
online only

Proof by Absurdity alias proof by contradiction

How the demand for consistency supports the law of the excluded middle

Help Me Learn/Teach;

  1. Algebra
    words before symbols - direct & indirect use of formula, numerical versus algebraic solutions - what is a variable (more words)
  2. Arithmetic
    - exercises
    - with fractions
    - videos on primes, lcm, gcm,lcd, square roots etc
  3. Calculus - geometric preview, algebraic preview,
    3 study guides,
    much more
  4. Complex numbers
    -starter lesson with java applet - easy consequences for trig & vectors in the plane
  5. Education
    - Empirical Course Design & Delivery
  6. Fractions
    - alone
    - by rote
    - with algebra
    - videos
  1. Functions - introduction
    hindsight - composition aka
    substitution
    -
  2. Geometry, Euclidean - Correspondence of trianglesTriangle construction,  duplication & Isometry - Failure of ASA & the // line postulate - angle sum in triangles -// grams - Triangle Similarity
  3. Geometry- Analytic - functions, polynomials, complex numbers, unit circle trigonometry
  4. Logic
    - First Steps -
    Symbols in Logic -
     Occurrence & Truth Tables - Indirect Reason -Indirect Reason More
  5. Proportionality
    - Definition - Direct & Indirect Use - Numerical versus Algebraic Solutions
  6. Real Analysis
    - Decimal View of concepts and of proofs


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