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 |
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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
|
whenever x1
and x2 are both in
the interval [a,b] Suppose
|
Further suppose xj
< wn < xj+1 .
Then all sums of the form
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.
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)].
n
å
j = 0
f(wj)·(xj+1-xj)
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.
www.whyslopes.com
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 onlyProof by Absurdity alias proof by contradiction
How the demand for consistency supports the law of the excluded middle
Help Me Learn/Teach;
- Algebra
words before symbols - direct & indirect use of formula, numerical versus algebraic solutions - what is a variable (more words) - Arithmetic
- exercises
- with fractions
- videos on primes, lcm, gcm,lcd, square roots etc - Calculus - geometric
preview, algebraic preview,
3 study guides,
much more - Complex numbers
-starter lesson with java applet - easy consequences for trig & vectors in the plane - Education
- Empirical Course Design & Delivery - Fractions
- alone
- by rote
- with algebra
- videos
- Functions - introduction
hindsight - composition aka
substitution - - Geometry, Euclidean - Correspondence
of triangles, Triangle
construction, duplication & Isometry - Failure
of ASA & the // line postulate - angle
sum in triangles -//
grams - Triangle
Similarity
- Geometry- Analytic - functions, polynomials, complex numbers, unit circle trigonometry
- Logic
- First Steps -
Symbols in Logic -
Occurrence & Truth Tables - Indirect Reason -Indirect Reason More - Proportionality
- Definition - Direct & Indirect Use - Numerical versus Algebraic Solutions - Real Analysis
- Decimal View of concepts and of proofs