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YOU are better than YOU think. Show
yourself how:
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<| (o) (o) |>
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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.
Learn to read notes and textbooks like a lawyer, so that no nuance, no
subtlety and no clause escapes your attention |
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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.
| |
Sufficient Conditions
The following theorem asserts that a function with a bounded derivative on
interval is Lipschitz continuous on that interval.
Theorem G.5 [Lipschitz Continuity] Suppose f(x) is
continuous on an interval [a,b] and differentiable when a
< x < b. Further suppose |f¢(x)| £ K. Then
| |f(x2)-f(x1)| £ K·|x2-x1| |
|
whenever x1 and x2 are both in the interval
[a,b]. The proof is simple:
Proof.
By the mean-value theorem, if x1 < x2
then for some point c in the interval [x1,x2]
Ì [a,b],
|
f(x2)-f(x1)
x2-x1
|
= f¢(c) |
|
The latter implies
| | |
f(x2)-f(x1)
x2-x1
|
| = |f¢(c)| £ K |
|
and hence
| |f(x2)-f(x1)| £ K·|x2-x1| |
|
The case where x1 > x2 is similar.
A knowledge of the Lipschitz constant K is useful for unlimited error
control. That is, if d £ Ke then |x2-x1| £ d implies
| |f(x2)-f(x1)| £ K·|x2-x1| £ Kd = e. |
|
Now if the derivative or slope f¢(x)
is continuous on an interval [a,b], then the extreme value theorem
applied to f¢(x) (not f)
implies there are constants Mmax and Mmin
such that Mmin £ f¢(x) £ Mmax.
By the mean value theorem, this implies that K = max(|Mmin|,|Mmax|) is a Lipschitz constant (in fact the smallest possible)
for f(x) on the interval [a,b]. Indeed one can show
that
| Mmin(x2-x1) £
f(x2)-f(x1)
£ Mmax(x2-x1) |
|
whenever a £ x1 £ x2 £ b.
All this indicates how a knowledge of the slope governs the behavior of a
function.
Exercise. Suppose f(x) is differentiable at each point x
in the interval [a,b] and
| |f(x2)-f(x1)| £ K·|x2-x1| |
|
whenever x1 and x2 are both in the interval
[a,b]. Show |f¢(x)| £ K at each point x in the interval [a,b].
| |
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 only
Proof
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
|