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YOU are better than YOU think. Show
yourself how:
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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.
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Jumps and Limited Error Control
In some cases unlimited error control is not possible at
the point x = a. It fails in the following case:
There is an e > 0 such that for
every d > 0, there is some x satisfying
the condition
|
|x-a| < d and |f(x)-f(a)| > e. |
This means as the input x to the function
y = f(x) becomes a better approximation to the number a,
there is no guarantee the difference |f(x)-f(a)| will be
smaller than the error control target e.
This concept is illustrated by functions whose graphs have
a few jumps in them. The height of the largest jump near a
point x = a indicates how small the target tolerance e or
[1/2]·10-n can be in the discussion of error
control.
Again, unlimited error control is possible in the following circumstances:
For each target tolerance e > 0,
there is a tolerance d > 0 such that the condition
|
|x-a| < d and |f(x)-f(a)| £ e. |
These circumstances appear when f(x) is continuous at x = a.
Computations on machines with finite accuracy
precision arithmetic, restrict the number n of
decimals places that can be accurately
computed. Every computing machine which calculates to
finitely many binary or decimal places, suffers from such a limit.
Small discontinuities in calculations appear, except in
those case where exact arithmetic can be done. For example,
on a computing machine which computes to
at most n0 decimal places, the
existence of a rule of the form
|
|x-a| < |
1
2 |
|
1
10m |
implies |f(x)-f(a)| < |
1
2 |
|
1
10n |
|
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governing error cannot be guaranteed for n ³ n0 and
can be considered improbable for most functions evaluated
numerical by a computer. An exception is provided by
functions whose numerically values can be represented (or
encoded) exactly on a machine.
On a computing machine which computes to
at most n0 decimal places, the error control of a single
addition and multiplication are
guaranteed to only n0 binary (or decimal) places. Digits
beyond the n0 place are uncertain. If several such
calculations are done, with numbers in one calculation
being used in the next, errors accumulate and accuracy is
lost. The calculations in question may have to be reorganized
to improve accuracy.
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www.whyslopes.com
Volume 3, Why Slopes and More Math - Preview, starter &
further lessons for calculus to ease or avoid algebra shock in instruction
& self-instruction
Foreword, One Calculus preview and Online Chapters:
(V) signals video (RealPlayer Format) to
watchChapters 2 to 6: offer a very simple preview of calculus and a context
for earlier study of slopes and factored polynomials
Area Entrance & Hub
Foreword
Chapter Descriptions
1. Introduction
2. Calculus Starter Lesson
2. Second 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
Content Guide
Enriched material: The Appendices of Volume
3 are located in the Real
Analysis Area.
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.
Online Volume 2, Three Skills
for Algebra, Chapters 1 to 25 - skip 18., verbalizes and explains key
skills and concepts, those needed in calculus, again to make the hard easier.
A visual understanding of complex
numbers may help - serve as back ground info, in partial fraction
decomposition.
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