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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 |
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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.
Chapter 17
Area Approximation
Covering A Region by Squares
The areas of squares and rectangles may be calculated by calculating the product of the lengths of their sides. In the plane, the area of a bounded region S rectangular or not, may be approximated by covering the region S concerned with small squares, all of the same size, overlapping, if at all, only at their edges.
The example of an elliptic shaped region S is shown. Each covering by small squares gives three methods for approximating what the area A of the region should be.
FOOTNOTE: From a computational perspective, more than half-in but not completely in is not easy to define. This could be a matter of visual judgment - a step outside of the domain of rule-based mathematics. To give a mathematical algorithm, the toss of a coin might be sufficient, or a judgment could be made on how many of the four triangles formed by the diagonals are included completely in the region S.
Each of the above approximations is expected to improve as the squares are quartered (their sides halved) repeatedly and indefinitely. The latter would cause the lower estimate to increase, the upper estimate to decrease while the middle estimate together with the area A presumably approximate, remaining in between. Such halving results three sequences of numbers or quantities.
The area A should be the common, finite, limiting value L of the approximations as the sides of the covering squares become smaller (approach zero). This says how to compute the area A with an unlimited accuracy if a common, finite limiting value L exists for the approximations.Note: The lower estimates yield the increasing sequence, the upper estimate yield the decreasing sequence, and the middle estimates yield a sequence between the previous two.
The area of a region is defined by the methods for
approximating it. That is, the region has an area A = L if
and only if the
three numerical approximations described above all approach
a single finite limiting value L. This limiting L is
then called the area of the region. Otherwise, with some
disappointment perhaps, we may say that the area is not
defined. (Alternatively, we might define inner and outer
areas using the limiting values of the inner and outer
approximations and identify circumstances in which they are
equal.
FOOTNOTE: For instance, for some odd (pathological) region, inner and outer area approximations may approach an inner limit Linner and an outer limit Louter which are not equal. Regardless, these limiting values may define what is meant by inner or outer concepts of area. There has been a concern with identifying conditions in which inner and outer approximations etc approach the same limits. For more information, a very specialized (that is, not for everyone) book or course on area computation, more precisely, integration theory, is indicated.
www.whyslopes.com
Volume 3, Why Slopes and More Math -Foreword, One Calculus preview and Online Chapters: (V) signals video (RealPlayer Format) to watch
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.