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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 2.
Local High-Points or Maximums
High points of bumps and hills are called maximums.
As the skier Barbara moves up a hill (or a local bump), and then down the other side, the slope of her ski changes from positive on the uphill side to negative on the downhill side. At the very top of hill, her ski is horizontal and its slope is zero.7
7Footnote: What happens if Barbara goes up and over a sharp peak? As she gets to the top and pivots from the uphill to downhill side, the slope of her ski goes from positive to negative.
There can be several hills, with different and varying steepness on each side. From the slope of her ski, Barbara knows even without looking when she crosses over the top of a hill or a bump: the slope of her ski changes from positive to negative. The next diagram shows the sign of the slope (of the one ski) before, at and after a high point.
Knowledge of the sign of the slope does not provide all information about the
ski trail y = h(x). In particular, unless she measures and
records the height h(a) of each hilltop, she can not say which is
the highest. Every hilltop gives a local maximum for her height. It has a height
greater than or equal to ( ³ ) all nearby heights. A
point with a height greater than or equal to ( ³ )
all other heights, is called an absolute maximum for the portion of the
trail or curve y = h(x) being examined.
Remark (Advanced Material.) To be more precise, a point (x0,h(x0)) is a local maximum for the portion of a curve y = f(x) where a £ x £ b if These two conditions describe precisely what is meant in talking about all nearby heights. Note that in talking about numbers and quantities, a legalistic precision is required. Otherwise, tacit assumptions will be made differently by different writers and different readers.
- a £ x0 £ b. That is, x0 is in the interval being examined.
- There exists at least one interval (x0-d,x0+d) centered at x0 such that h(x) £ h(x0) if a £ x £ b and x0-d £ x £ x0+d.
Definition. (Highest of the High Points). A point with height greater than all other heights in a given portion of the trail, more precisely not less than all other heights in the portion, is called an absolute maximum for the portion or interval in question.
In the event of a tie, each of those points having or sharing the greatest height is called an absolute maximum.
Local Low-Points or Minimums
Low points of depressions are called minimums.
As with high points (hill tops or maximums), several depressions or valley bottoms may be met and crossed in following a ski trail. From the slope of her ski, Barbara again knows even with her eyes closed when she crosses the valley bottom: the slope of her ski changes from negative to positive.
Note again that unless she measures and records the height h(a) of the low points in each depression or hollow, she can not say which is the lowest. The bottom of each depression or hollow gives a local minimum for her height. It has a height less than or equal to all nearby heights.
According to this definition, all the points on a horizontal straight lines are local minimums. Excluding the word equal here would yield a strict local minimum: a height less than nearby heights. See the previous discussion of nearby heights.
Definition. A point with a height less than or equal to ( £ ) all other heights in a portion of the trail is called an absolute minimum for the portion of the trail or curve y = h(x) in question.
In the event of a tie, each of those points having or sharing the least height is called an absolute minimum.
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.