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 3. Slope Sign Analysis
[Play Video] 2¼ minutes: Slope Interpretation for a 2D ski hill y = f(x). (Appeared earlier)
Recapitulation (Some Repetition)
From the slope of her ski, Barbara can say whether her height h(x) is increasing, decreasing or remaining constant.
Remember from the changes in the slope sign, she can locate the high points (local maximums) and low points (local minimums). As she passes over a high point or through a low point, the slope of her ski may be zero. That is, it may become momentarily horizontal as she makes the transition at the top from going up to going down, or vice-versa. In summary, the following can be said, and needs to be remembered.
- From the slope of her ski, Barbara can feel if she is moving uphill, downhill or horizontally but she cannot feel how high she is.
- As she passes over a hill top, the slope of her ski goes from positive (+ve) to (-ve). Momentarily at the top, the ski will be horizontally and the slope value will pass through 0.
- As she passes through the low point of a depression, the slope of her ski changes from negative (-ve) to positive (+ve). Momentarily at the bottom, the ski slope may pass through the value 0.
A First Example
Describe the up-down behavior of the height function y = h(x) given the information (recorded by Barbara) in the following line diagram. It shows the sign of the slope m = g(x) = h¢(x) in different intervals.
Solution. The slope is negative, and therefore the height should be decreasing and the motion downhill between
- between x1 and x2, and also
- between x2 and x3.
- between x0 and x1,
- between x3 and x4, and
- between x5 and x6.
- between x4 and x5.
The two words should indicate an expectation based on our physical senses or experience, but not on mathematical considerations. The mathematical theorem justifying this expectation relies only on arithmetic-based definitions and considerations. The slope is also zero at the following points: x1,x2,x3,x4, and x5. The slope at the end points x0 and x6 is unknown.
The next diagram summarize our information about the hills y = h(x).
The arrows show where the height is increasing, decreasing or constant, but not the values of the height. Here
means or signals uphill motion or increasing height,
means or signals downhill motion or decreasing height), and
means or signals horizontal motion or constant height,
From the identification of the intervals where the slope is positive or negative, that is where the skier Barbara went up and down hill, there is enough information to locate the high and low points (hill tops and depression bottoms) on the trail:
- There is a high point or local maximum at x1.
- There is a low point or local minimum at x3.
- There is a left end-point low-point at x0
- There is a right end-point high point at x6
The Height is Uncertain
To show the height is not determined by sign analysis, the following diagrams sketch two simple trails with the same the slope sign behavior as above, but different heights.
The magnitude or steepness of the slope and not just its sign determines the shape of a curve y = h(x).
A Change in Notation
Instead of using h(x) in the previous pages, we could have used f(x). Instead of using m = g(x) = h¢(x), we could have written f¢(x). What notation is used is less important than the role the notation takes in describing ideas.
| y = h(x) Height Function |
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y = f(x) Some Function |
| m = g(x) Slope Function for h(x) |
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m = f¢(x) First Derivative for f(x) |
As in a play, a given role can be assumed by one of many actors, unless a important star is involved. Second derivatives will appear later.
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-instructionForeword, 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.