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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 21
Arrow Addition
The terms arrow and vector will be used interchangeably.
This applet (online only) illustrates the ideas in this chapter.
Map Addition Method
In navigation, drawing an arrow on a map from a point A to a point
B represents a linear
FOOTNOTE: Physically, the arrow from A to C should lie on the path a taut line or string would follow between the two points.displacement or movement between them, that is the tail point A and the head point B. To show a second displacement from the head point B, put the tail of one second arrow at B. The result of these two movements, is a nonlinear movement from a tail point A of the first arrow to the head point, say C, of the second arrow.
The straight arrow joining the tail point A of the first arrow to the head point C of the second is a third arrow called the sum of the first two. It represents a linear movement from the points A to the point C. The foregoing describes the head to tail map addition method for adding two arrows together when the head of one is at the tail of another.
Two of the solid arrows and dotted lines parallel to them in the above diagram form the sides and diagonal of a parallelogram. The three solid arrows form a triangle. Another triangle is formed by the dotted lines and the diagonal arrow of the parallelogram. The rotation of these triangles and the parallelogram will have a deep consequence in the following overlapping discussion of complex, sines and cosines.
Reading Guide
Too much may be said in this chapter. If you get lost in the details, read this chapter lightly or go on to the next chapter. One aim of this chapter was to make fuss about some technical details - gaps in the author's comprehension perhaps.
Parallelogram Addition Method
When two arrows have the same tail points, they determine a parallelogram as well. They can be added by moving without a change in direction, one the arrows to the head of the other. This gives the parallelogram method for adding or summing two arrows with the same tail points (or origins).
Observe the presence of a two triangles and a parallelogram.
The dotted lines indicate positions of two of the solid arrows, after a movement to the head of the other without a change of direction. We will describe each displacement as a parallel movement of one along the other. The arrows before and after movement altogether form the sides of a parallelogram. The addition of one arrow to a second is represented by the parallel movement of the tail point of one to the head of the other. The formation of the parallelogram implies that which is added to which is immaterial, the result will be the same. Either way, the solid arrow along the diagonal of the parallelogram gives the (linear) arrow sum of the other two.
Arrow Components
The above diagram shows how the arrow from A to B can be regarded as the parallelogram sum of a horizontal arrow and a vertical arrow. The horizontal and vertical arrows are respectively called the horizontal and vertical components of the initial arrow from A to B. (These components depend on the choice of directions for the so-called horizontal and vertical axes.) The initial arrow is the map and parallelogram sum of the two component arrows. In the representation of arrows, an arrow can be viewed as the map addition of its vertical component to horizontal component arrows. Here the tail of the vertical component is moved to the head of the horizontal component. (The arrow could be also be viewed as the map addition of the horizontal component to the vertical one.) Which map addition is shown on a diagram is immaterial.
Component Addition Method
The following diagram shows the map addition of two solid arrows, namely the tail to head addition of the arrow from B to C to the arrow from A to B gives the same result as parallelogram addition of (i) the sum of the vertical components to (ii) the sum of the horizontal components.
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.