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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 6
Slopes and Vertical Shifts
The vertical shift or motion of a curve y = f(x) adds a constant d (the displacement) to each point. It yields a new curve y = f(x)+d. The next diagram indicates an example.
A simple vertical shift or motion may change the height of a skier, but not the slope of his or her skis.
Constant Difference Theorem
What can be said for sure about two functions when they have the same slope (or derivative) everywhere? One response is given by the following assertion. [Constant Difference] If the functions f1(x) and f2(x) have the same non-infinite slope (that is, derivative) m = f¢1(x) = f¢2(x) at every x in an interval (a,b) then the difference
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Note the assertion says that there is a constant d. It does not say how to find it. Here is an analogy: Saying there is a needle in a haystack, does not say how to find it. Note also the word every. If there is a point x1 in the interval (a,b) where the slope is not defined then no conclusions can be drawn from the Constant Difference theorem.
Remark. If f1(x) and f2(x) satisfy the conditions (hypotheses) in the Constant Difference theorem with f2(c) = A and f2(c) = B at some point c in the interval (a,b) then at x = c,
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Remark. Mathematical assertions and theorems which say that a number (or limit) is not defined or does not exist actually mean that a finite number (or finite limit) does not exist. The vertical motion theorem given above applies only when the common slope is finite in the interval (a,b) of interest.
Different vertical displacements over different portions of the trail are possible. The next diagram gives examples of this. Upshifts have been made in the topmost curve of the previous diagram.
Between a and b, between b and c and between c and d, the two trails y = f1(x) and y = f2(x) shown have the same slope but not the same heights. At the ski jump and cliffs in the upper trail y = f1(x), the slope is not defined.
Where Slopes are Not Defined
On a ski trail y = h(x), there may be a few places where the slope is undefined or a single slope to the graph or trail does not exist. In the following diagram, the slope is undefined at the ski jump above the point x = a. At sharp peaks and kinks, a short ski may pivot or rotate while keeping in contacting with one point, the kink or peak on the trail. So a single slope to the trail cannot be defined there. Where the trail has some vertical jumps, the graph ceases to be the graph of a function. The slope is said to be undefined or not to exist here, even though we might say it is infinite, +¥ or -.Consider the next diagram. At the points a,b,c,d and e, the slope function or derivative m = h¢(x) cannot be given a single value. In general, a single value cannot be assigned to slopes at sharp changes in the direction of a curve y = h(x).
The study of generalized slopes or gradients replaces the discussion of a single slope to a point on a curve by the discussion the set of slopes to a point on a curve. This set based discussion is too complicated to be examined further here.
But a single slope may sometimes be defined before and after such kinks or sharp turns in the graph of a function.
The ski at the sharp peak is shown pivoting, that is rotating, as the ski passes over.1 In pivoting at the peak, a ski can have many orientations or slopes without intersecting the curve y = h(x) on either side.
Problem for Advanced Students. A graph with vertical segments is not the graph of a function, but it may be the image of a parameterized curve (x(t),y(t)) where t belongs to some interval. Show that if (xj(t),yj(t)) are two continuous curves parameterized by t Î [a,b], then (x1¢(t),y1¢(t)) = (x2¢(t),y2¢(t)) for all t Î (a,b) implies there are constants d1 and d2 such that (xj(t),yj(t)) = (xj(t),yj(t))+(d1,d2).1The slope values during this rotation form a set, the slope set. The value 0 belongs to this set.
This problem is both a consequence and an extension, a generalization, of the constant difference theorem just stated. It applies to some graphs with vertical segments.
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.