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 11
Graphing the Slope versus Position
While following a curve or trail y = f(x), the slope may
- increase - become more positive or less negative;
- remain constant; or
- decrease - become less positive or more negative.
- From a to b in the last diagram, the slope is increasing by becoming more positive.
- From b to d, the slope is decreasing by becoming less positive and then more negative with a change of sign when x = c.
- From d to e, the slope is increasing by becoming less negative.
Some Vocabulary. A curve y = f(x)
is said to be convex on an interval where its slope (derivative) is increasing
or, at least, not decreasing. The curve is said to be concave on an interval
where the slope is decreasing (or at least not increasing).
FOOTNOTE: Some calculus books replace the two terms convex and concave by the phrases concave-up and concave-down, respectively. A curve y = f(x) is said to concave-up on an interval where its slope (derivative) is increasing, or at least not decreasing. The curve is said to be concave-down on an interval where the slope is decreasing (or at least not increasing).
Along a curve or trail y = f(x), the slope may vary. At points where the slope switches from increasing to decreasing, the slope is a maximum. At points where the slope switches from decreasing to increasing the slope is a minimum. Points inside an interval where the slope switches from increasing to decreasing, or vice-versa, are called inflection points. In a calculus course, you may meet another definition first, but it should be equivalent to this one.
Slope of the Slope
For a graph y = f(x) of a first quantity versus a second, there is a slope or derivative. But this slope as it changes can also graph versus the second quantity. Therefore there will slope to this graph as well. The slope functionm = g(x) = f '(x)
for a function y = f(x) also has a slope function
z = g'(x) = f "(x).
The latter, the slope z of the slope m, is called the second derivative of the original function y = f(x). The following diagram provides an example. It indicates the links between the behavior of the three functions, the height y = f(x), the first slope (or derivative) m = f '(x) , and the second slope (slope to the slope, derivative of the derivative) z = f "(x).
For an explanation why the unit of z = f¢¢(x) is 1/meter, see the next chapter.
Tongue Twister: The slope z = f "(x) of the slope m = f'(x) is negative for x < x1 and positive for x > x1. Therefore the slope m = f '(x) is decreasing for x < x1 and increasing for x > x1. So the slope m = f '(x) has a minimum at x = x1. This corresponds to the inflection point at x = x1 on the graph y = f(x). There the slope has the greatest magnitude. The slope of the slope is called the derivative of the slope or more properly the second derivative of the function f(x).
Note the slope m = f '(x) changes sign at x2 and x3. These sign changes correspond to a high point and a low point on the graph of y = f(x).
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.