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.
x-Dependence of Slope
[Play Video] 1¾ minutes: Along a 2D ski trail, see how height y = f(x) and slope m = f'(x) both depend on the horizontal coordinate x.
As Barbara, the one-ski skier, moves along a trail y = h(x), both the height of the ski midpoint y and the slope m of the ski depend on its x coordinate. The slope of her ski at x = a, x = b, x = c and x = 2 in the following diagram are all different. The slope of her ski midpoint depends on her location.
- At each point, the slope m of the ski is
determined by x and the shape of the trail y
= h(x). That is, the slope depends on x
and the hill y = h(x). To signal
this dependence, we write m = g(x)
= h¢(x). As
said before, the slope m = g(x)
could be measured from a snapshot - freeze Barbara and
her ski in place.10
10Footnote The mathematical definition of m = h¢(x) appears later.
- The slope m = g(x) depends on the shape of the hills y = h(x). The slope when x = a is m = g(a) = h¢(a). The slope when x = b is m = g(b) = h¢(b), and so on.
- In most problems that you will meet or be shown, formulas for the slope m = g(x) can usually be obtained or derived from formulas for the height y = h(x). For each new type of function added to your knowledge, there will be a differentiation rule to be learnt.
- For a given formula or function y = h(x), rules of differentiation say how to obtain or derive a formula or function g(x) = h¢(x) from a formula for height h(x). Presumably, because of these rules for deriving or obtaining the slope m = g(x) = h¢(x) from formulas for h(x), the function g(x) = h¢(x) is also called the derivative, the first derivative. Note that the use of the word obtainable for slopes is not an accepted alternative to the term derivative.
- There are several simple rules for calculation slope functions or derivatives. These differentiation rules are given in the first instance for the cases where an expression for the height function h(x) involves polynomials, logarithms, exponentials, sines or cosines. Here for each operation (addition, subtraction, multiplication, division and composition) involving functions and yielding a new function or formula, there are additional rules, all of which appear to be very simple after, but not before, they have been mastered. Meeting and mastering these rules requires or instills an understanding of the algebraic way of writing and thinking. The algebraic way of writing and thinking is seen or required here in full strength.
Notations for Slopes and Derivatives
Calculus has several expressions for the slope m of the ski or hill function y = h(x). Some follow.
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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.