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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.
Area Calculation Problem
The selection of labels x and y for the horizontal and
vertical axes for most of graphs met so far is
arbitrary. It can be changed. The letters s and q
could have been used instead in all the previous graphs.
You should imagine this replacement, and the effect, if
any, it has on your knowledge or opinion of mathematics.
Problem: Suppose G(x) and h(x) are continuous at each point x in the interval [a,b]. Further suppose that the slope G¢(x) of G(x) satisfies G¢(x) = h(x) for a £ x £ b. Find a formula for the area A under the curve q = h(s) between s = a and s = b.The solution to this problem follows in three steps.
Step 1. Define an Area Function
First, introduce a function F(x) as follows. For each x between a and b inclusive let
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Here F(a) = 0 and F(b) = A is the area to be computed. Note that the area computation will be based on (a) finding the slope or derivative F¢(x) and then (b) observing how to obtain F(x) from a knowledge of F¢(x). The value of A = F(b) is required. It will be given by a formula involving the function G(x).
Step 2. Area Function Slope Calculation
Second, the following diagram leads to a formula for F¢(x).
In this figure, the area under q = h(s) from s = x to s = x+Dx is given by
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Since h(s) is assumed to be continuous on the interval [a,b], it is continuous at the point s1 = x. Therefore, for every whole number k, there is a number n such thatwhen |s-x| £ [1/2]·10-n. Now given k and such an n, if x £ s £ x+Dx and 0 £ Dx £ [1/2]·10-n then |s-x| £ |Dx| £ [1/2]·10-n as well. For such numbers s, it follows that |h(s)-h(x)| £ [1/2]·10-k. The latter in turn (see diagram) implies that the region B [between q = h(x) and q = h(s) above the s-interval from s = x to s = x+Dx] has an area
|h(s)-h(x)| £ 1
2·10-k This is equivalent to
|Area B| = |F(x+Dx) -F(x) -h(x)Dx| £ Dx · 1
2·10-k Therefore
ê
ê
êArea B
Dxê
ê
ê= ê
ê
ê|F(x+Dx) -F(x) -h(x)Dx
Dxê
ê
ê£ 1
2·10-k when 0 £ Dx £ [1/2]·10-n. Since the foregoing argument holds for every whole number k > 0, it implies that the limiting value of [(F(x+Dx) -F(x))/(Dx)] = h(x) when Dx > 0 approaches zero.
ê
ê
êArea B
Dxê
ê
ê= ê
ê
êF(x+Dx) -F(x)
Dx- h(x) ê
ê
ê£ 1
2·10-k
Step 3. Difference of Two Functions
Third, the previous discussion of vertical motions (and earthquakes) implies or suggests that for x in the interval [a,b], the difference F(x)-G(x) = C for some constant number C which does not depend on x. Recall F(a) = 0. This implies C = F(a)-G(a) = 0-G(a) = -G(a). Therefore, the constant value is given by C = -G(a) because F(a) = 0. Now F(x)-G(x) = C implies
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Remark. The latter formula is correct whenever G(x) is a function whose slope or derivative is h(x) for every x in the interval [a,b]. Given a formula for the function h(s) or h(x), the area calculation problem can be solved easily if methods of anti-differentiation can provide a G(x). For some h(x), this is possible. In particular, methods for anti-differentiation can be employed (sometimes) to find several functions G(x) whose derivative or slope on the interval x = a to x = b coincides everywhere with the slope h¢(x) of F(x). The problem statement above assumed one such function G(x) was available.
Methods for anti-differentiation or reversing slope calculations
say how to find possible
formulas for a function f(x) from a single formula for
its derivative (slope) m = f¢(x). These methods are ad hoc.
They do not work in all examples, but they do work in a large
number. Methods for finding or obtaining a
function from its derivative or slope lead to formulas for
the calculation of areas, volumes, weights, masses, forces,
totals etc met in geometrical, physical and some business
computations.
Remark. The computation of many geometric, physical and business quantities can be related to the computation of the area under the graph of some function q = h(s). The unit of area in these graphs is given by a product of the units of the horizontal and vertical coordinates q and s.
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.