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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
and study.
Learn to read notes and textbooks like a lawyer, so that no nuance, no
subtlety and no clause escapes your attention. |
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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.
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Linear Approximation
Skip (?) on first reading.
Suppose for a given number k > 0, there exist an n > 0 such that
|
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ê
ê
ê
|
L- |
Dy
Dx |
ê
ê
ê
|
£ e = |
1
2 |
·10-k |
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whenever |Dx| £ d = [1/2]·10-n,
then the following holds whenever the inequality |x2-x1| = |Dx| £ [1/2]·10-n is satisfied.
The difference L-[(Dy)/(Dx)] = c for
some number c with magnitude |c| £ e = [1/2]·10-k.
(The number c will depend on x2.)
The foregoing implies
and hence that
The latter in turn implies
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f(x2)-y1 = y2-y1 = LDx- cDx |
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and
and hence
|
f(x2) = y1+L(Dx)+an error |
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where the error is -cDx and its magnitude
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|c·Dx| £ |Dx|· |
1
2 |
·10-k £ |
1
2 |
·10-n· |
1
2 |
·10-k |
|
The last inequality provides information about the
error behavior in the
approximation of y = f(x2) by the linear function y = y1+L(Dx) = y1+mski(x2-x1). Since x2 is
arbitrary, the letter which plays it role is not important.
It can be replaced. In particular, x2 in the above
exposition can be replaced by a number x.
Theorem: [Consequences of a Non-Zero Slope] If
the slope m = f¢(x1) = L of f(x) at x = x1 is nonzero,
then there exist a d > 0 such that the sign
of f(x)-f(x1) equals the sign of L·(x-x1) whenever
|x-x1| £ d.
Proof: In the previous discussion, choose k such [1/2]10-k < |L| and let d = [1/2]10-n.
This theorem implies if m = f¢(x1) ¹ 0 then no interior
maximum nor minimum can occur at x = x1. Finding all
solutions x = a of the equation f¢(x) = 0 identifies
locations x = a at which interior maximums and minimums
might be found. The latter can also occur at points where
the slope or derivative f¢(x) is not defined. The points
x where
- f(x) is undefined, and
- f¢(x) is zero or undefined are called critical points. On
finite and infinite intervals, the
maximums and minimums of functions f(x) are located
- at critical points inside that interval, and/or
- at included endpoints.
So finding the critical points locates some, if
not all, of the maximums and minimums. This an extremum,
that is, a maximum and minimum locating principle for
functions.
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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.
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