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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.
| |
Slope Sign Analysis Continued
Sign Analysis with Divisors (Rational Functions)
[Play
Video] 5 minutes: (coming soon) Sign Analysis for
slope given by quotient of linear terms
The previous sign analysis could also help in the study of a function with the
slope which includes some of above factors as divisors. For instance, the slope
function (derivative) for another function y = h1(x),
could be given by
| m1 = g1(x)
= |
(x-1)(x-2)
(x-4)(x+3) |
|
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When x ¹ 4 and x ¹
-3, this new slope m1 = g1(x)
has the same sign as m = g(x) = (x-1)(x-2)(x-4)(x+3).
This suggests that sign analysis can be done by identifying those intervals
where factors and divisors are positive, negative or zero. Note
division by a negative divisor has the same effect on the sign of m as
multiplication by a negative factor.
When m = [(q(x))/(r(x))] is a ratio of two
polynomials, sign analysis may be done by factoring of both the numerator
(top) polynomial q(x) and the denominator (bottom) polynomial r(x)
into linear and quadratic factors. Here the quadratic factors which have real
roots should be replaced by a product of linear factors. The Fundamental
Theorem of Algebra (proven by Gauss), says in principle that any
polynomial can be expressed as a product of linear and quadratic factors. But
Galois theory, a specialized topic in advanced algebra, implies no single
exact formula or method, involving roots of real and complex numbers for the
factors, will suffice to factor a polynomial of degree n > 5. In
contrast, formulas involving such roots are known for polynomials of degree n
£ 5. With the advent of the computer, approximate
methods can also be used to approximately find the roots and factors of some
polynomials. Moreover, for some special kinds of polynomials of a fixed degree
n > 5, exact formulas involving roots of complex numbers can also be
obtained - Galois theory just implies that no one formula will work for all
polynomials of degree n > 5.
Exercises
For each of the following cases where the slope function m is given by
a simple formula, find the x coordinate of the high and low points for
the corresponding height function y = h(x).
- m = (x+2)(x-5)/(x+1)
for -.5 £ x £
4
- m =(x+2))/[(x-5)(x+1))]
for -.5 £ x £
4
Pay attention to the end points of each interval. Each end point of an interval
may be a low or a high point, that is a minimum or maximum.
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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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