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tutorfinder.com.au
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findatutor.ca
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use or become a tutor at your own risk
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
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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Chapter 1. Introduction
A Calculus Preview
Slopes for the graphs of straight lines, that is, linear functions y = mx+b
are met in high school algebra or trigonometry. Many problems involving the
slopes of linear functions can often be resolved by setting up and solving two
linear equations in two unknowns.
Slopes for the graphs of both linear and nonlinear curves y = f(x)
are met in late high school or early college calculus courses along with rules
for their calculation. In calculus, slopes are called derivatives. Formulas for
slopes are obtained or derived from formulas for curves y = f(x).
A simple geometric interpretation of slopes follows. The graph of a function y
= f(x) gives a two-dimensional trail through hills and valleys. A
skier in crossing such two or three dimensional hills is aware of the slope of
the ground and how this slope changes. The skier in question can tell when or
where the uphill and downhill sections are located from the slope of a ski. This
represents the first easily visualized physical or geometry interpretation of
slopes. Further examples will be given.
Rules for differentiation (slope calculation) give formulas for the slopes of
functions y = f(x). In the opposite direction, formulas for
functions y = f(x) may in some instances be found by
reversing the methods of slope calculation, a process called
anti-differentiation or integration. Finding a function f(x) from
a knowledge of its slope etc., leads to and justifies common formulas for the
perimeters, areas of regions in the plane, the length of curves and the volumes,
weights and masses of solids.
Other Books
The following why slopes chapters complement what is usually written
in algebra and calculus texts about the calculation of slopes and their
geometrical or physical interpretation. Their aim is to explain in a simple way
why slope calculation (differentiation rules) and the reversal of the slope
calculation process (anti-differentiation rules) are of interest. The rules for
differentiation and anti-differentiation are somewhat involved. But it is
possible without them to grasp clearly many of the ideas and motivations for
slope-related computations.
Most of the material below may fit between the definition of slopes for
straight lines in a high school algebra or trig course and the calculation of
slopes for nonlinear functions in calculus courses. The remaining material may
be read in or along side a first or second course on calculus or read before by
gifted students (avid readers) still in school.
Remark: The following texts or others will supply the missing
details.
- Calculus with Analytic Geometry by D. G. Zill, PWS Publisher, 1985
- Calculus of One and Several Variables, by S. L. Salas and E. Hille
(John Wiley & Sons 1971 and 1974, ISBN 0-471-00956-3).
- Calculus by L. Bers (Holt, Rinehart and Winston 1969, SBN
03-065240-5).
The above books or others on calculus should be in a public library or
a school library. Just as two views are better than one, so are two calculus
books better than one. When the wording in one is obscure or not readily
understood, the slightly different description or ordering of the same topics
in the other may clarify matters.This advice applies even to the pages of this
book. A break from reading might also have the same effect.
Remark: The formal or proper presentation of mathematics requires no
diagrams and no physical interpretation/reasoning. But without diagrams and
without geometric or physical interpretations in examples, mathematical ideas
can be without motivation. The following pages put the motivation first.
Complex Numbers
The chapters on vectors and complex numbers appear after the last why
slope chapters, but they may be read before. Some of the chapters in this
book are like clothes in a suitcase: the order in which they are packed or
unpacked is often unimportant.
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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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