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tutorfinder.com.au
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USA: wiziq.com
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use or become a tutor at your own risk
YOU are better than YOU think. Show
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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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A Preview of Calculus and Calculus Courses
Several physical interpretations of slopes are summarized next.
- Slopes describe how fast the curve given by graphing one quantity versus
another, rises and falls.
Slopes in daily life occur in the
discussion of streets. A road or railway may rise three feet for
every 100 foot traveled horizontally. The grade or slope of the
road is then said to be 3%. An interesting or steep ski hill
may fall one meter for every three meters traveled - a slope of
-[1/3] or -33.3 percent. In buying and selling, there may be
an order charge plus a cost per unit for the total amount ordered.
The cost per unit is a slope. When one quantity is proportional to
second, the proportionality constant is also a slope. The cost of
purchase is often proportional to the amount bought. When apples
cost 25 cents each, ten more apples added to a purchase will cost
another 250 cents or 2.50 dollars.
- In general, a slope gives the change or increase in a
first quantity per or for each unit change in a second.
Mathematically the slope is expressed as the ratio of the
change in the first quantity to the change in a second.
Speed and velocities give the change in distance per unit
change in time. The acceleration which you feel as your
speed or velocity changes is mathematically represented by
the change in speed (or velocity) per unit change in time.
These quantities are all given by the slope of the graph of
one quantity or number versus another.
-
The slope at point on ski trail y = h(x) is first imagined to be given
by the slope of a small ski located at the point, more precisely
whose midpoint is located at the point.
This provides an initial image or definition of the slope to
a curve or ski trail, y = h(x). (The
mathematical definition to be preferred is much more precise
but less easily described. The
slope at a point is actually taken to be
the limiting value of numerical approximations to it. More
will be said about this.)
- In traversing the 2D (two dimensional) hill where
y = f(x), the slope m of the traveler's one ski changes with
(or depends on) its horizontal coordinate x. This gives a slope
function m = g(x). The notation m = g(x) signals that the
quantity m depends on the quantity x. Formulas for the slope
derivative function m = g(x) = h¢(x) can be obtained or derived
from simple formulas for the height function h(x), whenever the
latter are available or given. The prime in the notation h¢(x)
indicates that the formula for h¢(x) is obtained or derived
from the formula for h(x).
- A positive slope (when the horizontal
coordinate x is increasing) corresponds to the skier
going uphill. Similarly a negative slope means going
downhill. With this perspective, the slope of a ski will go
from positive to negative as it goes over a hill point. At
the top for one instant, its slope may be zero. When a ski
goes through a depression or a valley bottom, the slope of this ski is first
negative on the downhill side and then positive on the
uphill side. A skier may tell from the slope of a ski
when or
where he or she has crossed a
hilltop (maximum) or low point (minimum).
- A skier can recognize the intervals where the slope
is increasing, and the intervals where the slope is decreasing. A slope which is
becoming less negative or more positive as the skier moves
forward in the positive x directions is said to be
increasing. A slope which is becoming less positive or
more negative as the skier moves forward in the positive x
directions is said to be decreasing. On intervals where the
slope its slope is increasing, a function, y = h(x), is said
to be convex, and on intervals where its slope is
decreasing, a function, y = f(x), is said to be concave.
- Ski jumps and cliffs correspond to jumps or
discontinuities in the skiers trail y = h(x). Cross-sections of
rift valleys and plateaus further give examples of 2D ski
hills y = h(x) with ski jumps. Snow is assumed. At these
jumps the slope or derivative is not defined.
- At vertical drops
for instance, the slope is undefined. The slope of the ski
is further undefined or not determined by the trail at
kinks or sharp peaks where a short ski could pivot on its midpoint
without touching the trail on either side. For instance, the top of an
upside down V gives a sharp peak.
- Earthquakes, or
vertical motions up and down of 2D hills and curves,
suggest or imply that slope functions are not affected by the
upward and downward shifts of part of a curve. In
consequence, different hill shapes y = h(x) and y = f(x)
could have the same slope function m = g(x) = h¢(x) = f¢(x),
but different heights.
Yet (theorem) if the slope of a function y = h(x) is defined
everywhere on an interval, any other function with the same
slope will differ from y = h(x) by a constant vertical shift up or down,
in the interval (for why see the
advance material in the appendices).
The observation that two functions with the
same slope everywhere on an interval will differ by a
constant provides a key to the calculation of functions or
even their definition, from a knowledge of their slope.
Calculating functions from formulas for their slopes is
used to calculate area, volumes, and other
quantities.
-
Slopes, areas and volumes etc may be calculated or approximated
numerically by various methods. If the error in the
approximations tends to zero, the approximations approach
or converge to a limiting value. The limit should yield
the value of the number or quantity in question.
The question of what is a number or quantity may be
answered precisely by saying how it is calculated or how it
can be approximated with unlimited accuracy. Such
an answer often, if not always, gives the accepted
mathematical definition of the number or quantity in all
computational disciplines. See the chapters on slope, area
and velocity approximation.
-
In three dimensions, the direction of a perpendicular ^ to a
sledge which is flat against the trail surface, can be used
to locate slopes, hilltops, valley bottoms and mountain
passes between valleys. The perpendicular direction to the
sledge is vertical at hilltops, valley or depression
bottoms and at the high point of a pass between two valleys. Ski
jumps and sharp points on the terrain can be used to
represent the idea of discontinuity and the occasional
absence of tangent lines or planes. The rift valley in
Africa with its vertical sides, the Grand Canyon in North
America and holes or trenches (with vertical sides) dug
or found by road repair crews, give examples of three
dimensional discontinuities. Their cross-sections provide
examples of two dimensional ski jumps or discontinuities.
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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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