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YOU are better than YOU think. Show
yourself how:
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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
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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 9
About First Courses in Calculus
Physics, chemistry,
engineering disciplines and some business disciplines all
employ calculus concepts, to efficiently describe their
computations and theories.
Formulas for slope function m = g(x) = h¢(x) (derivative)
can be obtained by applying rules
for differentiation when the function y = h(x) is given by a
simple enough formula. Rules for differentiation along a
collection of functions to which they apply, are typically
explained in a 12 to 16 week first course on calculus.
The differentiation process can be reversed sometimes.
Given a formula for the slope m = g(x), ad hoc integration
(that is, anti-differentiation methods) may identify
functions h(x) with slope h¢(x) = g(x). This reversal
provides or justifies the common formulas for areas,
volumes, weights and masses met in geometry and physics.
Anti-differentiation methods (alias integration methods)
and its applications are typically discussed
briefly at the end of a first course in calculus, and then
discussed more completely in a second course.
A third course in calculus may talk about the slopes
(directional derivatives) of 3D hills z = h(x,y) in place of 2D
hills y = f(x). Here above a point (x,y) in the plane, the
height of the hill z = h(x,y) depends on the coordinates (x,y).
Imagine a sledge in place of a ski traveling across this terrain.
The sledge in its direction of travel, its longitudinal direction,
has a slope (rise/run). Perpendicular to the direction of travel,
that is across the sledge, there is another slope. This traverse
slope will be horizontal or zero if the sledge is directly uphill
or downhill in the direction of steepest ascent or descent.
Otherwise, it will be nonzero. At the top of a large smooth hill
or the bottom of large smooth depression), a sledge will be
horizontal. The longitudinal and traverse slopes of a horizontal
sledge are both zero. When the sledge is not horizontal, there are
directions of ascent and descent. So there are nearby points which
are higher or lower than the sledge height (or its midpoint). The
foregoing analysis holds for small hills and small depressions as
well, providing the sledge is made smaller or microscopic.
Imagine now that the sledge lies on the hill z = f(x,y) and
its direction is perpendicular to the y axis. Further
suppose that the x coordinate of the front-end of the
sledge is greater than that of the other end. The
longitudinal rise/run of the sledge then gives the slope of
the hill y = f(x,y) in the x direction. Rules for differentiation,
essentially mastered in a first course in calculus,
say how this x-slope (or x-direction derivative) of the
hill shape z = f(x,y) can be computed. The slope (or
derivative) of the hill in the y direction is defined and
computed similarly. As indicated, directional
slopes (or derivatives) are often the subject of a
third course.
The Limit Definition of Slope
Rules for slope computation, that is rules for obtaining g(x) = f'(x)
from formulas for f(x) are studied in the first weeks or months of a first
course on calculus ALONG with limits. Calculus do not use pictures of skiers
to define slopes to nonlinear formula or functions f(x). Limits are used
instead. Rules for the reversal of slope computation (anti-differentiation)
may be met say after three month of calculus. With these rules, formulas for a
functions can sometimes be more easily obtained from formula for their slopes.
In particular, by the use of functions, and anti-differentiation, formulas are
obtained in calculus for areas, weights and masses of common shapes and
objects.
See Chapter 14, Limits, Error Control
and Continuity and the essay Decimal
Insights besides this page
Suppose you are walking or skiing along a smooth path, a path with no
vertical steps or drops. At any point, the slope can be measured or estimated by
placing a straight rod (a ski?) on the path. The rod should be pivoting at the
point or provide a bridge between two points on the hill, on either side of the
point where the slope is to be found. Then the slope of the hill at the point is
approximately equal to the slope of the rod = its rise over its run. Shorter
rods are better for slope approximation than longer ones. A more precise
definition or explanation of how to compute slopes requires the notions of a
limit. Using a short rod (or ski) to estimate the slope at a point on a path is
enough to understand the first geometric or physical interpretation of slopes.
Units in Calculus
Unit are too often forgotten in teaching computation
The calculation of slope = rise/run results in a pure number when the rise
and run are measured in the same units. The practice in physics and chemistry
deal with quantities - numbers times a unit of measurement. If you keep the
units in formulas y=h(x) for the graph of one quantity versus another, the rise
and run will have different units. Speed and velocity are measured in terms of
distance/time. This leads to units such as 60 miles/hour or 100 Kilometer/hour
(the slash / is read per). In flow measurement, there 10 kilograms of matter
passing a given point per second. The graph of material passed (measured in
kilograms) versus time (measured in seconds) would have a slope =rise/run = (10
kilograms)/(1 second) = 10 * (kilograms/second).
If you are averse to kilograms, replace the 10 kilograms by 21 pounds.
In the case of distance/time or velocity, a positive speed corresponds to
forward motion, a zero speed to no motion, and a negative speed to backward
motion. Forward acceleration means the speed is increasing, and negative
acceleration (de-acceleration for the grammatically proper) means the speed is
decreasing.
To put the word increasing in proper perspective, a temperature changes from
5 to 10 degrees Celsius and a temperature change from -15 to -3 Celsius are both
temperature increases. The reverse changes would be decreases.
More About Calculus
The pictures above show how to interpret the sign of the slope, and the slope
behavior for a walk along a path. Now if you have a smooth enough curve y = h(x)
drawn in the coordinate plane, you can use a short line segment placed against
the curve to estimate the slope at each point of this 2D hill y = h(x). You can
imagine that when your x-coordinate is specified, the function h(x) gives a rule
or formula for computing the height.
When y = h(x) = a x + b, you should know from algebra courses how to compute
the slope m = rise/run = a. The symbol we use to represent the slope is not
important. For more complicated expressions for h(x) involving polynomials,
sines, cosines, logarithms and exponentials etc, there are rules (justified in
first courses on calculus) which say how to obtain or derive a formula
for the slope of the curve y = h(x) from a formula for h(x). Slopes in calculus
are called derivatives, presumably because they are derived. But slopes
are not called obtainables. The latter word is not in fashion.
First courses in calculus are devoted to slope or derivative computation and
slope or derivative interpretation. We have seen one interpretation above.
Velocities, acceleration and all rates of change all give examples of slopes to
graphs y = h(x) or s = f(t). Here the letters used to define the horizontal and
vertical axis variables are not important. Whenever you have a smooth-enough
graph or hill shape y = h(x) or s = f(t) given by a simple formula, the slope
may be derived from the formula, and if not, estimated from the graph. (Slopes
are called derivatives in calculus.)
Rules for differentiation (slope calculation) give formulas for the slopes of
functions y = f(x). The word differentiation presumably stems from the
use of differences (Delta x over Delta y) in the estimation of slopes.
In the opposite direction, formulas for functions or formula y = f(x) may in
some instances be found by reversing the methods of slope calculation, an ad hoc
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
and weights and masses of solids. This reversal of slope computation, may be met
at the end of a first course in calculus. Second courses in calculus describe
and explore this reversal, that is, the integration or anti-differentiation
process, in more detail, ad nauseum. Third courses in calculus will further
describe slope computation and the slope reversal process for three dimensional
hills z = h(x,y) in place of two dimensional hills y = h(x), among other topics.
Finally note that saying how to compute a number or quantity defines it ---
and serves as a computational definition. The computation should be feasible,
otherwise the number or quantity in question is left undefined. In calculus,
there are two main examples:
- Slopes or derivatives at a point are defined by the limiting value of
| mchord
= |
Dy
Dx |
= |
{ |
slope of a chord between two points on a graph of y = f(x)
|
- Areas of regions are defined by covering the regions with small squares or
thin rectangles, adding up the areas of the squares or rectangles within the
region, and finally taking the limit of such sums as the width of the
squares or rectangles tend to zero.
Reversal of the slope computation process simply gives a shortcut (formulas)
for the computation of some common areas. It avoids the repeated approximation
of area by covering a region by small squares or thin rectangles. It gives the
limiting value of this approximations -- the number or quantity it tends to.
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www.whyslopes.com
Volume 3, Why Slopes and More Math - Preview, starter &
further lessons for calculus to ease or avoid algebra shock in instruction
& self-instruction
Foreword, One Calculus preview and Online Chapters:
(V) signals video (RealPlayer Format) to
watchChapters 2 to 6: offer a very simple preview of calculus and a context
for earlier study of slopes and factored polynomials
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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