Chapter 8. Slope Interpretation
Volume 3, Why Slopes and More Math.
The paragraphs
below repeat ideas met in earlier chapters. Is that repetition plus or a minus, that
may depend on the reader.
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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.
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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.
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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.)
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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).
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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).
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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.
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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.
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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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Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
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traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
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McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
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Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
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Is your child able to add, subtract and multiply amounts
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Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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