Chapter 1. Introduction
Volume 3, Why Slopes and More Math.
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.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
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
Safety Messages and Questions: When and why should you face
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,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
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
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
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;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
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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