Chapter 12. Units and Slopes
Volume 3, Why Slopes and More Math.
First Derivatives
Suppose or let y = f( x). Then the units of the slope m = f¢( x) are given by the ratio of the units of y over those
of x. That is,
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units of m = units of f '( x) =
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units of y
units of x
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The slope is given by a real number if the units of x and y are equal.
The case where the units of y are distance and the units of x are time
gives units of velocity in the form
Many different ratios of units are possible and allowable in the
computation of slopes.
Second Derivatives
Now let z = g( x) = f '( x) be the slope function for y = f( x). The
units of the slope M = g¢( x) = f '( x) are
given by the ratio of the units of z over those of x. That is,
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units of z = f '( x)
units of x
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(units of z)×
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1
units of x
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units of y
units of x
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×
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1
units of x
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Therefore the units of
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f '( x) =
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units of y
[units of x]2
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The case where the units of the first quantity y is meters and the units
of the second quantity x is also meters gives units
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units of y
[units of x]2
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=
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1
meter
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This case appeared in the previous chapter.
The case where the units of the first quantity y is meters and the units
of the second quantity x is seconds gives units
units ofy
[units of x]2
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=
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meter
sec2
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This last situation occurs in the discussion of acceleration, that is,
the rate of change of speed or velocity respect to time. In one such
discussion below, the role of the quantity x will be played by a time t
and the role of the quantity y will be played by a distance d = f( t).
Again, various combinations of units can appear.
Units or Dimensions
In physics courses, in place of talking about units of distance, time
and mass, there is talk about dimensions of distance D, time T and mass M.
With this specialized use of the word dimension, the foregoing notes on
units can be rewritten as follows.
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dimensions of y
dimensions of x
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dimensions of y
[dimensions of x]2
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Finally in some physics books, the square brackets [ y] is shorthand for
the phrase units of y, or more precisely the dimension of y.
(There is a subtle difference between units and dimensions which will not
be discussed here.) The last statements can be rewritten one more time as
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For home-tutoring or -schooling, or for schools or colleges
with course content control: Secondary
Mathematics for Ages 11+, A Practical Approach.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
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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
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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
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McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
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Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
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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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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
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Algebra
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Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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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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