www.whyslopes.com
Volume 3, Why Slopes and More Math
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YOU are better than YOU think. Show yourself how:
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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
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Caution: Site advice is approximately
correct, for some circumstances, not all. . That leaves room for thought and
refinement.. |
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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 serve as back ground info for partial fraction decomposition.
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For a straight line graph of a first quantity Q1 versus a
second quantity Q2, the slope
| m = |
change in Q1
change in Q2 |
= |
DQ1
DQ2 |
= |
rise
run |
· |
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The units of the slope m are therefore given by
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units of Q1
units of Q2 |
= |
units of the rise
units of the run |
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That is, the units of slope is the ratio given by the unit of
rise (vertical axis) over the units of run (the horizontal axis).
Examples of ratios of units that often occur are:
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$
kg |
, |
km
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km
hr |
, |
miles
hr |
,
1, |
etc. |
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Observe that when Q1 and Q2 are measured
with the same units, the slope m has no units. It is given by a (real)
number. An example follows. The purpose of the unit 1 in the previous
list is to be a reminder of this. We may say that a real number has the unit 1
to include numbers in any discussion of quantities. Alternatively, we may say,
as convenient, that real numbers are without units. See next example.
Improper Units
The unit 1 is an improper unit. The units 1.0 radian = 1, the degree
1.0 degree = [(p)/180] and the one percentage 1.0% =
[1/100] are further examples of improper units, that is units that are multiples
of real numbers. (The discussion of improper units is unique to this author.
Units in computations have been a concern for chemists and physicists but
not so far for mathematicians.
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For help in calculus, explore
Volumes
2. Three Skills
for Algebra
and 3. Why
Slopes & More Math, and Calculus
Introduction site area. See how to learn or teach key skills and
concepts, some not all.
Foreword, One Calculus preview and Online Chapters:
(V) signals video (RealPlayer Format) to
watch
Area Intro
Foreword
Chapter Descriptions
1. Introduction
Cal. Preview (1983 lesson why slopes)
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.
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