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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Limit Definition of Slopes
The slope m to a curve y = f(x)
at x = x1 is defined by the limit
calculation
| f¢(x1)
= |
lim
Dx ®
0
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Dy
Dx |
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The right hand side of the above equation may be read as the
limit as Dx approaches
zero of the ratio [(Dy)/(Dx)].
Computation of right hand side limDx
® 0[(Dy)/(Dx)]
requires the existence of a number or quantity L with
the following property in the absence of units.
For each whole number k > 0, there exist an n
> 0 such that
|
ê
ê
ê |
L- |
Dy
Dx |
ê
ê
ê |
£ |
1
2 |
·10-k |
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if 0 < |Dx|
£ [1/2]·10-n.
The value Dx = 0 is
excluded as division by zero is not allowed.
Note that Dy = f(x2)-f(x1).
In the presence of units, the preceding requirement becomes
the following.
For each whole number k > 0, there exist an n
> 0 such that
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ê
ê
ê |
L- |
Dy
Dx
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ê
ê
ê |
£ |
1
2
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·10-k· |
units of y
units of x
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if 0 < |Dx|
£ [1/2]·10-n
·(units of x).
The Physical Limit
In the case of a slope, our initial conception was that
the slope of a short ski whose midpoint was placed at a
point (x1,f(x1))
of a curve y = f(x), would give the
slope of the curve there, at least approximately.
Physically, there might be some error in the placement. The
ski has to be short enough so that the slope of the curve y
= f(x) does not change too much. The idea of a
limit can be seen here (if you look hard enough) in the
requirement that the ski be short enough to lie on the curve
y = f(x) and not crossing several bumps
or oscillations in it.
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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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