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
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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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Inverse Functions and Exponentials
A well-known theory briefly described
The above geometric definition implies ln(1) = 0. It also
implies that ln(2) > 0.5 Note the rectangle of height
0.5 with base segment [1,2] has area 0.5. It also lies
strictly beneath the curve u = [1/(v)] where 1 £ v £ 2. Now mathematical induction implies ln(2n) = n ln(2) > n/2 (since 2n+1 = 2n ·2).
Now ln(4) = ln(2)+ln(2) > 0.5+0.5 = 1. The
continuity
FOOTNOTE: This conitnuity can be shown directly. It is also
a consequence of the differentiability of this function.
of ln(x) between x = 1
and x = 4 implies by the Intermediate Value Theorem there
is at least one number e such that
ln(e) = 1. The number value y of the exponential
function exp(x) can now be defined as the unique number
y satisfying the equation ln(y) = x.
This definition of exp(x) leads to the
property
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exp(x1) ·exp(x2) = exp(x1+x2) |
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The
stage is now set for derivation of the algebraic properties
of the exponential expressions ab and the logarithm
loga(b). That can include a discussion of roots and
powers for positive numbers.
Note that the number e is called the natural number. The infinite decimal expansion
of e begins with 2.718281828 ¼
Note that the digits 1828 appear twice in this otherwise
non-repeating decimal expansion. The number e is irrational.
The proof of that e is not rational, is another
intellectual mortgage.
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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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