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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Linear Approximation
Skip (?) on first reading.
Suppose for a given number k > 0, there exist an n > 0 such that
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ê
ê
ê
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L- |
Dy
Dx |
ê
ê
ê
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£ e = |
1
2 |
·10-k |
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whenever |Dx| £ d = [1/2]·10-n,
then the following holds whenever the inequality |x2-x1| = |Dx| £ [1/2]·10-n is satisfied.
The difference L-[(Dy)/(Dx)] = c for
some number c with magnitude |c| £ e = [1/2]·10-k.
(The number c will depend on x2.)
The foregoing implies
and hence that
The latter in turn implies
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f(x2)-y1 = y2-y1 = LDx- cDx |
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and
and hence
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f(x2) = y1+L(Dx)+an error |
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where the error is -cDx and its magnitude
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|c·Dx| £ |Dx|· |
1
2 |
·10-k £ |
1
2 |
·10-n· |
1
2 |
·10-k |
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The last inequality provides information about the
error behavior in the
approximation of y = f(x2) by the linear function y = y1+L(Dx) = y1+mski(x2-x1). Since x2 is
arbitrary, the letter which plays it role is not important.
It can be replaced. In particular, x2 in the above
exposition can be replaced by a number x.
Theorem: [Consequences of a Non-Zero Slope] If
the slope m = f¢(x1) = L of f(x) at x = x1 is nonzero,
then there exist a d > 0 such that the sign
of f(x)-f(x1) equals the sign of L·(x-x1) whenever
|x-x1| £ d.
Proof: In the previous discussion, choose k such [1/2]10-k < |L| and let d = [1/2]10-n.
This theorem implies if m = f¢(x1) ¹ 0 then no interior
maximum nor minimum can occur at x = x1. Finding all
solutions x = a of the equation f¢(x) = 0 identifies
locations x = a at which interior maximums and minimums
might be found. The latter can also occur at points where
the slope or derivative f¢(x) is not defined. The points
x where
- f(x) is undefined, and
- f¢(x) is zero or undefined are called critical points. On
finite and infinite intervals, the
maximums and minimums of functions f(x) are located
- at critical points inside that interval, and/or
- at included endpoints.
So finding the critical points locates some, if
not all, of the maximums and minimums. This an extremum,
that is, a maximum and minimum locating principle for
functions.
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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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