the
Real Analysis appendices of
Why Slopes
and
More Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
|
These Real Analysis appendices
continue the decimal
viewpoint of limits, continuity and convergence in chapter 14. and
this further lesson
A. What's Next
B. Pigeon Hole Principle
B. Bolzano-Weierstrass
C1. Triangle Inequality
C2. Triangle Inequality
C. More T.Inequality
D. Sets & Sequences
D. Monotone Sequences
E. Limits, Properties
E Limits & Error Control
F. Continuous Functions
F. Closed Range Thm
F. Intermediate Val. Thm
F. Compactness Thm
F. Equicontinuity Thm
F Extreme Value Thm
G. Rolle's Theorem etc
G. Mean Val. Thm.
G. Constant Difference Thm
G. Lipschitz Continuity I
PS: One Sided Range Theorems
G. Velocity Revisited
G. Sufficient Conditions
H. Riemann Sums Conv
H. Lipschitz Continuity II
Proofs of one-sided theorems could be of interest in the
study of 2D topology.
| |
Constant Difference Theorem Again
We are now in a position to restate and prove the constant
difference theorem:
Theorem G.3 [Constant Difference Theorem]
If f1(x) and f2(x) have the same
non-infinite slope (that is, derivative)
m = f¢1(x) = f¢2(x) at every x in an interval
(a,b) then the difference
is constant for a < x < b. That is, there is a
constant d such that
for every x in the interval (a,b). This number d
does not depend on x.
Proof of Constant Difference Theorem.
Put f(x) = f1(x)-f2(x). Then f¢(x) = f¢1(x)-f¢2(x) for
every x interior to the interval [a,b and f is
continuous on the interval. Now f¢(x) = 0 interior to the
interval since f¢1(x) = f¢2(x). Now suppose a < x1 £ b. (Suggestion take x1 = b on first reading.) The mean value
theorem applied to each subinterval [a,x1] implies there
exist a c in the interval [a,x1] such that
But f¢(c) = 0. Therefore f(x1) = f(a). The latter holds
for a < x1 £ b.
Recall the role of the constant difference theorem in the
chapter Slopes and Areas. This theorem implied
the area under a (continuous) curve y = f(x) from x = a to
x = b is given by G(b)-G(a) whenever G¢(x) = f(x) for
a < x < b and G(x) is continuous on [a,b].
| |
|
|
|
www.whyslopes.com
site
search
Parents: Help
your Child/Teen Learn covers Speaking
Skills, Reading
& Writing,
Preparing for Science &
Having Patience, etc
Math How-TOs
1. Arithmetic
2. Algebra
3. More
Algebra 4. Geometry
5 More
Geometry 6. Calculus
>> densely written
>> use as skill checklists
Online Volumes (orders)
1, Elements of Reason.
1996
1A. Pattern Based
Reason 1995
1B. Math
Curriculum Notes 1996
2. Three
Skills for Algebra 1995
3 .Why.Slopes.&.More.Math.1995
Skill &
Concept
Review or Development
1. Decimal
Arith - Video Based ]
2 Fractions
3. Fractions
with Units
3. Solving
Linear Equations -
making alg easier
4. Formulas
forwards & Backwards - unifying theme for Algebra
5. Proportionality,
Back- & For-wards - theme at work.
6. Logic
- Math Free, good for precision in work & studies
7. Euclidean-Geometry
(leanly)
8. Slopes
and Lines
9. Why
Study Slopes - a context
10. Quadratics
11 Polynomials
12 Factored
Polys - a context
13 Functions
- For-& Back -wards
14 Number Theory,
Richly
15. Exponents, Radicals
& logs.
16 Calculus
- Examples & Advice
17. Real
Analysis
18 Electric
Circuits Etc (So So)
19 Maps,
Similarity & Trig, (alt view)
20 Complex
numbers
21
Logic with Symbols+truth tables
22 Consistent
Story Telling
23. Even
More Logic
|
|