the
Real Analysis appendices of
Why Slopes
and
More Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
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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.
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Mean Value Theorem
Theorem G.2. [Mean Value Theorem] Assume f(x)
is
continuous on an interval [a,b] and
differentiable when a < x < b.
Then there exists a point c interior to the interval
[a,b] which satisfies
Proof of Mean Value Theorem.
Let
| G(x)
= f(x)- |
f(b)-f(a)
b-a
|
·(x-a) |
|
Then G(a) = f(a) and
| G(x)
= f(b)- |
f(b)-f(a)
b-a
|
·(b-a)
= f(b)-(f(b)-f(a))
= f(a) |
|
as well. Therefore by Rolle's theorem applied to the
function G(x) on the interval [a,b],
there exist a point c interior to the interval [a,b]
such that 0 = G¢(c).
But
| G¢(x)
= f¢(x)- |
f(b)-f(a)
b-a |
|
|
With this, 0 = G¢(c)
implies
Remark. If g(x) is
continuous on an interval [a,b] then the mean
value of the g(x) on the interval is the
unique number r = [1/(b-a)]òab
g(x) dx. It is the unique number
with the property that 0 = òab
[g(x)-r]
dx. Now if g(x) = f¢(x)
is the slope or derivative of a function then the mean-value
of the slope g(x) = f¢(x)
on the interval [a,b] is r = [(f(b)-f(a))/(b-a)].
This may explain the phrase Mean Value employed
above.
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