Mean Value TheoremTheorem G.2. [Mean Value Theorem] Assume f(x) iscontinuous 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
Let
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
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Presenting Appendices from Volume 3, Why Slopes and More Math, If the epsilon-delta viewpoint of limits, continuity and convergence is not yet comfortable, see Chapters 14 to 19 in Volume 3 are related. Proofs of one-sided theorems could be of interest in the study of 2D topology.
Vol 1A Logic Postscripts Proof by Absurdity alias proof by contradiction How the demand for consistency supports the law of the excluded middle Reality versus or with the aid of Imagination
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