Three forms of mathematical induction: Difference between revisions

Revision as of 22:45, 6 March 2006

Revision as of 22:46, 6 March 2006

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===Second form===

====Product rule====

* The usual product rule taught in [[calculus]] says

* The usual product rule taught in [[calculus]] says

::<math>(fg)' = f'g + g'f.\,</math>

::<math>(fg)' = f'g + g'f.\,</math>

:For a product of ''n'' functions, one has

:For a product of ''n'' functions, one has

::<math>(f_1 f_2 f_3 \cdots f_n)' \,</math>

:::<math>= (f_1' f_2 f_3 \cdots f_n)' \,</math>

::<math>= (f_1' f_2 f_3 \cdots f_n)' \,</math>

+ (f_1 f_2' f_3 \cdots f_n)

+ (f_1 f_2 f_3'\cdots f_n)+\cdots +(f_1 f_2 \cdots f_{n-1} f_n').\, </math>

:In each of the ''n'' terms, just one of the factors is a derivative; the others are not.

:Now observe four facts:

**If ''n'' = 1, then the proposition just says

:Now observe four facts:

:::<math>f' = f'.\,</math>

**If ''n'' = 1, then the proposition just says

::It can be said that just one factor is a derivative, and it is [[vacuous truth|vacuously true]] that the others are not, because there are not others!

:::<math>f' = f'.\,</math>

*To go from case ''n'' to case ''n'' + 1 is trivial if ''n''&nbsp;&ge;&nbsp;2, but impossible if ''n''&nbsp;=&nbsp;1.

::It can be said that just one factor is a derivative, and it is [[vacuous truth|vacuously true]] that the others are not, because there are not others!

**To go from case ''n'' to case ''n'' + 1, isone trivial ifmust ''nuse''&nbsp;&ge;&nbsp;2, butthe impossible ifcase ''n''&nbsp;=&nbsp;12.

**ToThe gosubstantial frompart caseof ''n''the toproof is the case ''n'' += 12, oneand mustthat ''use''is precisely the caseproduct rule ''n''&nbsp;=&nbsp;2that is taught differential calculus.

====Triangle inequality====

**The substantial part of the proof is the case ''n'' = 2, and that is precisely the product rule that is taught differential calculus.

[More examples to be added.]