As of 05Oct2006

    These are the Extra, spur-of-the-moment problems that we (any of us) come
    up with in our Advanced Calc class.

    Genesis has kindly agree to keep track of the problems.  I adjoin them to a
    text file which is linked from our course page

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EP0: Consider  an arbitrary differentiable function f.
Compute [d/dx]( x^f(x)).
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EP1:  We are given differentiable fncs H() and f(),  with
H() always positive.  Let

        H(x)   :=   b(x)^{f(x)}.

Compute H'(x).
Footnote:  Note that        H(x)   =   exp(f(x)*log(b(x))).
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EP2: Cook-up three relations A,B,C such that:

1: Relation A is reflexive and symmetric, but not transitive.
2: Relation B is reflexive and transitive, but not symmetric.
3: Relation C is symmetric and transitive, but not reflexive.
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EP3:   The  set of "justin numbers" is the set   1 + 3*|N.
Characterize the justin-irreducibles in terms of their prime
factorization over the integers.
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EP4: Characterize the justin-primes in terms of their prime factorization
over the integers.  (We showed in class that the J-Primes are a proper
subset of the J-irreducibles.)
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EP5:  Consider an infinite sequence, bvec, of reals.   Prove that bvec has
a monotone (infinite) subsequence.
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EP6:  Fix a posint N and set

	S   :=   Ceil(sqrt(N)).

Consider a length-N tuple (finite sequence)

	bvec = (b_1, ..., b_N)

of reals.  Prove that bvec has a MONOTONE subsequence of  length S.

Footnote:  The set of reals is irrelevant; any
totally-ordered set   (T, <)  will do.
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