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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