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 /--------------------------------------------------------\ EP0: Consider an arbitrary differentiable function f. Compute [d/dx]( x^f(x)). \________________________________________________________/ /--------------------------------------------------------\ 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))). \________________________________________________________/ /--------------------------------------------------------\ 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. \________________________________________________________/ /--------------------------------------------------------\ 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. \________________________________________________________/ /--------------------------------------------------------\ 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.) \________________________________________________________/ /--------------------------------------------------------\ EP5: Consider an infinite sequence, bvec, of reals. Prove that bvec has a monotone (infinite) subsequence. \________________________________________________________/ /--------------------------------------------------------\ 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. \________________________________________________________/