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  1. M

    Given an average cost function, calculate output level

    I think the responses so far are somewhat too bogged down in the details of computing the solution. Take a step back and look at the original problem, which is a business/economics question. Whate value of q minimizes the marginal cost? Standard calculus procedure: determine where (a)...
  2. M

    Integrate 1/x2+a2.dx=1

    The a^2 factor must be on the denominator, otherwise the integral is divergent both horizontally at infinity and vertically at zero. Just clearing up the interpretation. Since the original poster has access to the original problem anyway, this doesn't constitute helping him. ;) --Stuart...
  3. M

    Trouble with Limits

    Since the trouble was originally that you got 0/0 by plugging 6 in directly, you could also have used L'Hopital's rule. Of course, it is conceptually more elementary to factor the polynomial, but it may actually be less work to do 1 derivative than to do the polynomial long division. Just...
  4. M

    Adding trig functions for positive output

    It's absolutely not possible. Fourier theory says that if any function has a Fourier series (or Fourier transform) at all, then that transform is unique. Since a dc signal can be written as exp(i*0*t), this is its expression in terms of Fourier components. No other expression can possibly...
  5. M

    Dot product cross product, where did they come from?

    Responding mostly to mjsd: Pseudovectors are also sometimes called bivectors in the 3d case (in case someone is googling for further information). The main difference between wedge and cross is that wedge products produce multicomponent objects (essentially antisymmetric tensors). It is...
  6. M

    A quick Question about the set of Automorphisms of a field F

    I have at home a book called 'A Modern View of Geometry' (can't recall the author's name just now) in which the properties of projective spaces are gradually developed through a series of finite point models of the axioms. Hand in hand with this, the author sets up an algebraic structure, which...
  7. M

    ? about .999~=1

    @Hurkyl Got it! Thanks. I must be having a bad day, because it's obvious now that you mention it. Of course, the axioms are the same as for standard analysis when the new predicate "standard" is not used, so of course all the usual results follow whenever no distinction between standard and...
  8. M

    ? about .999~=1

    Thanks! I had forgotten this (or never new, perhaps), as I have been more interested in the surreals than the hyperreals lately. No special reason why, I just mentioned larger ordinals as an example of a possible extension. BTW, could you give me a reference on the hypernaturals indexing the...
  9. M

    ? about .999~=1

    While I fully agree with the correct subset of the arguments given above (you know who you are), it seems to me that there is a genuine ambiguity in the notation 0.999.... Caveat: Of course the real number system is defined axiomatically, so questions about the notation resolve ultimately not...
  10. M

    What are the condition of agency in the individual and the State?

    "Agency" means the ability to initiate action. It is often confused with "free will" which means a lack of constraint upon action. You have free will if you can decide for yourself whether to put on white socks or black socks this morning. You have agency if you have the power to put your...
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