Christopher Creutzig
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 Mar21 comment Prove or disprove my guess @Vim If you follow that link, you'll find examples that show that neither your nor the standard formulation is really rigorous and leaves out at least one precondition. Which I don't think I have seen pointed out anywhere else, but the examples make sense and yes, I did miss that point in the proof, come to think of it. Mar21 revised If f is a linear function of one variable,then how many points on the graph of the function are needed to specify the function? added 40 characters in body Mar21 comment If f is a linear function of one variable,then how many points on the graph of the function are needed to specify the function? No, that is one input. That is just how you deal with vector spaces: An element of the vector space is one value. (A vector can need infinitely many real numbers to be determined, so it doesn't make sense to always talk about those.) Mar21 comment Prove or disprove my guess @Vim That is not the usual formulation of l'Hospital's rule, see, e.g., mathworld.wolfram.com/LHospitalsRule.html or the wikipedia entry. I don't have a counterexample for the extension and it seems plausible; neither do I doubt that including it in the lemma simplifies things. But please don't be surprised if not everyone agrees that your proof directly follows from LHR. Mar20 revised If f is a linear function of one variable,then how many points on the graph of the function are needed to specify the function? added 798 characters in body Mar20 comment If f is a linear function of one variable,then how many points on the graph of the function are needed to specify the function? “Dimension” is outside the scope of the chapter on vector spaces? What exactly does that chapter talk about then, if it never gets around to vector spaces having bases? But I'll put a little more meat into the answer, all right. Mar20 answered If f is a linear function of one variable,then how many points on the graph of the function are needed to specify the function? Mar20 comment Prove or disprove my guess Well, strictly speaking, your question asks for the “why” of a claim that is wrong. The equality holds for $e^{-x}$, since $0=0$. You just can't use LHR to derive or prove it. Jan2 comment An Additional Rule for Calculus Your counterexample isn’t, since by definition $x^x=\exp(x\ln(x))$. You might need something like $_2F_1(1,x;1/2;x)$, although it feels like there should be simpler examples. Jan2 comment Show that $\lim_{n \rightarrow \infty} \left(\prod_{i=1}^{n} (a_i+1) \right)^{1/n}$ using Birkhoff Ergodic Theorem I assume the product goes to $n$, not $\infty$. As written, that limit exists if and only if the infinite product converges. In which case the limit is 0 or 1. Dec31 awarded Good Answer Dec30 revised Why do bases of infinite dimensional spaces need to be orthonormal? a_n must be inside the sum, since it depends on n Dec30 suggested approved edit on Why do bases of infinite dimensional spaces need to be orthonormal? Dec29 comment Chicken Problem from Terry Tao's blog Nice. But why must $u$, $v$, and $w$ be pairwise different? Dec29 comment What parts of a pure mathematics undergraduate curriculum have been discovered since 1964? We did Pollard’s rho method in second year (Germany), in a course focused on getting the computer do advanced computations. (Kind of a math-centered programming course, if you will, but still a math course, including the theoretical discussion. No complexity analysis or anything like that, though.) Dec28 comment Can we teach calculus without reals? While I’m a fan of differential algebra (which is the term I learned for what you described), it’s probably not a substitute for calculus in a school setting. You really want things like the mean value theorem, and without reals (or, if you take a rather complicated route, constructible reals), it’s just plain wrong. Dec28 comment Can we teach calculus without reals? I believe what you’re getting at here is what is known as “constructible reals.” They are, of course, countable. For a proper definition, though, you need to specify a language (in the CS sense) for describing the numbers, and in my experience, they are not really easy to work with. Dec26 comment Why do we need to assume continuity in the proof of the chain-rule? @Nick I’m not sure I see how it fails. What’s written there is pretty much the definition of being continuous at $x$, no continuity anywhere else is needed. Dec26 answered Finding Tangent Line to Graph Dec26 awarded Nice Answer