David Čepelík
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 Oct 31 awarded Popular Question Sep 13 asked Derivative of a multivariate function Sep 13 comment Convergence $\int_0^1{\log{(\arctan{}x)}\dfrac{\frac{\pi}{2}-\arcsin{x}}{(e^{1-x}-1)^a}}$ @Dr.MV OK, I'll try that! Sep 13 comment Convergence $\int_0^1{\log{(\arctan{}x)}\dfrac{\frac{\pi}{2}-\arcsin{x}}{(e^{1-x}-1)^a}}$ Thinking about it, it's probably the truth, as what we get is not a Taylor polynomial, right? We just use the TP of a little bit different function to get somewhere and then change the argument to finally arrive at an approximation, which works in this case, but it's not polynomial. Am I correct? Sep 13 comment Convergence $\int_0^1{\log{(\arctan{}x)}\dfrac{\frac{\pi}{2}-\arcsin{x}}{(e^{1-x}-1)^a}}$ I did see them and I tried. (Thanks, by the way, it's a really clever way to get around.) What I mean is; how can it be that we avoid the problems with infinite derivatives of ${\rm arcsin}\ x$ by substituting back and forth? From the definition of Taylor polynomial, I would not even guess this is remotely possible and would probably think that such growth just cannot be approximated by a polynomial. Sep 13 comment Convergence $\int_0^1{\log{(\arctan{}x)}\dfrac{\frac{\pi}{2}-\arcsin{x}}{(e^{1-x}-1)^a}}$ @Dr.MV, could you comment a little bit more on your expansion of ${\rm arcsin}\ x$ around $x = 1$? Is your approach purely "syntactic" or is there some analysis I don't get? I thought it wouldn't be possible to do the expansion at all. Sep 13 comment Convergence of $\int_0^1 \frac{\sin\big(\frac1{x^2}\big)}{\frac\pi2 -\ \rm{arccotg}\ x}$ It's good having that "peer reviewed" :). I'll update my post not to confuse anyone. Sep 13 comment Convergence of $\int_0^1 \frac{\sin\big(\frac1{x^2}\big)}{\frac\pi2 -\ \rm{arccotg}\ x}$ You're right. I was a little too fast about the 1st factor. In fact, I am not sure it converges at all. What I should've said is that $\sin(\frac1{x^2})/x^3$ can be written as $g(\omega(x))\omega'(x)$ for $g(y) := \sin y$, $\omega(x) = \frac1{x^2}$. Therefore, the primitive function at $(0, 1)$ is $-\cos(\frac1{x^2})$, which is bound. Not disturbing at all :). Sep 13 comment Convergence of $\int_0^1 \frac{\sin\big(\frac1{x^2}\big)}{\frac\pi2 -\ \rm{arccotg}\ x}$ never mind about that, what I regard the best is that I had somenone to bear the burden with me for a while :). You have a nice time! Sep 13 comment Convergence of $\int_0^1 \frac{\sin\big(\frac1{x^2}\big)}{\frac\pi2 -\ \rm{arccotg}\ x}$ Thank you for thorough explanation! Sep 13 accepted Convergence of $\int_0^1 \frac{\sin\big(\frac1{x^2}\big)}{\frac\pi2 -\ \rm{arccotg}\ x}$ Sep 13 comment Convergence of $\int_0^1 \frac{\sin\big(\frac1{x^2}\big)}{\frac\pi2 -\ \rm{arccotg}\ x}$ @eudes, thanks! That was the piece I missed. If you formulate that into an answer, I'll be glad to Czech the accept-mark ;) Sep 13 comment Convergence of $\int_0^1 \frac{\sin\big(\frac1{x^2}\big)}{\frac\pi2 -\ \rm{arccotg}\ x}$ @eudes, that may be it! That will simplify the calculation of the derivative. Great! Sep 13 revised Convergence of $\int_0^1 \frac{\sin\big(\frac1{x^2}\big)}{\frac\pi2 -\ \rm{arccotg}\ x}$ added 324 characters in body Sep 13 comment Convergence of $\int_0^1 \frac{\sin\big(\frac1{x^2}\big)}{\frac\pi2 -\ \rm{arccotg}\ x}$ @eudes, I'll try to use that. Sep 13 revised Convergence of $\int_0^1 \frac{\sin\big(\frac1{x^2}\big)}{\frac\pi2 -\ \rm{arccotg}\ x}$ f should converge, not g Sep 13 comment Convergence of $\int_0^1 \frac{\sin\big(\frac1{x^2}\big)}{\frac\pi2 -\ \rm{arccotg}\ x}$ Oh, sure. I'm sorry, I'll fix that. Sep 13 comment Convergence of $\int_0^1 \frac{\sin\big(\frac1{x^2}\big)}{\frac\pi2 -\ \rm{arccotg}\ x}$ Thanks! But how will that work for me? The function changes signs at $(0, 1)$, therefore, I cannot use the limit test. And the expansion can only be done for ${\rm arccotg} x$. Sep 13 asked Convergence of $\int_0^1 \frac{\sin\big(\frac1{x^2}\big)}{\frac\pi2 -\ \rm{arccotg}\ x}$ Sep 10 comment Function of distance to a moving object I'm a bit confused now :). Could you alter your question to describe the situation better? Furthermore, what's $d(t)$?