Category: Exercises and dialogues

25 November 2019
Breaking the harmony
I found this curious exercise on the book Problems and Theorems in Analysis I, by Pólya and Szegö, and I thought it was a nice problem to propose here, since its solution...
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14 November 2019
Non sequitur
In this post the behavior of sequences obtained by convolving two sequences is analyzed. In particular, we will give sufficient conditions for the convolution to be a null...
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22 October 2019
The revenge of Menelaus
This post is devoted to Menelaus's Theorem, named for the astronomer Menelaus of Alexandria. It is a very imporant theorem in geometry, relating the ratios of the segments...
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10 October 2019
My slant on asymptotes
In this post we will investigate the connection between slant asymptotes and the behavior of the derivative at \(\infty\) for differentiable real functions. We say that a...
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23 September 2019
Yet another triangle
As we have seen in another post, identities involving inverse tangent look much less mysterious if analyzed from a purely geometrical perspective. Here you have the chance to...
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26 July 2019
Feeling grilled?
Get pen and paper and draw a cartesian plane. Then draw, on it, a set of points with integer coordinates, such as \(A(1,1)\), \(B(1,0)\), \(C(2,0)\), \(D(3,2)\), and so forth......
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10 July 2019
The bare necessities
This is the first of a series of posts in which a problem is presented to the reader, who in invited to solve it by following a proposed path. The background knowledge required...
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9 July 2019
Rock and rolle
Continuing were we left on a previous post about non differentiable functions that however have a well defined right derivative, we want now to show that...
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6 July 2019
Ups and downs
We are given a function \(f(x)\), for which the right derivative \[f'_+(x) = \lim_{h\rightarrow 0^+}\frac{f(x+h)-f(x)}{h}\] is defined in \(\Bbb R\). We also know...
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9 June 2019
Monotonically speaking
The concepts of monotonicity and continuity are very different. However, if we consider a monotonic function whose domain is an interval, we can deduce that right and left limits...
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