Category Archives: Calculus

(Ir)rational Behavior of Calculation

Posted on October 18, 2011 by Vadim Kulikov

We had the following set of exercises with a group of secondary school students (in Ressun lukio). Assume that a and b are positive real numbers. Then assume one of the following: (i) a and b are both rational (ii) … Continue reading →

Posted in Algebra, Calculus, Education, Mathematics, Wednesday Problem | 1 Comment

0.999999……=1? (continuation)

Posted on July 10, 2011 by Vadim Kulikov

Earlier, I posted a pretending-to-be-funny discussion on this subject and now continue in a more formal fassion. Namely the dialogue in the previous post didn’t actually arrive to any conclusion as to whether 0.9999… is 1 or not. When you … Continue reading →

Posted in Calculus, Mathematics | 2 Comments

0.999999….. = 1?

Posted on July 4, 2011 by Vadim Kulikov

The question in the title is constantly discussed on web forums and at our department’s corridors; sometimes even by professional mathematicians and is in fact already quite seen and banal. In our department this question is often raised by the … Continue reading →

Posted in Calculus, Education, Mathematics, Philosophy, Popular | 5 Comments

Back To College

Posted on March 17, 2010 by Vadim Kulikov

Please comment your solutions, questions and remarks.. Show that the following inequality holds for all [tex]k>0[/tex] and [tex]x\in \mathbb{R}[/tex]: [tex]e^x>kx-k\ln k.[/tex] Level 1/5

Posted in Calculus, Mathematics, Wednesday Problem | 2 Comments

Solution To The Previous Problem

Posted on March 6, 2010 by Vadim Kulikov

Exceptionally I will present a solution to the previous Wednesday Problem. Only not shall I present any simplest solution but a cool one. The question was the following. Suppose a function [tex]f\colon \mathbb{R}\to\mathbb{R}[/tex] has the property that given any reals … Continue reading →

Posted in Calculus, Mathematics | 3 Comments

Intermediate Value Functions

Posted on March 3, 2010 by Vadim Kulikov

Please comment your solutions, questions and remarks.. A continuous function [tex]f\colon\mathbb{R}\to\mathbb{R}[/tex] on the real numbers has the following property: Given any two reals a<b, the function f gets all possible values between f(a) and f(b) on the interval [a,b] (i.e. … Continue reading →

Posted in Calculus, Mathematics, Wednesday Problem | 1 Comment

The Monk

Posted on January 13, 2010 by Vadim Kulikov

Please comment your solutions, questions and remarks.. The posts have been a bit advanced lately. Let us lighten the atmosphere by this riddle which admits a simple solution, though mathematicians tend to use calculus in solving it: An Indian monk … Continue reading →

Posted in Calculus, Recreation, Wednesday Problem | 2 Comments

Continuum Hypothesis II

Posted on January 10, 2010 by Vadim Kulikov

Differentiability of Space Filling Curves A Peano curve is a surjective (onto) function [tex]f\colon\mathbb{R}\to\mathbb{R}^2[/tex]. Apparently such an f cannot be smooth. To see this consider the restrictions of this function to closed intervals [tex]f\restriction [n,n+1][/tex]. By smoothness and compactness the … Continue reading →

Posted in Calculus, Foundations, Mathematics, Set Theory | Tagged | Leave a comment

Where Do Random Walks Lead

Posted on December 14, 2009 by Vadim Kulikov

Rami Luisto asked the following question: Pick randomly n unit vectors from the plane, [tex]x_1,\dots,x_n\in \mathbb{R}^2[/tex]. What is the probability that the sum of these vectors has magnitude less than or equal to one, [tex]|x_1+\cdots+x_n|\leqslant 1[/tex]? (magnitude = norm = … Continue reading →

Posted in Calculus, Mathematics, Probability | 3 Comments

Continued fractions and strange homeomorphisms.

Posted on November 21, 2009 by Vadim Kulikov

Usually one thinks of real numbers represented by a sequence (possibly infinite) of numbers between 0 and 9. For example [tex]34,140414\dots[/tex] One can use various bases like binary, hexadesimal, but the most common is decimal, although with the growing trend … Continue reading →

Posted in Calculus, Mathematics, Topology | 8 Comments