# Category Archives: Calculus

## (Ir)rational Behavior of Calculation

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)

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?

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

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

Posted in Calculus, Mathematics, Wednesday Problem | 2 Comments

## Solution To The Previous Problem

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 $$f\colon \mathbb{R}\to\mathbb{R}$$ has the property that given any reals … Continue reading

Posted in Calculus, Mathematics | 3 Comments

## Intermediate Value Functions

Please comment your solutions, questions and remarks.. A continuous function $$f\colon\mathbb{R}\to\mathbb{R}$$ 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

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

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

Rami Luisto asked the following question: Pick randomly n unit vectors from the plane, $$x_1,\dots,x_n\in \mathbb{R}^2$$. What is the probability that the sum of these vectors has magnitude less than or equal to one, $$|x_1+\cdots+x_n|\leqslant 1$$? (magnitude = norm = … Continue reading
Usually one thinks of real numbers represented by a sequence (possibly infinite) of numbers between 0 and 9. For example $$34,140414\dots$$ One can use various bases like binary, hexadesimal, but the most common is decimal, although with the growing trend … Continue reading