Mathematics Weekly Newsletter
Mathematics Weekly Newsletter

Top new questions this week:

What did mathematicians study as an undergraduate/graduate before modern mathematics such as modern algebra and analysis?

I am curious as to what mathematicians such as Leibnitz and Gauss and the Bernoulli's studied when they were students in university. I find it fascinating how we are taught calculus and abstract ...

(soft-question) (math-history) (education) (big-list) (mathematicians)  
asked by Johann Franklin 29 votes
answered by Geremia 7 votes

Can you cancel out a term if equal to zero?

quick question here: In my proofs class we had a problem that after a little work we end up with: $x(x-y)=(x+y)(x-y)$ where $ x = y $. Now, I know this is pretty basic, but my teacher said that for ...

asked by Descoladan 21 votes
answered by Lubin 80 votes

A function having limit at every point but continuous nowhere

Is there a function $\,f:\mathbb{R}\rightarrow\mathbb{R},\,$ that has a limit at every point but is continuous nowhere?

(calculus) (real-analysis) (analysis) (limits) (continuity)  
asked by Tal Porat 17 votes
answered by Yiorgos S. Smyrlis 13 votes

Explain $x^{x^{x^{{\cdots}}}} = \,\,3$

$$x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} =\quad 2$$ This equation has the answer $\sqrt{2}$ by taking $\log$ to both side. This answer is correct because I'd proved it by computing the equation repeatedly ...

(algebra-precalculus) (limits) (tetration) (power-towers)  
asked by off99555 14 votes
answered by Omnomnomnom 17 votes

Dividing by 2 numbers at once, what is the answer?

Let's say i have 4/1/5. or 4 divided by 1 divided by 5. Are there any rules that i am allowed to use to stop any mistakes?, for example this has 2 solutions, 4/5 , and 20. Edit: Thanks for your ...

(notation) (fractions)  
asked by It'sRainingMen 14 votes
answered by John 32 votes

Time-optimal control to the origin for two first order ODES - But wait, the node is unstable? Hard-mode active!

I want to find the time optimal control to the origin of the system: $$\dot{x}_1 = 3x_1+ x_2$$ $$\dot{x}_2 = 4x_1 + 3x_2 + u$$ where $|u|\leq 1$ I ran straight into the problem full strength, hit it ...

(optimization) (calculus-of-variations) (optimal-control)  
asked by wordsthatendinGRY 14 votes

Evaluation of $\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$

How to evaluate the following integral $$\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$$ It looks like beta function but Wolfram Alpha cannot evaluate it. So, I computed the numerical value of ...

(calculus) (real-analysis) (integration) (definite-integrals) (closed-form)  
asked by Venus 12 votes
answered by Anastasiya-Romanova 3 votes

Greatest hits from previous weeks:

Visually stunning math concepts which are easy to explain

Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain, but are ...

(soft-question) (education) (big-list)  
asked by Nicholas 432 votes
answered by congusbongus 405 votes

Demystify integration of $\int \frac{1}{x} \mathrm dx$

I've learned in my analysis class, that $$ \int \frac{1}{x} \mathrm dx = \ln(x). $$ I can live with that, and it's what I use when solving equations like that. But how can I solve this, without ...

(integration) (indefinite-integrals)  
asked by polemon 65 votes
answered by Mike 84 votes

Can you answer these?

Ratio of product from one point and minimum distance

Given points $A_0,A_1,\ldots,A_n$ in the plane, let $m$ denote the minimum distance among any two points. What is the minimum value of $$\dfrac{|A_0A_1|\cdot|A_0A_2|\cdot\ldots\cdot|A_0A_n|}{m^n}?$$ ...

(inequality) (contest-math) (combinatorial-geometry)  
asked by simmons 5 votes

$\int\limits_{a}^{b} f(x) dx = b \cdot f(b) - a \cdot f(a) - \int\limits_{f(a)}^{f(b)} f^{-1}(x) dx$ proof

I just wanted to ask, if my proof is correct. I haven't seen the equation before, but I think it's quite useful. Let $f$ be an bijective differentiable function. Then the inverse function $f^{-1}$ ...

asked by Rummelluff 10 votes

Question on the paper Donal F. Connon, "Some integrals involving the Stieltjes constants"

I'm reading Donal F. Connon, Some integrals involving the Stieltjes constants. It gives a definition of the generalized Stieltjes constants $\gamma_n(u)$ as coefficients in the Laurent series ...

(calculus) (definite-integrals) (special-functions) (zeta-functions) (stieltjes-constants)  
asked by Vladimir Reshetnikov 7 votes
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