updates
What is the definition of a cusp?
So there are 3 situations that a function is not differentiable 1. a vertical tangent, 2. a discontinuity and 3. at a cusp. Consider the function $f(x)=e^...
Apr 13, 2026
Archive: Updates
updates
How to determine whether an improper integral converges or diverges?
For example: $$\int_{9}^{\infty}\frac{1}{\sqrt{x^3+1}}\,dx$$ The answer is that it converges, but why? I am so confused about this kind of questions. I tr...
Apr 13, 2026
updates
Plot a single point on number line in interval notation
For example, I want to plot the solution set $\{3\}\cup (2, \infty$). How do I represent 3 as a single point?
Apr 13, 2026
updates
Why does AB+BC=AC when adding vectors?
The formula for adding two vectors, as defined in Kells' Analytical Geometry is $AB+BC=AC$ This makes sense since we're concerned with both the ...
Apr 13, 2026
updates
Universal formula to calculate rotating by angle
I am generating roads and buildings that belong to them and since I want the streets to be rotated and then connected with each other, I need to rotate bo...
Apr 13, 2026
updates
Sierpinski triangle formula: How to take into account for 0th power?
The formula to count Sierpinski triangle is 3^k-1 .It is good if you don't take the event when k=0.But how can you write a more precise formula that ...
Apr 13, 2026
updates
Rules of imaginary numbers
I know $i=\sqrt{ -1}$; what I don't get is the results you get from raising $i$ to an exponent: $i^1 = i$ makes sense since anything to the first is ...
Apr 13, 2026
updates
integral of $\text{abs}(\sin(x))$ explanation
I know form wolfram alpha: https://www.wolframalpha.com/input/?i=integral+of+abs(sin(x)) that the integral is $-\cos(x)\text{sgn}(\sin(x))$ where sgn$(x)$...
Apr 13, 2026
updates
Square of matrix product equals product of squares
$A,B$ are two $n \times n$ matrices. Prove that $(AB)^2 = A^2B^2$ if $AB=BA$. We have $(AB)_{ij} = \displaystyle\sum_{k=1}^n A_{ik}B_{kj}$, so $(AB)^2_{ij...
Apr 13, 2026
updates
$x^n =2($mod $13)$
Consider the congruence $x^n =2($mod $13)$. Then for which n does it have a solution? n= 5 n=6 n=7 n=8 Can we take any help from this fact $x^{12 }=1 ($ m...
Apr 13, 2026
