Given that $$\int_{-\infty}^\infty e^{-x^2} dx = \sqrt\pi,$$ evaluate $$\int_{-\infty}^\infty e^{-kx^2} dx$$ and $$\int_{-\infty}^\infty x^2 e^{-kx^2} dx....
It is usually not difficult to google search mathematical notions; for example, one can search (with quotation marks) the term "brunnian braid" and find t...
Use the definition of a limit to prove the following: $$\lim_{x\to -2}(x-3x^2)=-14.$$ Our definition: Let $L$ be a number and let $f(x)$ be a function whi...
I understand that SOH CAH TOA represents a ratio. However, is there any proof or explanation that someone without knowledge on series can understand on ho...
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I have this limit: $$ \lim_{x\to2}\frac{x^2}{2x-4}$$ and I know (by looking at Wolfram Alpha) that this function does not have a limit, but it has left an...
It's easy to find the concavity based on an equation, but for some reason I don't understand how you could figure it out just by looking at it. ...
By the chain rule, we know that the derivative of $f(g(x))$ is $f'(g(x))g'(x)$. Question: When is the derivative of $f(g(x))$ equal to $g(f'...
Given a 3x3x3 cube (like a rubik's cube) where each of the 27 cubes has a distinct number, how many unique permutations are possible? Simple rotation...
Which is bigger among (i) $\log_2 3$ and $\log _3 5$ (ii) $\log_2 3$ and $\log _3 11$. Trial: I use AM-GM inequality and get $$\dfrac{\log_e 2+\log_e 5}{2...
Since the logarithm is the reversed exponentiation, why does it need a distinct notation for it? Why can't we just ask: $$2^x=8$$ Instead of: $$\log_...
This is a sort of soft-question to which I can't find any satisfactory answer. At heart, I feel I have some need for a robust and well-motivated form...