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  • 0 posts edited
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  • 24 votes cast
Apr
22
awarded  Necromancer
Oct
19
comment Why is $\tan^{-1} (\tan(4 \pi/5)) = - \pi/5 $ not $4 \pi/5$?
Learn something about domain and image of a function.
Oct
14
comment Variance of Product of Uniform and Normal Random Variables
Find the distribution for Z=XY, and then calculate Var(Z).
Oct
12
comment Minimize $5\sqrt{36+x^2}+4(20-x)$ using Lagrange Multipliers
My fault, I should have clearly stated it as "equality constraints". Please read the wikipedia page for Lagrange multipliers. And then you'll see, you don't need that in this problem.
Oct
12
comment Minimize $5\sqrt{36+x^2}+4(20-x)$ using Lagrange Multipliers
Check when to use Lagrange multipliers. You don't have constraints in this problem.
Oct
12
comment How should I calculate $\lim_{n\rightarrow \infty} \frac{1^n+2^n+3^n+…+n^n}{n^n}$
Another proof for $e^{k-n} \geq (k/n)^n = e^{n \ln (k/n)}$: $\Leftarrow k-n \geq n \ln (k/n) \Leftarrow \frac{k}{n}-1 \geq \ln \frac{k}{n}$ $\Leftarrow$ for $x>0$, $x-1 \geq \ln x$ $\Leftarrow$ Let $y(x) = x-1-\ln x$, then the stationary point should satisfy $y'=1-\frac{1}{x}=0$; meanwhile $y''=x^{-2}>0$. Thus $y_{\min} = y(1) = 0$, for all $x>0$, $y(x) \geq 0$.
Oct
12
comment How should I calculate $\lim_{n\rightarrow \infty} \frac{1^n+2^n+3^n+…+n^n}{n^n}$
@JohnMa Did you? $(1/n)^n$ was substituted with $e^{1-n}$, $(2/n)^n$ with $e^{2-n}$, ..., $(\frac{n-1}{n})^n$ with $e^{n-1-n}$, and $(\frac{n}{n})^n$ with $e^{n-n}$.
Oct
12
comment How should I calculate $\lim_{n\rightarrow \infty} \frac{1^n+2^n+3^n+…+n^n}{n^n}$
@JohnMa For every term $(k/n)^n \leq e^{k-n}$, k = 1, 2, ..., n, so it should be "The limit should be no larger than ..."?
Oct
10
comment How do you find the Fourier transform of a function?
@BLAZE technically, yes.
Oct
10
comment How do you find the Fourier transform of a function?
@BLAZE if still cannot understand, you can always try the simplest case to help you. Just let $f(x,y)=x+y$, and then calculate $\int f(x,y)\; dx$.
Sep
18
comment Assistance with proof of $(AB)^T=B^T A^T$
Check a very simple case, say 1*2 & 2*3, you will find the problem.
Sep
17
comment Find equation of circle with reduced radius
What's the Radius of the larger circle?
Sep
13
revised why is frobenius norm of a matrix greater than or equal to the 2 norm?
added 780 characters in body
Sep
12
awarded  Yearling
Dec
20
awarded  Constituent
Dec
16
awarded  Caucus
Apr
11
answered Prove that $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$.
Apr
10
comment Prove that $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$.
Solve $f'(x)=e^x (x^2+5x+1) - 2x -2 = 0$, i.e. $e^x (x^2+5x+1) = 2x +2$. Because $e^x$ doesn't change the sign of $x^2+5x+1$, $y=e^x (x^2+5x+1)$ at most has two intersection points with the line $y=2x +2$. Thus $f(x)=0$ at most has three roots, and you have found them all.
Apr
10
comment Power calculation for simplification?
@RoyiNamir Not quite. But for further information about that, I suggest you read wikipedia pages about IEEE754, "float", and "double". If you have any problem on these issues, it may be better to ask it on stackoverflow.com
Apr
10
comment Power calculation for simplification?
@RoyiNamir 6.805647*10^38 - 0.000000406*10^38 = (6.805647-0.000000406) * 10^38. Please distinguish "multiplication" and "power" :)