9
votes
Find a rectangular equation from a parametric equation. Why is my approach wrong?
find the rectangular equation for $x = \dfrac{t+1}{t}$ and $y = \dfrac{t - 1}{t}$
$\;\dots\,$ the correct answer is $x^2 - y^2 = 4$
$x^2 - y^2 = 4\,$ is the correct answer for the parametrization $\,...
- 68.3k
8
votes
Accepted
Finding where in Ramsey's theorem one uses the Axiom of choice
Ramsey's theorem for countably infinite graphs (or even just Dedekind-infinite graphs) does not require choice. It's only when we try to formulate it for arbitrary infinite graphs that choice comes ...
- 218k
7
votes
Accepted
Finding the value of $\lim_{a\to \infty}\int_0^1 a^x x^a \,dx$
First, note that
$$
\begin{align}
\lim_{a\to\infty}\int_0^1a^xx^a\,\mathrm{d}x
&\le\lim_{a\to\infty}\int_0^1ax^a\,\mathrm{d}x\tag{1a}\\
&=\lim_{a\to\infty}\frac{a}{a+1}\tag{1b}\\[3pt]
&=1\...
- 326k
4
votes
Accepted
Find a rectangular equation from a parametric equation. Why is my approach wrong?
Notice that
$$x^{2}-y^{2}=\left(\frac{t+1}{t}\right)^{2}-\left(\frac{t-1}{t}\right)^{2}=\frac{4}{t},\quad t\not=0$$
But since solving for $y$ and $x$ we get
$$\frac{1}{x-1}=\frac{-1}{y-1}\iff y+x=2$$
...
- 2,321
4
votes
Accepted
Is this valid to prove that $2 \ \vert \ x^3 \Rightarrow 2 \ \vert \ x$?
This method is not valid. You showed that $2(4k^3)\neq x^3$ for all $k\in\mathbb Z$ correctly, but but to show that $2\nmid x^3$, you need to show that $2n\neq x^3$ for all integers $n$, not just ...
- 21.2k
4
votes
Misapplication of the divergence theorem when calculating a surface integral?
I've figured out my mistake, thanks to @Event Horizon. My impression from the Help Center page is that I shouldn't delete my question, so I'll outline what went wrong:
I should've applied the ...
- 71
3
votes
Accepted
Does this proof of the binomial expansion (a+b)^2 work?
Here is the modified version of your proof with few skipped steps added and properties mentioned.
\begin{align}
(x+y)^2 &=(x+y).(x+y) & \\
&=(x(x+y))+(y(x+y))&\text{(using property 4 ...
- 1,500
3
votes
Accepted
Correct my strategy about :$\frac{a^3}{2a^2+b^2}+\frac{b^3}{2b^2+c^2}+\frac{c^3}{2c^2+a^2}\geq\frac{a+b+c}{3}.$
I think your strategy is correct. Also, I expanded the expressions after substitution using Python and compared it to your expansion, and the two lists are equal, so at least you didn't make a typo.
...
- 383
3
votes
Accepted
How to prove that $\int_{-\infty}^{+\infty}\frac{1+x}{1+x^2}dx$ doesn't exist
Let
$$f(u,v)=\ln\left(\frac{1+v^2}{1+u^2}\right)$$
we will prove that
$$\lim_{(-u,v)\to(\infty,\infty)}f(u,v)$$
does not exist.
if we take $ (-u,u) $ we find
$$\lim_{(-u,u)\to(\infty,\infty)}f(u,u)=0$$...
- 62k
3
votes
Accepted
Proof that $\inf(A)=-\sup(-A)$
You wrote: For each $a \in A$:
$$ x \text{ is a lower bound of } A \iff x \leq a$$
This is wrong because the quantifier "For each $a \in A$" is in the wrong place: it
should not apply to the ...
- 416k
3
votes
Accepted
Are proofs in infinite axiom systems verifiable?
I think you've got a bit confused. You can certainly have infinitely big axiom schemas, and that's what Noah's reply in your linked question affirmed. But e.g. in ZFC (which has infinitely many axioms)...
- 34.3k
3
votes
Misapplication of the divergence theorem when calculating a surface integral?
Just for reinforcement, I thought it would be good to show that your work is correct for $\iint_S \vec F \cdot d\vec S$, and we can show this by working directly:
$$\vec r (x, y) = \langle x, y, \...
- 3,934
2
votes
Does this proof of the binomial expansion (a+b)^2 work?
Basically correct, but here are two remarks:
From $(x + y)(x + y)$ to $x(x + y) + y(x + y)$ you need the result $(a + b)\cdot c = a \cdot c + b \cdot c$ which can be proved from 1. 4. but you haven't ...
- 20.6k
2
votes
Accepted
(Expectation)Consider the joint density $f(x,y)=c(x−y)e^{−x}, 0 \le y \le x$.
$X$ and $Y$ are jointly distributed, so the total probability over the common support $0 \le Y \le X$ must be $1$. In other words, you cannot equate the constant $c$ to some function of $x$. That ...
- 112k
2
votes
Accepted
Computing $\pi_1$ of subset $X\subseteq \mathbb R^2$, $X$ union of 3 simp.conn. subspaces w/ simp.conn. pairwise intersection but empty intersection
I think the easiest way to calculate $\pi_1(X)\cong\mathbb Z$ is to construct the universal cover. Consider infinitely many disjoint copies of each $V_i$, denoted $\{V_i(k)\}_{k\in \mathbb Z}$. Then ...
- 25.7k
2
votes
Accepted
Proving $F(x)$ is a CDF using the PDF
What you have show in that their is a massive step in $F(x)$ at $x=2$.
That $F(2)-\lim\limits_{y\uparrow 2}F(2)=1/3$. The function is not left continuous.
This does not mean that it is not a CDF, ...
- 119k
2
votes
Is there a mistake in an old question on this site: "Where does the sum of sin(n) formula come from?"
hint
Yes, Abel's result seems to be false.
$$2S\sin(1)=\sum_{k=1}^n\Big((\cos(k-1)-\cos(k))+(\cos(k)-\cos(k+1))\Bigr)=$$
$$1-\cos(n)+\cos(1)-\cos(n+1)=$$
$$2\sin^2(\frac{n+1}{2})+\cos(1)-\cos(n)=$$
$$...
- 62k
2
votes
Combinatorics explanation of Inclusion-Exclusion Principle Exactly-$m$ Properties Formula $E_m=\sum_{j=m}^n(-1)^{j-m}\color{blue}{{j\choose m}}S_j$
Here we follow closely the proof of Exercise 3, chapter 2: Sieve methods from Enumerative Combinatrics, Vol. I by R. P. Stanley. To ease comparison I'll try to use OP's notation where appropriate.
...
- 94k
2
votes
Accepted
($C^1$ Function)False or true? (justify) a) Let $f: X \to \mathbb R$. If $f'(x) = 0, \forall x \in X$, then $f'$ is constant...
a) is False.
Let $$X=(-\infty,0)\cup (0,+\infty)$$
and $ f $ defined at $ X $, by
$$(\forall x<0)\;\; f(x)=-1$$
and
$$(\forall x>0)\;\;f(x)=1$$
$ f $ is differentiable at $ X $ and
$$(\forall x\...
- 62k
2
votes
Accepted
(convex function) Let $f: [a,b] \to \mathbb R$ twice differentiable, and $f''(x) \ge 0$, $\forall x \in [a,b]$....
Definition. a function $f$ on an interval $(a,b)$ is said to be
midpoint convex if $$f\left(\frac{x_1 + x_2} {2}\right) \le \frac{1}{2}[f(x_1) + f(x_2)] \tag{1}$$ for all $x_1, x_2 \in (a,b)$.
...
- 6,663
2
votes
Solving $\tan ^{-1}(\frac{1-x}{1+x})=\frac{1}{2}\tan ^{-1}(x)$
Apply $tan$ to both sides
$ \dfrac{1 - x}{1 + x} = \tan \left( \dfrac{1}{2} \tan^{-1}(x) \right) \\
= \dfrac{ x }{ \sqrt{1 + x^2} + 1 } $
From this,
$ (1 - x) (\sqrt{1 + x^2} + 1 ) = x (1 + x) $
$ (1 -...
- 8,756
2
votes
Solving $\tan ^{-1}(\frac{1-x}{1+x})=\frac{1}{2}\tan ^{-1}(x)$
Thank you to @Robin'sPremiumCoffee for a nice algebraic solution. Now the problem with your original method is that although:
$$\frac{1-\tan \theta}{1+\tan \theta} = \tan(\frac \pi 4 -\theta)$$
It is ...
- 861
2
votes
Accepted
Prove that is is not possible to define the connective $\land$ in terms of $\lnot$ and ↔
First thing you should notice is that you also get $\top$ and $\bot$ since $a\leftrightarrow a =\top$. Also notice that $(\neg a\leftrightarrow b)\equiv \neg(a\leftrightarrow b)$. We also can restrict ...
- 1,512
2
votes
Accepted
Proving that $\ast\ast\alpha=(-1)^{kn+k}\alpha$ for every $k$-form $\alpha$ on $\mathbb{R}^n$.
Seems like it's on the right track. But one way to streamline the presentation, and avoid the case work, is to write the sign as $\epsilon_{I,I^c}=\text{sgn}(I,I^c)$, the sign of the permutation $(1,\...
- 34.1k
2
votes
Accepted
Prove that $\{x\in A:P(x)\}$ is always a subset of $A$
It's by definition, no proof required. Let $x \in P$ be arbitrary, $x\in P \iff x\in A: P(x)$.
Hence, $P \subseteq A$. "Technically" there is a conjunction where i put the colon, but we can ...
2
votes
Accepted
Find $\frac{b}{a}$ of trigonometry system of equation.
This problem does not require any Math.
Instead, all that is required is visualization and meta-cheating.
The $\cos(a) + \cos(b) = 0$ constraint is equivalent to saying that the angles $a$ and $b$ are ...
- 21.6k
1
vote
$L^1$ and a.s. convergence
We have $\omega$-wise $$
2^{k/2} Z_k \not\to 0 \iff Z_k\not=0\quad \forall k\in\Bbb N \iff X^2 \ge 2^k \quad\forall k\in\Bbb N \iff X^2 = +\infty
$$
and so:
$$
\Bbb P(2^{k/2}Z_k \not\to 0) = \Bbb P( ...
- 4,996
1
vote
Let $x \in (0,1).$ How to show that $x \in \bigcup \left(0, \frac{n}{n+1}\right)$?
In order to show that $x$ is in the union;
$$\displaystyle\bigcup_{n=1}^{\infty}{\left(0,\frac{n}{n+1}\right)},$$
you'll have to show that $x$ is in one of the intervals of the form;
$$\left(0,\frac{n}...
- 923
1
vote
Accepted
How do I show that, if $fp=fq$ then $p=q$, in the context of $f$ being the counit of an adjunction and $p,q$ natural transformations?
I want to suggest an alternative solution to you, since your notation is rather involved. Given the adjunction $F\dashv G$ with its unit $\eta:1\rightarrow GF$ and counit $\varepsilon: FG \rightarrow ...
- 1,303
1
vote
How to prove that $\int_{-\infty}^{+\infty}\frac{1+x}{1+x^2}dx$ doesn't exist
Alternatively, by definition $$\int_{-\infty}^{+\infty}\frac{1+x}{1+x^{2}}\, {\rm d}x<+\infty \quad \text{ if and only if}\quad \begin{cases} \int_{-\infty}^{\varepsilon}\frac{1+x}{1+x^{2}}\,{\rm d}...
- 2,321
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