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Generalizing, we obtain, as required:
$\forall x (x\in \phi \rightarrow Q)$
Another approach: the 'vacuous truth' for $\forall$ is roughly the logical equivalent of an empty product being defined as 1 or an empty sum being defined as 0. Just as we want $\sum_{i=1}^{n+1} a_i = a_{n+1} + \sum_{i=1}^{n} a_i$ (and want t... | {
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# Finding a recursion formula
We have a sequence $$a_1=\sqrt{6}\\a_2=\sqrt{6+\sqrt{6}}\\a_3=\sqrt{6+\sqrt{6+\sqrt{6}}}\\...$$a)Find a recursion formula for $a_{n+1}$
b)Find a limit
Attempt: a) Tried finding the recursion formula: $$a_{n+1}=\sqrt{6+a_n}$$ I am not not sure about it because the problem does not say whe... | {
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# Choosing a substitution to evaluate $\int \frac{x+3}{\sqrt{x+2}}dx$
Is there any other value you can assign to the substitution variable to solve this integral?
$$\int \frac{x+3}{\sqrt{x+2}}dx$$
Substituting $u = x + 2$: $$du = dx; u +1 = x+3 ,$$ and we get this new integral that we can then split into two differe... | {
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• How did you come up with this solution? To me, it looks just incredibly clever; is there a specific rule to identify this kind of cases? – Johnny Bueti Jan 31 '16 at 12:15
• Very often if an integrand involves a radical, substituting by setting a new variable to be the radical expression improves the situation, and o... | {
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Let $\sqrt{x+2}=t\implies \frac{dx}{2\sqrt{x+2}}=dt$ or $dx=2t\ dt$ $$\int \frac{x+3}{\sqrt{x+2}}\ dx$$$$=\int \frac{t^2-2+3}{t}(2t\ dt)$$ $$=2\int (t^2+1)\ dt$$ $$=2\left(\frac{t^3}{3}+t\right)+C$$ $$=2\left(\frac{(x+2)^{3/2}}{3}+\sqrt{x+2}\right)+C$$ $$=\frac 23(x+5)\sqrt{x+2}+C$$
• Isn't $dt = \frac{1}{2\sqrt{x+2}}... | {
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# Can two distinct sets of N numbers between -1 and 1 have the same sum and sum of squares?
Is is possible to find two different sets of numbers $$\{ a_1, a_2, \dots, a_N\}$$ and $$\{ b_1, b_2, \dots, b_N\}$$ with $$a_i,b_i\in[-1,1]$$ such that $$\sum a_i = \sum b_i$$ and $$\sum a_i^2 = \sum b_i^2$$ are both true at t... | {
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However, for $$N = 3$$, we have
$$\frac{1}{2} - \frac{1}{2} + 0 = \frac{1}{\sqrt{3}} - \frac{1}{2\sqrt{3}} - \frac{1}{2\sqrt{3}} = 0 \tag{5}\label{eq5A}$$
$$\left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^2 + 0 = \left(\frac{1}{\sqrt{3}}\right)^2 + \left(\frac{1}{2\sqrt{3}}\right)^2 + \left(\frac{1}{2\sqrt{3}}\... | {
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You can easily extend this to $$N \gt 3$$. Also, you can also ensure that all $$|a_i|$$ and $$|b_i|$$ are unique, but the algebra becomes more complicated and messy.
• Thank you very much! Your answer made me realize that there should be an additional requirement for the elements in each set, which I unfortunately did... | {
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# Show that a vector $w \in V$ is in the span $(v_1, \dots , v_m)$
$$(2.A.10)$$ Suppose $$v_1, \dots , v_m$$ is a linearly independent list in $$V$$ and $$w \in V$$. Prove that if $$v_1 + w, \dots v_m + w$$ is a linearly dependent list, then $$w \in$$ span$$(v_1, \dots , v_m)$$.
Is $$w$$ being added to each vector in... | {
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# Is it true that a ring has no zero divisors iff the right and left cancellation laws hold?
This is the definition of zero divisor in Hungerford's Algebra:
A zero divisor is an element of $R$ which is BOTH a left and a right zero divisor.
It follows a statement: It is easy to verify that a ring $R$ has no zero divi... | {
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Suppose $ab = 0$ with $a, b \ne 0$. Either $ba = 0$ (which means $a$ and $b$ are zero-divisors), or $ba \ne 0$, in which case $ba$ is a zero-divisor because $a(ba) = 0$ and $(ba)b = 0$. | {
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# Thread: Proving a Trigonometric Identity
1. ## Proving a Trigonometric Identity
Hi everyone:
Is there any way to prove the following identity:
tan(A/2)=(sinA)/(1+cosA)
without drawing a diagram? I know that it is a basic trig identity, but I don't know how to prove it by manipulating hte formulas.
Thanks!
2. #... | {
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Introduce into the equation and you should solve your problem
Edit
In the time I took to write this I was beaten by 2 other forum users
I really need to take typing classes.
On the other hand, at least I provided a different method
5. Thank you so much!
6. Okay, Similar Question.
Prove that:
tan(A/2)=(1+sinA-cosA)... | {
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# Can you eat a 4-dimensional Rubik's Cube?
• Start by eating any piece except the central one
• Next, eat a piece orthogonally adjacent to the previously eaten piece
• (repeat)
• The last piece to get eaten in this way must be the centre piece
can you eat all the 81 pieces of a 3x3x3x3 Rubik's Hypercube?
To make th... | {
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If we denote by $$P$$ the path graph
T 0 1
o---o---o
on three vertices, then the graph $$\Gamma$$
• whose vertex set is the set of pieces in the 4D Rubik's cube, and
• for which two vertices share an edge if and only if the pieces are adjacent, is just $$P^{\square 4} = P \operatorname{\square} P \operatorname{... | {
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More generally:
If $$G$$ and $$H$$ are simple graphs with respective Hamiltonian paths $$(v_1, \ldots, v_k)$$ and $$(w_1, \ldots, w_\ell)$$, then $$((v_1, w_1), \ldots, (v_1, w_\ell), (v_2, w_\ell), \ldots (v_2, w_1), (w_3, v_1), \ldots, (v_k, w_\bullet)$$ is a Hamiltonian path on $$G \operatorname{\square} H$$ starti... | {
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# Proof verification: another convergent sequence proof
Note: Sorry, I posted this earlier with a glaringly obvious error - here's the improved version:
The statement I'm trying to prove is:
Let $(x_n)$ be a convergent sequence and $K \in \Bbb N$. Let $(y_n)$ be the sequence defined by $y_n = x_{n+K}$. Then $(y_n)$ ... | {
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Now let $\epsilon>0$ be arbitrary, and let $N\in\Bbb N$ be such that $|x_n-x^*|<\epsilon$ whenever $n\ge N$. $K\in\Bbb N$, so $K\ge 0$, and therefore $n+K\ge n\ge N$ whenever $n\ge N$. In particular, this ensures that $|x_{n+K}-x^*|<\epsilon$ whenever $n\ge N$. Since $\epsilon>0$ was arbitrary, this implies that $\lim\... | {
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# Existence of Solution to Differential Equations.
$f$ is locally Lipschitz in $y$ if for every $(t_0, y_0) \in (c,d) \times U$, there exists a neighborhood V of $(t_0, y_0)$, (i.e. $V = \{f(t,y) \in(c,d) \times U : ||t-t_0|<a \ \text{and} \ |y-y_0|\leq b\}$) and a constant K = K(V) such that $||f(t,x)-f(t,y)||\leq K|... | {
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• The steps in (a): First, find $M$, the maximum of $1+y^2$ on the interval $[-b,b]$. Can you do this? Then, find $K$ by using the fact that the Lipschitz constant of $1+y^2$ is just the maximum of the absolute value of its derivative on the interval $[-b,b]$. Can you find the value of this maximum? Then $\alpha$ comes... | {
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• Not quite. For one thing, $b$ is positive by the logic of the problem (you see $|y-y_0|\le b$ there), so there is no need to check negative values of $b$. More importantly, the relevant number here is the maximum of $\frac{b}{1+b^2}$, not where it is attained. This maximum is $\frac12$. Hence, the conclusion is that ... | {
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(a) By definition, $M = \max (1+y^2)$ on the interval $[−b,b]$. The Lipschitz constant $K$ is the maximum of $|(1+y^2)'|$ on the same interval. Hence, $\alpha<\min(1/(2b), b/(1+b^2),a)$.
(b) Since $b/(1+b^2)\le 1/2$ for any $b\ge 0$, any $\alpha$ allowed to us by the theorem must be less than $1/2$.
The value of $\alph... | {
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# Cube Root times Cube Root
• Nov 30th 2009, 10:03 AM
sharkman
Cube Root times Cube Root
cube root of 16(m^2)(n) multiplied by cube root of 27(m^2)(n)
I converted the problem to (16m^2n)^1/3 * (27m^2n)^/3.
Is this the correct way to write the original problem another way?
I finally got (432m^4n^2)^2/3.
Is this cor... | {
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Originally Posted by sharkman
The answer should be cubert{432m^4n^2}, right?
Don't be too shocked if your teacher marks it wrong because it has not been simplified as much as it could be. $m^4= m^3 m$ so $\sqrt[3]{m^4}= \sqrt[3]{m^3}\sqrt[3]{m}= m\sqrt[3]{m}$. Perhaps more importantly, $432= (16)(27)= (2^4)(3^3)= (2^3... | {
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# How to verify almost sure convergence?
Let $X_n \sim \text{Exp}(c^n)$, with $c > 0$. Let $Y_n = \min\{X_1, \ldots, X_n\}$. Find the distribution of $Y_n$ and study its convergence in distribution. Finally, show that $(Y_n)_{n \in \mathbb N}$ converges almost surely to some random variable for every $c$.
With the us... | {
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For $c>1$ you are basically right. Let $\epsilon>0.$ Then we have $P(Y_n>\epsilon) = e^{-\lambda_n\epsilon}$ where $\lambda_n = (c^{n+1}-c)/(c-1).$ Since $c>1,$ $\sum_{n=1}^\infty P(Y_n>\epsilon) <\infty,$ and thus by Borel-Cantelli, the probability $Y_n>\epsilon$ infinitely often is zero so $\limsup_n Y_n\le\epsilon$ ... | {
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So since $Y_m$ has rate $\lambda_y = \frac{c-c^{m+1}}{1-c}$ and $X_{m+n}$ has $\lambda_x = c^{m+n}$ so $$P(X_{m+n} <Y_m) = \frac{c^{m+n}}{c^{m+n} + \frac{c-c^{m+1}}{1-c}}$$ and then $$\sum_{n=1}^\infty P(X_{m+n} < Y_m) < \frac{1}{\frac{c-c^{m+1}}{1-c}} \sum_{n=1}^\infty c^{m+n}= \frac{c^m}{1-c^m}.$$
Now we can sum up ... | {
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# How to graph $x^2 -4x$?
I know about transformations and how to graph a function like $f(x) = x^2 - 2$. We just shift the graph 2 units down. But in this case, there's an $-4x$ in which the $x$ complicated everything for me. I understand that the graph will be a parabola for the degree of the function is 2, but I'm ... | {
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# Partial fractions
#### Petrus
##### Well-known member
Hello MHB,
I got stuck on this integrate
$$\displaystyle \int_0^{\infty}\frac{2x-4}{(x^2+1)(2x+1)}$$
and my progress
$$\displaystyle \int_0^{\infty} \frac{2x-4}{(x^2+1)(2x+1)} = \frac{ax+b}{x^2+1}+ \frac{c}{2x+1}$$
then I get these equation that I can't solve
an... | {
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#### Petrus
##### Well-known member
Re: partial fractions
My choice would be elimination. Try subtracting the third equation from the first, and this will eliminate $c$, then combine this result with the second equation and you have a 2X2 system in $a$ and $b$. Can you state this system?
I made it like a matrice and ... | {
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"openwebmath_score": 0.8400549292564392,
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$$\displaystyle \frac{2x-4}{(x^2+1)} = (2x+1)\frac{ax+b}{x^2+1}+ c$$
Now, having multiplied by our choice of term in the denominator, we plug in a value of x that makes this term zero. $$\displaystyle 2x+1=0$$ when $$\displaystyle x=-\frac{1}{2}$$, so plug in that value. The first term on the right becomes zero after ... | {
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"openwebmath_perplexity": 418.1120737288165,
"openwebmath_score": 0.8400549292564392,
"ta... |
# Probability of rolling to dice
If 2 fair dice are rolled together , what is the probability that the sum will be 9
1)Is the probability 4/36 (1/9); as no. of favorable cases are {(3,6);(6,3);(4,5);(5,4)} ?
2) Or is it 2/36 (1/18); as no. of favorable cases are {(3,6);(4,5)} the reason I am confused is that the que... | {
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"lm_q2_score": 0.8558511506439708,
"openwebmath_perplexity": 370.48851805647575,
"openwebmath_score": 0.6072463989257812,
... |
# A set $S$ has $n$ elements. How many ways we can choose subsets $P$ and $Q$ of $S$, so that $P \cap Q$ is $\emptyset$?
This is how far I could go..
example:
S = {1,2,3}
Unique possible subset pairs such that the intersection is {}
P = {} Q = {}
P = {} Q = {1}
P = {} Q = {2}
P = {} Q = {3}
P = {} Q = ... | {
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which works. But how do I simplify it?
Also, what would be a better approach for this problem?
• Why didn't you count $P=\{1\},Q=\{\}$?
– bof
May 16, 2015 at 23:42
• @bof: Since I am trying to count unique subset pairs. The pair $\left \{ \{1\}, \{\} \right \}$ is same as the pair $\left \{ \{\}, \{1\} \right \}$ May... | {
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"openwebmath_perplexity": 340.4451917258159,
"openwebmath_score": 0.8102799654006958,
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• What a nice answer! But when $S$ has $3$ elements, as Pragy has shown answer must be 17; while taking $n=3$ in this way gives $14$, so where is the problem?
– Sry
May 18, 2015 at 4:47
• In that list, $3$ items occur twice each. For example, one line (item 9) has $\{1\}$ then $\{2\}$. Three lines later the post lists ... | {
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Note that my solution above assumes that we count ordered pairs of such sets. If we seek to count unordered pairs, then there is a bit more we must do. Let $S(k,m)$ be the Stirling number of the $2^{nd}$ kind (i.e. $S(k,m)$ counts the number of partitions of $\{1,2,\cdots,k\}$ into exactly $m$ blocks). Then the set $\m... | {
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"lm_q1_score": 0.9871787853562028,
"lm_q1q2_score": 0.8448780993384236,
"lm_q2_score": 0.8558511506439708,
"openwebmath_perplexity": 340.4451917258159,
"openwebmath_score": 0.8102799654006958,
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$K= \mathbb{F}_2(\alpha)$ $\alpha$ root of $X^4+X+1 \in \mathbb{F}_2[X]$. Find degree and minimal polynomial
Question 1: Find $[K:\mathbb{F_2}]$
Idea: I have tried looking at the irreducibility of the polynomial, $X^4+X+1$ and have so far been unsuccessful. Is there another way to do this apart from using Eisenstein'... | {
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"lm_q1_score": 0.9744347920161259,
"lm_q1q2_score": 0.8448698236650416,
"lm_q2_score": 0.8670357735451834,
"openwebmath_perplexity": 159.7111273415104,
"openwebmath_score": 0.8905219435691833,
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Would really appreciate some help from you, thanks
• I don't think this is the first time irreducibility of $x^4+x+1$ over $\Bbb{F}_2$ has been handled on our site. You could search :-) Eisenstein cannot be applied, because there are no primes in $\Bbb{F}_2$. Nor in any other field. Fields have no primes - only units.... | {
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$$\begin{cases} \Bbb F_2[x]\to \Bbb F_2[\alpha]\\ x\mapsto \alpha\end{cases}$$
The map is clearly surjective, and by definition of the minimal polynomial and the fact that $\Bbb F_2[x]$ is a PID, we see the kernel is exactly $(x^4+x+1)$. But then the first isomorphism theorem for rings says that $\Bbb F_2[\alpha]\cong... | {
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• a dumb question : how do I jump from $\alpha$ is a root of some irreductible degree $4$ polynomial to $[\mathbb{F}_2(\alpha) : \mathbb{F}_2] = 4$ ? Mar 29 '16 at 17:21
• @user1952009 because the polynomial is irreducible, it means that $1,\alpha,\alpha^2,\alpha^3$ are linearly independent, so the space $\Bbb F_2(\alp... | {
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for the second part we have $\alpha^{-1}$ = $\alpha^{14}$ because $\alpha^{14} = \alpha^{3}+1$ and $\alpha \times \alpha^{14} =\alpha \times (\alpha^{3}+1)= \alpha^{4}+\alpha = 1$ because we know $\alpha^{4}+\alpha+1=0$. so the minimal polynomial with respect to $\mathbb{GF_2}$ containing $\alpha^{14}$ comes from produ... | {
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"lm_q2_score": 0.8670357735451834,
"openwebmath_perplexity": 159.7111273415104,
"openwebmath_score": 0.8905219435691833,
"tag... |
# Is the definition of the Riemann sum from Thomas' Calculus correct?
I'm having trouble with theoretical understanding of the Riemann sum with this explanation/definition from Thomas' Calculus. I checked Wikipedia and it seems to state virtually the same.:
On each subinterval we form the product $$f(c_k)*∆x_k$$. Thi... | {
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"openwebmath_score": 0.8009864687919617,
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Thank you.
• the Riemann sum approaches the signed area, which is $-\frac12+\frac12=0$ in your example; cf. the Wikipedia integral page May 24, 2020 at 23:17
• @J.W.Tanner what is a signed area? Negative? I looked into an online app to calculate sums and take the screen shot from here. And it says the area is approxim... | {
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# Solving Hamiltonian eigenvalue problem
I would like to solve an eigenvalue problem of a Hamiltonian. I was able to find the lowest eigenvalue by converting the Hamiltonian into a matrix and applying linear algebra eigenvalue techniques. But this method is extremely cumbersome and does not generalize to arbitrary-siz... | {
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You can solve this by referring to this question.
To estimate the eigenvalues of $$H\left( s \right) =\left( 1-s \right) H_0+sH_m=I-\left( 1-s \right) |\psi _N\rangle \langle \psi _N|-s|m\rangle \langle m|$$, we can only calculate the eigenvalues of $$\left( 1-s \right) |\psi _N\rangle \langle \psi _N|+s|m\rangle \lan... | {
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"lm_label": "1. YES\n2. YES",
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"lm_q1_score": 0.974434786819155,
"lm_q1q2_score": 0.8448698174851232,
"lm_q2_score": 0.8670357718273068,
"openwebmath_perplexity": 108.88494371026168,
"openwebmath_score": 0.982545018196106,
"tags... |
For any state in the span, you can think about a linear combination $$\alpha|a\rangle+\beta|b\rangle$$ and how $$H$$ acts on this. The outcome is always a state in the same span. Hence, we can talk about this as a two-dimensional subspace and just write out a $$2\times 2$$ matrix. It looks something like $$H_\text{sub}... | {
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Prove there are no prime numbers in the sequence $a_n=10017,100117,1001117,10011117, \dots$
Define a sequence as $a_n=10017,100117,1001117,10011117$. (The $nth$ term has $n$ ones after the two zeroes.)
I conjecture that there are no prime numbers in the sequence. I used wolfram to find the first few factorisations:
... | {
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"lm_q1_score": 0.9744347838494567,
"lm_q1q2_score": 0.8448698132363299,
"lm_q2_score": 0.8670357701094303,
"openwebmath_perplexity": 317.69015171432295,
"openwebmath_score": 0.7401050925254822,
"ta... |
• Can you explain the last line? – zz20s Mar 23 '16 at 13:55
• Thank you! Your answer makes a lot of sense. Can you explain why you decided to multiply by $9$? – zz20s Mar 23 '16 at 14:11
• @zz20s, to clear the denominators. – lhf Mar 23 '16 at 14:21
• When you have a number whose base-10 representation has long string... | {
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"openwebmath_score": 0.7401050925254822,
"ta... |
Scilab Home page | Wiki | Bug tracker | Forge | Mailing list archives | ATOMS | File exchange
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Ajuda do Scilab >> Biblioteca de Gráficos > 2d_plot > plotimplicit
# plotimplicit
Plots the (x,y) lines solving an implicit equation or Function(x,y)=0
### Syntax
plot... | {
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The bounds of the 1st plot drawn by plotimplicit(..) are set according to the bounds of the solutions of fun. Most often they are (much) narrower than x_grid and y_grid bounds.
plotOptions
List of plot() line-styling options used when plotting the solutions curves.
### Description
plotimplicit(fun, x_grid, y_grid) e... | {
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"openwebmath_score": 0.5431338548660278,
"tags": nul... |
// Increase the contours thickness afterwards:
gce().children.thickness = 2;
// Setting titles and grids
title("$3x^2 e^x - x y^2 + {{e^y}\over{(y^2 + 1)}} - 1 = 0$", "fontsize",4)
xgrid(color("grey"),1,7)
Overplotting:
clf
plotimplicit("x*sin(x) = y^2*cos(y)", [-2,2])
t1 = gca().title.text;
c1 = gce().children(1);... | {
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"tags": nul... |
Review question
Can we draw the graph of $\left| x + [x] \right|$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource
Ref: R5362
Solution
A function $f$ is defined on $\mathbb{R}$ by $\begin{equation*} f \colon x \to \left| x + [x] \righ... | {
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"openwebmath_score": 0.942801952362060... |
The notation $x \in \mathbb{R}_+$, $x \notin \mathbb{Z}_+$ means that the domain of $g$ is the real numbers greater than zero but excluding all the positive integers. Where the question asks for the ‘rule’ it means an algebraic definition of the function.
Remember that an inverse function does not exist for a function... | {
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"openwebmath_score": 0.942801952362060... |
Thus, the required rule is $g^{-1}(x) = x - \dfrac{[x]}{2}$, as before.
The graph of $y = g(x)$ is in blue, the line $y = x$ is a dashed line, and $y=g^{-1}(x)$ is in red.
Suppose someone objects that $g^{-1}(x) = x - \dfrac{[x]}{2}$ is not the inverse of $g$ as, for example, $\begin{equation*} g(g^{-1}(5.75)) = g\le... | {
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"openwebmath_perplexity": 78.56456086163674,
"openwebmath_score": 0.942801952362060... |
# Math Help - Evaluating Infinite Sums
1. ## Evaluating Infinite Sums
Evaluate the following infinite sums. (In most cases they are f(a) where a is some obvious number and f(x) given by some power series. To evaluate the various power series, manipulate them until some well-known power series emerge.)
[sum from n=0 ... | {
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"openwebmath_perplexity": 2235.3607863318994,
"openwebmath_score": 0.9766163229942322,
... |
# Interpolate on log scale
I have data in Mathematica that comes from y-log scale
Data = {{5.0, 23.87548081003781},
{6.94392523364486, 0.511639358262082},
{8.925233644859812, 0.23397526329810545},
{10.962616822429906, 0.16190746961888203},
{12.906542056074766, 0.17751810380557045},
{14.925233644859812, 0.256534458699... | {
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"openwebmath_score": 0.22834773361682892,
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# First positive root
Simple question but problem with NSolve.
I need help how to extract first positive root?
For example:
eq = -70.5 + 450.33 x^2 - 25 x^4;
NSolve[ eq == 0, x]
If I have an equation eq that I am not sure of polynomial order and I need to define all positive roots x[1], ..., x[2].
The most strai... | {
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Plot[ eq, {x, -4.5, 4.5}, PlotStyle -> Thick, AspectRatio -> 1/3, Epilog -> {
Darker @ Red, Thickness[0.005], Line[{{#1, 0}, {#2, 0}}]& @@@ intervals,
Green, PointSize[0.007], Point[{#, 0}] & /@ roots } ]
the same plot in pieces :
GraphicsRow[
Plot[ eq, {x, #1 - 0.3, #2 + 0.3}, PlotStyle -> Thickness[0.01], Epilog... | {
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• yes, but how to extract second, third ... separately?
– Pipe
Dec 18, 2012 at 21:35
• @Pipe you can use Sort instead of Min to get the ordered positive roots
– ssch
Dec 18, 2012 at 21:35
• @Nasser it is working but how to extract just second for example
– Pipe
Dec 18, 2012 at 21:39
• @Nasser Thank you very much, it is... | {
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# How to correctly enumerate all the schemes of this cube coloring problem?
This problem is the fifth question of 1996 Chinese High School Mathematics League or Chinese Mathematical Olympiad in Senior:
Choose several colors from the given six different colors to dye six faces of a cube, and dye each two faces with co... | {
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The results of the above codes are 198030 , 4080 and 215, but the reference answer is 230 (Maybe I didn't effectively exclude the same dyeing scheme after rotation). How to correctly list all the solutions to this problem?
f = Table[{i, Delete[Range[6], {{i}, {7 - i}}]}, {i, 6}];
g = Table[{i, 7 - i}, {i, 3}];
sol = V... | {
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• Are you working through all of the combinatorial exercises in a textbook? – JimB Feb 21 at 18:28
• @JimB Yes, but it's an exercise problem that I can't solve. It's different from the conventional problem without different color restrictions. I don't know how to solve this problem effectively at present. – A little mo... | {
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1. ChromaticPolynomial[g, k] gives colorings using exactly k colors, whereas you need to choose up to k = 6 colors
2. ChromaticPolynomial[g, k] considers graphs to be labeled, and so, for example, there are, according to ChromaticPolynomial, 2 colorings of the graph 1 •-• 2.
We could do this by "standard" combinatoria... | {
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In[1]:= Map[c, {1 •-• 2, 2 •-• 3}, {2}]
Out[1]:= {c[1] •-• c[2], c[2] •-• c[3]}
The question is then whether we wind up with a color connected to a color n the output. If so, then two adjacent vertices have been assigned the same color by c. We want to check that this is avoided. That is, we want to check that that... | {
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AllLabeledColorings[g_] := Select[AllUnrestrictedColorings[g], ColoringQ[g, #] & ]
### Modding out by vertex relabeling
Now comes the interesting part. We want to consider the action under reflections and rotations of the cube. Mathematically, we're modding out by the action of that symmetry group. Usually this is d... | {
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Function[h, KeySort @ KeyMap[Function[v, PermutationReplace[v, h]], c]] /@ AutG
We're going to package this into a function with parameter c then map over the list of colorings. Once we do, we want to delete equivalent, uh, equivalence classes (i.e. equivalence classes with the same elements) by DeleteDuplicates with... | {
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Fix a certain color on the top, leaving 5 options for the bottom, and $$(4-1)! = 6$$ colors for the remaining 4 sides, totaling 30 methods.
However, they have double-counted the configurations for the remaining 4 sides, as they have forgotten to account for the reflection that identifies two of the 4 sides.
The fact ... | {
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AutMod[g_, clist : {___Association}, autg_List : Null] :=
With[{AutG = Replace[autg, Null :> GroupElements[GraphAutomorphismGroup[g]]]},
DeleteDuplicates[
Function[c,
Function[h, KeySort @ KeyMap[Function[v, PermutationReplace[v, h]], c]] /@ AutG
] /@ clist,
ContainsExactly]
]
(If we were being more precise, we'd pro... | {
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Order your colors 1 through 6. Up to rotation + flipping (i.e. isometry), we can demand that the least-ranked color appearing be on the bottom. Now, up to isometry, there are 2 choices for the second-least-ranked color (which might be the same color!): opposite the least or adjacent to it. If it's adjacent, it cannot b... | {
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It's not an original answer, it's just a supplement to the answer of thorimur.
g1 = Graph[(Sort /@
Flatten[Map[
Thread[#[[1]] \[UndirectedEdge] #[[2]]] &, {{1, {2, 3, 4,
5}}, {2, {1, 3, 5, 6}}, {3, {1, 4, 2, 6}}, {4, {1, 3, 5,
6}}, {5, {1, 2, 4, 6}}, {6, {2, 3, 4, 5}}}]]) //
DeleteDuplicates,
UnrestrictedColoringQ[g_... | {
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Where num1 = num3, this conclusion is very useful. But one thing I'm confused about is that groups G1 and G2 are both groups of order 48, representing regular hexahedral groups. Why are num2 and num3 not equal? I want to know the underlying reasons for their different results.
Comparison with the results of standard a... | {
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• So, one reason num2 ≠ num3 is: despite being an isomorphic group to our graph automorphism group, FiniteGroupData["Octahedral", "PermutationGroupRepresentation"], knows nothing about how we've numbered our graph vertices. That information is particular to g1 and the groups derived from it. As such, when it tries to, ... | {
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# Meaning of "$\exists$" in "$\{y \in Y : \exists x \in X \text{ such that }f(x) = y\}$"
I came across this definition of the range of a function:
For a function $$f : X → Y$$, the range of $$f$$ is $$\{y \in Y : \exists x \in X \text{ such that }f(x) = y\},$$ i.e., the set of $$y$$-values such that $$y = f (x)$$ for... | {
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Set $$A$$ is populated precisely with the even integers:
• take some (any) integer, then double it; the result is a member of set $$A$$;
• repeat infinitely.
2. $$B=\{n \in\mathbb Z: \forall a\in\mathbb Z\;\,n=2a\}\\ =\text{the set of integers such that each one is double }\textit{every }\text{ integer}\\ =\emptyset.$... | {
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• I don't understand how the set you defined is ∅?! How does $\forall$ implies that y is simultaneously 0 and 1?? Oct 26 at 21:12
• No, never. @Prakhar. $\{y \in Y : \exists x \in X$ such that $f(x) = y\}$ is $\mathrm{Im}f$(range of $f$), but $\{y \in Y : \forall x \in X$ such that $f(x) = y\}$ is generally $\emptyset$... | {
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How do I get this matrix in Smith Normal Form? And, is Smith Normal Form unique?
As part of a larger problem, I want to compute the Smith Normal Form of $xI-B$ over $\mathbb{Q}[x]$ where $$B=\begin{pmatrix} 5 & 2 & -8 & -8 \\ -6 & -3 & 8 & 8 \\ -3 & -1 & 3 & 4 \\ 3 & 1 & -4 & -5\end{pmatrix}.$$
So I do some elementar... | {
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To expand my comment...Add column 2 to column 1. Subtract row 2 from row 1. Now you have a scalar in the (1,1) position -- rescale to 1.
$$\begin{pmatrix} x-3 & 0 & 0 & 0 \\ 0 & x+1 & 0 & 0 \\ 0 & 0 & x+1 & 0 \\ 0 & 0 & 0 & x+1\end{pmatrix} \sim \begin{pmatrix} x-3 & 0 & 0 & 0 \\ x+1 & x+1 & 0 & 0 \\ 0 & 0 & x+1 & 0 \... | {
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# Are polynomials with the same roots identical?
I know that polynomials can be refactored in terms of their roots. However, this must imply that two different polynomials have different roots (this is just what I think). So my question is: Are polynomials with the same roots identical? - if so, why?
A follow-up ques... | {
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• Aha okay, thanks. But then how come you can write a polynomial in terms of its roots? Like $(\lambda - a)(\lambda - d)-bc = 0$ can be written in terms of its roots $(\lambda - \lambda_1)(\lambda - \lambda_2) = 0$? Since having the same roots apparently does not imply that two polynomials are identical, using the root... | {
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The image above shows a simple $$Y=(X-1)(X-2)(X-3)$$ and an overlapping $$Y=-3(X-1)(X-2)(X-3)$$.
This helps show that manipulation made to an equation such as factoring may preserve the roots, but do not leave an equation with the the same nature, e.g. the end behaviour which might be important, is easily lost.
Edit ... | {
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• Of course for finite fields $\mathbb{F}$, the pigeonhole principle alone can say that there will be distinct polynomials which induce the same map $\mathbb{F}\to\mathbb{F}$. Because the number of such maps is finite, while the number of polynomials is infinite. – Jeppe Stig Nielsen Jun 8 '19 at 8:27
No, they aren't:... | {
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So: you have to count the roots with multiplicity in the algebraic closure.
No they are not, and it's easy to see why that is the case. You probably wouldn't consider $$f(x)=x$$ and $$f(x)=10x$$ to be identical even though they have the same root.
Let's start by considering polynomials with all their roots, real and ... | {
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• a different non-zero multiplier (A is different between the polynomials, when factored)
• repeated roots (one or more of P, Q,R will differ between the polynomials, when factored)
# What if we only allow real roots?
The polynomial can still only be factored one way as above. The only difference is, any B,C,D that i... | {
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Note that this last transformation might or might not change some of the coefficients in the equation from real to complex coefficients or vice-versa, depending what you do (see especially the last example where they don't). It may well change the complex roots of the polynomial. But it will not change, add or remove a... | {
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Arthur answered your question very nicely, but I'd like to tell you a much more general result that might pique your interest in a field of math called "algebraic geometry". So – if we are working in an algebraically closed field, say the complex numbers $$\mathbb{C}$$, then every polynomial in one variable splits comp... | {
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In particular, there's an important result in commutative algebra called Hilbert's Nullstellensatz, which I won't state in full generality here. But one corollary of it is that, if the roots of a complex polynomial $$p(t_1, ..., t_n)\in\mathbb{C}[t_1, ..., t_n]$$ are also roots of another complex polynomial $$q(t_1, ..... | {
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Proof: let $$p$$ and $$q$$ be as above: non-zero complex polynomials in $$n$$ variables with no repeated factors and which share the same roots. In particular, the roots of $$p$$ are also roots of $$q$$, so by the corollary to the nullstellensatz there is some $$k\in\mathbb{N}$$ and $$r\in\mathbb{C}[t_1,...,t_n]$$ such... | {
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Putting this together with the fact that $$q^k=rp$$ gives us $$q^k=q^{k-1}r'p$$, and dividing out gives $$q=r'p$$. Now, on the other hand, the roots of $$q$$ are also roots of $$p$$, and so we can go through exactly the same arguments as above to show that there is some polynomial $$s\in\mathbb{C}[t_1,...,t_n]$$ such t... | {
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# Math Help - prove or disprove direct sum question
1. ## prove or disprove direct sum question
Prove or disprove
$W, U_1, U_2$ are subspaces of $V$.
If $V=U_1 \oplus U_2$
then $W = (U_1 \cap W) \oplus (U_2 \cap W)$
Attempt:
False
let's say $dimV = 8, dimU_1 = dimU_2 = 4 (dimV = dimU_1 + dimU_2)$
and $dimW = 5$... | {
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Tonio
I truly hope you aren't grading my exam (;
i sohld get at least partial credit
5. The misconception that some students have, and that this problem is attempting to dispel, is that the direct sum is like a (set) union. But in fact, the direct sum of U1 and U2 contains a whole lot of elements that are not in U1 o... | {
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# Error in TransformedField
I am using TransformedField to convert a system of ODEs from Cartesian to polar coordinates:
TransformedField[
"Cartesian" -> "Polar",
{μ x1 - x2 - σ x1 (x1^2 + x2^2), x1 + μ x2 - σ x2 (x1^2 + x2^2)},
{x1, x2} -> {r, θ}
] // Simplify
and I get the result
{r μ - r^3 σ, r}
but I am pre... | {
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Now, you are dealing with a differential equation, and based on your expected answer I'll assume what you have is a first-order system and you are transforming the associated vector field (AKA the "right-hand side"). Finding solutions means find the integral curves of the vector field. This gives as a nice relationship... | {
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"lm_label": "1. YES\n2. YES",
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"lm_q1_score": 0.9591542829224747,
"lm_q1q2_score": 0.8448202837255983,
"lm_q2_score": 0.880797071719777,
"openwebmath_perplexity": 1371.313547085985,
"openwebmath_score": 0.6747981309890747,
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thetadot[x1_,x2_]:=(x1 (x1+μ x2-σ x2 (x1^2+x2^2)) - x2(μ x1-x2-σ x1 (x1^2+x2^2)))/r^2
We now make the substitution and simplify
thetadot[r Cos[t], r Sin[t]] // FullSimplify
This yields (does not match Mathematica, but see accepted answer)
$$\theta'= 1$$
I have asked this question before on this site in two diffe... | {
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"openwebmath_score": 0.6747981309890747,
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## Sincerely,
Wolfram Technology Group http://www.wolfram.com/support/
• Have you reported it to the Wolfram tech support? – Alexey Popkov Mar 7 at 22:48
• @AlexeyPopkov: I have not. I have had many issues with it when transforming between different methods. These days, I don't trust it and just create my own transfo... | {
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"lm_q1q2_score": 0.8448202837255983,
"lm_q2_score": 0.880797071719777,
"openwebmath_perplexity": 1371.313547085985,
"openwebmath_score": 0.6747981309890747,
"tags... |
# Probability Misconception
Hey, buddies :)
Recently people had discussion in Brilliant-Lounge on a probability problem which is:
In a family of 3 children, what is the probability that at least one will be a boy?
Some of them believe that $\frac 34$ is the correct answer while the others believe that the correct a... | {
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"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9591542887603537,
"lm_q1q2_score": 0.8448202828650959,
"lm_q2_score": 0.8807970654616711,
"openwebmath_perplexity": 2417.813163943126,
"openwebmath_score": 0.9637357592582703,
"tag... |
print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=... | {
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"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
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"lm_q1q2_score": 0.8448202828650959,
"lm_q2_score": 0.8807970654616711,
"openwebmath_perplexity": 2417.813163943126,
"openwebmath_score": 0.9637357592582703,
"tag... |
Simple conceptual learning condtitonal probability Probability Rocks By- YDL
- 3 years, 7 months ago
What do you guys think? I would be great if you participate in this discussion as you were playing a major role in the slack discussion. Thanks!
- 3 years, 7 months ago
The Family cares about getting Boy, not about ... | {
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"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9591542887603537,
"lm_q1q2_score": 0.8448202828650959,
"lm_q2_score": 0.8807970654616711,
"openwebmath_perplexity": 2417.813163943126,
"openwebmath_score": 0.9637357592582703,
"tag... |
PS: I am weak in probability
- 3 years, 7 months ago
It's only about the children, not the parents. So consider a family of 3 children (assuming total members as 3) and then find out the probability that at least one of them is a boy. Thanks!
Don't worry. Keep practicing. You will soon be a master in combinatorics. ... | {
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"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9591542887603537,
"lm_q1q2_score": 0.8448202828650959,
"lm_q2_score": 0.8807970654616711,
"openwebmath_perplexity": 2417.813163943126,
"openwebmath_score": 0.9637357592582703,
"tag... |
# Why tan x=x as x approaches 0?
• I
Hi! In one of my textbook i saw the relation tan(x) = x where x is very small value and expressed in radians. I want to know why its true and how it actually works. I would appreciate someone's help
mfb
Mentor
Did you draw a sketch? A small part of the circle can be approximated b... | {
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"lm_q1q2_score": 0.844817590225287,
"lm_q2_score": 0.8791467706759584,
"openwebmath_perplexity": 1083.5200391542446,
"openwebmath_score": 0.9414845705032349,
... |
You simply compare a value ##0.12## with its ##60-##fold value. But both are still in a very good approximation to ##\tan (x) \approx x##. Radians are the natural unit here, degree more because of historical reasons, habit and clarity for humans. The approximation ##\tan (x) \approx x## requires radians if taken numeri... | {
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"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9609517072737737,
"lm_q1q2_score": 0.844817590225287,
"lm_q2_score": 0.8791467706759584,
"openwebmath_perplexity": 1083.5200391542446,
"openwebmath_score": 0.9414845705032349,
... |
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