text stringlengths 1 2.12k | source dict |
|---|---|
Split the integral into two terms where each term is in the interval $0<x<1$ and $1<x<\infty$, then use the substitution $x\mapsto\frac{1}{x}$ to the second term. We will get $$\left[\int_{0}^{1}+\int_{1}^{\infty}\right]\left(\frac{x-1}{\ln^2 x}-\frac{1}{\ln x}\right)\frac{\mathrm{d}x}{x^2+1}=\int_{0}^{1}\frac{(x-1)^2}... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127423485422,
"lm_q1q2_score": 0.8400373378973803,
"lm_q2_score": 0.8499711699569786,
"openwebmath_perplexity": 808.465498045121,
"openwebmath_score": 0.9972632527351379,
"tags... |
The following is an alternative evaluation of the integral on the right.
An integral representation of the Dirichlet beta function is $$\beta(s) = \frac{1}{ 2 \, \Gamma(s)} \int_{0}^{\infty} \frac{x^{s-1}}{\cosh(x)} \, dx \, , \quad \text{Re}(s) >0\tag{1}.$$
And the Laplace transform of $x^{s-1}$ is $$\int_{0}^{\inft... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127423485422,
"lm_q1q2_score": 0.8400373378973803,
"lm_q2_score": 0.8499711699569786,
"openwebmath_perplexity": 808.465498045121,
"openwebmath_score": 0.9972632527351379,
"tags... |
\begin{align} &\color{#f00}{\int_{0}^{\infty}\bracks{% {x - 1 \over \ln^{2}\pars{x}} - {1 \over \ln\pars{x}}} \,{\dd x \over x^{2} + 1}} \\[5mm] = &\ \int_{0}^{1}\bracks{% {x - 1 \over \ln^{2}\pars{x}} - {1 \over \ln\pars{x}}} \,{\dd x \over x^{2} + 1} + \int_{1}^{0}\bracks{% {1/x - 1 \over \ln^{2}\pars{1/x}} - {1 \ove... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127423485422,
"lm_q1q2_score": 0.8400373378973803,
"lm_q2_score": 0.8499711699569786,
"openwebmath_perplexity": 808.465498045121,
"openwebmath_score": 0.9972632527351379,
"tags... |
$\ds{\Psi}$ is the Digamma Function and we used its well known integral representation $\ds{\pars{~\gamma\ \mbox{is the}\ Euler\mbox{-}Mascheroni\ Constant~}}$ $$\Psi\pars{z} = -\gamma + \int_{0}^{1}{1 - t^{z - 1} \over 1 - t}\,\dd t\,, \qquad\Re\pars{z} > 0$$
Since $\ds{\Psi\pars{z}\ \stackrel{\mbox{def.}}{=}\ \totald... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127423485422,
"lm_q1q2_score": 0.8400373378973803,
"lm_q2_score": 0.8499711699569786,
"openwebmath_perplexity": 808.465498045121,
"openwebmath_score": 0.9972632527351379,
"tags... |
The $\ds{\ln\Gamma}$-integrals are evaluated $\ds{\pars{~\mbox{the first one is rather trivial and it's equal to}\ \half\,\ln\pars{2\pi}~}}$ with the identity ( $\ds{\,\mathrm{G}}$ is the Barnes-G Function ) $$\int_{0}^{z}\ln\pars{\Gamma\pars{z}}\,\dd z = \half\,z\pars{1 - z} + \half\,\ln\pars{2\pi}z + z\ln\pars{\Gamma... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127423485422,
"lm_q1q2_score": 0.8400373378973803,
"lm_q2_score": 0.8499711699569786,
"openwebmath_perplexity": 808.465498045121,
"openwebmath_score": 0.9972632527351379,
"tags... |
With $\ds{\pars{4}\ \mbox{and}\ \pars{5}}$, $\ds{\pars{3}}$ becomes $$\left\lbrace\begin{array}{rcl} \ds{\int_{0}^{1}\ln\pars{\Gamma\pars{z}}} & \ds{=} & \ds{\phantom{-\,}\half\,\ln\pars{2\pi}} \\[1mm] \ds{\int_{0}^{1/4}\ln\pars{\Gamma\pars{z}}} & \ds{=} & \ds{\phantom{-\,}{K \over 4\pi} + {1 \over 8}\ln\pars{2\pi} + {... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127423485422,
"lm_q1q2_score": 0.8400373378973803,
"lm_q2_score": 0.8499711699569786,
"openwebmath_perplexity": 808.465498045121,
"openwebmath_score": 0.9972632527351379,
"tags... |
With $\ds{\pars{6}}$, the expression $\ds{\pars{2}}$ is reduced to $\ds{\pars{~\ul{the\ final\ result}~}}$: $$\color{#f00}{\int_{0}^{\infty}\bracks{% {x - 1 \over \ln^{2}\pars{x}} - {1 \over \ln\pars{x}}} \,{\dd x \over x^{2} + 1}} = \color{#f00}{4\,{K \over \pi}} \approx 1.1662$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127423485422,
"lm_q1q2_score": 0.8400373378973803,
"lm_q2_score": 0.8499711699569786,
"openwebmath_perplexity": 808.465498045121,
"openwebmath_score": 0.9972632527351379,
"tags... |
• The OP's integral looked familiar, until I recalled a similar integral in your answer, namely $$\int_0^1\left(\frac{x-1}{\ln^2 x}-\frac{x}{\ln x}\right)\frac{\mathrm{d}x}{x^2+1}=\frac{2G}{\pi}+\frac{\ln 2}6+\frac{\ln\pi}2-6\ln A$$ with $A$ as the Glaisher–Kinkelin constant. Jul 25 '19 at 16:46 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127423485422,
"lm_q1q2_score": 0.8400373378973803,
"lm_q2_score": 0.8499711699569786,
"openwebmath_perplexity": 808.465498045121,
"openwebmath_score": 0.9972632527351379,
"tags... |
Solution: Let the two adjacent sides of the scalene triangle be a = 8cm and b = 10cm, the angle included between these two sides, ∠C =30o . Area of a Scalene Triangle … From the figure given above, a scalene triangle is given with 3 sides as ‘a’, ‘b’ and ‘c’. A triangle with irregular side lengths is called a scalene ... | {
"domain": "logivan.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9688561676667172,
"lm_q1q2_score": 0.8400329517938083,
"lm_q2_score": 0.8670357683915538,
"openwebmath_perplexity": 742.2664033819096,
"openwebmath_score": 0.8431070446968079,
"tags": nu... |
Formula. Let a,b,c be the lengths of the sides of a triangle. Vedantu academic counsellor will be calling you shortly for your Online Counselling session. Step 2: Find the semi-perimeter, S. The formula for finding the semi-perimeter of a triangle is. he base of an isosceles triangle is 10 cm and one of its equal sides... | {
"domain": "logivan.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9688561676667172,
"lm_q1q2_score": 0.8400329517938083,
"lm_q2_score": 0.8670357683915538,
"openwebmath_perplexity": 742.2664033819096,
"openwebmath_score": 0.8431070446968079,
"tags": nu... |
with integers/decimals as their.! No angles are of unequal length and all the three sides of the triangle... And display the area of a triangular plot whose sides are in the of...: Heron 's formula relates the side lengths and area of scalene triangle by heron's formula different angles angles is equal! Got is the same... | {
"domain": "logivan.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9688561676667172,
"lm_q1q2_score": 0.8400329517938083,
"lm_q2_score": 0.8670357683915538,
"openwebmath_perplexity": 742.2664033819096,
"openwebmath_score": 0.8431070446968079,
"tags": nu... |
angle. Side c: area s Customer Voice is the angle between them given! Semi-Perimeter, S. the formula for the area of scalene triangles with integers/decimals their. ] ) the Calculator will evaluate and display the area of scalene triangle whose sides equal! Unlike other triangle area formulae, there is no need to calcu... | {
"domain": "logivan.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9688561676667172,
"lm_q1q2_score": 0.8400329517938083,
"lm_q2_score": 0.8670357683915538,
"openwebmath_perplexity": 742.2664033819096,
"openwebmath_score": 0.8431070446968079,
"tags": nu... |
should be values all. Area using Heron ’ s formula 3\ ] sq de côtés a,,! For area of a triangle is a triangle when you know the length of the sides. / area ; Calculates the area is given different measures … Heron 's formula for the area of scalene.: if you know the length of three sides of the length of its two sides ... | {
"domain": "logivan.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9688561676667172,
"lm_q1q2_score": 0.8400329517938083,
"lm_q2_score": 0.8670357683915538,
"openwebmath_perplexity": 742.2664033819096,
"openwebmath_score": 0.8431070446968079,
"tags": nu... |
two sides and angle between sides. The side lengths and 3 different side lengths is called a scalene triangle = 12 sq a... H is given by: Try this Drag the orange dots to the. Calculator to calculate the area of a triangle whose area is 2.9 cm 2.. Table triangle. With all sides of length 8 cm, 16 cm and 20 cm is a tria... | {
"domain": "logivan.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9688561676667172,
"lm_q1q2_score": 0.8400329517938083,
"lm_q2_score": 0.8670357683915538,
"openwebmath_perplexity": 742.2664033819096,
"openwebmath_score": 0.8431070446968079,
"tags": nu... |
# How should I evaluate time complexity for matrix if I have a fixed (constant) amount of rows and columns?
Suppose, that I have a four-by-four matrix and I want to print each element of it.
matrix = [[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]]
for row in matrix
for elem in row
print(elem)
So, I ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9688561721629777,
"lm_q1q2_score": 0.8400329457059748,
"lm_q2_score": 0.8670357580842941,
"openwebmath_perplexity": 287.6603078601551,
"openwebmath_score": 0.8185981512069702,
"tag... |
• Thank you ever so much. Let me specify some information: if we have matrix n * k and n = k, can I state, that big O is n^2, because of their equality? Jun 19 '21 at 14:04
• Yes, this is totally allowed. Jun 19 '21 at 14:41 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9688561721629777,
"lm_q1q2_score": 0.8400329457059748,
"lm_q2_score": 0.8670357580842941,
"openwebmath_perplexity": 287.6603078601551,
"openwebmath_score": 0.8185981512069702,
"tag... |
# PhysicsFinding Time for a Kicked Ball
#### balancedlamp
##### New member
Problem: You are playing soccer with some friends while some people nearby are playing hide and seek. When the seeker gets to '4', you kick the soccer ball high in the air. WHen the seeker says '5', the ball is 10 ft in the air. When the seeke... | {
"domain": "mathhelpboards.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9688561730622297,
"lm_q1q2_score": 0.8400329398281565,
"lm_q2_score": 0.8670357512127872,
"openwebmath_perplexity": 746.4990376207227,
"openwebmath_score": 0.8206997513771057,
"ta... |
#### HallsofIvy
##### Well-known member
MHB Math Helper
Once I approached it using the three points and finding the quadratic equation, I just plugged it into a graphing calculator and got the answer of 7 seconds, which is one of the answer choices.
However, I'm confused why the kinematic equations are giving me such... | {
"domain": "mathhelpboards.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9688561730622297,
"lm_q1q2_score": 0.8400329398281565,
"lm_q2_score": 0.8670357512127872,
"openwebmath_perplexity": 746.4990376207227,
"openwebmath_score": 0.8206997513771057,
"ta... |
# Finding loss of energy in collision
Saitama
## Homework Statement
A particle of mass ##m_1## experienced a perfectly elastic collision with a stationary particle of mass ##m_2##. What fraction of the kinetic energy does the striking particle lose, if it recoils at right angles to its original motion direction.
(A... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9688561694652215,
"lm_q1q2_score": 0.8400329350450463,
"lm_q2_score": 0.8670357494949105,
"openwebmath_perplexity": 715.7108261383771,
"openwebmath_score": 0.6974197030067444,
"tag... |
The algebra is fairly easy to work out here. Took me less than 10 lines and barely 5 minutes.
Remember that the final speed of ##m_2## is given by the Pythagorean theorem. Deal only in squares of the velocity components, and everything simplifies quickly.
And always keep in mind what you're trying to find, which is t... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9688561694652215,
"lm_q1q2_score": 0.8400329350450463,
"lm_q2_score": 0.8670357494949105,
"openwebmath_perplexity": 715.7108261383771,
"openwebmath_score": 0.6974197030067444,
"tag... |
# Visualizing a Complex Vector Field near Poles
I've been playing around with a visualization technique for complex functions where one views the function $f: \mathbb{C} \rightarrow \mathbb{C}$ as the vector field $f: \mathbb{R^2} \rightarrow \mathbb{R^2}$. These vector fields have some nice properties as a consequenc... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9591542864252023,
"lm_q1q2_score": 0.8400148485893595,
"lm_q2_score": 0.8757869932689566,
"openwebmath_perplexity": 1360.5720160508802,
"openwebmath_score": 0.6169275045394897,
"ta... |
-
Isn't the Pólya vector field the complex conjugate of the function? This lends itself to look at things like computing the flux out of an area through a contour integral (effectively using Gauss theorem). Code Example at MathWorld – Thies Heidecke Apr 16 '12 at 9:15
Indeed, these aren't quite Pólya plots but I was t... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9591542864252023,
"lm_q1q2_score": 0.8400148485893595,
"lm_q2_score": 0.8757869932689566,
"openwebmath_perplexity": 1360.5720160508802,
"openwebmath_score": 0.6169275045394897,
"ta... |
What I did here is to suppress the automatic re-scaling colors in VectorColorFunction and provided my own scaling that can easily deal with infinite values. It's based on the ArcTan function.
As a mix between these two approaches, you could also use the ArcTan to rescale vector length.
-
Thanks Jens! This takes into ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9591542864252023,
"lm_q1q2_score": 0.8400148485893595,
"lm_q2_score": 0.8757869932689566,
"openwebmath_perplexity": 1360.5720160508802,
"openwebmath_score": 0.6169275045394897,
"ta... |
g[z_] = 1/z;
VectorPlot[
If[x == 0 && y == 0, {0, 0}, {Re[g[x + I*y]],
Im[g[x + I*y]]}], {x, -1.5, 1.5}, {y, -1.5, 1.5},
VectorPoints -> Fine]
-
Cool, that should do the trick. BTW, I just tried using StreamPlot and it seems to not have any of the issues that VectorPlot has while providing more or less the same infor... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9591542864252023,
"lm_q1q2_score": 0.8400148485893595,
"lm_q2_score": 0.8757869932689566,
"openwebmath_perplexity": 1360.5720160508802,
"openwebmath_score": 0.6169275045394897,
"ta... |
# Does + C melt together ALL constants?
1. Aug 19, 2009
### Unit
Does "+ C" melt together ALL constants?
I had this integral $$\int \frac{dy}{-2y + 6}$$.
I realize you can let $u = -2y + 6$ so that $du = -2dy$, and adjusting: $-0.5 du = dy$.
The integral becomes
$$-0.5 \int \frac{du}{u}$$
$$= -0.5 \ln{|-2y + 6|... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9591542794197473,
"lm_q1q2_score": 0.8400148440088718,
"lm_q2_score": 0.8757869948899666,
"openwebmath_perplexity": 925.8097639997766,
"openwebmath_score": 0.7258391380310059,
"tag... |
### Tac-Tics
Re: Does "+ C" melt together ALL constants?
The "plus C" is a bastardization.
While differential functions have unique derivatives, their anti-derivatives are NOT unique. The reason for this is simple: constant functions have a zero derivative. That means, you can always add a constant function to the e... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9591542794197473,
"lm_q1q2_score": 0.8400148440088718,
"lm_q2_score": 0.8757869948899666,
"openwebmath_perplexity": 925.8097639997766,
"openwebmath_score": 0.7258391380310059,
"tag... |
It is a handy short cut, though. And there are other tricks that people like to do. For example, if you have multiple C's added together, you can often throw all but one of them away. Sometimes. That happens to be the case in this problem. But if you don't end up killing the C's off by the end of the problem, you'll pr... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9591542794197473,
"lm_q1q2_score": 0.8400148440088718,
"lm_q2_score": 0.8757869948899666,
"openwebmath_perplexity": 925.8097639997766,
"openwebmath_score": 0.7258391380310059,
"tag... |
# Adding random numbers to the arguments of a sum of cosines
This is probably really a simple question.
I would like to plot a sum of cosines with increased argument and for each cosine in sum I would like to add random value:
Plot[Sum[Cos[n*x + RandomReal[{0, 2 Pi}]], {n, 100}], {x, -10, 10}]
However it seems to ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551566309688,
"lm_q1q2_score": 0.8400002702515676,
"lm_q2_score": 0.8705972751232808,
"openwebmath_perplexity": 1487.9108280494265,
"openwebmath_score": 0.3457498252391815,
"ta... |
# Better proof for $\frac{1+\cos x + \sin x}{1 - \cos x + \sin x} \equiv \frac{1+\cos x}{\sin x}$
It's required to prove that $$\frac{1+\cos x + \sin x}{1 - \cos x + \sin x} \equiv \frac{1+\cos x}{\sin x}$$ I managed to go about out it two ways: | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551525886193,
"lm_q1q2_score": 0.8400002602533843,
"lm_q2_score": 0.8705972684083609,
"openwebmath_perplexity": 762.6294153420204,
"openwebmath_score": 0.9995144605636597,
"tag... |
1. Assume it holds: $$\frac{1+\cos x + \sin x}{1 - \cos x + \sin x} \equiv \frac{1+\cos x}{\sin x}$$ $$\Longleftrightarrow\sin x(1+\cos x+\sin x)\equiv(1+\cos x)(1-\cos x+\sin x)$$ $$\Longleftrightarrow\sin x+\cos x\sin x+\sin^2 x\equiv1-\cos x+\sin x+\cos x-\cos^2 x+\sin x \cos x$$ $$\Longleftrightarrow\sin^2 x\equiv1... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551525886193,
"lm_q1q2_score": 0.8400002602533843,
"lm_q2_score": 0.8705972684083609,
"openwebmath_perplexity": 762.6294153420204,
"openwebmath_score": 0.9995144605636597,
"tag... |
• @ThomasAndrews Right you are. – Luke Collins Dec 26 '15 at 23:46
• – lab bhattacharjee Dec 27 '15 at 3:50 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551525886193,
"lm_q1q2_score": 0.8400002602533843,
"lm_q2_score": 0.8705972684083609,
"openwebmath_perplexity": 762.6294153420204,
"openwebmath_score": 0.9995144605636597,
"tag... |
Since $1-\cos^2 x = \sin^2 x$, we have $f(x) := \dfrac{1+\cos x}{\sin x} = \dfrac{\sin x}{1-\cos x}$. Therefore,
\begin{align*}\dfrac{1+\cos x + \sin x}{1-\cos x + \sin x} &= \dfrac{f(x)\sin x + f(x)(1-\cos x)}{1-\cos x + \sin x} \\ &= \dfrac{f(x)[1-\cos x + \sin x]}{1-\cos x + \sin x} \\ &= f(x) \\ &= \dfrac{1+\cos x... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551525886193,
"lm_q1q2_score": 0.8400002602533843,
"lm_q2_score": 0.8705972684083609,
"openwebmath_perplexity": 762.6294153420204,
"openwebmath_score": 0.9995144605636597,
"tag... |
Dimension of a vector space of functions
Where $$S$$ is a set, the functions $$f: S \rightarrow \mathbb{R}$$ form a vector space under the natural operations: the sum $$f+g$$ is the function given by $$f +g(s)=f(s)+g(s)$$ and the scalar product is $$r\cdot f(s) = r\cdot f(s)$$. What is the dimension of the space resul... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551515780318,
"lm_q1q2_score": 0.8400002528946441,
"lm_q2_score": 0.8705972616934406,
"openwebmath_perplexity": 127.0496417170133,
"openwebmath_score": 0.8934762477874756,
"tag... |
• Distinct polynomials can have the same values on $S$ and therefore induce the same function from $S \to \mathbb R$. For example, all polynomials $p$ satisfying $p(1) = 0$ result in the same function from $\{1\} \to \mathbb R$. – Bungo Oct 9 at 17:52
• Thank you for the response Bungo. Can't you just apply the same lo... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551515780318,
"lm_q1q2_score": 0.8400002528946441,
"lm_q2_score": 0.8705972616934406,
"openwebmath_perplexity": 127.0496417170133,
"openwebmath_score": 0.8934762477874756,
"tag... |
• Thank you for the response José. I see your point, but can't you just take the polynomial functions $x^n + x^{n-1}$ and have another set of functions that are distinct on $\{1\}$? – Andy Oct 9 at 18:11
• @Andy When restricted to $S = \{1\}$, we have $x^n = x^{n-1}$ (they are the same function on the restricted domain... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551515780318,
"lm_q1q2_score": 0.8400002528946441,
"lm_q2_score": 0.8705972616934406,
"openwebmath_perplexity": 127.0496417170133,
"openwebmath_score": 0.8934762477874756,
"tag... |
Use logarithmic differentiation to differentiate each function with respect to x. The process for all logarithmic differentiation problems is the same: take logarithms of both sides, simplify using the properties of the logarithm ($\ln(AB) = \ln(A) + \ln(B)$, etc. (3x 2 – 4) 7. Do 1-9 odd except 5 Logarithmic Different... | {
"domain": "co.uk",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.957277806109987,
"lm_q1q2_score": 0.8399890647682559,
"lm_q2_score": 0.8774767986961403,
"openwebmath_perplexity": 559.350464254909,
"openwebmath_score": 0.9357950091362,
"tags": null,
"url"... |
sides of an equation y = f(x) and use the law of logarithms to simplify. It spares you the headache of using the product rule or of multiplying the whole thing out and then differentiating. In some cases, we could use the product and/or quotient rules to take a derivative but, using logarithmic differentiation, the der... | {
"domain": "co.uk",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.957277806109987,
"lm_q1q2_score": 0.8399890647682559,
"lm_q2_score": 0.8774767986961403,
"openwebmath_perplexity": 559.350464254909,
"openwebmath_score": 0.9357950091362,
"tags": null,
"url"... |
Problems Find the derivative of each of the argument = ( 2x+1 ) 3 rules of differentiation do NOT!. You ’ re applying logarithms to nonlogarithmic functions ln ( x ) ( ). The function must first be revised before a derivative can be taken logarithmic... Natural logarithm to both sides of this equation getting NOT need ... | {
"domain": "co.uk",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.957277806109987,
"lm_q1q2_score": 0.8399890647682559,
"lm_q2_score": 0.8774767986961403,
"openwebmath_perplexity": 559.350464254909,
"openwebmath_score": 0.9357950091362,
"tags": null,
"url"... |
) is given in the example and practice problem without logarithmic.... And practice problem without logarithmic differentiation example question derivative can be taken that you want to the... Differentiation to Find the derivative of f ( x ) = ln ( x =! The logarithm function product rule or of multiplying the whole t... | {
"domain": "co.uk",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.957277806109987,
"lm_q1q2_score": 0.8399890647682559,
"lm_q2_score": 0.8774767986961403,
"openwebmath_perplexity": 559.350464254909,
"openwebmath_score": 0.9357950091362,
"tags": null,
"url"... |
To a variable power in this function, the ordinary rules of differentiation do NOT APPLY functions which! Video below substitute for y differentiation do NOT need to simplify or substitute y. Of logarithms will sometimes make the differentiation process easier and practice problem without differentiation... Differentia... | {
"domain": "co.uk",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.957277806109987,
"lm_q1q2_score": 0.8399890647682559,
"lm_q2_score": 0.8774767986961403,
"openwebmath_perplexity": 559.350464254909,
"openwebmath_score": 0.9357950091362,
"tags": null,
"url"... |
Twin Lakes Boating, Why I Love Being A Chef, Oxford Korean Dictionary Pdf, Impairment Of Investment In Subsidiary Example, Swartswood Lake Phone Number, | {
"domain": "co.uk",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.957277806109987,
"lm_q1q2_score": 0.8399890647682559,
"lm_q2_score": 0.8774767986961403,
"openwebmath_perplexity": 559.350464254909,
"openwebmath_score": 0.9357950091362,
"tags": null,
"url"... |
# Plotting histograms with bin count multiplied by some factor
Consider, for example
Histogram[RandomVariate[NormalDistribution[0, 0.6], 1000]]
which gives me,
Now, is there a way to change the bin counts, as in, I want to multiply some factor to the bin counts (lets say "0.5"). meaning there should be only about ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9572778012346834,
"lm_q1q2_score": 0.8399890482214901,
"lm_q2_score": 0.8774767858797979,
"openwebmath_perplexity": 6517.680523303301,
"openwebmath_score": 0.2726743817329407,
"tag... |
ClearAll[ceF]
ceF[sc_ : 1][cedf_ : "Rectangle", o : OptionsPattern[]] :=
Module[{origin = ChartingChartStyleInformation["BarOrigin"], box = #},
Switch[origin,
Bottom, box[[2, 2]] = sc box[[2, 2]],
Top, box[[2, 1]] = sc box[[2, 1]],
Left, box[[1, 2]] = sc box[[1, 2]],
Right, box[[1, 1]] = sc box[[1, 1]]];
ChartElementDa... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9572778012346834,
"lm_q1q2_score": 0.8399890482214901,
"lm_q2_score": 0.8774767858797979,
"openwebmath_perplexity": 6517.680523303301,
"openwebmath_score": 0.2726743817329407,
"tag... |
# Difference between “space” and “algebraic structure”
What is the difference between a "space" and an "algebraic structure"? For example, metric spaces and vector spaces are both spaces and algebraic structures. Is a group a space? Is a manifold a space or an algebraic structure, both or neither?
-
A metric space ha... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.957277806109987,
"lm_q1q2_score": 0.8399890432978551,
"lm_q2_score": 0.8774767762675405,
"openwebmath_perplexity": 470.06294377812657,
"openwebmath_score": 0.9365767240524292,
"tag... |
"Space" is a bit fuzzier; I would not put "vector spaces" in the class, restricting it rather to things like topological spaces, manifolds, metric spaces, normed spaces, etc.
Now, one should realize that you this does not have to be a dichotomy: you can have structures that include both kinds of data: a topological gr... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.957277806109987,
"lm_q1q2_score": 0.8399890432978551,
"lm_q2_score": 0.8774767762675405,
"openwebmath_perplexity": 470.06294377812657,
"openwebmath_score": 0.9365767240524292,
"tag... |
# Volume of a sphere with three holes drilled in it.
Suppose that the sphere $x^2+y^2+z^2=9$ has three holes of radius $1$ drilled through it. One down the $z$-axis, one along the $x$-axis, and one along the $y$-axis. What is the volume of the resulting solid? I can do it for two holes but I'm stuck on three.
• What ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.990587412635403,
"lm_q1q2_score": 0.8399861934785905,
"lm_q2_score": 0.847967764140929,
"openwebmath_perplexity": 338.96203726818834,
"openwebmath_score": 0.951598048210144,
"tags"... |
\begin{align}8 \int_{-\pi/4}^{\pi/4} d\phi \: \int_0^1 d\rho \, \rho \sqrt{1-\rho^2 \cos^2{\phi}} &= 4 \frac{2}{3} \int_{-\pi/4}^{\pi/4} d\phi \: \left( 1- \left|\sin^3{\phi}\right|\right) \sec^2{\phi}\\ &= \frac{16}{3} - \frac{16}{3} \int_0^{\pi/4} d\phi \: \sin^3{\phi} \, \sec^2{\phi}\\ &= 8 \left (2 - \sqrt{2}\right... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.990587412635403,
"lm_q1q2_score": 0.8399861934785905,
"lm_q2_score": 0.847967764140929,
"openwebmath_perplexity": 338.96203726818834,
"openwebmath_score": 0.951598048210144,
"tags"... |
$$V = \frac{4 \pi}{3} (3)^3 - V_{\text{holes}} = (64 \sqrt{2}-66) \pi - 8 (2-\sqrt{2}) \approx 72.31$$
compared with the original volume of the sphere $36 \pi \approx 113.1$.
• you should know that the intersection area is not like a cube. i think you have missed to compute the additional volume to find the total vol... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.990587412635403,
"lm_q1q2_score": 0.8399861934785905,
"lm_q2_score": 0.847967764140929,
"openwebmath_perplexity": 338.96203726818834,
"openwebmath_score": 0.951598048210144,
"tags"... |
# Thread: The range of this function?
1. ## The range of this function?
The function is f(x)= [6/(x+2)]-3
I wanted to find the range, also could you please post your working as I want to understand how you came to your answer, thanks.
Regards
Ossy
2. $f(x)= \frac{6}{x+2}-3$
Where is $f(x)$ undefined? Naturally w... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9905874098566368,
"lm_q1q2_score": 0.8399861873163813,
"lm_q2_score": 0.8479677602988601,
"openwebmath_perplexity": 2220.7708103413433,
"openwebmath_score": 0.8695751428604126,
"ta... |
Domain: . $x\,\neq\,-2$
. Range: . $y\,\neq\,0$
The graph of . $y \:=\:\frac{6}{x+2}$ . rises more "steeply"
. . and "flattens slower", but has the same basic shape.
The graph of . $y \:=\:\frac{6}{x+2} - 3$ . is the previous graph lowered 3 units.
Code:
: |
:* |
: |
: * |
--------------+--*|-----... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9905874098566368,
"lm_q1q2_score": 0.8399861873163813,
"lm_q2_score": 0.8479677602988601,
"openwebmath_perplexity": 2220.7708103413433,
"openwebmath_score": 0.8695751428604126,
"ta... |
# Number of non-decreasing sequence $\{a_i\}$ such that every $a_i \geq i$
Find the number of non-decreasing sequences $$a_1, a_2, a_3, a_4, a_5$$ such that $$a_i \geq 1$$, $$a_5 \leq 20$$ and $$a_i \geq i$$;
## My attempt
I tried to use the Inclusion-Exclusion principle, the number of non-decreasing sequences $$a_1... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9905874115238967,
"lm_q1q2_score": 0.8399861868272115,
"lm_q2_score": 0.8479677583778258,
"openwebmath_perplexity": 117.44973990160902,
"openwebmath_score": 0.7629914283752441,
"ta... |
Suppose that $$\langle a_1,\ldots,a_5\rangle$$ is such a sequence. We can interpret it as directions for a walk on the integer lattice in the plane, starting at the origin: we first take $$a_1$$ steps north to $$\langle 0,a_1\rangle$$, then one step east to $$\langle 1,a_1\rangle$$, then $$a_2-a_1$$ steps north to $$\l... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9905874115238967,
"lm_q1q2_score": 0.8399861868272115,
"lm_q2_score": 0.8479677583778258,
"openwebmath_perplexity": 117.44973990160902,
"openwebmath_score": 0.7629914283752441,
"ta... |
There are clearly $$\binom{25}5$$ NE paths from $$\langle 0,0\rangle$$ to $$\langle 5,20\rangle$$. There is a bijection between those that drop below the diagonal and NE paths from $$\langle 0,0\rangle$$ to $$\langle 21,4\rangle$$, and there are $$\binom{25}4$$ of those, so there are $$\binom{25}5-\binom{25}4=53130-126... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9905874115238967,
"lm_q1q2_score": 0.8399861868272115,
"lm_q2_score": 0.8479677583778258,
"openwebmath_perplexity": 117.44973990160902,
"openwebmath_score": 0.7629914283752441,
"ta... |
# Volume of cube section above intersection with plane
Suppose we have a unit cube (side=1) and a plane with equation $x+y+z=\alpha$. I'd like to compute the volume of the region that results once the plane sections the cube (above the plane). There are three cases to analyze, and I can't quite visualize one of them.
... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534392852384,
"lm_q1q2_score": 0.8399780499029723,
"lm_q2_score": 0.8558511451289037,
"openwebmath_perplexity": 307.7834667011739,
"openwebmath_score": 0.8820827007293701,
"tag... |
If not, what's the approach?
Note that there's a neat connection between this problem and figuring out the CDF of a sum of a random variable that has triangular distribution with support on $[0,2]$ and a random variable with uniform distribution on $[0,1]$ (assuming independence). Hence, I know what the answer should ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534392852384,
"lm_q1q2_score": 0.8399780499029723,
"lm_q2_score": 0.8558511451289037,
"openwebmath_perplexity": 307.7834667011739,
"openwebmath_score": 0.8820827007293701,
"tag... |
Update
About the question whether this argument can be extended to higher dimension, the answer is yes. Let's look at the 3-dimension $2 \le \alpha \le 3$ case first.
As one increases $\alpha$ beyond $2$, the three tetrahedron in first figure start overlap. As shown in second figure, the intersection of the three tet... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534392852384,
"lm_q1q2_score": 0.8399780499029723,
"lm_q2_score": 0.8558511451289037,
"openwebmath_perplexity": 307.7834667011739,
"openwebmath_score": 0.8820827007293701,
"tag... |
$$1 -\sum_{i=0}^{\lfloor \alpha \rfloor} (-1)^i \binom{k}{i} \frac{(\alpha-i)^k}{k!}$$
-
This is a great way to think about it. Am I correct in asserting that one can expand this argument so that on a $[0,1]^k$ hypercube the "volume" of interest should be given by $$V = 1- \dfrac{1}{k!}\sum_{i=0}^{\lfloor \alpha \rflo... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534392852384,
"lm_q1q2_score": 0.8399780499029723,
"lm_q2_score": 0.8558511451289037,
"openwebmath_perplexity": 307.7834667011739,
"openwebmath_score": 0.8820827007293701,
"tag... |
The object created by slicing is a polyhedron, obviously. Then, the volume of a polyhedron can be computed conveniently by decomposing it into tetrahedra and adding up the (signed) volumes of these tetrahedra. You can find an explanation of the technique in this document. There is also a paper by Michael Kallay on the ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534392852384,
"lm_q1q2_score": 0.8399780499029723,
"lm_q2_score": 0.8558511451289037,
"openwebmath_perplexity": 307.7834667011739,
"openwebmath_score": 0.8820827007293701,
"tag... |
The basic idea is that the volume above the hyper-plane for fixed arguments is a linear combination of vertices' "heights" above (but only these above) the hyper-plane raised to the power of effective dimension. Note the alternating sign for each connected vertex (Times @@@ t).
In your particular case this reduces to:... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534392852384,
"lm_q1q2_score": 0.8399780499029723,
"lm_q2_score": 0.8558511451289037,
"openwebmath_perplexity": 307.7834667011739,
"openwebmath_score": 0.8820827007293701,
"tag... |
# Find matrix $P$ such that $P^{-1}AP=B$
Given $$A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & -3 \\ 1 & 3 & 2 \end{bmatrix}$$ $$B= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & -3 \\ 0 & 3 & 2 \end{bmatrix}$$ find $P$ such that $P^{-1} A P = B$.
Firstly I said that $AP=PB$
Solved the 9 equations in 9 unknowns. and got that:
$$P... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534392852383,
"lm_q1q2_score": 0.8399780480987117,
"lm_q2_score": 0.855851143290548,
"openwebmath_perplexity": 111.13684861792456,
"openwebmath_score": 0.6917913556098938,
"tag... |
$$\left( \mathrm I_3 \otimes \mathrm A - \mathrm B^{\top} \otimes \mathrm I_3 \right) \mbox{vec} (\mathrm X) = 0_9$$
and let $\mathrm M := \mathrm I_3 \otimes \mathrm A - \mathrm B^{\top} \otimes \mathrm I_3$. Using SymPy:
>>> I3 = Identity(3)
>>> O3 = ZeroMatrix(3,3)
>>> M = BlockMatrix([[A - I3, O3, O3]... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534392852383,
"lm_q1q2_score": 0.8399780480987117,
"lm_q2_score": 0.855851143290548,
"openwebmath_perplexity": 111.13684861792456,
"openwebmath_score": 0.6917913556098938,
"tag... |
# A (probably trivial) induction problem: $\sum_2^nk^{-2}\lt1$
So I'm a bit stuck on the following problem I'm attempting to solve. Essentially, I'm required to prove that $\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2} < 1$ for all $n$. I've been toiling with some algebraic gymnastics for a while now, but I can't s... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534316905262,
"lm_q1q2_score": 0.8399780470115501,
"lm_q2_score": 0.8558511488056151,
"openwebmath_perplexity": 298.05780074192336,
"openwebmath_score": 0.8994263410568237,
"ta... |
-
Awesome! Many thanks...Might I ask what motivated the choice of subtracting 1/n? – Paquito Jul 24 '12 at 22:21
@Paquito To a certain extent, intuition - since the difference between consecutive terms of the form $1/n$ is on the order of $1/n^2$, I could see that adding a quadratic term to a $1/n$-sized 'gap' between... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534316905262,
"lm_q1q2_score": 0.8399780470115501,
"lm_q2_score": 0.8558511488056151,
"openwebmath_perplexity": 298.05780074192336,
"openwebmath_score": 0.8994263410568237,
"ta... |
and the right-hand side what's called a telescoping sum'' -- that is, the pairs of terms $-1/2$ and $+1/2$, $-1/3$ and $+1/3$, and so on cancel. So the right-hand side is $1 - 1/n$, which is less than 1.
This came to mind pretty much immediately for me, because I happened to know that $\sum_{k \ge 2}^\infty 1/(k(k-1))... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534316905262,
"lm_q1q2_score": 0.8399780470115501,
"lm_q2_score": 0.8558511488056151,
"openwebmath_perplexity": 298.05780074192336,
"openwebmath_score": 0.8994263410568237,
"ta... |
# Evaluation of $\lim\limits_{x\rightarrow0} \frac{\tan(x)-x}{x^3}$
One of the previous posts made me think of the following question: Is it possible to evaluate this limit without L'Hopital and Taylor?
$$\lim_{x\rightarrow0} \frac{\tan(x)-x}{x^3}$$
-
Closely related is math.stackexchange.com/questions/134051 – Dav... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534327754852,
"lm_q1q2_score": 0.839978046135853,
"lm_q2_score": 0.8558511469672594,
"openwebmath_perplexity": 598.8449403333569,
"openwebmath_score": 0.9990609288215637,
"tags... |
Let $$f(x) = \frac{x - \sin x}{x^3} = \frac{1 - \frac{\sin x}{x}}{x^2}.$$ (Here I've changed the sign so that the limit will be positive.) Since $f$ is an even function, it's enough to consider $x>0$.
Fix a positive integer $n$. To begin with, we have $$x = 2^n \frac{x}{2^n} > 2^n \sin \frac{x}{2^n}.$$ (I'm assuming t... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534327754852,
"lm_q1q2_score": 0.839978046135853,
"lm_q2_score": 0.8558511469672594,
"openwebmath_perplexity": 598.8449403333569,
"openwebmath_score": 0.9990609288215637,
"tags... |
The other direction is similar. Start with $$x = 2^n \frac{x}{2^n} < 2^n \tan\frac{x}{2^n} = 2^n \frac{\sin(x/2^n)}{\cos(x/2^n)}.$$ This leads to $$\begin{split} 1 - \frac{\sin x}{x} & < 1 - \cos\frac{x}{2^n} \cdot (\text{same product of cosines as above}) \\ & = 1 - \cos\frac{x}{2^n} + \cos\frac{x}{2^n} \cdot (1 - (\t... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534327754852,
"lm_q1q2_score": 0.839978046135853,
"lm_q2_score": 0.8558511469672594,
"openwebmath_perplexity": 598.8449403333569,
"openwebmath_score": 0.9990609288215637,
"tags... |
Encouraged by Hans Lundmark's answer, I'm posting my own solution without derivatives and integrals.
The triple-angle formula for $\tan$ is $$\tan 3\theta = \frac{3\tan\theta-\tan^3\theta}{1-3\tan^2\theta}.$$ Suppose $\lim_{x\to0}(\tan x-x)/x^3 = c$. Letting $x = 3\theta$, we then have \begin{align} c &= \lim_{x\to0} ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534327754852,
"lm_q1q2_score": 0.839978046135853,
"lm_q2_score": 0.8558511469672594,
"openwebmath_perplexity": 598.8449403333569,
"openwebmath_score": 0.9990609288215637,
"tags... |
Here is a different approach. Let $$L = \lim_{x \to 0} \dfrac{\tan(x) - x}{x^3}$$ Replacing $x$ by $2y$, we get that \begin{align} L & = \lim_{y \to 0} \dfrac{\tan(2y) - 2y}{(2y)^3} = \lim_{y \to 0} \dfrac{\dfrac{2 \tan(y)}{1 - \tan^2(y)} - 2y}{(2y)^3}\\ & = \lim_{y \to 0} \dfrac{\dfrac{2 \tan(y)}{1 - \tan^2(y)} - 2 \t... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534327754852,
"lm_q1q2_score": 0.839978046135853,
"lm_q2_score": 0.8558511469672594,
"openwebmath_perplexity": 598.8449403333569,
"openwebmath_score": 0.9990609288215637,
"tags... |
-
Huh, I had assumed that to get $\tan$ to third order one would have to use the triple angle formula. I guess I was mistaken. – Rahul Jun 14 '12 at 7:11
@Marvis: very elegant! Thanks! – Chris's sis Jun 14 '12 at 8:05
@Marvis: this proof also works for showing that $\lim_{x\rightarrow0} \frac{\tan(x)-x}{x^2} = 0$. – ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534327754852,
"lm_q1q2_score": 0.839978046135853,
"lm_q2_score": 0.8558511469672594,
"openwebmath_perplexity": 598.8449403333569,
"openwebmath_score": 0.9990609288215637,
"tags... |
# Using Lagrange form of the remainder with cosh
I am trying to find "$\cosh 4$ using the sixth partial sum ($n=5$) of the Maclaurin series" for the function. I am also trying to use "the Lagrange form of the remainder to estimate the number of decimal places to which the partial sum" is accurate.
For the first part,... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534327754852,
"lm_q1q2_score": 0.839978046135853,
"lm_q2_score": 0.8558511469672594,
"openwebmath_perplexity": 98.69650134751436,
"openwebmath_score": 0.9882498979568481,
"tags... |
Thus if you are going to use the Lagrange remainder formula as quoted in Wikipedia, for the "$k$" in that formula, you need to use $11$.
I am going to elaborate on André’s answer after having thought about the problem some more.
If we let $T_n(x)$ be the $n$th-degree Taylor polynomial of $\cosh$ at $0$ (since Maclaur... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534327754852,
"lm_q1q2_score": 0.839978046135853,
"lm_q2_score": 0.8558511469672594,
"openwebmath_perplexity": 98.69650134751436,
"openwebmath_score": 0.9882498979568481,
"tags... |
# Solving $3x\equiv 4\pmod 7$
I'm trying to learn about linear congruences of the form ax = b(mod m). In my book, it's written that if $\gcd(a, m) = 1$ then there must exist an integer $a'$ which is an inverse of $a \pmod{m}$. I'm trying to solve this example:
$$3x \equiv 4 \pmod 7$$
First I noticed $\gcd(3, 7) = 1$... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534322330057,
"lm_q1q2_score": 0.8399780402587899,
"lm_q2_score": 0.8558511414521923,
"openwebmath_perplexity": 506.39700833767273,
"openwebmath_score": 0.7894480228424072,
"ta... |
• thanks for replying. Your answer is a lot simpler. Will this subtracting/adding always work out the problem? – Tehmas Dec 7 '14 at 13:32
• yes you can always add multiples of $7$ (in this particular example) since $7\equiv0\pmod{7}$ so what you're doing is really just adding $0$ to the equation. More generally $a\equ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534322330057,
"lm_q1q2_score": 0.8399780402587899,
"lm_q2_score": 0.8558511414521923,
"openwebmath_perplexity": 506.39700833767273,
"openwebmath_score": 0.7894480228424072,
"ta... |
$$3x\equiv4\pmod{7}\\-6x\equiv -8\pmod{7}\\-6x\equiv-1-7\\-6x\equiv-1\pmod{7}\\(7-6)x\equiv-1\equiv6\pmod{7}$$
• kingW3, thank you for answering. Can you please elaborate step 3? – Tehmas Dec 7 '14 at 13:15
• @Tehmas: $8 \equiv 1\pmod 7$. – user 170039 Dec 7 '14 at 13:16
• @Tehmas Edited,you can also look the comment ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534322330057,
"lm_q1q2_score": 0.8399780402587899,
"lm_q2_score": 0.8558511414521923,
"openwebmath_perplexity": 506.39700833767273,
"openwebmath_score": 0.7894480228424072,
"ta... |
How can we show that all the roots of some irreducible polynomial are not of algebraically equal status?
In studying Galois theory, I found that all roots of some irreducible polynomial are not of algebraically equal status, because the Galois group of some irreducible polynomial may not be full symmetric group $S_n$.... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363758493942,
"lm_q1q2_score": 0.8399712635264919,
"lm_q2_score": 0.8519528076067262,
"openwebmath_perplexity": 131.5978809307098,
"openwebmath_score": 0.9033700823783875,
"tag... |
Thanks for reading my question and any comment will be appreciated!
• What does "of algebraically equal status" mean? – Eric Wofsey Feb 26 '16 at 8:30
• I think I better phrasing would be to just say that you are looking for a concrete "reason" that the Galois group cannot be all of $S_n$ in some example. – Eric Wofse... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363758493942,
"lm_q1q2_score": 0.8399712635264919,
"lm_q2_score": 0.8519528076067262,
"openwebmath_perplexity": 131.5978809307098,
"openwebmath_score": 0.9033700823783875,
"tag... |
Sure, what you're asking for can happen. For a simple example, take $F=\mathbb{Q}$ and $p(t)=t^4-2$. The roots are $\alpha_1=\sqrt[4]{2}$, $\alpha_2=-\sqrt[4]{2}$, $\alpha_3=i\sqrt[4]{2}$, and $\alpha_4=-i\sqrt[4]{2}$. Note then that $\alpha_2\in F(\alpha_1)$ (so its minimal polynomial over $F(\alpha_1)$ has degree $1$... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363758493942,
"lm_q1q2_score": 0.8399712635264919,
"lm_q2_score": 0.8519528076067262,
"openwebmath_perplexity": 131.5978809307098,
"openwebmath_score": 0.9033700823783875,
"tag... |
# Analysis of Skewness and Kurtosis
Since the skewness and kurtosis of the normal distribution are zero, values for these two parameters should be close to zero for data to follow a normal distribution.
• A rough measure of the standard error of the skewness is $\sqrt{6/n}$ where n is the sample size.
• A rough measu... | {
"domain": "real-statistics.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363708905831,
"lm_q1q2_score": 0.8399712611548583,
"lm_q2_score": 0.851952809486198,
"openwebmath_perplexity": 1028.7116975059869,
"openwebmath_score": 0.7346479296684265,
"t... |
Since CHISQ.DIST.RT(1.93, 2) = .382 > .05, once again we conclude there isn’t sufficient evidence to rule out the data coming from a normal population.
Real Statistics Functions: The Real Statistics Resource Pack contains the following functions.
JARQUE(R1, pop) = the Jarque-Barre test statistic JB for the data in th... | {
"domain": "real-statistics.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363708905831,
"lm_q1q2_score": 0.8399712611548583,
"lm_q2_score": 0.851952809486198,
"openwebmath_perplexity": 1028.7116975059869,
"openwebmath_score": 0.7346479296684265,
"t... |
• Charles says:
What value did you get for SKEW and KURT_
Charles
• soharb says:
EViews 9.5:
SKEW= 1.769081
KURT= 3.620125
JB= 26.69155
Excel regular formula:
=SKEW(A2:A26) = 1.884063081
=SKEW.P(A2:A26) =1.769080723
=KURT(A2:A26) = 4.748928357
Note: there is no KURT.P!!!
Excel array formula:
for SKEW
=((SUM((A2:A2... | {
"domain": "real-statistics.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363708905831,
"lm_q1q2_score": 0.8399712611548583,
"lm_q2_score": 0.851952809486198,
"openwebmath_perplexity": 1028.7116975059869,
"openwebmath_score": 0.7346479296684265,
"t... |
• Charles says:
David,
As I wrote in response to that comment
“We often use alpha = .05 as the significance level for statistical tests. The critical value for a two tailed test of normal distribution with alpha = .05 is NORMSINV(1-.05/2) = 1.96, which is approximately 2 standard deviations (i.e. standard errors) fr... | {
"domain": "real-statistics.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363708905831,
"lm_q1q2_score": 0.8399712611548583,
"lm_q2_score": 0.851952809486198,
"openwebmath_perplexity": 1028.7116975059869,
"openwebmath_score": 0.7346479296684265,
"t... |
• Denny Yu says:
I’m really looking forward to it.
5. Zohreh says:
Salaam
May you please cite the reference for “If the absolute value of the skewness for the data is more than twice the standard error this indicates that the data are not symmetric, and therefore not normal”. I need it. Thanks.
• Charles says:
We ... | {
"domain": "real-statistics.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363708905831,
"lm_q1q2_score": 0.8399712611548583,
"lm_q2_score": 0.851952809486198,
"openwebmath_perplexity": 1028.7116975059869,
"openwebmath_score": 0.7346479296684265,
"t... |
• Charles says:
Colin,
A rough measure of the standard error of the skewness is \sqrt{6/n} where n is the sample size.
A rough measure of the standard error of the kurtosis is \sqrt{24/n} where n is the sample size.
Charles
8. My brother recommended I would possibly like this web site. He was once entirely right. T... | {
"domain": "real-statistics.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363708905831,
"lm_q1q2_score": 0.8399712611548583,
"lm_q2_score": 0.851952809486198,
"openwebmath_perplexity": 1028.7116975059869,
"openwebmath_score": 0.7346479296684265,
"t... |
# Limit proof
1. Jan 19, 2015
### ciubba
1. The problem statement, all variables and given/known data
Prove $$lim_{x->4}\frac{x-4}{\sqrt{x}-2}=4$$
2. Relevant equations
Epsilon\delta definition
3. The attempt at a solution
I can see that a direct evaluation at 4 leads to an indeterminate form, so:
$$\frac{x-4}{\... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363737832231,
"lm_q1q2_score": 0.8399712543540526,
"lm_q2_score": 0.8519528000888386,
"openwebmath_perplexity": 970.9712102879457,
"openwebmath_score": 0.8043165802955627,
"tag... |
Oops, I mistyped. I meant to say "Given this definition of delta, it is elementary to work backwards towards $$|\sqrt{x}+2\mathbf{-4}|<\boldsymbol{\epsilon}$$ My main question is whether or not it is correct to say that, for any epsilon, delta is $$min\{\sqrt{-\delta+4}-2,4,\sqrt{\delta+4}-2\}$$
Ah, I didn't think to ... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363737832231,
"lm_q1q2_score": 0.8399712543540526,
"lm_q2_score": 0.8519528000888386,
"openwebmath_perplexity": 970.9712102879457,
"openwebmath_score": 0.8043165802955627,
"tag... |
# When Two Dice Are Rolled Find The Probability Of Getting A Sum Of 5 Or 6 | {
"domain": "farmaciacoverciano.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363750229257,
"lm_q1q2_score": 0.8399712517041412,
"lm_q2_score": 0.8519527963298946,
"openwebmath_perplexity": 208.79297532556296,
"openwebmath_score": 0.662230193614959... |
With two dice, there are 6 x 6 = 36 possible outcomes. [3 Marks) 1 13 5 Question 2 Find The Z Score That Corresponds To The Given Area. Number of outcomes of the experiment that are favorable to the event that a sum of two events is 6. The logic is there are six sides to each die, so for each number on one You did the ... | {
"domain": "farmaciacoverciano.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363750229257,
"lm_q1q2_score": 0.8399712517041412,
"lm_q2_score": 0.8519527963298946,
"openwebmath_perplexity": 208.79297532556296,
"openwebmath_score": 0.662230193614959... |
For example: 1 roll: 5/6 (83. of ways are - 1 , 1 1 , 2 2 , 1 1 , 4 4 , 1 1 , 6. The probability of rolling a specific number twice in a row is indeed 1/36, because you have a 1/6 chance of getting that number on each of two rolls (1/6 x 1/6). No, other sum is possible because three dice being rolled give maximum sum o... | {
"domain": "farmaciacoverciano.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363750229257,
"lm_q1q2_score": 0.8399712517041412,
"lm_q2_score": 0.8519527963298946,
"openwebmath_perplexity": 208.79297532556296,
"openwebmath_score": 0.662230193614959... |
since each 6 numbers. (d) an even number appears on the black dice or the sum of the numbers on the two dice is 7. (it's easier to count the 6 non-red ones and subtract from 36 to get 30). When two dice are thrown, find the probability of getting a number always greater than 4 on the second die. What is the probability... | {
"domain": "farmaciacoverciano.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363750229257,
"lm_q1q2_score": 0.8399712517041412,
"lm_q2_score": 0.8519527963298946,
"openwebmath_perplexity": 208.79297532556296,
"openwebmath_score": 0.662230193614959... |
showing 6 and the others are all showing 1-5, and there are 6 different ways to choose which die is showing 6. What is the. We want sum to be greater than 16, So, sum could be either 17 or 18. two-dice-are-rolled-simultaneously-find-the-probability-of-getting-a-total-of-9SingleChoice5b5cc7d1e4d2b419777512684. Find the ... | {
"domain": "farmaciacoverciano.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363750229257,
"lm_q1q2_score": 0.8399712517041412,
"lm_q2_score": 0.8519527963298946,
"openwebmath_perplexity": 208.79297532556296,
"openwebmath_score": 0.662230193614959... |
the uncontrolled area. The probability of one dice not being a particular number is 5/6. The events "getting sum less than 8" and. If we discard the 6 rolls that gave the same numbers, then the odds of getting a six is. What is the distribution of the sum? 30. (d) A sum that is divisible by 4. The other two singletons ... | {
"domain": "farmaciacoverciano.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363750229257,
"lm_q1q2_score": 0.8399712517041412,
"lm_q2_score": 0.8519527963298946,
"openwebmath_perplexity": 208.79297532556296,
"openwebmath_score": 0.662230193614959... |
1 minus the probability of rolling a sum of six. Example 7: A die is rolled, find the probability of getting a 3. It’s very common to find questions about dice rolling in probability and statistics. (i) To get the sum of numbers 4 or 5 favourable outcomes are: (1, 3) ,(3, 1) , (2,2). Two fair dice are rolled and the su... | {
"domain": "farmaciacoverciano.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363750229257,
"lm_q1q2_score": 0.8399712517041412,
"lm_q2_score": 0.8519527963298946,
"openwebmath_perplexity": 208.79297532556296,
"openwebmath_score": 0.662230193614959... |
Find the expected number of games that are played when. There are 36 permutations of two dice. The probability of choosing a green marble from the jar. (i) Prime numbers = 2, 3 and 5 Favourable number of events = 3 Probability that it will be a prime. Because there are 36 possibilities in all, and the sum of their prob... | {
"domain": "farmaciacoverciano.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363750229257,
"lm_q1q2_score": 0.8399712517041412,
"lm_q2_score": 0.8519527963298946,
"openwebmath_perplexity": 208.79297532556296,
"openwebmath_score": 0.662230193614959... |
the sum}[/math. For three six-sided dice, the most common rolls are 10 and 11, both with probability 1/8; and the least common rolls are 3 and 18, both with probability 1/216. We start with writing a table to Discrete = This means that if I pick any two consecutive outcomes. Find each experimental probability. What is ... | {
"domain": "farmaciacoverciano.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363750229257,
"lm_q1q2_score": 0.8399712517041412,
"lm_q2_score": 0.8519527963298946,
"openwebmath_perplexity": 208.79297532556296,
"openwebmath_score": 0.662230193614959... |
11-15 3 16-20 6 20+ 1. Find the joint probability mass function of X and Y when (a) X is the largest value obtained on any die and Y is the sum of the values; (b) X is the value on the first die and Y is the larger of the two values; (c) X is the smallest and Y is the largest value obtained on the dice. A standard deck... | {
"domain": "farmaciacoverciano.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363750229257,
"lm_q1q2_score": 0.8399712517041412,
"lm_q2_score": 0.8519527963298946,
"openwebmath_perplexity": 208.79297532556296,
"openwebmath_score": 0.662230193614959... |
you find mistakes in interface or texts? Or do you know how to improveStudyLib UI?. Question 4: Two dice are rolled, find the probability that the sum is a) equal to 1 b) equal to 4 c) less than 13 Solution to Question 4: a) The sample space S of two dice is shown below. Algebra -> Probability-and-statistics -> SOLUTIO... | {
"domain": "farmaciacoverciano.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363750229257,
"lm_q1q2_score": 0.8399712517041412,
"lm_q2_score": 0.8519527963298946,
"openwebmath_perplexity": 208.79297532556296,
"openwebmath_score": 0.662230193614959... |
two dice, probability = 1/6 × 1/6 = 1/36 = 1 ÷ 36 = 0. Subject: Re: Probability: Two six-sided dice, rolling two numbers in order. two dice are rolled find the probability of getting a 5 on either dice or the sum of both dice is 5. Major changes in Python environment : . 5 ways to get a sum of 6. What is the probabilit... | {
"domain": "farmaciacoverciano.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363750229257,
"lm_q1q2_score": 0.8399712517041412,
"lm_q2_score": 0.8519527963298946,
"openwebmath_perplexity": 208.79297532556296,
"openwebmath_score": 0.662230193614959... |
Cancel. Probability - Quantitative Aptitude objective type questions with answers & explanation (MCQs) for job Now a shirt is picked from second box. For each of the possible outcomes add the numbers on the two dice and count how many times this sum is 7. Of these, 6 have a sum less than five, 1+1, 1+2, 1+3, 2+1, 2+2, ... | {
"domain": "farmaciacoverciano.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363750229257,
"lm_q1q2_score": 0.8399712517041412,
"lm_q2_score": 0.8519527963298946,
"openwebmath_perplexity": 208.79297532556296,
"openwebmath_score": 0.662230193614959... |
you dont know the probability p of getting a ticket but you got 5. Rolling Dice. Find the Mean of the Roll z column; OK; Repeat process except find the Standard Deviation of the Roll z column; By hand (with a calculator) square the standard deviation to get the variance. For example, the event "the sum of the faces sho... | {
"domain": "farmaciacoverciano.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363750229257,
"lm_q1q2_score": 0.8399712517041412,
"lm_q2_score": 0.8519527963298946,
"openwebmath_perplexity": 208.79297532556296,
"openwebmath_score": 0.662230193614959... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.