content string | quality_label int64 | meta string | all-MiniLM-L6-v2_embedding list | doc_id int64 | unique_id string |
|---|---|---|---|---|---|
Demonstration
Hi There might be a better way to do this, but IŽll give one solution. Note: $x-\frac{x^{2}}{2}$ is the first two terms of the maclaurin series of $ln(1+x)$ . Anyway, let $f(x)=x-\frac{x^{2}}{2}-ln
(1+x)$ , then we want to show that $f(x)<0$ for all $x>0$. Solution: $f$ is continous and differentiable for... | 5 | [
0.0311279296875,
-0.08935546875,
0.0966796875,
0.03466796875,
0.01251220703125,
-0.044189453125,
-0.01495361328125,
0.0206298828125,
0.0230712890625,
0.01031494140625,
0.078125,
0.007476806640625,
-0.00701904296875,
0.07080078125,
-0.08837890625,
0.01251220703125,
0.042236328125,
-... | 6,900 | 6900 | |
Is there an example of a formally smooth morphism which is not smooth?
up vote 10 down vote favorite
8
A morphism of schemes is formally smooth and locally of finite presentation iff it is smooth.
What happens if we drop the finitely presented hypothesis? Of course, locally of finite presentation is part of smoothnes... | 5 | [
0.026123046875,
0.0712890625,
0.00933837890625,
0.0279541015625,
-0.0380859375,
-0.046875,
0.0198974609375,
-0.03173828125,
0.035888671875,
0.017333984375,
0.0223388671875,
-0.0250244140625,
0.02392578125,
0.080078125,
0.020263671875,
-0.00144195556640625,
-0.0341796875,
0.01037597... | 6,901 | 6901 | |
How many equivalent classes there are?
June 26th 2011, 08:12 AM
godelproof
How many equivalent classes there are?
$\forall a, b \in \mathbb{R}$, $a\sim b$ iff $a$ and $b$ differ in finitely many digits. (So for example $3.14\sim 502.441$ but $3.14sim \pi$). Then this equivalent relation $\sim$ partitions $\
... | 5 | [
-0.06982421875,
-0.080078125,
-0.04052734375,
-0.07373046875,
-0.03759765625,
0.059814453125,
0.017578125,
-0.0301513671875,
0.03564453125,
-0.06494140625,
-0.007659912109375,
-0.0654296875,
0.04541015625,
0.0172119140625,
0.051025390625,
-0.10205078125,
-0.0380859375,
-0.065917968... | 6,902 | 6902 | |
Surds (length of a side of a rhombus)
June 5th 2010, 11:34 PM #1
Junior Member
Surds (length of a side of a rhombus)
Hello !
You are close to the answer. You have the same one, written in a different form, but you need some more work to get to their actual answer :
$BC^2 = 18 \frac{4}{5} + 4 \sqrt{5} -... | 4 | [
-0.01348876953125,
0.039794921875,
0.043212890625,
-0.000335693359375,
-0.01495361328125,
0.018310546875,
-0.09521484375,
0.076171875,
-0.01708984375,
-0.032958984375,
0.041259765625,
-0.01025390625,
0.0025482177734375,
0.013916015625,
-0.07568359375,
0.0062255859375,
-0.019287109375... | 6,903 | 6903 | |
An integral, partial fractions
July 19th 2008, 07:04 PM #1
An integral, partial fractions
This one is a really tough one for me. I'm betting an integration via partial fractions : $\int \frac{2a^2x-6a+1}{a^2x^2-6ax+10}dx$ with $a eq 0$.
I could reach that the integral is worth $\int 1+\frac{-a^2x^2+6ax-9+2a^2... | 5 | [
-0.057861328125,
0.00933837890625,
0.07080078125,
-0.007293701171875,
0.041748046875,
0.06494140625,
-0.08349609375,
0.0203857421875,
-0.00162506103515625,
0.0181884765625,
0.047119140625,
-0.0146484375,
0.00958251953125,
0.0556640625,
-0.021728515625,
0.0189208984375,
0.07421875,
... | 6,904 | 6904 | |
Implicit Differentiation & Indefinite Integral :(
February 9th 2009, 05:44 PM #1
Newbie
Joined
Feb 2009
Posts
5
Implicit Differentiation & Indefinite Integral :(
Help guys - I have loads of these to do and I can't get my head to work. I would really appreciate some detailed help so I can do the rest with... | 5 | [
-0.048095703125,
-0.043212890625,
0.0341796875,
-0.0263671875,
0.055908203125,
-0.01318359375,
-0.05322265625,
-0.034423828125,
-0.042724609375,
0.046142578125,
0.0810546875,
-0.00799560546875,
-0.01458740234375,
-0.041748046875,
0.0281982421875,
0.047607421875,
-0.0240478515625,
0... | 6,905 | 6905 | |
Choosing 3 of 6 Colors
Date: 03/03/2000 at 00:33:35
From: Cheyenne Marcy
Subject: Different ways to pick 3 crayons out of 6
My math problem of the week is this:
Miss Davis asked her students each to draw a picture describing their
hobby. She asked them to use only three colors. Patrick had a box of
crayons containi... | 4 | [
0.01397705078125,
-0.04833984375,
0.01470947265625,
-0.06640625,
0.0030059814453125,
0.04052734375,
0.1044921875,
-0.031494140625,
-0.0257568359375,
0.048095703125,
-0.05712890625,
0.0108642578125,
0.044677734375,
-0.00213623046875,
0.01055908203125,
0.0859375,
-0.1123046875,
-0.01... | 6,906 | 6906 | |
Limited Area, Unlimited Perimeter
Associated Topics || Dr. Math Home || Search Dr. Math
Limited Area, Unlimited Perimeter
... | 5 | [
0.10498046875,
0.0625,
-0.039794921875,
-0.005340576171875,
0.0284423828125,
-0.0225830078125,
-0.036376953125,
0.09814453125,
-0.01409912109375,
0.0006103515625,
-0.00933837890625,
-0.004241943359375,
-0.0159912109375,
0.06298828125,
0.052734375,
0.0262451171875,
0.01007080078125,
... | 6,907 | 6907 | |
Elementary power rule of derivatives/Proof
Talk0
872pages on
this wiki
$\frac{d}{dx}x^n=nx^{n-1}$, for n is a natural number.
Proof by Induction
Prerequisites
• The product rule of derivatives, $\frac{d}{dx}uv=uv'+vu'$
Proof
The proof proceeds by mathematical induction. Take the base case k=0. Then:
$... | 5 | [
-0.0196533203125,
0.049560546875,
0.0478515625,
0.07861328125,
-0.01708984375,
-0.07666015625,
0.0125732421875,
-0.0390625,
0.05224609375,
0.08349609375,
0.0201416015625,
-0.0213623046875,
0.07177734375,
0.02392578125,
-0.044921875,
0.038330078125,
-0.05712890625,
0.061279296875,
... | 6,908 | 6908 | |
Logarithmic Diff.....
January 27th 2007, 05:01 PM #1
Senior Member
Joined
Jul 2006
From
Shabu City
Posts
381
Logarithmic Diff.....
Find the derivative of the function
$f(x) = x^{\sqrt{x}}$; $x>0$
Do i have to use logarithmic differentiation?
The answer according to the back of book is....... | 5 | [
0.0908203125,
-0.044677734375,
0.048828125,
0.0000032782554626464844,
-0.006195068359375,
-0.1396484375,
0.01116943359375,
0.00738525390625,
0.005950927734375,
0.051513671875,
0.0966796875,
0.011962890625,
0.03515625,
0.037353515625,
0.01904296875,
0.0390625,
-0.07177734375,
-0.030... | 6,909 | 6909 | |
Why is TopGrp the category of topological groups and continous homomorphisms protomodular?
up vote 8 down vote favorite
1
Why is TopGrp the category of topological groups and continous homomorphisms protomodular? I know it is, and I have several indirect proofs, but am not able to prove this directly by showing that t... | 4 | [
-0.09326171875,
0.0142822265625,
0.0206298828125,
-0.0162353515625,
0.08544921875,
0.0302734375,
-0.058837890625,
0.07861328125,
-0.00013637542724609375,
-0.033447265625,
-0.0057373046875,
-0.08740234375,
-0.08740234375,
-0.07080078125,
-0.002777099609375,
-0.00579833984375,
-0.05371... | 6,910 | 6910 | |
bra
Narrow Search
Earth and space science
Mathematics
Sort by:
Per page:
Now showing results 1-10 of 18
In this activity, students look at the distribution of aluminum foil balls arranged in a circle on the floor, and compare them to the distribution of gamma-ray bursts on the sky. This activity uses
Gamma-ray Bu... | 4 | [
0.00927734375,
-0.0172119140625,
0.05908203125,
0.048095703125,
-0.0191650390625,
-0.06787109375,
0.050537109375,
0.060302734375,
0.006561279296875,
0.019775390625,
0.01031494140625,
0.0205078125,
-0.0013275146484375,
0.033447265625,
-0.00104522705078125,
-0.02294921875,
-0.002807617... | 6,911 | 6911 | |
Exam Geometry Question
April 22nd 2009, 05:38 PM #1
Senior Member
Joined
Jul 2008
Posts
347
In the diagram, triangle ABC is equilateral, the length of the sides being a. Base line AB is produced in both directions to D and E respectively such that AD = AB = a, while E is the
intersection of line AB pr... | 4 | [
0.000125885009765625,
0.0234375,
0.0242919921875,
-0.07861328125,
-0.0654296875,
0.01507568359375,
0.0037384033203125,
0.06787109375,
0.0218505859375,
0.043701171875,
0.02197265625,
-0.05029296875,
0.016845703125,
-0.0247802734375,
-0.0030975341796875,
0.007476806640625,
-0.112792968... | 6,912 | 6912 | |
Math Forum Discussions
Rupert
Posts: 3,806
Registered: 12/6/04
Re: JSH: End of an error
Posted: Dec 8, 2006 10:44 PM
Rupert wrote:
> jstevh@msn.com wrote:
> > The widespread usage of flawed ideas in number theory has put me in the
> > position of not just proving the errors and getting publication of
> > proof in a ... | 5 | [
-0.1279296875,
0.07861328125,
0.076171875,
0.040283203125,
0.06689453125,
0.0069580078125,
-0.0003261566162109375,
0.00482177734375,
-0.04296875,
0.0244140625,
0.080078125,
-0.007049560546875,
0.09033203125,
0.032958984375,
-0.01434326171875,
-0.0159912109375,
-0.006256103515625,
0... | 6,913 | 6913 | |
Understanding Parametric Equations
Date: 04/02/98 at 20:29:40
From: Kristen Demergian
Subject: Parametric Equations
Dear Dr. Math,
I'm a junior at Granite Hills High School In El Cajon. For honors
Pre-Calculus we are required to write an essay on an advanced
calculus concept that relates to the real world. I have o... | 4 | [
-0.0986328125,
-0.0732421875,
-0.0751953125,
-0.048828125,
-0.056884765625,
0.02587890625,
-0.0130615234375,
0.0810546875,
-0.03271484375,
0.039306640625,
0.05908203125,
0.04052734375,
0.0262451171875,
0.0634765625,
0.0361328125,
0.08740234375,
-0.07080078125,
-0.10986328125,
-0.... | 6,914 | 6914 | |
Area of Inscribed Circle
Date: 12/01/98 at 13:24:52
From: Anna
Subject: Trigonometry
The question is: Find the area of the circle inscribed in a triangle
a, b, c. The hint is: Draw lines to the center of the circle and use
Heron's law.
I would appreciate it if you could help me!
Anna
Date: 12/01/98 at 15:41:37
F... | 4 | [
-0.052978515625,
0.123046875,
-0.0245361328125,
-0.0213623046875,
-0.02294921875,
-0.0020751953125,
0.0390625,
0.09326171875,
0.01287841796875,
-0.008056640625,
-0.00787353515625,
-0.12890625,
-0.01361083984375,
0.05712890625,
0.050537109375,
0.06591796875,
-0.057861328125,
0.01086... | 6,915 | 6915 | |
Measuring the Area of a Region of Sky
Date: 05/30/2003 at 11:31:28
From: Randy
Subject: Area of a Circle in Degrees
Dr. Math,
I enjoy reading your answers, and feel you are offering a great
service. Keep up the good work.
I'm learning astronomy and have been trying understand the geometry of
the sky. My question is... | 5 | [
0.1103515625,
-0.05859375,
-0.034912109375,
0.045166015625,
0.042236328125,
-0.0859375,
-0.031494140625,
0.035888671875,
0.01031494140625,
-0.00640869140625,
-0.06005859375,
-0.01519775390625,
0.039794921875,
-0.0322265625,
0.049560546875,
-0.032470703125,
0.0146484375,
-0.01770019... | 6,916 | 6916 | |
PROOF of the Directional Derivative
What's the PROOF of the Directional Derivative of a 3-variable function? DuF(x,y,z)=(Fx,Fy,Fz).(U1,U2,U3) where U=(U1,U2,U3) is a unit vector.
Assuming that F is differentiable at the given point, $(x_0, y_0. z_0)$ then it can be approximated by the linear function $F_x(x_0,y_0,z_)(... | 4 | [
-0.0159912109375,
-0.0419921875,
-0.06201171875,
-0.0517578125,
0.006103515625,
0.00250244140625,
0.03173828125,
-0.019287109375,
-0.06494140625,
0.045654296875,
0.000698089599609375,
0.0042724609375,
0.0556640625,
0.0159912109375,
-0.010986328125,
-0.0140380859375,
-0.11474609375,
... | 6,917 | 6917 | |
Analytical Mechanics
240 ANALYTICAL MECHANICS
external and those which are internal to the system. Let F/ denote the resultant of the external forces acting on the particle, and F/' denote the resultant of the internal forces acting upon it, due to its connection with the rest of the system. Then
... | 4 | [
-0.064453125,
-0.007476806640625,
0.0072021484375,
0.052734375,
0.02880859375,
-0.02001953125,
-0.06982421875,
0.11474609375,
0.0693359375,
0.07373046875,
0.0537109375,
-0.035888671875,
0.0194091796875,
0.017333984375,
-0.0634765625,
-0.05908203125,
-0.0478515625,
-0.00038909912109... | 6,918 | 6918 | |
power of a block triangular matrix
up vote 1 down vote favorite
1
I have a matrice in the form :
$$M = \begin{pmatrix} A & 0 & 0 \\\ B & A & 0 \\\ C & D & A \end{pmatrix} $$
where $A,B,C,D$ are diagonalizable square matrice and I want to determine
$$M^{\infty}:=\lim_{n\rightarrow \infty} M^n$$
in function of A,B,C... | 4 | [
-0.049072265625,
0.043212890625,
-0.1396484375,
0.04248046875,
-0.02490234375,
-0.01007080078125,
-0.0654296875,
-0.04150390625,
0.005218505859375,
-0.0031890869140625,
0.072265625,
0.0230712890625,
-0.02783203125,
-0.01239013671875,
0.0206298828125,
0.062255859375,
0.01348876953125,... | 6,919 | 6919 | |
Formulae of standard set theory interpreted in suitable segments of L
up vote 5 down vote favorite
In a letter dated 16 October Georg Kreisel asked me the question that follows. He agreed to my idea of posting the question here.
I’d be grateful if you could recommend an exposition of formulae of standard set theory i... | 5 | [
0.021728515625,
-0.083984375,
0.005584716796875,
-0.0654296875,
0.08544921875,
0.1650390625,
-0.056396484375,
0.05078125,
0.0130615234375,
-0.041259765625,
-0.017333984375,
0.024169921875,
-0.03076171875,
-0.005706787109375,
0.02001953125,
-0.00127410888671875,
0.00191497802734375,
... | 6,920 | 6920 | |
Questions Answered Correctly
Date: 06/11/2003 at 04:17:28
From: Rachelle
Subject: Contest
There are 30 multiple-choice questions in a contest. Five marks are
awarded for each correct answer, and three marks are deducted for each
incorrect answer. No marks are awarded for questions left unanswered.
If a student score... | 4 | [
0.0238037109375,
0.045166015625,
-0.04248046875,
-0.025634765625,
-0.05615234375,
0.10400390625,
0.0128173828125,
0.0419921875,
-0.08935546875,
0.07568359375,
-0.00994873046875,
-0.0257568359375,
0.08203125,
0.078125,
-0.01123046875,
0.0299072265625,
-0.0264892578125,
-0.0143432617... | 6,921 | 6921 | |
calculus linear approximations
October 24th 2006, 11:04 AM #1
Junior Member
Joined
Oct 2006
Posts
71
calculus linear approximations
(a)Find the linear approximation L1(x) of the function f(x)=sqrt(x+3) at a=1
(b)use linear approximate that you founf in part(a) to approximate sqrt(3.98)
(c)scetch ... | 5 | [
-0.0036468505859375,
-0.0108642578125,
0.031494140625,
-0.0859375,
-0.0184326171875,
-0.0191650390625,
-0.01531982421875,
0.0294189453125,
0.0098876953125,
0.0791015625,
0.058349609375,
-0.02587890625,
0.057373046875,
0.022705078125,
0.004119873046875,
-0.0247802734375,
-0.0251464843... | 6,922 | 6922 | |
Ohm's Law and Battery Life
Date: 08/18/99 at 17:04:12
From: Scott Stongle
Subject: Ohm's law
If you have a 12 V battery that has 275 cold-cranking amps and you put
a load of 12 V at 7.2 A on it, how long should that battery run the
load? Also, I was wondering if I need to figure the resistance of the
system before I... | 4 | [
-0.0027618408203125,
0.126953125,
-0.0174560546875,
-0.0181884765625,
-0.07470703125,
-0.005615234375,
-0.0294189453125,
0.06298828125,
-0.033203125,
0.11083984375,
-0.024169921875,
0.0262451171875,
0.052978515625,
-0.025146484375,
0.0194091796875,
0.06884765625,
-0.03857421875,
-0... | 6,923 | 6923 | |
Using Parameters
July 9th 2009, 09:39 PM
Stroodle
Using Parameters
A question from my text is:
The simultaneous equations $x+2y+3z=13,\;-x-3y+2z=2$ and $-x-4y+7z=17$ have infinitely many solutions. Describe these solutions through the use of a parameter.
I understand how to do this. I just don't unders... | 4 | [
-0.09619140625,
0.0247802734375,
-0.09619140625,
0.002655029296875,
-0.002410888671875,
0.051025390625,
0.057373046875,
-0.0238037109375,
-0.05078125,
-0.0419921875,
0.0703125,
-0.0147705078125,
0.05419921875,
-0.00023746490478515625,
-0.0196533203125,
-0.008056640625,
-0.05517578125... | 6,924 | 6924 | |
Optimization
February 3rd 2008, 10:17 PM
Truthbetold
Optimization
You operate a tour service that offers the following rates:
$200 per person if 50 people ( the minimumn number to book the tour) go on the tour.
For each addiational person up to a maximum of 80 people total, the rate per person is reduc... | 5 | [
-0.0003986358642578125,
0.07275390625,
-0.0244140625,
-0.021484375,
-0.08740234375,
-0.009033203125,
0.051025390625,
0.1025390625,
-0.0419921875,
0.0703125,
0.013916015625,
-0.003814697265625,
0.09033203125,
0.0615234375,
-0.00677490234375,
0.040283203125,
0.026611328125,
-0.003387... | 6,925 | 6925 | |
s
Kickstarting R - ANOVAs
ANOVAs
For more complex hypotheses, we'll have to go to ANOVAs. As a pretty artifical example, we'll combine the test performed in the t-test example with another test in a simple ANOVA using aov(), askin... | 4 | [
0.0576171875,
0.05224609375,
-0.015380859375,
0.0213623046875,
0.03955078125,
0.0177001953125,
-0.09912109375,
0.07080078125,
0.04296875,
0.09619140625,
0.0859375,
0.0269775390625,
-0.055419921875,
0.05712890625,
-0.033935546875,
-0.040771484375,
0.003265380859375,
-0.0030517578125... | 6,926 | 6926 | |
Tensor Calculus
Date: 10/10/2002 at 07:55:07
From: Lachieze-Rey
Subject: Tensor calculus
Could you please remind me of the general definition of a determinant
for a tensor with more than 2 indices, like T_m1_m2..._mk ?
Thank you.
Date: 11/30/2002 at 01:48:17
From: Doctor Nitrogen
Subject: Re: Tensor calculus
Hi, ... | 4 | [
0.01202392578125,
-0.09814453125,
-0.001068115234375,
-0.047119140625,
0.01324462890625,
0.103515625,
-0.0062255859375,
0.029541015625,
-0.058349609375,
-0.025390625,
-0.013916015625,
-0.009033203125,
-0.040283203125,
0.0888671875,
-0.0234375,
0.0184326171875,
-0.033203125,
0.04882... | 6,927 | 6927 | |
Length of the Curve
April 17th 2009, 01:59 PM #1
Newbie
Joined
Oct 2008
Posts
18
Length of the Curve
Find the Length of The Curve
y = ln(1 - x2)
0 < x < (1/7)
I got the derivative: (-2x)/(1-x^2)
Then i put it in the formula : sqr*(1+((-2x)/(1-x^2))^2)
My problem seems to be simpli... | 5 | [
0.0546875,
-0.038818359375,
0.08984375,
0.0213623046875,
0.0341796875,
-0.11572265625,
-0.12158203125,
0.06787109375,
-0.04150390625,
0.0177001953125,
0.06787109375,
0.007049560546875,
-0.005828857421875,
0.05419921875,
-0.08447265625,
0.031005859375,
-0.0225830078125,
-0.120605468... | 6,928 | 6928 | |
Transfer
HeatTransfer
From OpenFOAMWiki
1 heatTransfer
This page presents a small study performed on a particular conjugate heat transfer problem. There have been considered two different approaches: the complete solution, where a solid and a fluid
domain are taken into account; the approximate solution, where only ... | 4 | [
-0.087890625,
0.03564453125,
-0.03564453125,
0.0025787353515625,
0.01611328125,
-0.0986328125,
-0.0125732421875,
-0.0089111328125,
-0.031982421875,
-0.0361328125,
-0.0003681182861328125,
-0.0634765625,
0.056396484375,
-0.046142578125,
0.00060272216796875,
-0.05029296875,
-0.080078125... | 6,929 | 6929 | |
Questions about Logs and similar type word problems
March 10th 2010, 04:58 PM #1
Newbie
Joined
Mar 2010
Posts
2
Questions about Logs and similar type word problems
Hey guys I have a couple of questions about the topic stated above that I simply do not understand.
If you could help, it would be greatl... | 4 | [
-0.015625,
0.0146484375,
0.0174560546875,
0.00177001953125,
0.0096435546875,
-0.06787109375,
0.03955078125,
0.007568359375,
0.0260009765625,
0.03662109375,
0.03857421875,
-0.039794921875,
0.032958984375,
0.00946044921875,
0.032958984375,
0.048828125,
-0.08203125,
-0.04638671875,
... | 6,930 | 6930 | |
Basic Counting Principles
September 23rd 2010, 07:10 AM
DBA
Basic Counting Principles
Given nine different English books, seven different French books, and five different German books.
How many ways are there to make a row of three books in which exactly one language is missing (the order of the three books... | 4 | [
-0.0191650390625,
-0.0654296875,
0.0072021484375,
-0.031982421875,
-0.0306396484375,
0.0830078125,
0.0255126953125,
0.0029296875,
0.0625,
0.05810546875,
-0.0147705078125,
-0.035888671875,
0.021728515625,
-0.0244140625,
-0.03271484375,
-0.036376953125,
-0.1376953125,
0.0286865234375... | 6,931 | 6931 | |
Prove a direct sum...
April 12th 2010, 05:52 PM #1
Newbie
Joined
Feb 2010
Posts
11
Prove a direct sum...
Here's the question as an image because it's easier to format:
Please note I'll just write + to mean the plus with the circle around it (direct sum). + is just a normal addition.
My work so... | 4 | [
-0.061767578125,
0.06103515625,
0.007354736328125,
0.08740234375,
0.000804901123046875,
-0.00665283203125,
0.09814453125,
-0.0125732421875,
0.03466796875,
-0.00592041015625,
0.0029296875,
-0.058837890625,
0.001373291015625,
0.03759765625,
-0.09130859375,
-0.051513671875,
-0.044921875... | 6,932 | 6932 | |
What do I need to do with this equation?
20x -10 = 6+12x Maths was never my best subject! Thanks!
20x -10 = 6+12x You have to solve for x, so you need to isolate x, to do that, you can subtract 12x from both sides: $20x-10-12x =6+12x-12x$ and you will get -> $8x-10=6$, now add 10 to both sides
and you get rid of that ... | 4 | [
-0.037841796875,
0.08203125,
0.08935546875,
0.006103515625,
-0.00579833984375,
0.0267333984375,
0.01129150390625,
0.0322265625,
-0.054931640625,
-0.0235595703125,
0.0361328125,
0.0107421875,
0.0859375,
0.022216796875,
-0.027099609375,
0.061279296875,
-0.06640625,
0.0859375,
-0.07... | 6,933 | 6933 | |
help solve 1
May 19th 2010, 10:17 PM #1
Junior Member
Joined
May 2010
Posts
39
help solve 1
The polynomial p(x)is given by p(x)=ax3 +16x2+cx – 120, where a and c are constants. The three zeros of p(x)are –2, 3 and α. Find the value of α.
with working out please
$P(x) = ax^3 + 16x^2 + cx - 120$.... | 4 | [
-0.06591796875,
0.056884765625,
-0.052001953125,
-0.037841796875,
0.029296875,
0.017822265625,
0.0458984375,
-0.0074462890625,
-0.0673828125,
0.03369140625,
0.0966796875,
-0.07177734375,
-0.00457763671875,
0.01422119140625,
0.0018157958984375,
0.0267333984375,
-0.048583984375,
-0.0... | 6,934 | 6934 | |
spanning set for R^2
September 8th 2008, 09:39 PM
apple12
spanning set for R^2
First, can anyone explain to me the meaning of spanning set? I am a little confused with it meaning. :(
Then please show me how to do the below questions. Thank you very much.
Determine whether any of the following sets are ... | 5 | [
0.0673828125,
-0.0196533203125,
-0.12451171875,
-0.0079345703125,
0.040283203125,
0.0224609375,
-0.001922607421875,
-0.01153564453125,
0.0123291015625,
-0.0546875,
0.0089111328125,
0.03369140625,
0.0498046875,
-0.042236328125,
-0.07373046875,
-0.06640625,
-0.0252685546875,
-0.08349... | 6,935 | 6935 | |
Intergration....plz help
March 28th 2008, 07:10 AM
sweetchocolat113
Intergration....plz help
Intergrate....
1. x^5-2x+5 diveded by x^3 dx
2. √x (1+√x)^3 dx
3. (1-3x)^9 dx
4. sin2xcosxdx
5. x + 2x^3 diveded by (x^4 + x^2)^3 dx
March 28th 2008, 07:38 AM
ThePerfectHacker
$(x^5-2x+5)/x^3 ... | 5 | [
-0.07421875,
-0.046142578125,
0.0771484375,
0.048828125,
0.05712890625,
-0.06689453125,
0.0174560546875,
-0.017822265625,
-0.01220703125,
0.01031494140625,
0.076171875,
-0.01519775390625,
0.02490234375,
0.0069580078125,
-0.05810546875,
0.03955078125,
0.01519775390625,
0.00473022460... | 6,936 | 6936 | |
inequality
Find the solutions set of the inequality $|x-2|<\frac{1}{x}$ , where x is not 0 . THanks .
it's quite obvious that $xe0,$ that's an assumption that it must be implicit and that you should know. well then, by a well known property your inequality equals $-\frac{1}{x}<x-2<\frac{1}{x},$ so
you have to solve ea... | 4 | [
-0.034423828125,
-0.049072265625,
0.044677734375,
0.059326171875,
0.00445556640625,
-0.0108642578125,
-0.034912109375,
0.0986328125,
0.013916015625,
-0.002532958984375,
0.014404296875,
-0.107421875,
0.07958984375,
0.01153564453125,
-0.0152587890625,
-0.0101318359375,
0.043701171875,
... | 6,937 | 6937 | |
proof of a property of the Chi-Squared distribution
June 4th 2011, 08:07 AM #1
Junior Member
Joined
Mar 2009
Posts
49
proof of a property of the Chi-Squared distribution
Hi
I was wondering if someone could please provide me with the proof of the following property of a Chi-Squared distribution.
... | 4 | [
-0.0390625,
-0.0296630859375,
0.1044921875,
-0.0086669921875,
0.015869140625,
-0.026123046875,
-0.00124359130859375,
0.037109375,
-0.007476806640625,
0.08349609375,
0.047119140625,
0.03955078125,
0.0654296875,
-0.091796875,
-0.00634765625,
0.03955078125,
0.06103515625,
-0.034667968... | 6,938 | 6938 | |
Is this equation Correct and can it be simplified/ solved for Vmax
March 22nd 2013, 10:08 AM
calltronics
Is this equation Correct and can it be simplified/ solved for Vmax
Good day to you all,
I have the following fixed values: -
Acceleration = 1000 mm/s^2 - (A[a])
Deceleration = -400 mm/s^2 - (A[d... | 4 | [
0.00823974609375,
0.044921875,
0.01531982421875,
-0.09228515625,
-0.023681640625,
0.022216796875,
-0.083984375,
0.06298828125,
-0.068359375,
0.039794921875,
0.061279296875,
0.0242919921875,
0.0216064453125,
0.040771484375,
-0.032470703125,
0.0283203125,
0.11083984375,
0.05200195312... | 6,939 | 6939 | |
factoring
Why factor? The quadratic formula will answer that for you. You don't even need to use the entire formula!
Yeah, the quadratic formula will work and is simpler to explain. As far as factoring goes, here is what to do. 10x^2+9x-9=0 First you want to find what two numbers who's sum is 9 and when multiplied
equ... | 4 | [
-0.0294189453125,
0.033935546875,
0.06640625,
0.0029449462890625,
0.045654296875,
0.119140625,
0.00445556640625,
-0.0086669921875,
-0.0576171875,
-0.046875,
0.11181640625,
-0.044921875,
0.07861328125,
0.0576171875,
-0.0517578125,
-0.00457763671875,
0.00823974609375,
0.044189453125,... | 6,940 | 6940 | |
Having difficulty with linear equations
June 3rd 2012, 04:28 PM
astuart
Having difficulty with linear equations
Hello,
I've been out of school for over 6 years so these questions are probably relatively simple, but I'm just not grasping the concept here. I'm doing a bridging course for a university degree, ... | 4 | [
-0.05419921875,
0.0184326171875,
0.006805419921875,
-0.0263671875,
-0.052978515625,
0.07177734375,
-0.02978515625,
0.00128173828125,
-0.0010833740234375,
0.08837890625,
0.1064453125,
-0.051513671875,
0.053466796875,
-0.00153350830078125,
-0.0186767578125,
0.043701171875,
-0.126953125... | 6,941 | 6941 | |
Finding max value of a 3-variable function
October 28th 2009, 12:07 AM #1
Newbie
Joined
Sep 2009
Posts
20
Finding max value of a 3-variable function
Find the maximum value of f(x,y,z) = x^2 + xy + y^2 + yz + z^2 on the unit sphere.
I don't know whether I should parametrize the unit sphere and use a ... | 4 | [
0.0196533203125,
-0.045166015625,
-0.03466796875,
-0.060302734375,
-0.037841796875,
-0.02880859375,
-0.0169677734375,
-0.00799560546875,
-0.08984375,
0.03857421875,
-0.0223388671875,
0.06689453125,
0.01507568359375,
0.0380859375,
-0.01031494140625,
0.042724609375,
-0.044921875,
-0.... | 6,942 | 6942 | |
Help with integral using ln
June 1st 2009, 07:36 PM #1
Member
Joined
May 2009
Posts
211
Help with integral using ln
Well, i was assigned to do this problem but i haven't yet covered the ln formula and my professor still wants me to do it.
Can someone please show me how to solve this particular integr... | 5 | [
0.095703125,
-0.056396484375,
0.072265625,
0.0240478515625,
-0.04638671875,
-0.056884765625,
-0.03466796875,
-0.025390625,
0.01019287109375,
-0.042724609375,
0.01220703125,
0.03759765625,
-0.017578125,
0.05126953125,
-0.0654296875,
0.0419921875,
-0.0341796875,
0.0048828125,
-0.07... | 6,943 | 6943 | |
Matrix Formula
January 18th 2007, 01:54 PM #1
Newbie
Joined
Jan 2007
Posts
3
Matrix Formula
I have this:
A=any square matrix over any field F
p(X)=any polynomial over F
P=any invertible matrix over F
B=inverse(P)AP=P^(-1)AP
I need to show that p(B)=inverse(P)p(A)P=P^(-1)p(A)P.
Ho... | 5 | [
-0.0174560546875,
0.0224609375,
-0.0174560546875,
0.0654296875,
-0.036376953125,
-0.021240234375,
0.040771484375,
-0.04248046875,
-0.006317138671875,
0.01361083984375,
0.00921630859375,
0.0012664794921875,
0.006744384765625,
-0.0240478515625,
-0.031494140625,
0.1533203125,
-0.0163574... | 6,944 | 6944 | |
Coefficient in the expansion
What is the coefficient of x^6 in the expansion (1+x+x^2)^9. Explain please
Count the number of ways to rearrange each of these string. $111xxxxxx,~1111xxxxx^2,~ 11111xxx^2x^2,~ \&~111111x^2x^2x^2$ For example: the string $1111xxxxx^2$ can be rearranged in $\frac{9!}{(1!)
(4!)(4!)}$ ways. ... | 4 | [
0.04638671875,
-0.022705078125,
0.019775390625,
-0.00689697265625,
-0.0272216796875,
0.11181640625,
0.020751953125,
0.023681640625,
-0.0240478515625,
-0.0294189453125,
0.0257568359375,
0.0016021728515625,
-0.005096435546875,
-0.076171875,
-0.0703125,
0.06591796875,
-0.01953125,
0.0... | 6,945 | 6945 | |
Proof by induction inequality
October 4th 2012, 05:06 PM #1
Junior Member
Joined
Jul 2011
Posts
28
Proof by induction inequality
Prove $\sum_{i=1}^n \frac{1}{i^2} \le 2 - \frac{1}{n}$ for all $n \epsilon \mathbb{N}$
Proof: i) Show P(1) holds.
LHS = $\sum_{i=1}^1 \frac{1}{i^2} = \frac{1}{1} = 1$... | 5 | [
-0.051513671875,
0.0498046875,
0.1123046875,
0.0302734375,
0.00104522705078125,
-0.0257568359375,
0.0517578125,
0.010498046875,
0.01458740234375,
-0.00150299072265625,
-0.1484375,
-0.0234375,
0.07568359375,
0.042236328125,
-0.05224609375,
-0.046142578125,
0.158203125,
0.00306701660... | 6,946 | 6946 | |
Cockroft Walton Voltage Multipliers
The classic multistage diode/capacitor voltage multipler, popularized by Cockroft and Walton, is probably the most popular means of generating high voltages at low currents at low cost. It is used in
virtually every television set made to generate the 20-30 kV second anode accelerat... | 4 | [
-0.0810546875,
0.0308837890625,
-0.01129150390625,
-0.060791015625,
-0.04541015625,
0.009521484375,
-0.078125,
0.054931640625,
0.04052734375,
0.003448486328125,
0.0181884765625,
0.005859375,
-0.0101318359375,
0.051513671875,
-0.06982421875,
-0.001434326171875,
-0.0272216796875,
0.0... | 6,947 | 6947 | |
Maximizing the Volume of a Cylinder
Associated Topics || Dr. Math Home || Search Dr. Math
Maximizing the Volume of a Cylinder
... | 4 | [
0.03857421875,
0.083984375,
0.0027313232421875,
-0.0673828125,
-0.0294189453125,
-0.003326416015625,
-0.0164794921875,
0.1669921875,
-0.06884765625,
0.00445556640625,
-0.0703125,
0.07177734375,
-0.005950927734375,
0.1103515625,
-0.035400390625,
0.0169677734375,
0.005157470703125,
0... | 6,948 | 6948 | |
A First Course in the Finite Element Method Using Algor
More About This Textbook
Overview
Daryl Logan's clear and easy to understand text provides a thorough treatment of the finite element method and how to apply it to solve practical physical problems in engineering. Concepts are
presented simply, making it underst... | 5 | [
-0.07373046875,
0.142578125,
0.049560546875,
0.0157470703125,
-0.06884765625,
-0.0888671875,
0.0220947265625,
-0.0186767578125,
-0.0986328125,
0.01025390625,
0.031982421875,
-0.04736328125,
0.034912109375,
-0.059814453125,
-0.064453125,
-0.0150146484375,
-0.05517578125,
-0.04223632... | 6,949 | 6949 | |
Circle tangent to the line.
January 14th 2010, 02:52 AM #1
Newbie
Joined
Nov 2009
Posts
17
Circle tangent to the line.
Find the eq. of the circle:
"The circle is tangent to the line x-y=2 at the point (4,2) and the center is on the x-axis".
Can someone help me with this one, I really don't know ... | 4 | [
0.00008916854858398438,
0.021484375,
-0.0712890625,
0.02392578125,
0.07177734375,
0.02685546875,
0.0150146484375,
0.166015625,
0.04931640625,
-0.0264892578125,
0.027587890625,
-0.0174560546875,
-0.0242919921875,
-0.00118255615234375,
-0.0052490234375,
0.11669921875,
-0.07080078125,
... | 6,950 | 6950 | |
Cochran's Theorem
Next: Order Statistics Up: Multivariate Normal Distribution Previous: The role of c.g.f.   Contents
This is a very important theorem which allows us to decompose sums of squares into several quadratic forms and identify their distributions and establish their independence. It can be used to... | 5 | [
-0.0732421875,
0.0005340576171875,
0.056640625,
-0.055908203125,
0.00311279296875,
0.123046875,
-0.01123046875,
-0.0223388671875,
0.0223388671875,
0.087890625,
0.0751953125,
0.1162109375,
0.0306396484375,
0.001983642578125,
-0.039794921875,
0.024658203125,
-0.006561279296875,
0.028... | 6,951 | 6951 | |
Converting a polar equation to a rectangular equation
December 4th 2012, 03:55 AM
beastcoast515
Converting a polar equation to a rectangular equation
I was given the polar equation: r + 2cos(theta) = 3sin(theta)
Which I converted (I believe correctly) to x^2+y^2=-2x+3y yet my professor circled my answer and... | 4 | [
-0.09326171875,
0.01080322265625,
-0.12353515625,
0.03759765625,
-0.0128173828125,
0.052978515625,
-0.07861328125,
-0.0223388671875,
0.0185546875,
0.01373291015625,
0.078125,
0.0167236328125,
0.0228271484375,
0.08740234375,
0.0888671875,
0.01806640625,
-0.0810546875,
-0.00372314453... | 6,952 | 6952 | |
How to find out of planes are perpendicular
April 22nd 2009, 11:51 AM
supersaiyan
How to find out if planes are perpendicular
Determine if planes in each pair are parallel, perpendicular, or neither.
$\pi_{1}: 4x-5y+z-9=0$
$\pi_{2}: 2x-9y+z-2=0$
This is what I did
n1 :[4, -5, 1]
n2:[2, -9, 1]... | 4 | [
-0.07421875,
-0.0308837890625,
-0.035400390625,
-0.06640625,
0.0234375,
0.06591796875,
-0.0322265625,
0.0191650390625,
0.009521484375,
0.00004673004150390625,
0.0234375,
0.0272216796875,
0.00012302398681640625,
0.01220703125,
-0.07080078125,
0.00115203857421875,
-0.03173828125,
-0.... | 6,953 | 6953 | |
Graph of a function
February 22nd 2009, 12:04 AM #1
Junior Member
Joined
Feb 2009
Posts
34
List all the asymptotes of the graph of the function and the approximate coordinates of each local extremum.
f(x)=x(2^x)
I've had a hard time trying to solve this graph , I used my graphing calculator, but... | 5 | [
0.041748046875,
-0.0478515625,
0.00191497802734375,
0.01361083984375,
0.08251953125,
-0.103515625,
0.0009918212890625,
0.142578125,
-0.003662109375,
0.025390625,
0.07666015625,
-0.0625,
0.0673828125,
0.044189453125,
0.01287841796875,
0.0146484375,
-0.035400390625,
-0.158203125,
0... | 6,954 | 6954 | |
Homework Help
Posted by Anonymous on Monday, September 10, 2012 at 2:11pm.
I;m not sure if i am doing this right...I keep getting a negative intersection point, which doesn't seem possible.
Please help!
a) find the equatoin of line 1 which passes through (1,3) and (9,7)
I get y=3+1/2(x+1)or y=1/2x+2.5
b) the line ... | 4 | [
-0.1005859375,
0.018798828125,
0.01519775390625,
-0.0791015625,
-0.01470947265625,
0.01141357421875,
0.0218505859375,
0.0859375,
-0.0250244140625,
-0.0225830078125,
0.0849609375,
-0.0908203125,
-0.01068115234375,
0.046875,
-0.1279296875,
0.013427734375,
-0.0380859375,
-0.0922851562... | 6,955 | 6955 | |
Multiplying imaginary factors of a polynomial
August 20th 2010, 06:33 AM #1
Senior Member
Joined
Apr 2010
Posts
487
[Solved] Multiplying imaginary factors of a polynomial
The problem:
$(x - e^{\frac{i\pi}{4}})(x - e^{\frac{-i\pi}{4}})$
My attempt:
I used the distributive rule to expand it, a... | 4 | [
-0.01385498046875,
0.048583984375,
0.037841796875,
0.05029296875,
0.004638671875,
-0.017578125,
-0.028076171875,
0.033203125,
0.052001953125,
0.03173828125,
0.1064453125,
-0.0859375,
0.053466796875,
0.08642578125,
0.0400390625,
0.0034942626953125,
0.01611328125,
-0.0281982421875,
... | 6,956 | 6956 | |
[racket-dev] floor-log/base, ceiling-log/base, from Neil Toronto's recent commit
From: Neil Toronto (neil.toronto at gmail.com)
Date: Wed Feb 1 23:10:16 EST 2012
On 02/01/2012 05:58 PM, John Boyle wrote:
> I happened to observe this commit from today by Neil Toronto:
>
> http://git.racket-lang.org/plt/commitdiff/47fc... | 5 | [
-0.012939453125,
0.048583984375,
-0.05224609375,
-0.04736328125,
0.016845703125,
-0.06884765625,
0.0076904296875,
0.0537109375,
-0.0009765625,
0.04150390625,
-0.09912109375,
0.0184326171875,
0.040771484375,
0.020751953125,
0.035888671875,
0.072265625,
-0.01043701171875,
0.060302734... | 6,957 | 6957 | |
Gaussian Elimination. Question is attached.
August 11th 2008, 02:14 AM #1
Newbie
Joined
Aug 2008
Posts
18
Gaussian Elimination. Question is attached.
When I work out from the piovt rows 2, 3 & 4 work out the same and I not sure about x,y,z &w being non negative, some of my values are negative.
The r... | 4 | [
-0.05859375,
0.0068359375,
-0.078125,
0.045654296875,
0.032958984375,
-0.025390625,
0.0262451171875,
-0.027587890625,
-0.07763671875,
0.0296630859375,
0.04150390625,
0.0286865234375,
0.068359375,
0.06640625,
-0.06396484375,
0.0751953125,
-0.080078125,
-0.072265625,
-0.02697753906... | 6,958 | 6958 | |
Differentiate an integral
Anybody know how to differentiate: Int[0,x](f(k)(x-k))dk with respect to x?
No worries: Differentiation under the integral sign - Wikipedia, the free encyclopedia I may be the first person to get a degree using nothing but wikipedia...
Hello, Actually, this is not so difficult oO If F is a p... | 5 | [
0.03857421875,
0.0322265625,
0.0283203125,
-0.013427734375,
-0.02978515625,
-0.007415771484375,
0.0291748046875,
-0.03759765625,
0.0107421875,
0.10546875,
0.0556640625,
-0.01092529296875,
0.029296875,
-0.021240234375,
-0.0234375,
0.0145263671875,
-0.0228271484375,
0.05126953125,
... | 6,959 | 6959 | |
Differentiable submanifold
April 29th 2010, 12:35 PM
AlexanderW
Differentiable submanifold
Hello.
Let $M \subset \mathbb R^n$ be a differentiable submanifold (with the induced metrics from $\mathbb R^n$), $q \in \mathbb R^n\setminus M$,
and $g:M\to \mathbb R, \quad x \mapsto \left \| q-x \right \|$ has... | 5 | [
-0.0030364990234375,
0.045654296875,
0.04052734375,
-0.10107421875,
-0.04150390625,
0.08642578125,
0.09716796875,
0.0947265625,
-0.0023193359375,
0.0250244140625,
0.051513671875,
0.032958984375,
0.025634765625,
-0.0157470703125,
-0.044921875,
0.0927734375,
0.057861328125,
0.0122070... | 6,960 | 6960 | |
logarithmic problem still struggling
February 1st 2007, 11:01 AM
kcsteven
logarithmic problem still struggling
I have worked this out different ways and I am not comming up with the correct answer, although I have gotten many answers. Here it is:
solve for x
9/7log3x=8
(The 3 is the base but I don't ... | 4 | [
0.0299072265625,
0.0299072265625,
-0.0283203125,
0.019287109375,
-0.022216796875,
-0.044921875,
0.06787109375,
0.0125732421875,
0.0390625,
0.08203125,
0.0179443359375,
-0.06884765625,
0.02099609375,
0.06787109375,
0.05419921875,
0.019775390625,
-0.15625,
0.06298828125,
-0.1176757... | 6,961 | 6961 | |
Real Analysis - Uniform Convergence
May 11th 2013, 03:56 PM
blmalikov
Real Analysis - Uniform Convergence
Let $f_n(x):=\frac{1}{n\sqrt{2\pi}}\exp{{\frac{-x^2}{2n^2}}}$
and $f(x):=0$
$f_n(x) \rightarrow f(x)$ pointwise.
$\forall n\in\omega\int_{-\infty}^{\infty}f_n(x)dx = 1$
and $\int_{-\infty}... | 4 | [
-0.07177734375,
-0.01446533203125,
0.031494140625,
0.035888671875,
0.05517578125,
0.0120849609375,
0.048828125,
0.09814453125,
-0.047119140625,
-0.04296875,
0.06884765625,
0.05224609375,
0.015625,
-0.0673828125,
-0.0031890869140625,
-0.083984375,
0.032470703125,
-0.1083984375,
0.... | 6,962 | 6962 | |
Inverse Functions of polynomials
September 15th 2008, 12:52 PM #1
Newbie
Joined
Sep 2008
Posts
4
Inverse Functions of polynomials
Hey I was hoping you guys could help me out.
I have to find the inverse of
f(x)= 3 + x^2 + tan((y x pi)/2))
I know how to do inverses, but I cant figure out ho... | 5 | [
-0.05615234375,
-0.002044677734375,
0.07421875,
0.051025390625,
0.0478515625,
-0.058837890625,
0.03759765625,
-0.01507568359375,
-0.0211181640625,
0.043701171875,
0.08251953125,
-0.10986328125,
-0.0308837890625,
-0.004119873046875,
0.0174560546875,
0.0208740234375,
-0.0556640625,
0... | 6,963 | 6963 | |
Solving Third-Degree Equations
Date: 09/24/2001 at 11:03:28
From: De Kort
Subject: Re: Solve third-degree equations
I'm trying to solve x^3 + 2x^2 - 3x - 4 = 0
p = -13/3
q = -38/27
To calculate t I need to take the square root of 4p^3 + 27q^2, which
is -272... Of course I can consider t and u to be complex numbers... | 5 | [
-0.07470703125,
0.0164794921875,
-0.048828125,
0.032470703125,
-0.039794921875,
0.0179443359375,
0.0234375,
-0.032470703125,
0.0277099609375,
-0.005859375,
0.07080078125,
-0.036865234375,
-0.005096435546875,
0.055419921875,
0.040771484375,
-0.042724609375,
-0.1123046875,
-0.0011749... | 6,964 | 6964 | |
s of
Examples of Analysis of Variance and Covariance
C. P. Doncaster and A. J. H. Davey
This page presents example datasets and outputs for analysis of varianc... | 5 | [
0.021240234375,
-0.01953125,
-0.01123046875,
0.05419921875,
-0.048583984375,
0.0283203125,
-0.1005859375,
0.0017547607421875,
0.03857421875,
0.064453125,
0.01434326171875,
0.00579833984375,
0.007080078125,
0.01416015625,
-0.0341796875,
-0.0087890625,
0.048095703125,
-0.025512695312... | 6,965 | 6965 | |
Area between Curves
March 15th 2011, 01:30 PM #1
Member
Joined
Sep 2009
From
Ontario
Posts
162
Area between Curves
Hey I have a calc exam Thursday and have been practicing for it. I came across this area problem that has me stumped, usually area is no problem for me but I can't seem to figure this on... | 5 | [
0.04833984375,
0.001922607421875,
0.040283203125,
-0.004150390625,
-0.030517578125,
-0.10791015625,
-0.00433349609375,
0.0294189453125,
-0.01336669921875,
-0.0712890625,
-0.00775146484375,
-0.0086669921875,
0.0341796875,
0.03857421875,
-0.0054931640625,
0.046875,
-0.033935546875,
-... | 6,966 | 6966 | |
Algebraic Fractions question
October 3rd 2006, 10:12 AM
neo658
Algebraic Fractions question
I'v been stuck on this question and I dont know how to solve it. I have to use the remainder theorem to find the answer for (4x³ - 5x² + 3x - 14) / (x² + 2x - 1).
Since I can't expand (x² + 2x - 1) so I cannot find a... | 4 | [
-0.04052734375,
0.015380859375,
0.03662109375,
0.05615234375,
0.01318359375,
0.0040283203125,
0.00714111328125,
-0.0031890869140625,
0.0302734375,
-0.06787109375,
0.0751953125,
-0.051513671875,
0.02978515625,
0.037841796875,
-0.026611328125,
0.0189208984375,
-0.036376953125,
-0.005... | 6,967 | 6967 | |
Upper and Lower Bounds of the Total Surface Area of Convex Polytopes that Partition a Hypercube
up vote 0 down vote favorite
Let $C$ be a hypercube in $\mathbb{R}^D$ with edge length of $L$. Let $\mathcal{P}_1,\ldots,\mathcal{P}_K$ be $K$ convex polytopes that partition $C$. Let $S_k$ be the surface area of the polyt... | 5 | [
0.0693359375,
-0.03759765625,
-0.00677490234375,
-0.087890625,
0.022216796875,
0.0205078125,
0.038818359375,
0.06396484375,
-0.01470947265625,
0.0186767578125,
-0.1474609375,
0.0038604736328125,
0.0245361328125,
-0.0079345703125,
-0.06298828125,
-0.0113525390625,
0.0693359375,
0.00... | 6,968 | 6968 | |
What is the intersection of the following lines...
y - 3x + 2 = 0 and -3x + 4y -1 = 0 Thanks!
The easiest thing to do when to already have a variable isolated (in this case y) is to use the method of substitution $(1) y-3x+2=0$ $(2)-3x+4y-1=0$ Isolate y in the first equation $y=3x-2$ Plug
that value into the second eq... | 4 | [
-0.054931640625,
-0.006988525390625,
-0.03955078125,
0.017333984375,
0.000023365020751953125,
0.0279541015625,
0.0244140625,
0.0791015625,
-0.0419921875,
-0.09228515625,
0.01953125,
-0.0184326171875,
0.0771484375,
-0.057861328125,
-0.06298828125,
0.0712890625,
-0.1337890625,
-0.067... | 6,969 | 6969 | |
Distributive Property
Date: 10/13/2001 at 08:10:12
From: Susan Kinney
Subject: Distributive property
We are learning the distributive property in school. Why do we have
to learn this, and how will we use it, and when?
Susan
Date: 10/13/2001 at 22:47:25
From: Doctor Peterson
Subject: Re: Distributive property
Hi,... | 4 | [
-0.0257568359375,
0.07421875,
0.091796875,
0.01483154296875,
-0.06689453125,
0.0380859375,
0.011962890625,
-0.00994873046875,
0.00124359130859375,
0.064453125,
-0.0166015625,
-0.01458740234375,
0.0263671875,
-0.01287841796875,
0.033935546875,
0.04638671875,
-0.044189453125,
0.14550... | 6,970 | 6970 | |
integrate: x/(sqrrt(1-9x^2)
${\color{blue}u = 1 -9x^{2}}$ $du = -18x dx \Rightarrow {\color{red}x dx = -\frac{du}{18}}$ Looking at your integral: $\int \frac{{\color{red}xdx}}{\sqrt{{\color{blue}1-9x^{2}}}}$
To integrate $\int \frac{x}{\sqrt{1-9x^2}}\,dx$, you would need to make a u-substitution: $u=1-9x^2$. As a resu... | 5 | [
0.0198974609375,
-0.0023040771484375,
0.08154296875,
-0.0167236328125,
-0.0272216796875,
-0.011962890625,
-0.06689453125,
0.038818359375,
0.00101470947265625,
-0.05810546875,
0.027587890625,
-0.0186767578125,
-0.00156402587890625,
0.047119140625,
-0.0966796875,
-0.036865234375,
0.016... | 6,971 | 6971 | |
Math Help
September 4th 2012, 06:01 PM #1
Newbie
Joined
Jan 2011
Posts
11
Limits.
"Find the limit or explain why it does not exist"
I'm having trouble with these problems, I can't factor/these into a different form.
Having trouble with the algebra.
I've just started calculus and we hav... | 5 | [
-0.0771484375,
-0.0242919921875,
0.0081787109375,
0.052490234375,
0.044921875,
0.03466796875,
-0.01385498046875,
-0.076171875,
0.0032806396484375,
-0.0023956298828125,
0.06640625,
-0.10498046875,
-0.0303955078125,
0.0849609375,
-0.003753662109375,
-0.025634765625,
-0.04150390625,
0... | 6,972 | 6972 | |
Stuck on a trig limit
Is there no possible way to rearrange it? I was told to always try that when a denominator went to 0, which of course I can't seem to do on this question.. Also the answer is negative infinity, not
positive.
I didn't notice the "right hand limit" part. There are several ways to rearrange the expr... | 5 | [
-0.091796875,
0.000316619873046875,
0.03271484375,
0.046875,
0.00946044921875,
0.046875,
0.0123291015625,
-0.07177734375,
-0.004669189453125,
0.048583984375,
0.0289306640625,
-0.021484375,
0.019775390625,
0.010986328125,
0.011474609375,
-0.056640625,
0.0634765625,
0.0458984375,
0... | 6,973 | 6973 | |
Expected values and variance
February 21st 2009, 10:09 AM
mistykz
Expected values and variance
So, the problem is:
X is a random variable with E(x) = 100 and V(x) = 15. Find:
a) E(x^2)
b) E(3x+10)
c) E(-x)
d) V(-x)
e) std dev(-x)
For part a), i used the equation that V(x) = E(x^2) -... | 4 | [
-0.044677734375,
0.05859375,
0.0625,
-0.08642578125,
0.002288818359375,
0.02734375,
0.0615234375,
0.0225830078125,
0.0242919921875,
0.039306640625,
0.1298828125,
-0.04541015625,
0.01806640625,
0.056884765625,
-0.0264892578125,
-0.0257568359375,
0.05517578125,
-0.026123046875,
-0.... | 6,974 | 6974 | |
Proof by Induction : Inequalities
February 15th 2010, 08:10 AM
firebio
Proof by Induction : Inequalities
Proof by induction that
$<br /> 1+ \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1} {\sqrt{n}}< 2\sqrt{n}<br /> <br />$
K+1 Step
$<br /> 1+ \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1} {... | 5 | [
-0.00274658203125,
-0.0242919921875,
0.07568359375,
0.0576171875,
-0.052978515625,
-0.03564453125,
-0.051513671875,
-0.00176239013671875,
0.038818359375,
0.04248046875,
-0.1572265625,
-0.056884765625,
0.056884765625,
-0.01495361328125,
-0.07763671875,
-0.00433349609375,
0.060546875,
... | 6,975 | 6975 | |
The Shortest Crease
Date: 12/29/97 at 03:14:48
From: Kathy
Subject: The shortest crease
Q: A paper which is 6 units one side and 25 units another side. Put
the shorter side to face you. Fold the lower right corner to the left
side, making a crease. Our task is to find the minimum crease by
folding the right corner t... | 5 | [
-0.025634765625,
0.1376953125,
0.041259765625,
-0.0216064453125,
-0.038818359375,
0.035400390625,
-0.022216796875,
0.09228515625,
-0.07373046875,
0.0771484375,
-0.0291748046875,
0.0021514892578125,
0.002532958984375,
0.1025390625,
-0.10205078125,
0.041748046875,
-0.1416015625,
0.05... | 6,976 | 6976 | |
First order non-linear ODE
October 16th 2012, 03:22 PM
lillybeans
First order non-linear ODE
Hi everyone,
x^2y'-20+x^2y^2=0; y(1)=1
The question asks me to use substitution
y=1/u*du/dx to solve
So far I have
y'=-1/u^2*du/dx+1/u*d^2u/dx^2
problem is when I sub in for the x^2y^2 term ... | 5 | [
-0.039306640625,
-0.00110626220703125,
0.0966796875,
0.06787109375,
0.041259765625,
-0.0419921875,
-0.0703125,
-0.06982421875,
0.0169677734375,
-0.0216064453125,
0.10595703125,
0.047119140625,
0.01116943359375,
0.07861328125,
-0.042724609375,
0.060302734375,
-0.01470947265625,
-0.0... | 6,977 | 6977 | |
(Series) Prove the following:
September 29th 2010, 10:13 AM #1
(Series) Prove the following:
Show that when n is even,
$<br /> 1^3-2^3+3^3-...-n^3 \equiv 1^3 + 2^3 + 3^3 +...+ n^3 - 16(1^3 + 2^3 + 3^3 +...+(\frac{n}{2})^3) \equiv \sum\limits_{r=1}^{n} r^3 - 16 \sum\limits_{r=1}^{\frac{n}{2}} r^3$
I don'... | 4 | [
-0.0693359375,
0.041259765625,
-0.0400390625,
0.07177734375,
-0.0203857421875,
-0.05078125,
-0.02392578125,
-0.0205078125,
0.0091552734375,
-0.033447265625,
-0.134765625,
-0.038818359375,
0.0274658203125,
0.057373046875,
-0.0025634765625,
-0.0654296875,
0.052978515625,
-0.012145996... | 6,978 | 6978 | |
Prove-vector
Prove that the planes 2x-3y+z=4, x+4y-z=7 and 3x-10y+3z=1 meet in a straight line
Hello, kingkaisai2! Quote: Prove that the planes $\begin{array}{ccc}2x-3y+z\,=\,4 \\ x+4y-z\,=\,7 \\ 3x-10y+3z\,=\,1\end{array}\;\;\hdots\;?$ We have no idea what we're supposed to prove . . . Its
derminant: . $\begin{vmatri... | 5 | [
-0.0869140625,
0.0091552734375,
-0.06982421875,
-0.006500244140625,
0.0625,
0.0361328125,
0.07958984375,
0.041748046875,
-0.07568359375,
0.01007080078125,
-0.01190185546875,
-0.054931640625,
0.01519775390625,
0.0162353515625,
-0.0634765625,
-0.04541015625,
-0.10107421875,
-0.050537... | 6,979 | 6979 | |
root three irrational
Hi One way is to suppose that exist p and q integers with no common prime factor such as $\sqrt{3} = \frac{p}{q}$ Then $3 = \frac{p^2}{q^2}$ $p^2 = 3 q^2$ which means that 3 divides p² Therefore 3
divides p because - if p=3k+1 then p² = 9k²+6k+1 = 3(3k²+2k)+1 cannot be divided by 3 - if p=3k+2 the... | 4 | [
0.000713348388671875,
-0.032958984375,
-0.09228515625,
0.047119140625,
-0.00018310546875,
0.0849609375,
-0.000606536865234375,
-0.023193359375,
0.07861328125,
-0.048828125,
-0.035888671875,
0.0042724609375,
0.05419921875,
0.10791015625,
-0.005950927734375,
-0.123046875,
-0.0986328125... | 6,980 | 6980 | |
Counting Techniques
Date: 12/28/97 at 10:09:37
From: Kate
Subject: Counting Techniques
Problem: Find the number of positive divisors of 17640, inclusive of
1 and 17640 itself.
I have no idea how to do this. Please help me.
Thanks.
Date: 01/03/98 at 09:01:58
From: Doctor Mitteldorf
Subject: Re: Counting Techniques... | 4 | [
-0.05517578125,
0.00933837890625,
-0.037841796875,
-0.008056640625,
-0.034912109375,
0.0322265625,
0.033447265625,
0.0732421875,
-0.06201171875,
0.017822265625,
-0.04248046875,
0.04345703125,
0.0166015625,
0.0228271484375,
-0.1025390625,
-0.0157470703125,
-0.0380859375,
0.032226562... | 6,981 | 6981 | |
Polar equations, limited information given
May 23rd 2010, 04:45 AM
BG5965
Polar equations, limited information given
Hi, for polar equations, you usually need r and theta, in this case, I am only given one bit of information, how do I do it?
1) Convert these polar equations to Cartesian:
a] $r = 4$
... | 4 | [
-0.0033416748046875,
0.0084228515625,
-0.06103515625,
-0.006500244140625,
0.0380859375,
0.00193023681640625,
-0.0230712890625,
0.018310546875,
-0.0017547607421875,
-0.0264892578125,
0.048828125,
0.028076171875,
0.032470703125,
0.11474609375,
0.000335693359375,
0.061279296875,
-0.0634... | 6,982 | 6982 | |
A few integration problems
November 7th 2006, 09:01 PM #1
Newbie
Joined
Mar 2006
Posts
15
A few integration problems
I tried some of these problems and came up with some answers, but not 100% sure about it, please help me check them.
$1. \int \sin^3(5x)\cos(5x)dx$
$2. \int \tan^2(x)dx$
$3.... | 5 | [
-0.0380859375,
-0.0159912109375,
0.11279296875,
0.00811767578125,
0.0238037109375,
0.06005859375,
0.03662109375,
-0.045166015625,
-0.0150146484375,
0.01361083984375,
0.057861328125,
-0.0732421875,
-0.034423828125,
0.007354736328125,
-0.02880859375,
0.00982666015625,
0.037841796875,
... | 6,983 | 6983 | |
Last Project Problem Help Please!!!
October 13th 2007, 10:07 AM
rlarach
Last Project Problem Help Please!!!
:confused::confused::confused::confused:
Consider a collection of piles of bananas consisting of one pile of 9 bananas, one pile of 6, and one pile of 2. Such a collection, denoted as C1, could be exp... | 4 | [
-0.0673828125,
0.0703125,
-0.016845703125,
-0.0194091796875,
0.0087890625,
0.043212890625,
-0.033447265625,
0.0419921875,
-0.0306396484375,
0.0301513671875,
0.037109375,
-0.173828125,
-0.0301513671875,
-0.043701171875,
-0.03369140625,
0.044921875,
-0.049072265625,
-0.0177001953125,... | 6,984 | 6984 | |
Riemann Sums and Integrals
August 29th 2009, 08:35 AM
Billy Maize
Riemann Sums and Integrals
Question: Find the integral of the equation 2x+1 from 0 to 3 using Riemann Sums.
Please show me how to do this, I keep on getting 9 as the answer, but it is actually 12. (Worried)
August 29th 2009, 08:52 AM
eXist
... | 4 | [
-0.036865234375,
0.035400390625,
0.08056640625,
0.01446533203125,
0.0208740234375,
-0.0014801025390625,
-0.0174560546875,
0.04638671875,
0.0208740234375,
-0.09716796875,
-0.03125,
-0.0751953125,
0.0185546875,
-0.007110595703125,
-0.0615234375,
-0.09716796875,
-0.0196533203125,
0.01... | 6,985 | 6985 | |
quick question with simplifying rational expressions
July 6th 2008, 05:06 PM #1
Newbie
Joined
Jul 2008
Posts
24
quick question with simplifying rational expressions
I just need some confirmation to whether or not these are right,
can I simplify $-\frac{3x^2}{2x}$ into $\frac{3x}{2}$?
can $\frac... | 4 | [
-0.04345703125,
-0.00994873046875,
0.021728515625,
0.0517578125,
-0.004791259765625,
-0.0186767578125,
0.0150146484375,
-0.06591796875,
0.0625,
0.0096435546875,
0.0986328125,
-0.07568359375,
0.0159912109375,
0.047607421875,
0.0267333984375,
0.00823974609375,
-0.04443359375,
0.05151... | 6,986 | 6986 | |
A Calculus Question
Associated Topics || Dr. Math Home || Search Dr. Math
A Calculus Question
Date: 02/09/98 at 1... | 5 | [
0.0810546875,
-0.0142822265625,
-0.025146484375,
0.04931640625,
-0.003021240234375,
-0.06591796875,
-0.043701171875,
0.0250244140625,
-0.047119140625,
0.0264892578125,
0.06640625,
0.0140380859375,
0.052978515625,
0.0771484375,
0.010986328125,
0.03857421875,
0.016845703125,
-0.09277... | 6,987 | 6987 | |
Proof of the Feuerbach Theorem
Date: 03/14/2000 at 13:52:26
From: Stefa Ben Ari
Subject: Feuerbach theorem
Please submit the proof of the Feuerbach theorem (the nine point
circle is tangent to the incircle and the circumcircle of a triangle.)
Thank you in advance,
Stefa
Date: 03/15/2000 at 13:12:57
From: Doctor F... | 4 | [
-0.08544921875,
0.08642578125,
-0.0703125,
0.057373046875,
0.0673828125,
0.012451171875,
0.00897216796875,
0.1552734375,
0.0859375,
-0.091796875,
0.022216796875,
-0.0228271484375,
0.00555419921875,
0.0186767578125,
0.057373046875,
-0.057861328125,
-0.04931640625,
-0.003997802734375... | 6,988 | 6988 | |
Distance to the Horizon
Date: 09/18/97 at 23:08:17
From: Tony Turner
Subject: Geometry/Trig
If a 6-foot man is standing on the beach at sea level looking straight
out to sea, how far can he see - i.e. what is the distance from the
man to the horizon?
The radius of the Earth is approximately 3963.21 miles. Also, thi... | 4 | [
0.05712890625,
0.05517578125,
-0.015625,
-0.025634765625,
0.026611328125,
0.042236328125,
-0.0205078125,
0.0303955078125,
0.00154876708984375,
-0.00872802734375,
-0.045654296875,
-0.0341796875,
-0.043701171875,
0.041259765625,
0.060791015625,
0.00537109375,
-0.08251953125,
0.050292... | 6,989 | 6989 | |
Coming up with an epsilon value
August 5th 2007, 11:42 AM
spiritualfields
Coming up with an epsilon value
I'm having a little problem with this. Here is the problem statement:
Quote:
Use the graph of the function f given in figure 3.21 to estimate a number epsilon such that: If 0 < |x - 1| < 0.5, then ... | 4 | [
0.01611328125,
0.059814453125,
0.1064453125,
-0.0086669921875,
0.023681640625,
-0.016845703125,
0.036865234375,
0.017822265625,
0.0245361328125,
-0.00811767578125,
0.023193359375,
-0.06689453125,
0.06982421875,
-0.00640869140625,
0.03564453125,
0.0035858154296875,
0.00677490234375,
... | 6,990 | 6990 | |
Math Help
June 5th 2007, 01:00 PM #1
Junior Member
Joined
Mar 2007
Posts
63
work done
A quantity of gas with an initial volume of 1 cubic foot and a pressure of 2500 pounds per square foot expands to a volume of 3 cubic feet. Find the work done by the gas for the given volume.
Assume that the pressu... | 5 | [
-0.054931640625,
0.02197265625,
-0.0040283203125,
0.01806640625,
-0.0281982421875,
-0.06884765625,
-0.0218505859375,
-0.03271484375,
0.004241943359375,
0.06787109375,
-0.029296875,
0.036865234375,
-0.02978515625,
0.06396484375,
-0.0263671875,
0.041015625,
-0.0238037109375,
0.065917... | 6,991 | 6991 | |
ength
Length Scales in Physics
John Baez
Feburary 12, 2010
It's important when learning physics to get an idea of how big different things are. How big is an atomic nucleus? An atom? A cell? A planet? A galaxy? Often it's good enough to just have a rough
idea.
If you are thinking about small things, its good to kno... | 5 | [
0.0281982421875,
-0.0038299560546875,
-0.064453125,
0.0252685546875,
0.00014591217041015625,
0.01263427734375,
-0.005340576171875,
0.142578125,
0.0218505859375,
-0.005523681640625,
-0.006134033203125,
-0.0703125,
-0.0308837890625,
0.00201416015625,
0.04541015625,
-0.052001953125,
0.0... | 6,992 | 6992 | |
Solving Rational Inequalities
August 11th 2011, 04:43 PM #1
Junior Member
Joined
Aug 2011
Posts
37
Solving Rational Inequalities
Solve the inequality: 2x-10/x >x+5
I tried it and got an unfactorable polynomial: x^2+3x+10, but the answer says it's x<-5 and -2<x<0.
Thanks in advance!
Re: Solving R... | 4 | [
-0.07861328125,
-0.0113525390625,
0.078125,
0.00787353515625,
0.026611328125,
0.055908203125,
-0.058349609375,
-0.0162353515625,
-0.048583984375,
-0.0269775390625,
0.036376953125,
-0.033203125,
0.05322265625,
0.08837890625,
0.01397705078125,
0.0556640625,
0.06494140625,
-0.03173828... | 6,993 | 6993 | |
Newton-Raphson Approximation
September 23rd 2010, 10:17 PM #1
Senior Member
Joined
Jul 2009
From
Singapore
Posts
338
Newton-Raphson Approximation
A student, asked to find a root of the equation
$f(x)\equiv x^3-14x^2+49x-8=0$
did not notice the solution x=8 but chose, instead, to use Newton's ... | 5 | [
-0.0341796875,
0.0203857421875,
0.060302734375,
0.04443359375,
0.012451171875,
-0.056884765625,
-0.05712890625,
0.119140625,
0.0262451171875,
0.06591796875,
0.06884765625,
0.013671875,
-0.000031948089599609375,
0.057861328125,
-0.00762939453125,
-0.030517578125,
-0.0262451171875,
-... | 6,994 | 6994 | |
Prove that the Image of a Continuous Function on a Dense Set is Dense
June 20th 2010, 10:55 AM
ragnar
Prove that the Image of a Continuous Function on a Dense Set is Dense
I've been working on this for days and have made no progress, so any advice would be appreciated.
Definition: Let $R$ be an ordered fiel... | 4 | [
0.0277099609375,
0.039306640625,
0.050048828125,
-0.006256103515625,
-0.01055908203125,
0.08544921875,
0.10302734375,
0.0172119140625,
0.0074462890625,
-0.0025787353515625,
-0.0546875,
-0.028564453125,
0.04150390625,
0.1328125,
0.10595703125,
-0.0205078125,
0.072265625,
-0.00096130... | 6,995 | 6995 | |
qaudratic eqautions
February 25th 2008, 10:28 AM #9
Newbie
February 25th 2008, 10:08 AM #8
February 25th 2008, 10:02 AM #7
Newbie
February 25th 2008, 09:59 AM #6
February 25th 2008, 09:55 AM #5
Newbie
February 25th 2008, 09:51 AM #4
February 25th 2008, 09:48 AM #3
Newbie
February 25th 2008, 09:43 AM #2
February... | 4 | [
-0.01507568359375,
0.0576171875,
0.044677734375,
0.0167236328125,
0.0047607421875,
0.107421875,
0.01153564453125,
0.0146484375,
-0.050048828125,
-0.016845703125,
0.1083984375,
0.051025390625,
0.049560546875,
0.0732421875,
-0.01275634765625,
0.0791015625,
-0.11572265625,
0.001007080... | 6,996 | 6996 | |
Meridians onf surface of revolution.
September 30th 2012, 09:29 PM
bluesboy91
Meridians onf surface of revolution.
Hi Guys,
I have a homework questions that I am stuck on, it's the last in the set and I can't do it!
It reads:
Prove that the meridians are geodesics when the curve: (x(u),0,u) is rev... | 4 | [
-0.0155029296875,
0.00225830078125,
0.03564453125,
-0.07958984375,
-0.001800537109375,
0.0009765625,
0.0849609375,
-0.02685546875,
0.0228271484375,
-0.043701171875,
0.006011962890625,
0.009765625,
-0.03564453125,
-0.00634765625,
-0.051025390625,
-0.0712890625,
-0.08056640625,
-0.00... | 6,997 | 6997 | |
How to prove this hyperbola question
February 19th 2006, 07:12 AM
totalnewbie
How to prove this hyperbola question
How to prove that hyperbola y=4/x determines a triangle which area is 8 when I draw tangent point at whatever point on this hyperbola and when I consider that x-axis, y-axis and tangent line form
... | 5 | [
0.0751953125,
-0.021484375,
0.033447265625,
-0.0478515625,
0.006927490234375,
-0.040771484375,
0.007110595703125,
0.10546875,
0.044677734375,
0.0174560546875,
0.07177734375,
-0.0306396484375,
0.0269775390625,
0.033935546875,
0.01953125,
0.1728515625,
0.0087890625,
-0.08056640625,
... | 6,998 | 6998 | |
Integral powers of sine and cosine
Question : $\int sin^4.cos^2x.dx$ Attempt : I reached a step where >> $\frac{1}{8} \int (1-2cos2x+cos^2x).(1+cos2x)$ How to expand this ?
$\int{sin^4xcos^2x}dx=\int{sin^2xsin^2xcos^2x}dx$ $=\int{\left(1-cos^2x\right)\left(1-cos^2x\right)cos^2x}dx$ $=\int{\left(1-2cos^2x+cos^4x\right)... | 5 | [
0.036865234375,
0.03515625,
0.021728515625,
0.046875,
-0.000545501708984375,
-0.0291748046875,
-0.0181884765625,
0.0166015625,
0.0079345703125,
0.053466796875,
-0.004638671875,
-0.018798828125,
0.0206298828125,
0.0400390625,
-0.0101318359375,
0.01190185546875,
0.04052734375,
0.0174... | 6,999 | 6999 |