i realised after i said it , sorry
#help-17
vast shale
sand lodge
dw
crystal drum
is there any other way I could approach this?
crystal drum
yes
sand lodge
maybe we can try finding the non shaded part
square - 4 corners + quartercircle = 2 non shaded part
check me on that
I'm not too sure
gray crescent
The solution is pretty complicated, see here: https://math.stackexchange.com/questions/2468453/quarter-circle-and-circle-inscribed-in-square
vast shale
i also have another idea
lets say we name the small pieces left x1 and x2 what's left from the corner. then the total area of the square would be shaded+corner +2x2 + notshaded+2x1+corner
notshaded = circle - shaded , and that removes shaded from the equation
lol i was wrong
sand lodge
squre - qcircle = corner + x + 2z
circle - qcircle = x - (corner + 2(corner-z)) = x - 3corner + 2z
qcircle + x = 2z + corner
where z is the small piece
idk these are the only equation I could come up with but It is likely one of these is just a rearrangement of the other leading to infinitey many solutions
vocal sleetBOT
@crystal drum Has your question been resolved?
main totem
how come the derivative is taken differently in these cases?
vocal sleetBOT
how come the derivative is taken differently in these cases?
main totem
heres the original problem im working with for the 1st picture btw
main totem
x^(constant) vs. (constant)^x?
yes
paper depot
i mean these are different functions arent they lmao
main totem
but now I understand since you've put it that way
but n is an unknown
so I got confused
paper depot
n is constant
vocal sleetBOT
@main totem Has your question been resolved?
vocal sleetBOT
vocal sleetBOT
Available help channel!
honest ridge
Technical question
Do you know a decent online tool for solid geometry?
I can't find any on my own =(
vocal sleetBOT
*Technical question*
Do you know a decent online tool for solid geometry?
*I can...
scenic ravine
I believe geogebra is good for graphing in $R^3$ if that's what you mean
twin meteorBOT
physicsrocks
honest ridge
I believe geogebra is good for graphing in $R^3$ if that's what you mean
I was looking more for a tool for doing 3d diagrams.
scenic ravine
wdym, you mean lines in 3d or more like pyramids, sphere, or a CAD tool?
wary mantle
GeoGebra can also do 3D
honest ridge
wdym, you mean lines in 3d or more like pyramids, sphere, or a CAD tool?
I was recently trying to solve some problems about solids, especially cuboids and their diagonals. The tool I was looking for would be for getting a diagram of a solid and some lines in 3d.
scenic ravine
I was recently trying to solve some problems about solids, especially cuboids an...
Yeah, geogebra can do tha and much more
honest ridge
I'll try this, thanks
scenic ravine
for instance, I just did this using geogebra , I'm assuming you want to do somthing simlar (To the cube), for instance
honest ridge
for instance, I just did this using geogebra , I'm assuming you want to do somt...
That would be perfect
thanks again
.close
lost timber
vocal sleetBOT
Available help channel!
honest ridge
Earlier I asked this question:
Given all of the roots of a quadratic equation (in its basic form) and the value of Δ , is it possible to find at least one set of values for a, b, c , for which the equation is true?
I was instructed to use Vieta's formulas, but I don't know how they could fit into the solution.
vocal sleetBOT
Earlier I asked this question:
**Given all of the roots of a quadratic equation ...
hushed pewter
Do you even need Δ if you already have the roots?
paper depot
yeah sounds like redundant info
paper depot
Earlier I asked this question:
**Given all of the roots of a quadratic equation ...
expand a(x-r)(x-s), where:
- r and s are the roots
- a is your favorite number, or 42069 if you can't decide, or 1 if you're boring.
that'll get you your coefficients.
honest ridge
that'll get you your coefficients.
Is that really it?
paper depot
sure is.
vivid merlin
why the time emojis im confused
paper depot
why the time emojis im confused
probably trying to spell 42069 with them
vivid merlin
average redditor comedy
honest ridge
I'll go with 31415
paper depot
sure is
honest ridge
expand a(x-r)(x-s), where:
- r and s are the roots
- a is your favorite number, ...
I have just one more question: HOW do I expand this to include b and c ?
obtuse sierra
comparing coefficients
honest ridge
by expand i mean apply the distributive law...
Can you link me to anything explaining this? After "the distributive law"went through a translator, it was complete nonsense.
paper depot
what language are you translating into?
honest ridge
what language are you translating into?
Polish. I just have a hard time matching conjecture/theorem/ ect. names.
honest ridge
expand a(x-r)(x-s), where:
- r and s are the roots
- a is your favorite number, ...
ann, I didn't ask, if r and s are the roots, what is x then?
vocal sleetBOT
ann, I didn't ask, if r and s are the roots, what is x then?
paper depot
do you have concrete numbers for r, s and D
i just realized that the discriminant is in fact not redundant like i claimed earlier
honest ridge
do you have concrete numbers for r, s and D
No, I wanted a general solution, also D means delta?
paper depot
it means discriminant, whichever way you like best to denote it.
the key idea is: the equation a(x-r)(x-s) = 0 has solutions x=r and x=s.
therefore it is the equation you're looking for, except it is in the wrong form.
honest ridge
therefore it is the equation you're looking for, except it is in the wrong form.
And here comes my question, How do I get b and c from this, since from what I know the distributive law does not touch this topic even once?
paper depot
??? you literally have a product here that you can (& should) distribute
(x-r)(x-s) = x^2 - rx - sx + rs
= x^2 - (r+s)x + rs
honest ridge
= x^2 - (r+s)x + rs
That's quite logical, but how can I get b and c from this? (if my questions seem dumb, that's because I learned about this topic just from casual research)
paper depot
i have to go to sleep now sorry
gloomy nova
vocal sleetBOT
gloomy nova
yoo
bold wasp
connect TR
gloomy nova
I find distance between Tr?
bold wasp
I find distance between Tr?
yep
you slice the shape into triangles
and heron's formula
hit that up on google
gloomy nova
what is heron formula
bold wasp
what is heron formula
it's a formula where you can calculate the area of a triangle using the three sides of them
instead of a base and height
gloomy nova
Oh alr
desert moth
vocal sleetBOT
desert moth
(x,y,z)
shouldnt the z coordinate reflect the distance from z axis?
and distance cannot be negative
correct?
unique swan
How did my question go in but was not ever assigned by the bot. It's all red text/
desert moth
How did my question go in but was not ever assigned by the bot. It's all red tex...
network issue?
civic otter
shouldnt the z coordinate reflect the distance from z axis?
No, the z coordinate represents the height/altitude wrt the plane z = 0 (or xy plane)
desert moth
No, the z coordinate represents the height/altitude wrt the plane z = 0 (or xy p...
oh i see now
unique swan
Ah, the red text signifies it did not go through. Okay, sorry for the interruption. Thank You.
civic otter
Don't worry, we're here to help
desert moth
No, the z coordinate represents the height/altitude wrt the plane z = 0 (or xy p...
this should be correct?
civic otter
Yep
last harbor
simple question: starting statistics, wanna know what the actual difference of the average and mean is. am i overthinking it? i know how to calc both but in this case apparently it's different i've been told.
vocal sleetBOT
simple question: starting statistics, wanna know what the actual difference of t...
urban edge
There are different types of mean but the one you are most familiar with is called the arithmetic mean
This is the one where you sum up all terms and then divide by the number of terms
This is also usually considered the average
However average also has different meanings
Sometimes, the median is considered the average or the mode is, but those tend to not be used as often
there are different types of means, idk if you me to get into them
last harbor
there are different types of means, idk if you me to get into them
could i give you a scenario? I think i'm referring to the sample mean based on what i've looked up
however i can't tell if a sample mean and average by definition are the same
chrome raptor
You're asking about the difference between the sample mean and the population mean
Aren't you?
$$\overline{X}\neq \mu$$
twin meteorBOT
Categorist
last harbor
no, not really, i haven't gotten into population mean yet, but i know a sample mean is a portion of the population one
but that may be helpful. let me phrase it this way
if a teacher receives 5 quiz scores going 100,100,100,0,0, the mean score is 60; i agreed with this
if the same quiz is given to a second class of 5 and the scores go 60,60,58,62,60, wouldn't the mean score still be 60?
urban edge
Yes
last harbor
ok, in that case then to calc the mean you don't just divide by the sample size but by the whole population one
or both are just 60 so i can't say that
hold on
urban edge
or both are just 60 so i can't say that
Why not?
(100+100+100+0+0)/5=60
And also (60+60+58+62+60)/5=60
They both have an arithmetic mean of 60
last harbor
ok now i see where i'm getting confused
urban edge
In this case, your sample means are the two classes separately, the population mean would be the whole school
last harbor
ok, let me just confirm this then;
-
the sample mean would be the same as the average (or arithmetic mean, if they would be synonyms?)
-
to calculate said mean, you just add up all terms and divide by sample size (so just arithmetic mean again)
chrome raptor
These are two means which are coincidentally equal
urban edge
Arithmetic mean is the act of taking the sum of the numbers you are working with and dividing by the number of numbers you are working with
For example, the arithmetic mean of the four numbers 5, 10, 16, and 1 is
(5+10+16+1)/4=32/4=8
Yes, sample mean and population mean are both arithmetic means
Sample mean refers to a subset of the population you are working with
last harbor
ah so the sample mean is just a fancy term for average of a sample data set
urban edge
For example, taking the average height of people in china is a sample mean
last harbor
the whole world would be population
chrome raptor
ok, let me just confirm this then;
1. the sample mean would be the same as the ...
Yes, in statistics is not that common to use the term "average". We say mean or arithmetic mean when we want to emphasize. "Average" is correct but more natural speech
last harbor
Statisticians love vocabulary hahaha
Thanks for the help though, I finally understand why it's worded like that in my homework
.solved
oop
.close
pseudo shale
I've been using anki to memorize calc formulas but in this formula there is the variable a, and I was wondering what I would input for it
vocal sleetBOT
I've been using anki to memorize calc formulas but in this formula there is the ...
calm hinge
a constant or an expression that isn't in function of x
vocal sleetBOT
vocal sleetBOT
Available help channel!
woven shell
help with this please
vocal sleetBOT
help with this please
solemn cape
If Q is the mid point of PR then 2QR=PR
solemn cape
2x+18=5x+9
calm hinge
What is a midpoint?
solemn cape
Solve for x
woven shell
Solve for x
thats it?
solemn cape
Ye
woven shell
alr ty
solemn cape
What is a midpoint?
It’s the point in the middle of a segment
woven shell
Solve for x
x is 3 right?
solemn cape
No
calm hinge
It’s the point in the middle of a segment
I wanted them to answer so as not to give them the answer without them understanding what they were doing
solemn cape
Oh sorry thought you asked for it
woven shell
2x+18=5x+9
2x+18=5x+9 = - 2x on both sides = 18=3x+9 - subtract 9 - 9= 3x - devide by 3, x = 3
woven shell
so x = 3?
solemn cape
Yes
solemn cape
What how !??
karmic imp
It wanted the length
Of QR
So you plug in 3 for into x + 9
To find the value of QR
vocal sleetBOT
Available help channel!
valid carbon
vocal sleetBOT
jolly epoch
hi
valid carbon
I don't understand how they've used the binomial expression here?
jolly epoch
you can do
1 - Pr(10) - Pr(11)
because there’s 2 outcomes
either yes (76%) or no (24%)
and the trials are independent of one another
X~Bi(11, 76)
so we can say this is binomially distributed
valid carbon
sooo my use here is incorrect?
jolly epoch
this is the formul
what you have done only accounts for P(X=10)
you need to consider P(X=11)
so you can do: 1 - Pr(X=10) - Pr(X=11)
because Pr(X<10) = 1 - Pr(X>=10)
vast shale
The internal dimension of a lidless wooden box is 48 cm x 38 cm x 31 cm.If the thickness of the box is 1 cm,find the volume of the wood
vocal sleetBOT
The internal dimension of a lidless wooden box is 48 cm x 38 cm x 31 cm.If the t...
vast shale
The internal dimension of a lidless wooden box is 48 cm x 38 cm x 31 cm.If the thickness of the box is 1 cm,find the volume of the wood
vocal sleetBOT
@vast shale Has your question been resolved?
vast shale
The internal dimension of a lidless wooden box is 48 cm x 38 cm x 31 cm.If the thickness of the box is 1 cm,find the volume of the wood
livid tapir
wait tf happened to my answer
vocal sleetBOT
@old quest Has your question been resolved?
sick owl
How do you derange identical objects like in this question
vocal sleetBOT
How do you derange identical objects like in this question
vocal sleetBOT
@sick owl Has your question been resolved?
sick owl
<@&286206848099549185>
untold surge
N(6)/2 where
N(m) means number of rearrangements of 1,2,…,m having no fix point
Which can be calculated by
Inclusion-exclusion principle
sick owl
Inclusion-exclusion principle
Can we not solve by D6 minus(some things)??
untold surge
Inclusion-exclusion principle
$N(m)=\sum_{k=0}^{m}(-1)^{k}\binom{m}{k}(m-k)!=m!(\sum_{k=0}^{m}\frac{(-1)^{k}}{k!}$
twin meteorBOT
Cogwheels of the mind
untold surge
I don’t know what D_6 is
sick owl
**Cogwheels of the mind**
this is what d6 is
6!(1-1/1! + 1/2! - 1/3!....1/6!)
But we use that when we don't want anyone on its original place right?
Here there are two 'A's
untold surge
Oh I know what the issue is wait a second
First viewing two A as A1 and A2, you have D_6 many
Back to two A, there are forbidden cases counted
Like A_2 under A_1
sick owl
Yes excatly
untold surge
So minus (5!+5!-4!)
Is the answer
The thing in the parentheses is
N(A_2 under A_1)+N(A_1 under A_2)-N(A_2 under A_1 also A_1 under A_2)
sick owl
So fixing any one?
sick owl
265-240+24=49
untold surge
So D_6 minus it
sick owl
Answer is given 84
sick owl
Yeah 5!+5!-4! Is how many cases where at least one A is put under A
I dont get it
Are we subtracting case where we keep one A on its own place?
Or one A always on ANOTHER A ?
untold surge
Shit i miscalculated
Should be
I forgot when A_2 is under A_1, other elements still have no fix point
Wait a min
sick owl
Ye
untold surge
Hey can you do me a favor
Can you calculate D_5 and D_4 for me
I am checking my answer
untold surge
Correct now
sick owl
:D
untold surge
(D_6-D_4)/2 -D_5
Okay so it should be
So for now when we say an arrangement we mean an arrangements of A_1,A_2, BHRT having no fix points
So D_6 many arrangements
What we are looking for is
N(arrangements such that A_2 is not under A_1, and A_1 is not under A_2)/2
=(N(arrangements)-N(A_1 under A_2 or A_2 under A_1))/2
sick owl
Alr
untold surge
Looks too long,
sick owl
Wait
untold surge
Say p: A_1 under A_2
q: A_2 under A_1
untold surge
No. What we are looking for is (N(not p and not q))/2
sick owl
Why /2?
untold surge
Because given an arrangement, having no A_2 under A_1, no A_1 under A_2, we need to view A_1 and A_2 as same A again
sick owl
Right right my bad
sick owl
Say p: A_1 under A_2
q: A_2 under A_1
Continue
sick owl
yes
untold surge
=(D_6-N(p and q)-N(p and not q)-N(q and not p))/2
sick owl
Got it
untold surge
Now N(p and q)=N(A_2 under A_1 and A_1 under A_2)=D_4
N(p and not q)=N(q and not p)=number of cases where exactly one A is under A, other five elements having no fix point=D_5
So together
(D_6-D_4-2D_5)/2
sick owl
Understood 👍
untold surge
👍
sick owl
Thanks a lot mate!
untold surge
Np
sick owl
You be big brain
untold surge
🤣
sick owl
.close
vocal sleetBOT
twin meteorBOT
BrotmitHonig
twin meteorBOT
aysob_ay212
jolly epoch
then you wanna find out the pair of numbers which multiply to ac and add to b
in this case what mulitplies to -3 and adds to -6
unfortunately there is no nice number so we will resort to the quadratic formula
nope
can you apply this to the quadratic formula?
hard atlas
remember that for pq the coefficient of x^2 has to be 1
twin meteorBOT
aysob_ay212
jolly epoch
you want to find what multiplies to ac
and adds to b
the coefficients
then you could factorise by grouping
hard atlas
(this is vieta and is basically just educated guessing)
jolly epoch
a = 3, b = -6, c = -1
jolly epoch
didn't know it was called vieta, interesting
anyways, we will use the quadratic formula (we should only use this method as a last resort)
$(-b +- \sqrt(b^2 - 4ac))/2a$
bruh
hard atlas
$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
twin meteorBOT
Denascite
jolly epoch
thank u
hard atlas
or pq but again remember that for pq the coefficient of x^2 needs to be 1
jolly epoch
$\frac{-(-6)\pm\sqrt{(-6)^2-4(3)(-1)}}{2(3)}$
twin meteorBOT
aysob_ay212
jolly epoch
which part
hard atlas
$-\frac p 2 \pm \sqrt{\left(-\frac p 2\right)^2-q}$
jolly epoch
**Denascite**
do you understand this
twin meteorBOT
Denascite
jolly epoch
maybe denascite can help you with this because i don't understand their method
and you do
hard atlas
pq is for x^2+px+q=0
and?
then p=something negative
but first we need to divide by 3
so x^2-2x-1/3=0
now p=-2
q=-1/3
and now plug in
jolly epoch
the imaginary second lol
twin meteorBOT
hard atlas
yes
although lets get used to writing a closed form
instead of just throwing it into a calculator
the minus before the 1 should be gone
(-1)^2 is 1
wait something is wrong here
1 +- sqrt(4/3)
or 1+-2/sqrt3
you're welcome
vocal sleetBOT
vocal sleetBOT
Available help channel!
past orchid
so I have a question hat was on my national examination... I really don't know how to approach it
vocal sleetBOT
so I have a question hat was on my national examination... I really don't know h...
vast shale
Find the Least common multiple LCM
For 72 and 80
Then divide by 80
That would be the number of laps Tanui ran before meeting Kipkoech at the starting line
vocal sleetBOT
@past orchid Has your question been resolved?
past orchid
.close
umbral snow
vocal sleetBOT
umbral snow
guys is here mistake?
this dispersion make no sense to me
its like inverse of what it should be
or thats the point?
paper depot
are you unhappy with how the axes are chosen
umbral snow
No its good but
paper depot
as far as i can tell you have income on the horizontal axis and expenses on the vertical
umbral snow
2003: Spending (potrošnja) 65.6 and Income (dohodak) 62.0
but on the chart its inverse
paper depot
ah, so someone fucked it up then
umbral snow
fr
paper depot
btw didn't croatia switch to euro recently
paper depot
can you remind me what jednostavna means
umbral snow
Simple Linear regresion
next is multiple linear regresion
and test statistaks in 5 days 😬
so big mistake in chart
gritty zenith
Anti-CCP flag
vocal sleetBOT
twin meteorBOT
guinearW
vocal sleetBOT
What step are you on?
1. I don't know where to begin
2. I have begun but got stuck midway
3. I got an answer but I'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
small quarry
looking...
First, u need to figure out the pattern
there's x^4, x^3, x^2... to x^0
twin meteorBOT
guinearW
twin meteorBOT
Fossil
small quarry
better
twin meteorBOT
guinearW
small quarry
oh
vocal sleetBOT
quick estuary
i just know that as identity
this one is simpler
what u can do here
is take sum of squares till 144
then subtract 2 times sum of squares of even numbers
to find that just take 4 common from each term
so for even terms
2^2+4^2+6^2.....
u can rewrite as
4(1^2+2^2+3^2...)
and apply formula again
kind swift
quick estuary
it does
its due to practice and observation tbh
because you have to use whats given
and its not directly applicable here
so u have to manipulate
the summation
vocal sleetBOT
Available help channel!
pallid cairn
vocal sleetBOT
pallid cairn
In the final step, how do you get from t^4 = -2 to t = -1 on the right side? Because if I try to 4 √-2 I get a domain error
paper depot
how do you get from t^4 = -2 to t = -1
you do not
you acknowledge that t^4 = -2 has no real solutions, while t^4 = 1 is responsible for both t=1 and t=-1
vocal sleetBOT
keen ermine
vocal sleetBOT
Available help channel!
stable carbon
Define:
Use induction to show that:
vocal sleetBOT
Define:
Use induction to show that:
stable carbon
First(1) we prove n=1 that s_1=1/2 = 1 - 1/2
Then we assume that its true for any k wich means that s_k = 1- 1/2^k
But im really wondering about this:
Which is included in the solution, how do they arrive at this? And how do i solve the question?
round plover
if s_n is the sum of 1/2^i for i = 1 to n
then s_(n+1) is the sum of 1/2^i for i = 1 to n+1, which is just the previous sum with an extra term
stable carbon
yes
round plover
\begin{align*}
s_n&=\frac{1}{2}+\frac{1}{2^2}+\cdots+\frac{1}{2^n}\
s_{n+1}&=\frac{1}{2}+\frac{1}{2^2}+\cdots+\frac{1}{2^n}+\frac{1}{2^{n+1}}\
s_{n+1}&=\left(\frac{1}{2}+\frac{1}{2^2}+\cdots+\frac{1}{2^n}\right)+\frac{1}{2^{n+1}}\
s_{n+1}&=s_n+\frac{1}{2^{n+1}}
\end{align*}
twin meteorBOT
Desync
stable carbon
Ok, that makes sense I supose
round plover
replace s_n with the induction hypothesis
and try rearrange the expression into 1-(1/2^{n+1})
stable carbon
ok, but why does s_n = 1 - 1/2^n dissapear
round plover
what do you mean
stable carbon
Yeah idk, the question was stupid
If we continue with the calculations
how do we arrive at s_n = 1 - 1/2^n
round plover
you showed it holds for the base case n=1
stable carbon
yes
round plover
you assume it holds for arbitrary n in the second part
that's how induction works
we want to show s_{n+1} = 1 - 1/(2^{n+1})
round plover
**Desync**
use this, and replace s_n by 1 - 1/2^n
round plover
we want to show s_{n+1} = 1 - 1/(2^{n+1})
and try rearrange it into this
stable carbon
ok, so we replace s_n before our fraction on the right side because s_n is equal to 1 - 1/2^n which is given to us in the task
twin meteorBOT
Phil
stable carbon
right @round plover ?
round plover
apart from the broken exponent yes
round plover
you've already used it
stable carbon
Yeah, im asking in general
round plover
you replaced s_n by 1 - 1/(2^n)
stable carbon
Yeah
round plover
yes
stable carbon
ok, good
I will do some work on my own, thank you so much
actually @round plover, sorry to bother you but i have a few questions about some basic calculation
round plover
okay
round plover
multiply top and bottom by 2
stable carbon
2k*2 = 2k+1???
round plover
\begin{align*}
\frac{1}{2^k}&=\frac{1}{2^k}\times\frac{2}{2}\
&=\frac{2}{2^k\times2}\
&=\frac{2}{2^{k+1}}
\end{align*}
twin meteorBOT
Desync
round plover
and yes, think about what an exponent means
2^k is 2 * 2 * ... * 2 k times
so if you multiply it by 2 again, it's k+1 many 2s being multiplied
so it's 2^(k+1)
round plover
can you write that with more brackets
stable carbon
yes
round plover
I don't know what that expression is exactly
stable carbon
$$ 1-frac{2}{2^{k+1}} + 2^(k+1) $$
twin meteorBOT
Phil
stable carbon
Yeah, how does it get removed
round plover
we only ever use it in the denominator of a fraction
stable carbon
I still have 2 / 2(k+1)
round plover
\begin{align*}
s_{n+1}&=s_n+\frac{1}{2^{n+1}}\
&=\left(1-\frac{1}{2^{n}}\right)+\frac{1}{2^{n+1}}\
\end{align*}
twin meteorBOT
Desync
round plover
do you follow this
twin meteorBOT
Phil
vocal sleetBOT
@stable carbon Has your question been resolved?
restive vector
Can anyone help me solve this?
vocal sleetBOT
Can anyone help me solve this?
lilac plaza
What the heck
restive vector
What the heck
LOL thats what I said
and its supposed to be "review from last semester" Ik damn well we aint learn dat
lilac plaza
I have never seen limit problems like that in my life
bright yew
its quite straightforward really
lim x->inf e^(-x) = 0
once you know that it should be easy
here's a hint if you don't get why that's useful: ||divide the numerator and denominator by e^x||
restive vector
here's a hint if you don't get why that's useful: ||divide the numerator and den...
wym by this... can I show you how I did it and you can tell me if i'm wrong
bright yew
wym by this... can I show you how I did it and you can tell me if i'm wrong
yeah go ahead
twin meteorBOT
nebula40
restive vector
yeah go ahead
bright yew
yes just apply (a^x)/(a^y) = a^(x-y) to get what I just wrote
restive vector
ahh okay, I got it, thank you!!
restive vector
yes just apply (a^x)/(a^y) = a^(x-y) to get what I just wrote
how do I go about solving this one?
bright yew
how do I go about solving this one?
$= x(3 - 7 \frac{\ln(x)}{x})$
twin meteorBOT
nebula40
bright yew
you need to know/prove that lim x->inf (ln(x))/(x) = 0
I have to go now, so if you still have any questions someone else can answer them
restive vector
**nebula40**
how did you get to this
just take out the x?
Can someone help me with this
vocal sleetBOT
@restive vector Has your question been resolved?
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@restive vector Has your question been resolved?
vocal sleetBOT
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Available help channel!
iron crown
vocal sleetBOT
iron crown
Are both a and b inconsistent matrices because the last row in each holds a contradiction?
Both last rows result in ■ = 0
vocal sleetBOT
@iron crown Has your question been resolved?
vocal sleetBOT
@iron crown Has your question been resolved?
iron crown
<@&286206848099549185>
glossy maple
(b) is consistent, because you get a contradiction (nonzero value equals 0)
But that's not the case for (a)
It might help if you write down an example of such a matrix, and convert it into a system of equations.
round plover
to add to this, remember that only the last column of an augmented matrix is constant
the others are coefficients of your variables
vocal sleetBOT
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iron crown
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compact zealot
Having issues figuring out how I should go about plotting and creating the graph here
vocal sleetBOT
Having issues figuring out how I should go about plotting and creating the graph...
twin meteorBOT
Huntifer
round plover
you've dropped a minus on the left
round plover
then you should have -1 on the right
round plover
your second equation looks good
compact zealot
Would it be more effective to not multiply by negative one then and leave y as a negative
?
Or do you have to
But anyways so it should actually be
$y\ge-1+4x$
twin meteorBOT
Huntifer
round plover
Would it be more effective to not multiply by negative one then and leave y as a...
I normally replace inequalities with equalities to avoid having to think about this kind of thing
to just find the boundary lines for the required region
then figure out which way the inequality should have pointed (by sampling points if it's not clear from inspection)
round plover
just look at 4x - y = 1
rearrange that easily to 4x - 1 = y
then figure out if any of the manipulations would have flipped anything afterwards
or sample points; for instance, plugging in (0,0) into the original inequality gives 4*0 - 0 <= 1 which is satisfied
so the origin is on the required side of the line y= 4x - 1
compact zealot
Yeah
That makes sense
And you can see 4x means that its a positive slope if you're thinking of it as y=mx+b
Then plugging (0,0) or a point in the accepted region lets you test it
I was just forgetting the sign flip thing
round plover
yeah, you can always check it after and just use =s during the rearranging
vocal sleetBOT
@compact zealot Has your question been resolved?
round plover
a) this channel is occupied, you should open a new one #❓how-to-get-help
b) you should not ping individuals for help
compact zealot
.close
muted tapir
vocal sleetBOT
muted tapir
my friend and I were discussing whether the answer should be 0 or 2
whats the correct interpretation
vocal sleetBOT
@muted tapir Has your question been resolved?
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@muted tapir Has your question been resolved?
vocal sleetBOT
@muted tapir Has your question been resolved?
twin meteorBOT
mild flower
so if you go forwards along either of the curves the integral is 1
and if you go backwards it's therefore -1
so what happens if you go forwards on C1 and then backwards on C2?
mossy latch
vocal sleetBOT
mossy latch
How do I find the question for area of rectangle
mild flower
if you put the vertical lines at x = a and x = -a
what is the area of the rectangle?
mossy latch
?
mossy latch
if you put the vertical lines at x = a and x = -a
What’s the a
mild flower
anything
try it with x = 1 first
but get a formula for the area for arbitrary values of a
mild flower
ok try it with x = 1 first
see the vertical blue lines? those are at x = 1 and x = -1
what's the area of the rectangle?
mossy latch
what's the area of the rectangle?
Or horizontal length?
mossy latch
2 is the base yes
1?
mossy latch
9?
mossy latch
how high does the rectangle go? in words
Nine
mild flower
where did you get 9
mossy latch
From the formula of the parbola
mild flower
how high is the point in red?
how high is the point in blue?
mossy latch
how high is the point in red?
how high is the point in blue?
Yo me it looks like blue is 1 and red is 1.5
As a rough estimate
mild flower
i'm not looking for a rough estimate here
how would you even make a rough estimate?? the y axis isn't labeled
mossy latch
I thought a = 1 = xcould be anything
mild flower
okay, that was partially my fault; we're not using a for right now. we're trying to evaluate one specific rectangle
the blue point and the red point are both on the parabola
mossy latch
Am I just overthinking this and it’s easy
Cause forming the equation is the only hard part of optimisation
mild flower
often that is the hardest part yes
mossy latch
Do u just substitute 1 and 0 as x to find y which is the height
mild flower
the red dot doesn't even really matter for us, it's the blue dot that matters
so yes we can put x=1 into the eqn to find the height which is y
mossy latch
I got 8,75
mild flower
yes great
so what about arbitrary x? what's the formula for the area of the rectangle for any x, not specifically 1?
mossy latch
Xy
vocal sleetBOT
<#359052604149465088>
cerulean meadow
Congruence
in the adjoining figure AB=AD and CB=CD
Prove that TRIANGLE-ABC=~TRIANGLE-ADC
.close
vocal sleetBOT
vocal sleetBOT
Available help channel!
civic otter
What can you say about the blue point if it belongs to the parabola?
civic otter
how high is the point in red?
how high is the point in blue?
This one here
Ops sorry I thought you were who asked the exercise
vocal sleetBOT
@civic otter Has your question been resolved?
chrome raptor
.close
plain aurora
Is there proper notation for the "English Alphabet set"?
vocal sleetBOT
Is there proper notation for the "English Alphabet set"?
plain aurora
That is, letters from ${a,b,c,...,z}$
twin meteorBOT
Kalgar
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split wind
I don't think there is?
split wind
I feel like this set pops up too often for it to not
well, it's nothing really special mathematically...
chilly pawn
vocal sleetBOT
chilly pawn
!status
vocal sleetBOT
What step are you on?
1. I don't know where to begin
2. I have begun but got stuck midway
3. I got an answer but I'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
1
muted tapir
I have a proof of exhaustion in mind if you'd like to hear about it
vocal sleetBOT
@chilly pawn Has your question been resolved?
chrome raptor
Have you tried induction? Where are you stuck? @chilly pawn
untold surge
Just use x^mdx=(1/(m+1))d(x^(m+1)) to obtain a recursive relation I(m,n)=(1/(m+1))((In(x)^n)x^(m+1)-nI(m,n-1))
Until n reduces to 0
vocal sleetBOT
@chilly pawn Has your question been resolved?
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@chilly pawn Has your question been resolved?
chilly pawn
guys but is there a easier way of doing this? im finding hard to understand
forest stump
I have a question is ax = b (mod n) equal to x = b (mod n).
Because I saw from a different source that 11m + 4 = 3 (mod 5) => 11m = -1 (mod 5) => m = -1 (mod 5).
chilly pawn
I dont understand the 1/m+1 from where did it came btw
this way of you differentiating by dx^m+1 is fabulous but Idont understand where the 1/m+1 came from btw
<@&286206848099549185>
bleak sapphire
It's just a constant from when you differentiate x^m+1
chilly pawn
why and how ;w;
chilly pawn
is this enough to prove that is necessary to use integration by parts n times? I thought about using induction
bleak sapphire
The question itself is phrased strangely imo
chilly pawn
It's just a constant from when you differentiate x^m+1
yeah true, now I realized
im still analyzing cogwheels picture, one sec
bleak sapphire
Like how would you prove that IBP n times is necessary
bleak sapphire
It probably just means 'do it/show it can be done' that way
young blaze
Just use x^mdx=(1/(m+1))d(x^(m+1)) to obtain a recursive relation I(m,n)=(1/(m+1...
If I were to elaborate on Cogwheels of the mind's hint:
Let (u = (\ln x)^n) and (dv = x^m dx). Then (du = n (\ln x)^{n-1} \frac{1}{x} dx) and (v = \frac{1}{m+1} x^{m+1}).
The integral becomes
[\frac{1}{m+1} x^{m+1} (\ln x)^n - \frac{n}{m+1} \int x^m (\ln x)^{n-1} dx.]
Observe that the integral term (\int x^m (\ln x)^{n-1} dx) appears on the right-hand side. This new integral has the same form as the original integral (\int x^m (\ln x)^n dx), but the exponent (n) has been reduced by 1 to (n-1).
If (n = 0), the integral can be solved immediately without the need for integration by parts. Otherwise, applying the same logic recursively, it can be seen that the integral will yield a series of new integrals in which the exponent (n) decreases by 1 at each step. This will continue until (n) is reduced to 0, at which point the integral can be solved directly. Therefore, it is necessary to use integration by parts (n) times to solve the integral (\int x^m (\ln x)^n dx).
twin meteorBOT
adzetto
untold surge
Thanks
chilly pawn
is this enough to prove that we need to integrate n times? <@&286206848099549185>
@young blaze @untold surge sorry for the ping
young blaze
is this enough to prove that we need to integrate n times? <@&286206848099549185...
Yeap. To clerify, you can write "For (n > 0, \in \mathbb{Z}), (I(m, n)) can be expressed in terms of (I(m, n-1)). Thus, to find an explicit expression for (I(m, n)), one would need to apply the method of integration by parts (n) times until (I(m, 0)) is reached, which can be directly integrated as (I(m, 0) = \int x^m dx = \frac{x^{m+1}}{m+1}.)"
twin meteorBOT
adzetto
chilly pawn
Yeap. To clerify, you can write "For \(n > 0, \in \mathbb{Z}\), \(I(m, n)\) can ...
tysm adzetto for the clarification, but this only works when n and m are positive right?
y bso confusing is proving things when I lack too much theory, Im doing my best here
young blaze
tysm adzetto for the clarification, but this only works when n and m are positiv...
actually integral briefly can be expressed as (\log ^{n+1}(x) ((-m-1) \log (x))^{-n-1} (-\Gamma (n+1,(-m) -1) \log (x)))) (says wolfram), but for example, "(15/6) times need to apply integral by parts", it seems meaningless to me. So for example what does (\pi) times mean? Maybe it's the integer part of the real part of (n), but it's not necessary. Let it simply be (n\in \mathbb{Z}_{>0}).
twin meteorBOT
adzetto
cloud flicker
Yes
chilly pawn
actually integral briefly can be expressed as \(\log ^{n+1}(x) ((-m-1) \log (x))...
tysm for the input, ty adzetto
.close
solemn tendon
can someone explain how (2n)!/n!n! = 2nCn
and how (2n)!/(n+1)!(n+1)! = 2nC(n+1)
young blaze
can someone explain how (2n)!/n!n! = 2nCn
and how (2n)!/(n+1)!(n+1)! = 2nC(n+1)
Use definition of binomial coefficient.
loud heath
what does $\binom{n}{k}$ mean to you?
twin meteorBOT
giannis_money
solemn tendon
what does $\binom{n}{k}$ mean to you?
n!/k!(n-k)!?
loud heath
yes
vocal sleetBOT
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Available help channel!
unborn prism
The minimum distance of the points on the function from the coordinate origin???
f(x)=√4x+7
vocal sleetBOT
The minimum distance of the points on the function from the coordinate origin???...
paper depot
graphing won't really help much, will it
sand lodge
won't it
I feel like it'll be very obviois
I mean idk if u can't see it immediately
paper depot
@unborn prism you mean $f(x) = \sqrt{4x + 7}$, right?
twin meteorBOT
Ann
unborn prism
<@550391700288438323> you mean $f(x) = \sqrt{4x + 7}$, right?
Yeah
paper depot
and you want to find the distance from the origin to the closest point on this graph, yes?
unborn prism
yes
untold surge
Quadratic
unborn prism
d= sqrt[(x2 - x1)^2 + (y2-y1)^2]
How should I do it when y^2 is available?
The closest distance between a point and a line is: |ax+by+c| / sqrt[a^2+b^2]
I tried with these formulas....
civic otter
The closest distance between a point and a line is: |ax+by+c| / sqrt[a^2+b^2]
You don't have a line, so you can't use this
civic otter
d= sqrt[(x2 - x1)^2 + (y2-y1)^2]
Yes, this is correct
What are your two points?
One is the origin, (0, 0) right?
civic otter
Well, before that we have to write the distance formula
Call a generic point P belonging to that curve f(x)
Let's say its x-coordinate is t for example
Then, what's the value of its y-coordinate?
unborn prism
And y = sqrt[4t+7]
unborn prism
But how can I make sure that the point is the closest one?
civic otter
Optimization should ring you a bell about derivatives
unborn prism
With min of the equation?
unborn prism
I got sqrt[t^2+4t+7]: Delta<0
civic otter
Do you remember vertex formulas for parabolas? If not, don't worry
unborn prism
Is it y = a(x - h)2 + k??
civic otter
x_V = -b/(2a)
This one, for the x-coordinate of the vertex
unborn prism
Oh yeah
civic otter
Nice, I think we can use this instead of derivatives
So, as you wrote before, d(t) = sqrt[t² + 4t + 7], wasn't it?
unborn prism
-Delta/4a for the min?
civic otter
So, as you wrote before, d(t) = sqrt[t² + 4t + 7], wasn't it?
Do you now agree that minimize that is perfectly equivalent to minimize its square? @unborn prism
civic otter
-Delta/4a for the min?
Don't run (you still don't have a parabola)
unborn prism
Do you now agree that minimize that is perfectly equivalent to minimize its squa...
Wdym?
civic otter
If I want to maximize something with a square root, isn't it the same as maximize it but without the square root?
This because the square root is increasing, hence a point of maximum of the inside will result in a point of maximum also with sqrt
unborn prism
If I want to maximize something with a square root, isn't it the same as maximiz...
Yes
civic otter
Ok perfect
We can therefore minimise the polynomial t² + 4t + 7
What is the value of t that makes it reach a minimum?
unborn prism
3
civic otter
What is the value of *t* that makes it reach a minimum?
It is the abscissa of the vertex, isn't it? @unborn prism
civic otter
It is the abscissa of the vertex, isn't it? <@550391700288438323>
Hence -b/(2a) which is in our case -4/(2•1) so t = -2
civic otter
Shouldn't we calculate Ys?
In this case yes, I'll explain you in a while why, but usually for thus type of exercises no, because they ask us for the values of t (which allows you to find the coordinate of the point) and not the value of the distance
unborn prism
By the way it's without the minus
Oh why??
unborn prism
In this case yes, I'll explain you in a while why, but usually for thus type of ...
Don't we want the min of function t² + 4t + 7 to find the lowest value of t??
civic otter
Oh why??
Because that's the formula (you can get it by plugging -b/(2a) onto the equation ax² + bx + c)
civic otter
Don't we want the min of function t² + 4t + 7 to find the lowest value of t??
Minimum is the y value, point of minimum is the x (or here t) value where the minimum happens to be
But since this is a very particular case, I would prefer to explain it with derivatives if you agree
unborn prism
Oh you'r right!
unborn prism
Because that's the formula (you can get it by plugging -b/(2a) onto the equation...
It gives -delta/4a...
unborn prism
But since this is a very particular case, I would prefer to explain it with deri...
OK
unborn prism
Minimum is the y value, point of minimum is the x (or here t) value where the mi...
t1+t2= -b/2a???
civic otter
Btw are you sure you copied the exercise correctly from the book? Because I've never seen an exercise like this where the usual way goes wrong
civic otter
t1+t2= -b/2a???
Yeah in a sense , but forget about parabolas right now,we don't need them any more if we do derivatives
unborn prism
Btw are you sure you copied the exercise correctly from the book? Because I've n...
Sure the options are:
- sqrt[5]
- sqrt[3]
- 49/16
- 7/4
civic otter
Sure the options are:
1) sqrt[5]
2) sqrt[3]
3) 49/16
4) 7/4
Ahn you wanted the value of the distance, not the point okok
I misread the exercise sorry
civic otter
I got sqrt[t^2+4t+7]: Delta<0
But until here it was all ok
Now you simply have to calculate the derivative of that function d(t)
You should get $$d'(t) = \frac{t + 2}{\sqrt{t^2 + 4t + 7}}$$
twin meteorBOT
Alberto Z.
unborn prism
You should get $$d'(t) = \frac{t + 2}{\sqrt{t^2 + 4t + 7}}$$
I didn't get it !!!!
civic otter
I didn't get it !!!!
I suppose you know how to calculate a derivative, don't you?
If not you should revise how to compute them
At least power rule and chain rule
unborn prism
I suppose you know how to calculate a derivative, don't you?
Oh no I havn't learned that
civic otter
Oh my god, how are you supposed to do this exercise?
What has your teacher explained to you so far about derivatives?
unborn prism
Oh my god, how are you supposed to do this exercise?
I don't even know how it works!!!
civic otter
Ok, then I'm sorry to say that I don't know how to approach this exercise, if not in this way
But probably other helpers here know that
unborn prism
I just know parabola and 2nd degree function.
civic otter
Let's see if someone can help you better than me, fir sure there are a lot
civic otter
The minimum distance of the points on the function from the coordinate origin???...
<@&286206848099549185>
unborn prism
OK thank you BTW 🙏
civic otter
You're welcome 🤗
chrome raptor
First
For finding minimum distance, usually one does not minimize the distance function d but, instead, the square of the distance function d²
Minimizing d² is easier than minimizing d, because you get rid of the square root
So you want to minimize d²(x,y)=x²+y²
the square of the distance from point (x,y) to the origin
But, y=√(4x+7)
So you can express d² as a single-variable function
d²(x)=x²+4x+7
It's a parabola, so it has a minimum at its vertex
You can calculate it serveral ways
vocal sleetBOT
@unborn prism Has your question been resolved?
vocal sleetBOT
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Available help channel!
placid pier
vocal sleetBOT
placid pier
how do i calculate the degrees with these given infos?
ruby vapor
how do i calculate the degrees with these given infos?
You're given alpha and beta, from the diagram we know that alpha+beta+eta = 180 degrees, so we can use that to find eta. After that, we can use that alpha and epsilon correspond to each other and so on.
wary mantle
Scheitelwinkel
placid pier
You're given alpha and beta, from the diagram we know that alpha+beta+eta = 180 ...
i just started with trig, how do i know whiich ones correspond to each other?