livid helm
how do i solve for radius
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timid leaf
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sly sierra
not all good
sly sierra
^ obviously 0 is a zero of this one (verify by plugging it in)
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chrome steppe
Bungo's right.. the y-intercept occurs when x = 0, and since all the terms have an x...
marble carbon
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inner osprey
seems suspicious
inner osprey
ur multiplication is fairly incorrect
helps to take out common factors when multiplying large numbers together
inner osprey
ur multiplication is fairly incorrect
(for example, 900 * 200 = 180000 not 1800)
inner osprey
helps to take out common factors when multiplying large numbers together
(in this case, i'd recommend dividing both sides of your equation by 450)
marble carbon
marble carbon
Ain't no way I'm this bad at math and taking an honors course 💀💀
inner osprey
do u know how to multiply numbers together when they have multiple 0s in them
u can multiply them as usual without the zeroes, then add back on those zeroes
marble carbon
Oh my lord I realized my mistake
inner osprey
so like __2__00 * __5__000 would be 2 * 5 = 10... but also with the 5 zeroes from 200 and 5000
so 1000000 (six zeroes)
marble carbon
After realizing my stupidity, x = 112 🙏🙏
inner osprey
show again?
marble carbon
Realized 450 x 600 is not 27 000 😃
inner osprey
so like __**2**__00 * __**5**__000 would be __**2**__ * __**5**__ = 10... but al...
^
and when i said "i'd recommend dividing both sides of your equation by 450" earlier,
i was referring to this line
we have a 900 factor on the left, and a 450 factor on the right
so we can divide both sides by 450 to make our lives either
marble carbon
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trail owl
So I got something called Factorization of a trinomial and the next is what I apparently don't understand and I just need an explanation of how it works to make my math homework and English isn't my native language so forgive the inaccuracies I might have while spelling
Factor the trinomial X^2-4x-45. .The square root is obtained to the quadratic term (x^2) and two numbers are found that added together give the coefficient of the linear term (-4); and multiplied by the independent term (45) -45=(-9)(5)-4=-9+5 The factorization of the trinomial X^4X-45 is as follows: X^4X-45=(X-9)(X+5)
this is just an example so could anyone please explain it to me? All I have is something that mightn't be correct
the trinomial I have which was displayed as an example in class was the next one X^2-12x+32=(X-4X)(X+6) I'm not entirely sure if it's truly like that or it's another thing that I missed since I did pay attention to the explanation of the teacher but I still didn't understand anything since I can't interpret Spanish as greatly as English. I would add an image but it's in Spanish
vocal sleetBOT
So I got something called Factorization of a trinomial and the next is what I ap...
livid tapir
,w (x - 4x)(x + 6) simplify
twin meteorBOT
trail owl
Solo ocupo la explicacion de como se hace no necesito nada mas
livid tapir
vez que si dice 4x so son iguales
Pero
Ok, lo que pasa es que si tienes una equación quadratica $ax^2 + bx + c$, a veces se puede factorizar como $a(x - r)(x - s)$, dónde $a, b, c$ son numeros reales, y también $r, s$.
twin meteorBOT
992qqoloy
twin meteorBOT
992qqoloy
livid tapir
En todas estas expresiones $a = 1$
twin meteorBOT
992qqoloy
livid tapir
Asi que $b = -(r +s)$ y $c = rs$
twin meteorBOT
992qqoloy
livid tapir
Asi que para factorizar, puedes intentar algunos numeros r, s cuales que el producto sea igual a $c$, asta que $-(r + s)$ también sea igual a $b$
twin meteorBOT
992qqoloy
livid tapir
Por ejemplo con la primera expresión quadratica
8*4 = 32
Y -(8 + 4) = -12
Asi que con esa información, saves que $(x - 4)(x - 8)$ es igual a $x^2 - 12x + 32$
twin meteorBOT
992qqoloy
livid tapir
Pudieras haber intentado 1*(-32) o 8*(-4),etc. Tambíen
trail owl
Tiene sentido
livid tapir
Pero $-(1 - 32) = - 31$ y$ - (8 - 4) = -4$ asi que no trabajan porque no valen a $'b'$
twin meteorBOT
992qqoloy
livid tapir
Ahora intentalo con $x^2 + 16x + 63$ 
twin meteorBOT
992qqoloy
trail owl
Ahi vere
livid tapir
Por ejemplo -3*-21 = 63 pero no trabaja con b 
trail owl
Muchas gracias por tu ayuda dios te bendiga
livid tapir
De nada
trail owl
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vast shale
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vast shale
where did he get 2x^2/2 from?
i thought it would've been x^2/2?
oh wait
💀
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raven badger
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raven badger
I have no idea where to start. I tried to express it in the form ax^n+b^n-1 .. and subsituted the function in. it didnt get me anywhere
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raven badger
<@&286206848099549185>
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maiden sphinx
Usually a good place to start with these kinds of problems is to start plugging in values for x and seeing what you can find.
main knot
Usually a good place to start with these kinds of problems is to start plugging ...
he has to find P(x) polynomials for which the equation abov is true
p(x)2+1 = p(x2+1)
it's pretty complicated
maiden sphinx
It’s usually less complicated than it seems
You can immediately find that p(1) = 1 and p(2) = 2
So you might conjecture that p(x) = x works, which it does
maiden sphinx
The hard part is to prove what doesn’t work
maiden sphinx
You need an equation
P(x) = x is an equation…
main knot
no it's a function it's not really the same you need some way to show every P(x) that meets the criteria
maiden sphinx
yes that's what I'm trying to explain
if $P(x) = x$ then $P(x^2 + 1) = x^2 + 1 = (P(x))^2 + 1$
twin meteorBOT
Awesam
main knot
There doesn't seem to be any good info, and I can't really understand much of it anyways but it has something to do with the combined functions idk much more than that.
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vast shale
why does LCM include all the ements
vocal sleetBOT
why does LCM include all the ements
hardy vector
This comes directly from the definition
If it didn't include all factors it won't be a multiple
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marsh orbit
30 60 90 Triangle.
ABC is Right Angle Triangle AB⊥BC
AD Is angle bisector of BAC
Given DC=2BD
Calculate angle C
inner osprey
as it would turn out, ABC is a 30-60-90 triangle with the given information
and so is ABD
which is pretty interesting since that's apparently what you're supposed to find 🤔
marsh orbit
which is pretty interesting since that's apparently what you're supposed to find...
Lmao
i know but how to write it?
I have a test and ofcourse it wouldnt tell me what Triangel is this,
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hoary cypress
how to solve this eqn for x
k1 - k2(x) = dx/dt
vocal sleetBOT
how to solve this eqn for x
k1 - k2(x) = dx/dt
hoary cypress
k1 and k2 are constants
hoary cypress
dt=dx/(k1-k2x)
how do i find integral of that
whole gale
by integrating both sides
signal pendant
Substitute k1-k2x=u
whole gale
why
hoary cypress
ok thanks
signal pendant
I know
whole gale
so integral 1/x = ln|x|
whole gale
its power is -1
if degree of x in the denominator is 1
well carry on
you can proceed in a way which you find convenient
hoary cypress
you can proceed in a way which you find convenient
what was the other way u were talking abt
whole gale
ill tell you once you solve it
hoary cypress
i solved it
signal pendant
They both are really the same, the integral will be a logarithmic function and you can guess ln(k1-k2x)/-k2
whole gale
yes
signal pendant
Its just the chain rule in reverse
whole gale
my bad if you got confused
hoary cypress
Its just the chain rule in reverse
but somehow convinved myself it was wrong
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tawdry turret
Given: Quadrilateral ABCD is a square.
Quadrilateral EBGF is a square (see diagram).
I need to prove that EG || AC
tawdry turret
So I thought of passing a diagonal in the square EBGF
And then continuing it to D
But, I realized, I can't force it to both be the diagonal of EBGF and ABCD
If I could, then it would have formed a 90° angle with both EG and AC and that's it
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tawdry turret
<@&286206848099549185>
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dusty nova
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dusty nova
Why does the general solution just have x^2
Wouldn't it be some constant multiplied by it?
sullen notch
x^2 gets you the inhomogeneous term
The other terms in the solution are for the homogeneous equation so you can multiply them by whatever (since putting them into the LHS gives 0)
regal bane
Wouldn't it be some constant multiplied by it?
I suggest trying 2x² in the left side, to see it doesn't give the right
dusty nova
Oh right so c1 and c2 would just adapt to if it were 18x^2 or any x^2 term?
sullen notch
Not really.. c1 and c2 can be any numbers, with no constraints
That’s because you change the RHS of the DE to 0, those last two terms are the solutions to it
And since the equation is linear, you can use the superposition principle to add any multiple of the homogeneous solutions
dusty nova
And since the equation is linear, you can use the superposition principle to add...
appreciate it
Thank you. Wasn't sure if the multiple added needed to have a constant to show it could be any multiple.. idk
but that sounds fine then
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cobalt wren
Hi I'm just strugling with this problem and dont really know how or what format to derive the non linear ODEs in?
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Hi I'm just strugling with this problem and dont really know how or what format ...
cobalt wren
This is the step I am working on and confused about
I get as far as balancing froces and confused where to go
<@&286206848099549185>
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true cliff
how do u
true cliff
solve the definite integral of
x^2cosx/1+e^x
considering the upper limit is pi/2 and lower limit is -pi/2
opal obsidian
what have you tried?
inner osprey
x^2cosx/1+e^x
hint for solving integral like this: ||what happens when x is replaced with -x?||
true cliff
what have you tried?
im lost as fuck
true cliff
hint for solving integral like this: ||what happens when x is replaced with -x?|...
$\frac{x^2cos(x)}{1+e^x}$
twin meteorBOT
sin
twin meteorBOT
sin
twin meteorBOT
inner osprey
if u are unfamiliar with IBP then this is def not smth u should be looking at
true cliff
if u are unfamiliar with IBP then this is def not smth u should be looking at
wtf is ibp
inner osprey
the process used to solve such integral ^
true cliff
interesting
inner osprey
"integration by parts" maybe you know it by another name
true cliff
what does it say
inner osprey
internet will help you here
u may know u-substitution is reverse chain rule
IBP is reverse product rule
kind of
inner osprey
hint for solving integral like this: ||what happens when x is replaced with -x?|...
.
vocal sleetBOT
rose shard
I’m currently in the inductive step
I was thinking on inserting these two based on our assumption (IH)
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rose shard
Same issue
vocal sleetBOT
Same issue
rose shard
I have arrived at this I don’t know how I can simplify it further For my induction proof
edgy sapphire
adding powers of 2 gets you close to the next power of 2
it's sort of just a thing to remember
edgy sapphire
yea the sum of powers of 2 lets you simplify that sum
does this help understand the backwards thing?
edgy sapphire
okok
rose shard
You see
I just inserted k=i for the first element in the sun
And the end element in the sum where k=j-1
edgy sapphire
alright what part are we at
rose shard
Here I say, we could write it as two times the sum of two to the power of Jay minus K
edgy sapphire
right
rose shard
And then we get to this point
And the task is again to show this
And now I’m showing it holds for K plus one as well
edgy sapphire
hm I'm not seeing how to get the right inequality you want to show in the end, it seems like it's sometimes less
rose shard
edgy sapphire
but usually the simplificaiton is that a sum of 2^n from anything to k is greater than 2^k and less than 2^(k+1)
rose shard
Here is the whole thing by the way maybe I’ve done something wrong
rose shard
but usually the simplificaiton is that a sum of 2^n from anything to k is greate...
🤔 hmmmm
edgy sapphire
oh it's really jank omg
so like the base case was j-i=1, so for induction you're allowed to assume j-i >= 2
so you can make a tighter restriction if you turn the giant summation into its largest 2 terms
like this
that should all work out
rose shard
before I do that I was just wondering can I not use geometric series to simplify the sum?
edgy sapphire
uh that should work too, if it gives something pretty simple at the end
rose shard
Yeah, I’m not quite sure how😂
I’m going to watch a video on it real quick
So ours is 2^1 + … + 2^(j-i)
edgy sapphire
2^i + ...
rose shard
2^i?
edgy sapphire
adding powers of 2 gets you close to the next power of 2
it's just this thing, so it should simplify to 2^(j+i+1)-2 after all the math
edgy sapphire
a should be 2 here since 2^1 is the first term
but yea then it's good and should make the induction work out in the end
edgy sapphire
r is 2, the ratio
edgy sapphire
right
rose shard
Don’t think it’s right
When assuming i is a variable, I get this
Or maybe it is right and then I just have to argue that then it’s higher
Even though I don’t arrive at that form that you sent which would be nice
To end the proof like that
edgy sapphire
the wolframalpha result looks right
rose shard
it's just this thing, so it should simplify to 2^(j+i+1)-2 after all the math
But I don’t get this
edgy sapphire
that's this part but I forgot to put a 2* in front
rose shard
So how do I connect it to the proof in the end here?
Because I wrote tij is greater
edgy sapphire
set it up >= 2^(j-i+1)-1 which was the starting goal
and you can move terms around to show it's true
rose shard
But have I technically not written it wrong then?
Because I have written t_i,j is greater than what we’ve just all this we’ve done
Where is it correct and just need to do something like this
edgy sapphire
yea this works because the goal is tij >= 2^... -1, and so far you have tij >= ... >= >... in a long chain
so if you end at that part you're good
rose shard
And maybe it should be equal instead of a greater than chain or what do you think?
edgy sapphire
I wouldn't call that right side t_ij, it's just a lower bound the problem gave
rose shard
Such that you know the difference that right now I am just rewriting the equation, and then the bound in the end
edgy sapphire
like the big picture is prove tij > lower bound, and the proof is tij > > > > > > > > lower bound
rose shard
I wouldn't call that right side t_ij, it's just a lower bound the problem gave
True tin is greater than that bound or equal but it isn’t tij
rose shard
Such that you know the difference that right now I am just rewriting the equatio...
What did you think about this?
edgy sapphire
the bottom row looks good to me
rose shard
Perfect and I should just argue that because you have the same exponent except the plus one part but in the left side you have four multiplied by it hence it outgrows?
edgy sapphire
eh it's not that it outgrows it has to be true for all j-i >= 2, you can split 4*2^(j-i) into 2^(j-i+1) + 2^(j-i+1) to make the comparison easier
rose shard
Wait damn what
Would that not be Nice to
Too
Then, in the exponent you could see it’s now +2 instead of only +1
edgy sapphire
uh I'd replace i with x when using wolframalpha
yea the left side is definitely a bigger exponent, but proving the >= is a little annoying
if you rearrange a bunch ofterms it'll have 2^... on the left and 2(j-i) on the right
rose shard
edgy sapphire
which is kind of a giant standard induction problem on it's own (?) but here I'd just say it works when you graph it out
edgy sapphire
yea there's not really a great way to algebraically prove it here, but graphically youcan show it works for (j-i) > 1 so it's valid
the "perfectly mathematical" way would be with another entire induction problem https://www.quora.com/How-do-you-prove-2-n-n
rose shard
Yeah, I’m definitely not doing a whole other proof
I already spent so much time on this
😂
edgy sapphire
yea totally
rose shard
Then I’d rather go down
Fail the course
But I just don’t understand shouldn’t it be for all ING
I and j
l=j-k
And it has to work for all l>=0
Oh and when its 0 we hit the base case
Makes sense
edgy sapphire
yea it has to work for all L and L=j-i so that's sort of our single variable
edgy sapphire
just "graphing both sides shows the left larger than the right for (j-i)>1, so we're done"
rose shard
rose shard
just "graphing both sides shows the left larger than the right for (j-i)>1, so w...
Yeah, okay I can’t just say visually you can see that the 2^(j-i+2) dominates the 2^(j-i+1)
rose shard
Yeah, okay is there a better term?
rose shard
I'm just saying larger
What do you think about my conclusion?
edgy sapphire
👍 looks done
rose shard
rose shard
👍 looks done
Awesome thank you for guiding me through this
♥️♥️🤞
At least I’ve done what I could now I’m gonna go to sleep it’s like almost 5 (midnight)
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winged burrow
Hi
winged burrow
,rccw
twin meteorBOT
winged burrow
What?
timber wagon
7/0 isn't equal to 7
timber wagon
How are you dividing by 0?
winged burrow
Ohh u mean its not the same thing?
timber wagon
Yeah
winged burrow
7 and 7/0 is different?
timber wagon
7/0 is not defined
main knot
try in your calculator
timber wagon
,w 7/0
twin meteorBOT
winged burrow
try in your calculator
It says math error
timber wagon
What does that tell you?
timber wagon
that's strange
It's not, it's using the answer we get from calculus
timber wagon
Can you see that they both have the same x coordinate?
winged burrow
How
timber wagon
And that is when the line is vertical
winged burrow
Ohh
timber wagon
Another way you can see that is thru the gradient you found
winged burrow
And that is when the line is vertical
What has vertical has to do with
timber wagon
It's undefined and gradient = tan(x) where x is the angle the line makes with the x axis
Tan(x) is undefined at x = 90
timber wagon
Yeah it does
timber wagon
Nvm
timber wagon
Your answer I'll just be x=0
timber wagon
I made an error mb
winged burrow
Ohh
winged burrow
Your answer I'll just be x=0
X=0?
timber wagon
Yeah
winged burrow
But
timber wagon
If the line is perpendicular to x axis then it'll have only 1 value of x hence it should be x=something
winged burrow
How should I show my working
If I can't 🔌 in the values
timber wagon
And then say that it means it's a line parallel to y axis
winged burrow
Just show that your slope is undefined
How
winged burrow
If I can't 🔌 in the values
.
timber wagon
It's just defined that way
timber wagon
You just need to say this stuff
winged burrow
If the line is perpendicular to x axis then it'll have only 1 value of x hence i...
I didn't got this part
@timber wagon
<@&286206848099549185>
main knot
the line is on x=0 and has no slope bro
main knot
x=0..
winged burrow
If the line is perpendicular to x axis then it'll have only 1 value of x hence i...
I didn't got his this part.And he told me to write it like a answer
Like this
main knot
well this is jsut a weird case where when you do the method to calculate slope it's an issue
so that's why he's saying to do that
you kind of jsut have to infer that x=0 here
winged burrow
No like as a answer
winged burrow
Can I say that as 7/0 is undefined x=0
main knot
honestly that's just not necessary
winged burrow
you can type some thing like
Then??
main knot
you can jsut say since there is no change on the x axis we can infer that the line has no slope and x=0
or just type x=0
if you want to say 7/0 is undefined so x=0 you can
I'd just write x=0
winged burrow
Ohh
winged burrow
you can jsut say since there is no change on the x axis we can infer that the li...
Can I draw and show it to u
To see if I got it
main knot
ok
winged burrow
Ohh
But the straight line is y axis
Not x
I mean both are straight but x axis is horizontal
winged burrow
Oh wait I think I got it
There is change in y axis becuz there is -3 and 4
But there is no change in x-axis as there isn't any value and therefore as there isn't any value there is no change of gradient
winged burrow
but there is a change in y
Yea but to use the formula we need both changes
white acorn
for along x axis u get 0/whatever u choose so gradient=0
winged burrow
Yea
white acorn
since u get anything/0
winged burrow
Then the whole gradient is 0?
white acorn
So gradient for y=0 or y=c is undefined
white acorn
Then the whole gradient is 0?
no, the whole gradient is undefined
It's rise over run right
rise is infinity, run is 0
so u get undefined
main knot
it's not infinity
white acorn
it's not infinity
okay, constant.
main knot
it's between points -3 and 4
winged burrow
It's rise over run right
What is rise/run now
white acorn
u can't change undefined to 0
white acorn
I just learn change in y axis and change in x axis
same thing lol
yeah so change in y axis can be any constant
but the change in x is 0
So it's undefined
winged burrow
yeah so change in y axis can be any constant
How constant?
white acorn
IT DOES NOT BECOME 0
white acorn
How constant?
any number is a constant as long as there's no variable like x or y or whatever
winged burrow
How y axis is constant then but x axis is not
white acorn
How y axis is constant then but x axis is not
constant means number
Like 1
Or 2
Or 3
winged burrow
Ohh
winged burrow
Then why run is constant but rise is always 0
white acorn
Then why run is constant but rise is always 0
nonono
winged burrow
That's called a constant
Ohh
main knot
idk what you guys are arguing about honestly
main knot
there's no slope and x=0 there's really nothing to it
white acorn
there's no slope and x=0 there's really nothing to it
yeah
anything/0 is undefined
Same for gradient
winged burrow
Or when there's no change in y
U mean when there's no change in y
And has change in x.Then rise is constant and run is 0
white acorn
U mean when there's no change in y
And has change in x.Then rise is constant an...
opposite
y is rise
x is run
winged burrow
Or when there's no change in y
.
winged burrow
I took it like this when there's no change in y it's rise is 0 and when there is change in y.Y is constant.Same goes for x when there is change in x.X is constant and when there is no change in x.X=0
white acorn
straight up is undefined, horizontal is 0
winged burrow
Did I grasp it right
winged burrow
straight up is undefined, horizontal is 0
Always?
winged burrow
I took it like this when there's no change in y it's rise is 0 and when there i...
And x means run
white acorn
And x means run
yup
winged burrow
I took it like this when there's no change in y it's rise is 0 and when there i...
Did I grasp it right
winged burrow
0
main knot
lol
winged burrow
Umm
white acorn
is it straight up
main knot
it does not exist dude
white acorn
if it's straight up what is it
.
white acorn
straight up is undefined, horizontal is 0
.
white acorn
A visual illustration would be perfect
okay, if there is no change in x, the graph is straight up. Like
main knot
winged burrow
okay, if there is no change in x, the graph is straight up. Like
Like it will keep going upp
white acorn
winged burrow
Forever
winged burrow
Ohh and its always?
winged burrow
Are u in class
white acorn
Are u in class
Yes 💀💀
winged burrow
Nice
white acorn
I'm not the teacher but I got special rights👍👍
winged burrow
I'm not the teacher but I got special rights👍👍
Ohh prefect?
white acorn
Ohh prefect?
no lol okay wait
Do u understand
If I give u the points (-1,196) and (-1,814) what's the gradient
winged burrow
Umm
white acorn
okay is there a change in y
winged burrow
Yea
white acorn
Is there a change in x
white acorn
so what is it
white acorn
when y changes and x has no change
.
winged burrow
Constant
white acorn
when y changes and x has no change
When x has no change but y has change, the gradient is undefined
winged burrow
it's undefined
I was going to say but over thought and said this 💀
white acorn
I was going to say but over thought and said this 💀
💀
don't overthink
try (1,28) and (1,-28)
what's the gradient
winged burrow
When x has no change but y has change, the gradient is undefined
Ohh and when x has change and y has no change?
white acorn
Ohh and when x has change and y has no change?
0
winged burrow
try (1,28) and (1,-28)
Oh I can assume it as x or y right
white acorn
cuz u get 0/change in x
white acorn
try (1,28) and (1,-28)
What's the gradient
winged burrow
The gradient is undefined
winged burrow
Cuz x has no change and y does
white acorn
Here the gradient is 0
Correct
winged burrow
Cuz has x has change and y doesn't
When we do not get any value of x but get of y we state that as undefined but when we do get a value of x but not y we can say that it is 0
@white acorn
.close
vocal sleetBOT
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Available help channel!
mossy latch
what are the shapes of each graph?
vocal sleetBOT
what are the shapes of each graph?
mossy latch
btw this is maths casue its statistics
hardy vector
what are the shapes of each graph?
They seem to have use kernel density estimate to get the graph
mossy latch
They seem to have use kernel density estimate to get the graph
What
I don’t think I’m at that level
hardy vector
If you want you could describe the shape
mossy latch
If you want you could describe the shape
As?
hardy vector
somewhat like a normal distribution or logistic distribution
mossy latch
somewhat like a normal distribution or logistic distribution
It’s not really a normal distribution
Cause mean doesn’t equal mode
Or it has to be perfectly symmetrical
hardy vector
key word somewhat like. Take any real world data, it is almost definitely going to follow a normal distribution
*not going
vocal sleetBOT
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mossy latch
<@&286206848099549185>
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mossy latch
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sudden moon
How do I test for point symmetry not about the origin, and line symmetry not relative to just the x and y axis algebraicly
vocal sleetBOT
How do I test for point symmetry not about the origin, and line symmetry not rel...
sudden moon
I know that even function => line symmetry with the y axis
and odd function => point symmetry about the origin
but if the function is not symmetric about the origin or the y and x axis how would I test it algebraicly
vocal sleetBOT
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merry prawn
this is a solution to a problem, i dont' understand why i need to take the transpose to check if the determinant is zero
vocal sleetBOT
this is a solution to a problem, i dont' understand why i need to take the trans...
obsidian stream
You don't need to, it's just there because they probably expect you are more comfortable with row operations
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tawny nacelle
there’s no product of functions here
Integration by parts is not needed
The - in front of the integral is distributed
The 3e^3x gets the -, and the -6e^x becomes a +
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vast shale
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vast shale
im having a big difficulty with continuity
if someone could just help me understand it w the help of these questions would be much appreciated
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livid void
Im not understanding this at all
wraith python
Im not understanding this at all
What are you having difficulty understanding?
livid void
What are you having difficulty understanding?
It seems like f(x) would be negative 1
vocal sleetBOT
It seems like f(x) would be negative 1
livid void
I dont get how its 0 in the question on the left
livid void
lim?
tight sonnet
limit
livid void
also so is that always the case?
tight sonnet
as x go more and more left y is close and closer to 0
and thats how you evaluate it
livid void
Theres no reference to limits in the chapter for "Rational Functions and their graphs"
tight sonnet
so forget about a term "limit"
that is asking when x is going left to "- infinity"
you are looking for the height its getting close to
and this height here is 0
do you understand?
wraith python
Are you studying end behaviors?
livid void
It was covered in a earlier chapter
wraith python
In the context of the question, you can substitute the y-axis with f(x).
The question is asking you what is the value of function relative to the y-axis as x goes to -infinity.
You can visually see that the curve is approaching the x-axis, or y = 0.
livid void
I think the other 2 lines are whats making me confused about the -inf one
Is it just always f(x)->0 when x->inf or -inf?
wraith python
No. Have you learned about asymptotes?
livid void
Yeah, but I had trouble understanding them. I seem to have trouble understanding arrow notation.
wraith python
If you look at this graph, there is a horizontal asymptote at y = 1.
The end behavior of f(x) approaches y = 1 as x goes to both -infinity and +infinity.
So it doesn't always go to y = 0.
Sometimes you can have a slanted asymptote. In those cases, the end behavior would go towards positive or negative infinity, depending on the slope of the slanted asymptote.
livid void
So I know how to find them, but there was no real reference to a question like this in the book.
wraith python
You do have to use your imagination a bit with regard to end behavior.
livid void
Thats why I posted the example because nothing showed the bottom line or how moving it horizontally affected it
like the right line seems to pass 0 so what would its f(x)->? equal
https://cdn.discordapp.com/attachments/903463353417072641/1215791437396840570/image.png?ex=65fe0902&is=65eb9402&hm=1f3794713ba91db328baf37f991818f0e9eb1f7f76e982e51aec3a29e5d22ee2&
wraith python
It would also be f(x)=0
hard crest
Yes
wraith python
You will eventually learn how to make those types of graph like so
At this point, you are probably only supposed to develop a basic understanding of what end behavior is.
Evaluating the values of end behaviors, aka limits as x -> +inf and x -> -inf, comes much later in Calculus.
livid void
hm, alright then. Guess I get the general idea then.
wraith python
Let me add on to the concept of end behaviors a bit.
You will find that the end behaviors of functions will usually do one of three things.
And you will find that certain types of polynomial functions will exhibit one or more of these end behaviors.
livid void
what would be a case of y not equaling 0 y -> to something not 0 then?
hard crest
The limit ?
wraith python
f(x) = x^2 would be one example.
Here you can see that the end behavior of f(x) = x^2 goes to +infinity as x-> -infinity and x-> +infinity.
hard crest
This one too
livid void
But like a rational function that x->inf but not f(x)->0 is what I mean
olive heron
two numbers x and y are chosen so that -2<x<5 and -3<y<6
what is the probability that x+y > 5
olive heron
Is there a fast way to calculate it other than brute forcing?
pallid zenith
uniform?
olive heron
what?
olive heron
yes
regal bane
Are these real numbers, or do they have to be an integer?
olive heron
integer
pallid zenith
integer
maybe you could build a table
its a little large so i wouldnt do it by hand unless you dont fill everything in
but you could kind of box out the places where x+y = 5
olive heron
but you could kind of box out the places where x+y = 5
than do the same with 6, 7, 8, 9, 10, and 11?
regal bane
There's a nice pattern, and you will catch onto it after seeing x+y = 5
pallid zenith
than do the same with 6, 7, 8, 9, 10, and 11?
hmm? no
here, lemme see if i can throw it together
but, i mean
lets say we fix y
and let x vary
so, lets let y be as big as it can be
y=6
whats the smallest x such that x+y>5 still
olive heron
whats the smallest x such that x+y>5 still
-1
pallid zenith
now let y=5
olive heron
smallest is 1
olive heron
2
olive heron
yes
pallid zenith
imagine we laid out all the values like, a table
x values going across the top
y values going down
maybe you say...y increasing downward, x increasing rightward
this is going to draw a barrier that kind of looks like a staircase
and its gonna separate one side where x+y>5 and one side where it doesnt
pallid zenith
you might imagine iterating this as far forward in your head as you can and ask how small you can make y such that there is an x that can provide x+y>5
pallid zenith
y | x
6 0 1 2 3 4 5
5 1 2 3 4 5
4 2 3 4 5
3 3 4 5
2 4 5
1 5
halfway
lemme see here
i suck with excel but maybe i can do something
,w Table[x + y, {y, -3, 6}, {x, -2, 5}]
twin meteorBOT
olive heron
hi
pallid zenith
if you had this on an exam or something, you wouldnt draw this whole table
the red x's, those are values we specifically identified as not working
for example, look at the row 'y=6'
you said that x=-1 won't work
and likewise y=5, x=0 ...
so, this is more or less it
one side is a region where x+y > 5
the other, it doesnt
olive heron
ok
olive heron
8x5-5
pallid zenith
so since the x are the ones = 5, its the squares on the bottem right
squares that what
olive heron
shaded area
pallid zenith
i just mean, identify the region is all
sure, yea, what area is this
i mean combinations in this region have what property
olive heron
the amount of squares per row is the y value -1?
pallid zenith
what? no
the problem is asking us to do what
identify the probability that some integers in that range will satisfy x+y > 5 right
olive heron
yes
pallid zenith
you drew a black region up here
whats true in the black region
we add x and y
and what
olive heron
its bigger than 5
pallid zenith
,calc 10/48
twin meteorBOT
Result:
mental falcon
0.20833333333333
20% ish
pallid zenith
ish
olive heron
There's a nice pattern, and you will catch onto it after seeing x+y = 5
is there like an algebraic pattern fomula thing?
pallid zenith
you could try deriving one
I mean, we know where the edge of the red x area is gonna be
if you want x+y > a
then the line will be along the combos where x+y = a + 1
we can probably figure out where this intersects with the edges of the region ...
regal bane
I see you guys made a triangle. The top row has 1 entry. The second row has 2 entries. The third has 3 entries...
If only we could quickly solve 1 + 2 + 3 +...
hmm
HMMMMMM
pallid zenith
If only we could quickly solve 1 + 2 + 3 +...
famous unsolved problems in mathematics
regal bane
I guess we'll never know
pallid zenith
this seems like a good problem to keep chewing over tbh
you need to incorporate the ranges youre given somehow
olive heron
regal bane
Google "triangular numbers" for a bit of extra learning imo
You won't need them here, but there's a pattern for sure
olive heron
authaeuhtua
pallid zenith
authaeuhtua
so, just know theres a formula
1+2+....+n = T_n where you can always get what T_n is
i think this is jumping forward
but maybe me and kaynex are thinking about the problem differently
to me: its a rectangular region, and you have a nice diagonal line cutting through it
if you can identify where that happens in terms of the ranges and the given quantity the sum needs to be greater than, youre super in business
pallid zenith
authaeuhtua
because the problem can be phrased in terms of this
but given a<x<b and c<y<d, requiring that x+y > s
then whats n?
its pretty generalized so its a pain but you can totally do it
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gentle umbra
i need help with solving these type of problems
