it shouldn't
#help-26
cinder sequoia
ivory moon
Probably just the first part would be correct
cinder sequoia
technically it's the first part divided by the derivative of x+1
which doesn't do anything since it's division by 1
sharp dew
I'm terrified
ivory moon
This one doesn’t need U-sub
sharp dew
by the complexity of the operation
cinder sequoia
reverse chain rule is just a specialized case of u sub
sharp dew
This one doesn’t need U-sub
then how can I solve it?
sharp dew
reverse chain rule is not a official thing
ivory moon
then how can I solve it?
Except if you make U=x+1
sharp dew
Except if you make U=x+1
is that U sub
ivory moon
Wdym is it not?
cinder sequoia
yes but technically unnecessary the moment you realize du = dx
ivory moon
Yeah I meant it’s unneeded here
thorny flameBOT
77²
sharp dew
I do not like the simplied result
sudden temple
this is the more generalized version
sudden temple
I do not like the simplied result
it is just came from U-sub
cinder sequoia
cause I do not know how it works
look up u sub
that's all it is
reverse chain rule is technically just a u-sub shortcut
ivory moon
cause I do not know how it works
You just substitute an expression as a variable so that it looks nicer
sharp dew
You just substitute an expression as a variable so that it looks nicer
when I substitute x+1 by U
I should learn about u-sub
sudden temple
how are you even solving integrals without knowing u-sub?
bro's a G tbh
ivory moon
how are you even solving integrals without knowing u-sub?
bro's a G tbh
Probably easier ones?
sudden temple
Probably easier ones?
like just single function?
ivory moon
how are you even solving integrals without knowing u-sub?
bro's a G tbh
This problem doesn’t really count as a problem that really needs U sub
ivory moon
like just single function?
Ones without fractions and stuff
sudden temple
Ones without fractions and stuff
yeah one without composite functions
ivory moon
Yeah
This is one that can be easily solved just by looking if you’re used to integrals.
neon iron
is there a nice way to denest:
topaz sinewBOT
is there a nice way to denest:
neon iron
$\sqrt{2 - \sqrt{2 + \sqrt{2}}}$
thorny flameBOT
!Kiz__
coarse tusk
Isn’t there like a denesting formula
thorny flameBOT
ƒ(Why am. I here)=I don't Know
neon iron
there is but the general one involves galoise theory
wary tulip
hi kiz
neon iron
Just spam that you should be good
wait but it already looks annoying to do
coarse tusk
It does
neon iron
as in it doesn't look like it has any "compact" way
coarse tusk
Where’s the problem from?
neon iron
,w a + b = 2 and sqrt(2) = 2 sqrt(ab)
thorny flameBOT
neon iron
Where’s the problem from?
it was a sub problem to one of the things i was doing
coarse tusk
Right
thorny flameBOT
coarse tusk
Ugh
I’ll be honest this doesn’t seem very doable
Atleast it probably doesn’t have a good answer
neon iron
,w a+b=2, 2sqrt(ab)=sqrt(2+sqrt(2))
oh yeah you went for the main thing
i was doing the inner bit first
yeah okay i have to show that is greater than 9/20
which it clearly is
coarse tusk
,w sqrt(2-sqrt(2+sqrt(2))) - 9/20
thorny flameBOT
coarse tusk
Seems like it isn’t…
coarse tusk
!xy
topaz sinewBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
neon iron
and yes that is thepremise of theq uestion
in this case i'd have to show that it is less than 9/20
i just did this a whole while back aso i didn't remember my answer
neon iron
i don't have the question tbh
it's really just showing that radical is
either less than 9/20 (which us howed with WA)
coarse tusk
i don't have the question tbh
That doesn’t seem like a good idea…
neon iron
how to show it analytically?
coarse tusk
how to show it analytically?
Just square both sides repeatedly
Should be good enough
neon iron
right that's what i did
and the other way was be an engineer and use the approximation of sqrt(2)
so okay there's no nice way to simplify that nested radical i'm guessing?
coarse tusk
Probably not
neon iron
i see, thanks
neon iron
Probably not
at what point does it become "obvious"
now that i think about it i used a bit of the fundamental theorem of engineering lmao alongside squaring (probably)
squaring leads you to
,, 6 + \sqrt{2} - 4 \sqrt{2 + \sqrt{2}} \text{ vs. } \qty(\frac{9}{20})^4
thorny flameBOT
!Kiz__
neon iron
or if you want: ${\color{red}6 + \sqrt{2}} \text{ vs. } {\color{green}\qty(\frac{9}{20})^4 + 4 \sqrt{2 + \sqrt{2}}}$
thorny flameBOT
!Kiz__
neon iron
i'm guessing comparing the values of this requires a calculator?
and there's no algebraic way to do this
topaz sinewBOT
@neon iron Has your question been resolved?
topaz sinewBOT
@neon iron Has your question been resolved?
neon iron
nah
odd flex
not sure if i'd do the same even if i knew it 
my calculator knows it to 10,000 decimal points
topaz sinewBOT
@neon iron Has your question been resolved?
neon iron
frfr
sudden temple
lemme give you an advice
neon iron
originally i had to show why $\frac {9}{20} > \sqrt{2 - \sqrt{2 + \sqrt{2}}}$
thorny flameBOT
!Kiz__
neon iron
i squared twice to get here but that surely doesn't help
sudden temple
**!Kiz\_\_**
yeah here
just square once and isolate the rads and other numbers
and then square after that
if you just double square keeping them on same sides, it will be very messy to interpret
which side is bigger
high void
which side is bigger
the right side is bigger
9/20 is smaller than 1
you need to prove the other side is bigger than 1
sudden temple
,w compute sqrt(2-sqrt(2+sqrt2))
thorny flameBOT
sudden temple
,w compute 9/20
thorny flameBOT
sudden temple
nw
neon iron
which side is bigger
okay but
you'd still need to know what sqrt(2) is to some precision right?
,calc (2-0.2024)^2
thorny flameBOT
Result:
3.23136576
@neon iron Has your question been resolved?
brave fox
Today i learned that knowing that something is possible and actually doing it in practice are 2 very different things
1.41² = 1.9881
1.84² = 3.3856
sqrt(2 + sqrt(2)) > sqrt(2 + sqrt(1.9881)) = sqrt(2 + 1.41) = sqrt(3.41) > sqrt(3.3856) = 1.84
sqrt(2 - sqrt(2 + sqrt(2))) < sqrt(2 - 1.84) = sqrt(0.16) = 0.4 < 9/20 < 0.45```@neon iron Has your question been resolved?
neon iron
Today i learned that knowing that something is possible and actually doing it in...
i can’t use the approximation tbh
sqrt(2) = 1.41 approximation , that is
brave fox
why not?
this is a rigorous proof
and at no point does it say sqrt(2) = 1.41 or even sqrt(2) ≈ 1.41
ripe crane
.close
thorny flameBOT
topaz sinewBOT
@neon iron Has your question been resolved?
neon iron
and at no point does it say sqrt(2) = 1.41 or even sqrt(2) ≈ 1.41
you’ve established a lower bound on sqrt(2) which was 1.41
it’s all just using an approximation which i don’t want to
anyway just repeatedly squaring should give:
A) ((2- 0.2025)^2 -2)^2
B) 2
Clearly B is bigger
brave fox
"clearly"
you also have to prove that the number isn't negative before you square it
specifically the difference
otherwise it's gonna break inequality
neon iron
you also have to prove that the number isn't negative before you square it
it’s obvious that the number isn’t negative whenever i squared it
by inspection
neon iron
"clearly"
you can multiply by hand it’s trivial
if need be
or you can bound it appropriately
it looks like it’s below 1.6-1.7
no where did i have to use 1.41 or 1.84 or whatever
brave fox
by inspection
it's non-negative not by inspection, but because that's how squaring works
hold on
neon iron
it's non-negative not by inspection, but because that's how squaring works
????????
squaring would not preserve the inequality
so whenever i squared i made sure
it was non-negative before squaring
and even if it was negative i just have to note that an inequality isn’t preserved so it’s still trivial
just reverse the final result how many ever times
brave fox
so you had to prove separately that 2 - sqrt(2+sqrt(2)) is non-negative?
if it was negative then the inequality wouldn't hold ... for obvious reasons
neon iron
so you had to prove separately that 2 - sqrt(2+sqrt(2)) is non-negative?
sqrt(2) < 2
brave fox
ok
brave fox
sqrt(2) < 2
you still have to put an upper bound for sqrt(2) then
neon iron
sqrt(2) is less than 2 because sqrt(4) = 2 and you know sqrt(x) is monotonically increasing
so eh it’s not a “fancy find” with the upper bound, it’s mostly utilizing the fact that sqrt(x) is increasing on its domain
if i argued that way then i don’t need any upper bound
it follows from the definition of an increasing function:
x_0 < x_1 -> f(x_0) < f(x_1)
brave fox
(and the fact that 4 > 2)
neon iron
that’s exactly what i said :) but yes
brave fox
you aren't wrong
novel yoke
,w Limit[[Sqrt[2]-Sqrt[1+cos(2x)]] ÷ [1-cos(x)],x->0]
topaz sinewBOT
,w Limit[[Sqrt[2]-Sqrt[1+cos(2x)]] ÷ [1-cos(x)],x->0]
thorny flameBOT
novel yoke
Why do i get this
novel yoke
Oh
ivory sorrel
Let me check
novel yoke
I wanna know how to get sqrt(2)
sudden temple
Why do i get this
how did you cancelled the sine?
novel yoke
how did you cancelled the sine?
Sorry i was supposed to erased it, bad habit
sudden temple
Sorry i was supposed to erased it, bad habit
kk
on the second step
that's gotta be -cos2x
novel yoke
that's gotta be -cos2x
denominator or numerator?
sudden temple
numerator
sudden temple
I don't see any other mistake
novel yoke
I don't see any other mistake
I still dont get sqrt(2)
novel yoke
yeah it comes root 2
Oh?
cunning kayak
(when I solved it)
so there is nothing wrong with answer key
and you got 1/root 2
novel yoke
(when I solved it)
Can i see your steps?
novel yoke
Ohh
novel yoke
and L hospital
Dang those two are overpowered
cunning kayak
cunning kayak
Dang those two are overpowered
can you remember a thing which I send?
you can replace like this
when it's limit is ending to 0
novel yoke
Hmm
cunning kayak
almost all of the above are derived from taylor series
you can apply it if it's in MCQ but if it's subjective
then you might not get marks for that
novel yoke
you can apply it if it's in MCQ but if it's subjective
Dang hm
novel yoke
Yo
novel yoke
Thankyou so much
fallow igloo
!nosols
topaz sinewBOT
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
novel yoke
What about the 1-cosx
Isnt it supposed to be 2sin²A/2?
cunning kayak
What about the 1-cosx
Isnt it supposed to be 2sin²A/2?
no
novel yoke
Long story short, they did
novel yoke
(1-cosx)(1+cosx) = 1-cos^2x = sin^2x
Ohhh
topaz sinewBOT
topaz sinewBOT
Available help channel!
near mantle
Can someone give me a hint? Only a hint
I dunno how to start. I have the fig
near mantle
Yeah, so what would be the relation of area between them?
Maybe I am missing a key concept here.... Similarity and Areas
?
icy sky
what do you know about medians and centroids
near mantle
Well, I know the centroid divided the median in 1:3 ratio and what centroid is and what median is, plus I understood that YX is parrallel to BC here thus GYX is similar to GBC
near mantle
Well, I can't seem to establish relation between the heights but I think I have a relation between BC and XY
BC = 2XY
According to midpoint Theorm
icy sky
draw the altitudes from BC to G and A
topaz sinewBOT
@near mantle Has your question been resolved?
near mantle
draw the altitudes from BC to G and A
Then?
icy sky
you should see a pair of similar triangles
near mantle
you should see a pair of similar triangles
Sorry to ping you so much
but I can't find those similar triangles
<@&286206848099549185>
near mantle
Can someone give me a hint? Only a hint
I dunno how to start. I have the fig
.
Well, I know the centroid divided the median in 1:3 ratio and what centroid is and what median is, plus I understood that YX is parrallel to BC here thus GYX is similar to GBC
Well, I can't seem to establish relation between the heights but I think I have a relation between BC and XY
BC = 2XY
According to midpoint Theorm
reef adder
Well, I know the centroid divided the median in 1:3 ratio and what centroid is a...
centroid divides the median in 2:1 ratio
Also i think BC = 2XY is wrong since Y is the mid- point of BE, to apply mid-point theorem Y should have been the mid-point of BG (same explanation for X)
reef adder
hmm you are right about we should start from proving similar triangles
near mantle
I can't find any relations, and I just found out the one I found is wrong
near mantle
hmm you are right about we should start from proving similar triangles
Only similar triangles I found are GYX and GBC
reef adder
Only similar triangles I found are GYX and GBC
ok great
now try to apply this... area(GYX)/area(GBC) = square of the ratio of their corresponding sides...
Then you will get realtion between the area of GYX aand GBC
after than establish and relation between area of triangle ABC and GBC
reef adder
take hint from this and try to establish that relation between area of ABC and GBC
near mantle
Hmmm... I'll try
reef adder
every step has a reason here... since we have the area of ABC, which is 480, and we have found GYZ and GBC similar... and it is easy to establish relation b/w ABC and GBC than ABC and GYZ... so indirectly we can find area of GYZ
topaz sinewBOT
@near mantle Has your question been resolved?
topaz sinewBOT
frosty iron
"Let there exist a sequence: 1, -2, -1, 4, 17, 46, ... Find the 10th number in this sequence"
sudden temple
2nd order difference seems interesting
topaz sinewBOT
topaz sinewBOT
Available help channel!
dull ravine
(k+1)/(k-3)=(4k-2)/(k+1)
topaz sinewBOT
(k+1)/(k-3)=(4k-2)/(k+1)
tardy lantern
Can I see your work?
mild oasis
Its -6 not + 6
tardy lantern
line 4-7
dull ravine
line 4-7
Can i ask if this works
(-3k+1)(k-5)
When equating for zero
I get -3k+1=0
And -3k=-1... Oh ok ty i got it
.close
topaz sinewBOT
tardy lantern
yh its right 😄
topaz sinewBOT
Available help channel!
neon iron
Hey
neon iron
Can somebody help me with ungrouping
I need to do it with these numbers! 11 36 53 10 95 0
It's a new term to me since im not english!
inner wren
!da2a
topaz sinewBOT
No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/
neon iron
ungroup these numbers 11 36 53 10 95 0
inner wren
!original
topaz sinewBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
neon iron
neon iron
i dont have an example
inner wren
Please provid more context
neon iron
roblox
inner wren
11
36
53
10
95
0
this should be correct
neon iron
LOOL
dusky thicket
this should be correct
Try to not get your self into this bc its a puzzle lol
I think he missed it up lol
neon iron
snitchg
dusky thicket
snitchg
Am not a snitch just telling you to not involve people that will not know about that kinda of puzzles lol
topaz sinewBOT
Available help channel!
wintry estuary
hello i am currently learning a chapter in my textbook "sketching gradient functions" and this is something im not understanding
topaz sinewBOT
hello i am currently learning a chapter in my textbook "sketching gradient funct...
wintry estuary
these are the 2 graphs
so the (x+7)^3 has a graph where the gradient is postive for the whole graph right
wintry estuary
hello i am currently learning a chapter in my textbook "sketching gradient funct...
how come the graph of the gradient function is in the negative x axis if the gradient of the original function is positive
To rephrase
if the gradient of f(x) is positive for the entire graph
wouldnt the value of x in the graph of f'(x) be positve for the entire graph too?
pearl patio
how come the graph of the gradient function is in the negative x axis if the gra...
the gradient sort of tells you how fast the fuction is decreasing and increasing at each point
when the gradient is negative, the function in red is decreasing, and when it is positive, the function in red is increasing
as you can see in your graph
note that deceasing from a positive value to a positive value is still decreasing! so it is possible that the function is positive but decreasing
wintry estuary
Ah okay I see I had a misconception that it had to be strictly positive
thanks for the explanation !
.close
topaz sinewBOT
topaz sinewBOT
Available help channel!
fast tundra
Greetings. I am currently attempting to solve this H.W
Q/ Prove that [(7 + 5ω²) / (7ω + 5) - (3 - 2ω) / (3ω² - 2)] ⁴ = 9
topaz sinewBOT
Greetings. I am currently attempting to solve this H.W
Q/ Prove that [(7 + 5ω²) ...
grim sable
omega
neon iron
i mean
grim sable
but yes
neon iron
okay
neon iron
it doesnt
remote agate
yeah im not solving this
fast tundra
Wait please. I have some steps noted
Might help
Q/ Prove that [(7 + 5ω²) / (7ω + 5) - (3 - 2ω) / (3ω² - 2)] ⁴ = 9
A/
I began by putting ω³ instead of 1 for both of 7 in (7+5ω²) and 3 in (3 - 2ω)
So, they become:
Term 1
- Numerator: (7ω³ + 5 ω²)
- Denominator: (7ω + 5)
Term 2 - Numerator: (3ω³ - 2ω)
- Denominator: (3ω² - 2)
We pick common factor ω² for the first term.
First Term
We simplify the denominator and numerator - Numerator: (7ω + 5)
- Denominator: (7ω + 5)
We pick common factor ω for the 2nd term
Second Term
- Numerator: (3ω² - 2)
- Denominator: (3ω² - 2)
After simplifying I am stuck at this step...
First Term
Becomes ω²
Second Term
Becomes ω
So, [ω²-ω]⁴
Now... what do I do? No matter how I solve this it still doesn't give me 9
I am assuming there is some formula I am not aware of
cursive patrol
Greetings. I am currently attempting to solve this H.W
Q/ Prove that [(7 + 5ω²) ...
so..
wait this is a different person
use this
sudden temple
Any other way?
Our Mr. didn't mention that omega squared = 1/omega
$1+\omega+{\omega}^{2}=0$
thorny flameBOT
77²
fast tundra
**77²**
Yep he did write this down
But it doesn't give me 9
Either he forgot this is a question for our next lecture, or I wasn't paying attention.
But I did rewind each lecture carefully just to see if I missed a tiny bit
cursive patrol
Any other way?
Our Mr. didn't mention that omega squared = 1/omega
this is extremely easy way
omega^3 = 1
divide both sides by omega
fast tundra
Mhm I know. But he'd get suspicious and tell me "OOOOO! You cheaaated huh?!"
cursive patrol
so.. what did your teacher tell you
because any other method will only make this problem harder than it needs to be
cursive patrol
omega^3 = 1
divide both sides by omega
this is a staple fact in problems involving cube roots of unity
fast tundra
so.. what did your teacher tell you
Only notes he mentioned were
- ω³ = 1
- And ω + ω²+ 1 = 0
Or formulas*
sudden temple
Only notes he mentioned were
- ω³ = 1
- And ω + ω²+ 1 = 0
in that case use that 2nd identity
there's not much calculation anyway
sudden temple
But it doesn't give me 9
what did you do btw?
sudden temple
Q/ Prove that [(7 + 5ω²) / (7ω + 5) - (3 - 2ω) / (3ω² - 2)] ⁴ = 9
A/
I began by ...
actually I am going now
but as a piece of hint:- Convert that omega square in the end to linear and then square it
cursive patrol
Only notes he mentioned were
- ω³ = 1
- And ω + ω²+ 1 = 0
yes ok perfect
ω³ = 1 is all you need. there is nothing stopping you from rewriting it as ω² = 1/ω
topaz sinewBOT
@fast tundra Has your question been resolved?
sudden barn
2 questions :
1 ) Why is tan theta = slope
2 ) how do i prove that tan theta = slope
topaz sinewBOT
2 questions :
1 ) Why is tan theta = slope
2 ) how do i prove that tan theta = s...
weary ledge
2 questions :
1 ) Why is tan theta = slope
2 ) how do i prove that tan theta = s...
use this: slope is delta y over delta x
check what is tan(theta)
sudden barn
sure, give me a second
weary ledge
btw you should make sure theta is the angle between line and x axis
sudden barn
gotcha
sudden barn
sudden barn
btw you should make sure theta is the angle between line and x axis
this is what i have for now
im not sure if i got it right or not but this should be right
weary ledge
tan(theta) = BD/AD
sudden barn
oh yeah thats the hyp mb
weary ledge
which is same as delta y over delta x
sudden barn
oh.
that is very much easy to understand
no idea what it took me that long
thank you!
weary ledge
haha np
sudden barn
.close
topaz sinewBOT
topaz sinewBOT
Available help channel!
amber locust
In the Oxyz space, given points A(1;1;2), B(1;0;3), and C(2;0;1), why does vector(AB).vector(AC) = vector(0)?
I know that vector(AB) = (0;-1;1), vector(AC) = (1;-1;-1), and vector(BC) = (1;0;-2) but I still can’t understand why
topaz sinewBOT
In the Oxyz space, given points A(1;1;2), B(1;0;3), and C(2;0;1), why does vecto...
amber locust
you mean the a.b.cos(a,b)?
wooden osprey
yeah, but thats not ideal
jade thunder
No the dot product is just multiply entrywise and add them up
wooden osprey
use the definition
amber locust
that's interesting I was only taught a.b.cos(a,b) ima look that up
AB . AC = 0.1 + (-1).(-1) + 1.(-1) = 0
ah
amber locust
use the definition
Thanks, I was so confused about how they identified the right angle using just coordinates
opal vault
In the Oxyz space, given points A(1;1;2), B(1;0;3), and C(2;0;1), why does vecto...
ah and small correction, vector(AB).vector(AC) = scalar 0
amber locust
oh so vector 0 is called scalar 0?
wooden osprey
yes, dot product does not output a vector, it outputs a scalar
wooden osprey
oh so vector 0 is called scalar 0?
no
opal vault
oh so vector 0 is called scalar 0?
no vector 0 is (0,0,0)
scalar 0 is... 0
the real number 0
amber locust
Ah ye right I got it
Forgot the product doesnt have direction
And I was confused cause I thought you guys call vector 0 scalar in English haha now i gotta look up what scalar is thanks yall
.close
topaz sinewBOT
topaz sinewBOT
Available help channel!
neon iron
topaz sinewBOT
topaz sinewBOT
@neon iron Has your question been resolved?
marble cliff
The area of a rectangle is 3x^2-5x-2
Determine an expression that represents the Perimeter (do not use quadratic formula)
topaz sinewBOT
The area of a rectangle is 3x^2-5x-2
Determine an expression that represents the...
pseudo sonnet
what have you tried?
marble cliff
So
Basicslly
The area of a rectangle is given by the formula Area = length * width.
In this case the area is 3x^2 - 5x - 2
pearl trench
The area of a rectangle is 3x^2-5x-2
Determine an expression that represents the...
hm, so its $$3x^2-5x-2$$
thorny flameBOT
Greydawn Dewer
marble cliff
For a rectangle, the perimeter is given by the formula
Perimeter = 2 * (length + width)
marble cliff
**Greydawn Dewer**
Yes
pearl trench
yup
pseudo sonnet
good!
marble cliff
We need to factor right so
The factored form of 3x^2 - 5x - 2 is (3x + 1)(x - 2).
we can identify the length and width of the rectangle so basically the
Length = 3x + 1
Width = x - 2
fair thorn
^
marble cliff
Now I calculate the perimeter using the length and width:
Perimeter = 2 * (length + width)
Perimeter = 2 * ((3x + 1) + (x - 2))
Perimeter = 2 * (4x - 1)
Perimeter = 8x - 2
marble cliff
not necessarily... <:eeveethink:601029006418116608>
Oh
pseudo sonnet
mhm
fair thorn
that factorizes to give the above expression too
pearl trench
we can identify the length and width of the rectangle so basically the
Length =...
not exactly yeah
pearl trench
I don’t follow?
its a way to represent the value
pearl trench
we can identify the length and width of the rectangle so basically the
Length =...
this approach is basic, but nearly fine, heh
pearl trench
So in that case c would represent it?
c would be a constat req to solve for the values
which would be surely determined by the orignal equation, factored or not
I cant specify it directly as then my chances of being wrong will be more, so yeah
marble cliff
we can identify the length and width of the rectangle so basically the
Length =...
Ohhh okay would you rewrite or should I keep it the same
As you said it’s more basic but does it work?
pearl trench
As you said it’s more basic but does it work?
it may, its an straight forward one, usually works for not so twisted questions
marble cliff
Okay
marble cliff
Now I calculate the perimeter using the length and width:
Perimeter = 2 * (lengt...
What about this
marble cliff
Yes
marble cliff
Thank you
pearl trench
nice
marble cliff
So the following question says
State an equivalent expression to the perimeter in the question before
pearl trench
So the following question says
ok, actually proceed, then i will speak
marble cliff
How would I do that
pearl trench
" equivalent expression to the perimeter" that should be 8x-2?
I mean, the original question said, represent it as a expression
that representation also means that the expression is equivalent
so yeah
and there are 2 methods to do this, first the method you tried
second
some kind of approximation like
marble cliff
that representation also means that the expression is equivalent
So I would just right that it’s equivalent already
marble cliff
and there are 2 methods to do this, first the method you tried
???
pearl trench
So I would just right that it’s equivalent already
yeah
like that or so
can you send a picture of the question?
I may understand it more correctly
and even so, the expression that you derived, is rather right along side, so the equivalency should be satisfied
pearl trench
???
the second one is to transfer the orignial equation into a different factored form
marble cliff
marble cliff
Yeah idk how to do that
pearl trench
Yeah idk how to do that
oh well, the second method would be to do it more like:
$$ (ax+b) (cx+d) = 3x^2 - 5x - 2 $$
thorny flameBOT
Greydawn Dewer
pearl trench
ac=3
ad+bc=−5
bd=−2
and for the factored form as you said
l=3x+b
w=x+d
and so on
this method is based on assumptions and on different forms, this may be long, but if you dont find a specific method first for the question, use this assuming-method with different forms
oh yeah, @marble cliff
factor 8x-2
try it
marble cliff
Okay
pearl trench
it can be the other form you thought of
marble cliff
I’m trying it right now one second
pearl trench
ok
and sure do use the method I described, it would be helpful if it fits to you
a try wont hurt
tawny widget
February 2024 had 5 Thursdays. What is the next year that will also have 5 Thursdays in February?
topaz sinewBOT
February 2024 had 5 Thursdays. What is the next year that will also have 5 Thurs...
tawny widget
Can anyone help with this? Thanks
icy sky
first off how can a february have 5 thursdays
pearl trench
first off how can a february have 5 thursdays
lol
viral turtle
first off how can a february have 5 thursdays
1st, 8th, 15th, 22nd, 29th
clever shore
Leap Year, I suppose.
viral turtle
Leap year
pearl trench
ah yeah
icy sky
can yall let OP answer
tawny widget
Why is it 2?
icy sky
its not
pearl trench
Why is it 2?
he did
tawny widget
Is it actually
pearl trench
Is it actually
idk
indigo island
February 2024 had 5 Thursdays. What is the next year that will also have 5 Thurs...
so first consider how does the february have 5 thrusdays
clever shore
can yall let OP answer
I assumed you were confused with how it would even be possible. My bad.
tawny widget
Is there like a way to do these questions
viral turtle
Is it actually
We can see that a day will only repeat 5 times if it’s the first of a leap year, so we know it’ll be some multiple of 4
indigo island
Is there like a way to do these questions
its a common sense question
you just gotta think
neon iron
bro you still there?
indigo island
the answer is 4 since a leap year is every 4 years once
pearl trench
man
icy sky
the answer is 4 since a leap year is every 4 years once
EXTREMELY LOUD INCORRECT BUZZER NOISE
tawny widget
Wait
stuck hearth
just cuz its a leap year
tawny widget
5 thursdays though
stuck hearth
doesnt mean it has 5 thurs
indigo island
**EXTREMELY LOUD INCORRECT BUZZER NOISE**
aw man
tawny widget
Exactly
icy sky
feb 2024 started on a thursday, note rhat it is a leap year
what day does feb 2025 start?
pearl trench
the answer is 4 since a leap year is every 4 years once
bro
2028 has thus on monday
and its a interesting pattern
check the calendars people
pearl trench
analysis, lets go
pearl trench
so feb of 24, 28th was on wed, thats why thus was on thus
coarse tusk
can everyone let one helper answer PLEASE
pearl trench
25 as 28 on fri as leap year shifts the dates by 1
tawny widget
Alright lmao
pearl trench
Alright lmao
do you want intuition or direct answer?
by direct answer, I mean direct method
even so
just open a calendar and observe
every year the 28 shifts by 1 day
icy sky
just open a calendar and observe
ah yes, let me get my calendar at a math competition
tawny widget
Direct
pearl trench
ah yes, let me get my calendar at a math competition
intuition,lol
tawny widget
Ill have the answer anyway in a moment
pearl trench
ok then, wait
pearl trench
Ill have the answer anyway in a moment
hm
tawny widget
As in I’ll receive the correct answers anyway
pearl trench
@tawny widget
Days of the week shift by 1 day each year and by 2 days in leap years.
note this
thats what I was observing
coarse tusk
do you know WHY this happens?
tawny widget
Okay
Because it’s a leap year?
I actually didn’t know that
I thought it was random
So shifts by 7 days
Uncritical
In total*
coarse tusk
you want the days to shift by some multiple of 7 days, yes
pearl trench
you want the days to shift by **some multiple** of 7 days, yes
becuase 365%7 = 1
that one extra day
so yeah
pearl trench
Because it’s a leap year?
you need to absorb the intuition for every question, dont run for the direct answer
pearl trench
but try for more close approximated approaches
pearl trench
Let me do it
yeah, try, if any problems, report here
tawny widget
Wait what
pearl trench
you, and @viral turtle , the repitition of the days by 1 and on leap year by 2, so for 28 to be back on wed on a leap year so 29 is thus.... we need to have the days increment by a multiple of 7 on the leap year
pearl trench
you, and <@1266959415152611348> , the repitition of the days by 1 and on leap ye...
from 2024, yeah
wait
WAIT
I GOT IT
DANG
pearl trench
see
if we want 7 days to be repeated again for the 28th to be on wed for 29th to be thus for the leap year
we.. need to account that 2-day repition for the leap year
pearl trench
Do you just count the dates until you reach 7
the thing is
you just need to account for, that when will the days when added with the +2 benefactor for the leap year be equal to a factor of 7
so for the day to be exactly 29th
tawny widget
Yes?
coarse tusk
in one cycle of four years, starting from a leap year, exactly 5 days are added
so you would need 7 cycles for it to get to the same configuration again
tawny widget
But its only 1 leap year right
coarse tusk
yes
but there are also three normal years in between
which also add 1 each to the shift of days
pearl trench
WAIT
tawny widget
So how many years are there in total
pearl trench
2024, its thus, 2028 its tue, 2032, its sun
coarse tusk
@pearl trench if you haven't solved the problem yet can you please stop trying to explain to somebody else how to solve it
pearl trench
<@756149983866257492> if you haven't solved the problem yet can you please stop ...
I.. am explaining it to him to how to approach it
well I will stop I guess, if you say so
coarse tusk
well you're just making haphazard statements randomly about things that you found
which isn't benefitting OP at all
@tawny widget did you understand why a normal year drifts the days by one day, and a leap year drifts the days by 2 days?
coarse tusk
you don't want it to drift 7 days, you want it to drift by a multiple of 7 days
that's a very important distinction
coarse tusk
Do you count the leap year you start in
do you?
tawny widget
But isn’t 7 the first one?
coarse tusk
think about it
coarse tusk
But isn’t 7 the first one?
it is, but you will realise that drifting 7 days precisely on a leap year is impossible
tawny widget
So…
coarse tusk
yes
tawny widget
4 years
coarse tusk
which means in a cycle from one leap year to another, we gain exactly 5 days of drift.
tawny widget
You need 14 days of drifting?
coarse tusk
so how many cycles do we have to go through before our drift becomes a multiple of 7 days?
stuck hearth
there is another rule for leap year
But if the year can be divided by 100 as well as four, it's not a leap year.
tawny widget
Uhh
coarse tusk
oh yeah this is important too
although it doesn't come up in this problem specifically
coarse tusk
so what year will that correspond to?
@tawny widget first answer this
coarse tusk
32 what?
tawny widget
Years?
coarse tusk
why?
tawny widget
7 cycles is 28 years
coarse tusk
precisely
coarse tusk
woah woah woah
we already got 35 days drift
in 7 cycles (i.e 28 years)
why are we going further?
pearl trench
I think it should be correct
Can I try now?
tawny widget
Okay
pearl trench
to give you something?
pearl trench
which corresponds to 2024+28 = 2052
are you done?
well ok
so, just tell me, how many repeats would you require to be back on monday, if you went back by 2 days
like first to saturday, then to thrusday, then to tuesday and so on
tawny widget
Wait did you like get a different answer?
tawny widget
Oh ok
pearl trench
so, just tell me, how many repeats would you require to be back on monday, if yo...
just answer this
tawny widget
I think I just had to write each of the numbers and formulas down
I think I understand now
Wait can you help out my friend with his question if you have time?
Help 18
pearl trench
hm ok
pearl trench
I think I just had to write each of the numbers and formulas down
but you could think it as days too
tawny widget
but you could think it as days too
That’s what I meant
pearl trench
Are they allowed to ping you?
yeah yeah
@tawny widget first let me get over with you
no matter how many days you go back at a time you just need 7 repeats
with that
as you need 7 leap years
you need to go 7 leaps ahead 2024
that is 2052
convert that in in single years by 7 * 4
its 28
so yeah
done
pearl trench
<@756149983866257492> if you haven't solved the problem yet can you please stop ...
I envy you though, but still, hah
topaz sinewBOT
@tawny widget Has your question been resolved?
topaz sinewBOT
topaz sinewBOT
Available help channel!
tawdry marten
topaz sinewBOT
dense pulsar
Where are you stuck? Do you have any working?
tawdry marten
I have no idea at all I tried to do root3 on n^3=x
dense pulsar
ah
meager kelp
We know that n = 20 from combining the 2 equations, from here substitute into the first equation
tawdry marten
And use that value on next one but didn't work
dense pulsar
for questions like these, you need to notice that we can divide both equations and end up with cancellations that favour us
tawdry marten
We know that n = 20 from combining the 2 equations, from here substitute into th...
How I don't understand
dense pulsar
divide the corresponding sides of the equations with each other
meager kelp
#15 is the same skill in that sense
dense pulsar
do you follow, @tawdry marten ?
pearl trench
How I don't understand
$$n^3 = x \implies n^3 \cdot n = xn \implies n^4 = xn $$
meager kelp
But even simpler
thorny flameBOT
Greydawn Dewer
dense pulsar
guys it gets confusing if yall just barge in trying to help 😭
pearl trench
guys it gets confusing if yall just barge in trying to help 😭
ah sorry
pearl trench
I wont do anything now
dense pulsar
it's okay!
pearl trench
**Greydawn Dewer**
but thats just the basic idea of what he said @tawdry marten
pearl trench
it's okay!
alright alright
neon iron
meager kelp
Does the SAT tend to ask questions like this
tawdry marten
Yes
meager kelp
Where u can find a neat cancellation to solve the problem in very few steps
pearl trench
So x is 8000 no?
can I just write it all then?
tawdry marten
Sorry but I am still no getting anything
pearl trench
If $$ n = 20 $$
thorny flameBOT
Greydawn Dewer
dense pulsar
Sorry but I am still no getting anything
n^3=x (1)
n^4=20x (2)
if we divide (2) by (1)
n^4/n^3=20x/x
correct?
tawdry marten
**Greydawn Dewer**
What is this
pearl trench
$$ (20)^3 = x \implies 20 \cdot 20 \cdot 20 = x \implies 8000 = x $$
thorny flameBOT
Greydawn Dewer
tawdry marten
?
pearl trench
What is this
n to be made out as 20
dense pulsar
greydawn, i don't think they know how you got n
tawdry marten
dense pulsar
that's what im trying to explain
pearl trench
greydawn, i don't think they know how you got n
bro, I just did it before
dense pulsar
okay good, this works too
pearl trench
**Greydawn Dewer**
@tawdry marten thisis how we get n
dense pulsar
now substitute for n^3
pearl trench
$$n^4 = xn \implies n^4 = 20x = xn \implies 20x = xn \implies 20 = n $$
thorny flameBOT
Greydawn Dewer
dense pulsar
phew
tawdry marten
It's x
tawdry marten
Ahh okay
dense pulsar
tawdry marten
.close
topaz sinewBOT
pearl trench
lets go
topaz sinewBOT
Available help channel!
exotic lance
does anyone know how to write this problem in sigma notation? I tried and failed to figure it out for like 30 minutes before brute forcing the problem by actually writing the whole formula out
topaz sinewBOT
does anyone know how to write this problem in sigma notation? I tried and failed...
@exotic lance Has your question been resolved?
bold urchin
does anyone know how to write this problem in sigma notation? I tried and failed...
what’s your definition of upper and lower sum? it should include sigma notation.
exotic lance
i SHOULD include sigma notation, but my problem is that I can't write it out correctly
bold urchin
i SHOULD include sigma notation, but my problem is that I can't write it out cor...
what is your book or class definition for the upper and lower sum before applying it to this problem is my question.
exotic lance
you mean it's not consistent what the notation is??
bold urchin
no no i just want you to give me your definition for the upper and lower sum
with a sigma notation in it preferably
exotic lance
well the upper sum, hold on
bold urchin
something like that
bold urchin
yeah sure
exotic lance
or red is upper sum blue is lower sum
bold urchin
blue is lower red is upper, do you understand sigma notation?
exotic lance
I'm trying to understand it
from what I understand, i=_ is the start, n is the end, then you have deltaX * f(x)
bold urchin
yeah you sum the small rectangle of surface area based on index i
does something like this make sense?
exotic lance
that does make sense
i starts at 1, the function increments n times
if the function has an i in it, the i goes up each time
by some amount
exotic lance
f(xi)deltaX?
bold urchin
you have 2 choice to make here the x_i and the x_i star
i’ll draw something to make it clearer
exotic lance
thank you
bold urchin
the x_i are between a and b the interval where we want to estimate area. a common choice is to make them evenly spaced between x_0 = a and x_n
so first you need to specify how to write that partition
exotic lance
I think I understand so far
exotic lance
I'm not sure I follow that
bold urchin
well we want $\sum_{i=1}^{n} base \times height$ for the rectangles
so here is the base of the rectangles
the delta x_i
here n=4
thorny flameBOT
pola_touche
bold urchin
it’s not an 1 it’s an a
bold urchin
you can compute it if you want. this choice for the x_i create a evenly spaced partition between a and b
exotic lance
so b and a refer to the endpoints of this
bold urchin
so if we have the base now we want to have the height of the rectangles for the approx. again uou can choose anything between x_{i-1} and x_i for x_i^{*}.
then evalutate f(x_i^{*}) to get the height of the rectangle
exotic lance
right, so base is deltaX and height if f(x_i), right?
bold urchin
then you get back to this
exotic lance
the summation of f(x_i)*deltaX
bold urchin
right, so base is deltaX and height if f(x_i), right?
not just x_i
exotic lance
how would I write it down?
thorny flameBOT
pola_touche
bold urchin
for each i
exotic lance
what is that xi*?
bold urchin
what is that xi*?
an extra point between each x_{i-1} and x_i that we evaluate f with to create height of the rectangle
if you choose this x_i* so that it maximise f between x_{i-1} and x_i , then the approx will overestimate the area. similarly if you choose x_i* so it minimize f then it will underestimate
that’s what i call upper and lower sums
exotic lance
I see
bold urchin
here i chose the midpoints. there is a bunch of choice possible, but the basic one are the upper and lower sum
because they give bounds on the error of the approximation
exotic lance
right
bold urchin
does anyone know how to write this problem in sigma notation? I tried and failed...
now back to this
exotic lance
yes please
bold urchin
you just combine what we did
like this one is the lower sum x_i* = x_i because the function is decreasing
between a=0 and b=1
exotic lance
n = 5 because we're not going to x=5 but we are doing the cycle 5 times
bold urchin
it’s how precise the approx is
exotic lance
man this is really hard for me to wrap my head around