okay so i got y = e(base x) -5
#help-36
sturdy flax
e^x, not e base x.
e is the base.
and yes, the -5 is not part of the exponent, absolutely correct.
sturdy flax
,rccw
soft zealotBOT
sturdy flax
that 5 is looking mighty like an S, but yes that's alright.
frail parcel
sorry yes its a 5
im writing fast as i have alot to do and little time
alos my teacher said thers gonna be a quetsion on the test saying
"what is the diffrence betweeen log and natrual log and what are the bases"
ik the base of natural log is is e and for log its 10
however i dont know what to say for the diffrence
sturdy flax
there's no other difference, really, other than that both are scaled relative to each other.
fundamentally, they are still logarithms and play by every single log rule.
there is one difference that only matters in calculus, so that's not in consideration here I believe.
I suppose ln(x) grows faster than log_10(x), and that's really all that can be said is different.
sturdy flax
anything else then?
sturdy flax
alright, all the best then!
frail parcel
but i will keep this open for a few mins just incase
thank you so much
i appreatcitce
sturdy flax
don't mention it.
frail parcel
i have to solve for x so how would i do log x = log 14 - log 2
i could just look at the key but i wanna learn it
sturdy flax
do you know about the quotient/difference rule of logarithms?
sturdy flax
$\log(a) - \log(b) = \log(\frac{a}{b})$, \textbf{provided that the base of both logs are the same.}
soft zealotBOT
Yukari
frail parcel
that was NOT me
sturdy flax
yes.
frail parcel
so if its addition its multipy right
sturdy flax
correct.
final saddleBOT
final saddleBOT
Available help channel!
north elm
final saddleBOT
radiant igloo
!status
final saddleBOT
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 was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
gimme a sec, just typin what ive understood so far
I chatgped the question and tried to understand the logic,
150x=1 unit of work
((x+8)/2)(150+150+(x+8-1)-4)=150x because apparently the logic of 150*1 worker day = 1 day's work plus summation all the way to the number of workers at x+8 days = the whole work unit=150x?
equating gets u 25?
This is all seems rather simple but would've never occured to me to take individual days' work unit to summate it, I just want to know of other ways to approach the same problem
radiant igloo
I chatgped the question and tried to understand the logic,
150x=1 unit of work
...
!noai
final saddleBOT
I chatgped the question and tried to understand the logic,
150x=1 unit of work
...
Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).
somber fog
I chatgped the question and tried to understand the logic,
150x=1 unit of work
...
We can help you to do without it.
north elm
We can help you to do without it.
tysm <3
the answer is right tho (its given in the back of my tb)
somber fog
tysm <3
The problem it’s simple you need to choose one unit of the work and do a arithmetic progression.
Do you understand?
radiant igloo
the answer is right tho (its given in the back of my tb)
if you argue with gpt enough its gonna give the wrong answer that is why it is best to review from actual ppl/sources
north elm
The problem it’s simple you need to choose one unit of the work and do a arithme...
I guess so, but is that the only approach?
somber fog
I guess so, but is that the only approach?
Yeah, but you can use the logic.
north elm
very well then, i'll just fixate that into my mind, tysm @somber fog @radiant igloo
somber fog
very well then, i'll just fixate that into my mind, tysm <@1071832799486673006> ...
You are welcome! Have a nice day.
!done
final saddleBOT
If you are done with this channel, please mark your problem as solved by typing .close
north elm
.close
final saddleBOT
radiant igloo
very well then, i'll just fixate that into my mind, tysm <@1071832799486673006> ...
🔥
final saddleBOT
Available help channel!
robust sequoia
is this even possible
final saddleBOT
is this even possible
desert mantle
by writing $\pi(x)=\sum_{p<x} 1$ you can write it as a sum of simple integrals at least
soft zealotBOT
Denascite
final saddleBOT
@robust sequoia Has your question been resolved?
robust sequoia
**Denascite**
this series has no closed forms so there's no way to use it no ?
I tried using an approximation which is π(x) = x/ln(x) but I keep getting the exponential integral function
or some weird integrals like int from -inf to inf of 1/(usinh(u)) du
<@&286206848099549185>
tender pollen
i think civil service pigeon did this
tender pollen
<@&286206848099549185>
first of all you can change the lower bound to ||2|| because ||pi = 0 for x<2||
sturdy cypress
i think civil service pigeon did this
ban csp
final saddleBOT
final saddleBOT
Available help channel!
loud sundial
i think civil service pigeon did this
My friend sent it to me in insta DMs when my food was microwaving lol
loud sundial
.close
final saddleBOT
final saddleBOT
Available help channel!
tough grail
Hi
tough grail
Wait
worldly spruce
Hello!
tough grail
prisma lance
Hello
blissful meadow
,rccw
soft zealotBOT
tough grail
Please help me with the first question
prisma lance
Search the value of angle
proper dagger
final tangle
What have you tried
tough grail
Nothing
final tangle
introduce a variable,
apply definition of angle bisector
tough grail
I was not able to think anything
tough grail
This was the question
final tangle
It's not clear, do you still need help or not?
And don't attempt to ping the entire server
somber fog
This was the question
Do you have more questions?
final saddleBOT
@tough grail Has your question been resolved?
final saddleBOT
final saddleBOT
Available help channel!
dim flume
I have a question regarding directional derivatives. Why is it that if we’re interested in a derivative of a multi variable function f(x,y) = z going in the direction of some unit vector u = <u1,u2>, we compute the operation
u1 · f_x + u2 · f_y
final saddleBOT
I have a question regarding directional derivatives. Why is it that if we’re int...
dim flume
While also making sure that the vector u is normalized
What’s the intuition behind all this?
Specifically the operation
onyx peak
Have you looked at the vids on khanacademy?
If you're looking for intuition, i think its best to have the visuals in front of you
dim flume
Not really tbh but I’d appreciate an explanation in this channel tbh because Its almost always the case that I have questions
vital crag
Not really tbh but I’d appreciate an explanation in this channel tbh because Its...
watch the videos first, then ask questions
final saddleBOT
@dim flume Has your question been resolved?
dim flume
Yeah I didn’t get it lol
dim flume
https://www.khanacademy.org/math/multivariable-calculus/multivariable-derivative...
It just made things more complicated for me ngl
vital crag
It just made things more complicated for me ngl
then ask your question here
vital crag
What’s the intuition behind all this?
the video answers this
dim flume
then ask your question here
It’s just gonna be the same initial question
But the videos approach didn’t really help
Or explanation
vital crag
It just made things more complicated for me ngl
yes that's the typical way math goes
you need a foundation of math to build on top of
vital crag
Or explanation
explanation of what exactly
wanting "intution" for something is too vague to answer here
that video is the best starting point
did you watch it yet
final saddleBOT
@dim flume Has your question been resolved?
sick olive
final saddleBOT
sick olive
Idk how to solve
And how is measure of angle BAD = measure of angle BEC
<@&286206848099549185>
sick olive
Ik how to solve just how is measure BAD = measure angle BEC
oak nimbus
I hate geometry
somber fog
<@&286206848099549185>
!15m
final saddleBOT
<@&286206848099549185>
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
timber leaf
Find arc BD
sick olive
Wdym arc bd
timber leaf
You got BEC=27
sick olive
Yes
timber leaf
Arc BD = arc BC, but do you know why
timber leaf
Yup perpendicular to chord
oak nimbus
Ye that true
timber leaf
Um could you help wit some more of them
Then you find angle ABD
In the right triangle ABD
Do you see why its right triangle?
sick olive
Yeah
sick olive
Wait why do we find abd
timber leaf
Just answer me bro everything link
sick olive
How do we find abd
sick olive
Ok got it
rare girder
Wait why do we find abd
in geometry, you try to find a path of deductions that lead you to your goal. there are very few cases where you can obtain the goal in one step
timber leaf
BAD = BEC, since subtending equal arc. Then ABD = ? In the triangle ABD?
rare girder
it helps to find as much information as possible
sick olive
Can’t you just set BAD and cEB as equal
timber leaf
Good thing is to trace from beginning
timber leaf
Can’t you just set BAD and cEB as equal
It does, but you need to reasoning why
sick olive
Because the arcs are equal
sick olive
Yes
timber leaf
Can you find angle ABD?
sick olive
No?
timber leaf
Angle ABD subtending to arc AD, but angle ADC also subtending to arc AC. And arc AC= arc AD
sick olive
sick olive
Sorry I’m stupid
sick olive
Yes
timber leaf
But you already found BAD previously
Which equals to BEC since subtending equal arcs
timber leaf
So what is ABD?
sick olive
63 is abd
timber leaf
Correct
timber leaf
Angle ABD subtending to arc AD, but angle ADC also subtending to arc AC. And arc...
Now read this
Arc AD= arc AC since arc BC = BD on two semi arcs AB
rare girder
we already agreed that ABD is a right triangle, but how do we know?
sick olive
Yes
sick olive
we already agreed that ABD is a right triangle, but how do we know?
Are u asking me
rare girder
yeah
sick olive
Because it looks 90 degrees
rare girder
ahhh we can't just say it looks like it
these diagrams don't have to be drawn to scale
sick olive
Uh
timber leaf
Wrong...
sick olive
I’m rlly sorry
timber leaf
It is AB
rare girder
you don't need to apologize
vague marten
lmao
sick olive
Why AB
vague marten
Wait guys do you verify it’s 90? I haven’t taken geometry in a long time
sick olive
Yes
timber leaf
Wait guys do you verify it’s 90? I haven’t taken geometry in a long time
Everything needs to be verified to become a rigorous statement
vague marten
I know but i did the questions based on arcs
vague marten
Is it because the line cd is perpendicular to the top of line ab so it instantly makes it a 90 degree angle right?
rare girder
no
sick olive
So its subtending arc AB
I thoguht they have to be directly across
rare girder
I thoguht they have to be directly across
take this image and drag P and Q to opposite ends of the circle, so that PQ is a diameter
sick olive
Ok
rare girder
then the relationship between angle PAQ and the arc PQ not containing A (the red arc) is still the same
timber leaf
Is it because the line cd is perpendicular to the top of line ab so it instantly...
Wdym?
rare girder
the length of the red arc is still twice the measure of angle PAQ
even if the arc is a semicircle
sick olive
take this image and drag P and Q to opposite ends of the circle, so that PQ is a...
Wouldn’t PQ be 180?
sick olive
I thoguht they have to be directly across
But it doesn’t look 180
rare girder
you drag the points in your mind, but the image stays the same
vague marten
I just thought that since point B is a cross A and the line CD is perpendicular an D is connected to that line we know that angle D would be 90 no? Just based on logic
sick olive
Smth like this?
rare girder
yup
rare girder
of course it doesn't
timber leaf
I just thought that since point B is a cross A and the line CD is perpendicular ...
That's intuitive
timber leaf
Is it because the line cd is perpendicular to the top of line ab so it instantly...
And CD perpendicular to AB doesnt imply D is 90 degree anyways
rare girder
in this case, the arc that angle ADB is subtending is the left semicircle AB, not containing D
with the points (D,A,B) as the analogue of the points in the diagram (A,P,Q)
sick olive
Can we just move AB
rare girder
A and B are fixed on a diameter of the circle, so absolutely not
sick olive
I’m stupid
vague marten
rare girder
!nogpt
final saddleBOT
Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).
timber leaf
!noai
final saddleBOT
Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).
vague marten
Not answering just proving my point as long as cd is perpendicular its correct
sick olive
WHY IS arc BD even work
rare girder
Not answering just proving my point as long as cd is perpendicular its correct
it doesn't prove anything, it's spam and doesn't belong here
sick olive
How is AB 2x or BDA?
vague marten
Alr whatever yall lol
timber leaf
Not answering just proving my point as long as cd is perpendicular its correct
If you are not contributing and giving false information, please get out!
rare girder
How is AB 2x or BDA?
it's a specification of the more general theorem, whose name I forget
sick olive
But AB doesn’t look like it’s subtending ADB
rare girder
in red, draw the left arc AB
sick olive
Wait is the arc the diameter?
sick olive
Is the arc the half circle
rare girder
yes, in this case
sick olive
Ok I drew it
rare girder
look at your drawing and the first instance of your APQ image
or the one you drew on top of, it's whatever
sick olive
rare girder
the diagram you're working with is a specification of the APQ one
sick olive
Do we use CE
rare girder
no
sick olive
How do we move AB
rare girder
why do you want to move AB?
sick olive
Cuz we moved PQ
rare girder
it's because of the instructions provided in the question itself
A is free, B is entirely dependent on A because AB must be a diameter of the circle
C is free, but D is determined since CD is perpendicular to AB
E is completely free, but we specify an angle regarding E
sick olive
I still don’t get how AB is an arc for BD
rare girder
I gotta sketch it then
sick olive
Srry
rare girder
you don't need to apologize I just meant that I might take a second in doing so
sick olive
Ty
rare girder
rare girder
this is a totally accurate to-scale picture of a circle with radius 1
sick olive
But still could you explain it
rare girder
the theorem this questions demands that we understand is the relationship between the measure of angle ADB and the length of the arc AB
rare girder
no
sick olive
Where POQ is double the angle measure
rare girder
the origin is fine, you can include it, but it's equivalent to the one I drew, only if the circle has radius 1
i.e. if it's a unit circle
rare girder
sure there is
I just didn't note the center of the circle
the angle ADB is notated
rare girder
the location of the points A, B, and D are completely irrelevant; they may be anywhere, but the relationship is the same
rare girder
the chord AB wasn't relevant just so you know
but it is definitely good to draw extra lines, and extend line segments
sick olive
How does it help ADC
rare girder
we decided we already knew what one of the angles in the triangle ABD were, and we've just decided that it is indeed a right triangle
I think it was ABD anyway
rare girder
as angles?
rare girder
I don't see it, can you explain?
rare girder
they wanted you to find ABD for the same reason you could have
they thought it'd be both feasible and useful to know
or if anything, it's feasible, and may result in further deductions
sick olive
Idk how it helps
rare girder
what is another deduction you can make, knowing what you currently know?
rare girder
that sounds familiar
rare girder
sounds real
rare girder
it sounds like you did it differently than what minhh and I were going for
we all did it differently actually
sick olive
O
rare girder
I'd say the triangle similarity thing is cooler
rare girder
if the question is answered, yes
rare girder
nah
sick olive
Like point to the arc
rare girder
like the angle bisector ADB intersects at the midpoint of arc AB
I know what you're saying
but no, the placement of D is arbitrary
rare girder
you type .close
sick olive
.close
final saddleBOT
sick olive
Thank you again
final saddleBOT
Available help channel!
grizzled socket
hey
next thorn
grizzled socket
.close
final saddleBOT
final saddleBOT
Available help channel!
trail shale
Is there any other away to do this problem other than brute force
final saddleBOT
Is there any other away to do this problem other than brute force
trail shale
3^3
worldly spruce
Exactly
trail shale
and area scales by 3^2 and the sides scale by 3
worldly spruce
So $V_{\text{big}} = 27 \cdot V_{\text{small}}$
soft zealotBOT
trail shale
That makes sense
trail shale
Just find a set of dimensions which make 20 cubic inches of volume
So you have to “brute force” that and just memorize factors
worldly spruce
Well I mean this isn't really brute force
It asks you to find a set of dimensions
There's infinitely many
You just pick one
For example
Obviously 1 * 1 * 20 works
For the small box
trail shale
Yeah so 3 3 60
trail shale
But that doesn’t give the volume of 540
worldly spruce
You aren't trying random things for one to work
Pretty sure 3 * 3 * 60 = 540
,calc 3360
soft zealotBOT
Result:
540
I mean
If you want to really move away from this
We know the new volume must be 540
So set up $l \cdot w \cdot h = 540$
soft zealotBOT
worldly spruce
Right
And then we know l, w, h > 0 since they are a measurement of length
And you can freely pick two
And the last you can solve for
trail shale
👍
worldly spruce
🙂
trail shale
Ok, tysm for answering my question
worldly spruce
👋
final saddleBOT
final saddleBOT
Available help channel!
frail parcel
can anyone help me understand problem 37 and 42? (they are more problems like 42 such as questions 43 and 44)
final saddleBOT
can anyone help me understand problem 37 and 42? (they are more problems like 42...
vital crag
can anyone help me understand problem 37 and 42? (they are more problems like 42...
Write 125 = 5^3 and use exponent rules
timber leaf
Also 25=5^2
timber leaf
can anyone help me understand problem 37 and 42? (they are more problems like 42...
42 is log rule
Loga+logb=log(a*b) if same log base
25 is 5^2, correct?
@frail parcel
frail parcel
correct
frail parcel
yea
frail parcel
correct
timber leaf
but by exponent rule, if a^b = a^c, then b = c
frail parcel
wait no
frail parcel
this is equivalent to 5^3(4x), if you see
this isn’t right
timber leaf
why not?
frail parcel
how is 125^4x = to 125(4x)
timber leaf
125 = 5^3, correct?
frail parcel
yes
frail parcel
oh yes cuz you multiply exponent
frail parcel
yes
timber leaf
So what have you done so far? 25^x-1=125^4x
frail parcel
so x-1 = 4x?
timber leaf
Make 25 into 5^2 and 125 into 5^3
frail parcel
but we need the same a
timber leaf
Yes, it is 5^2(x-1)=5^3(4x)
frail parcel
sturdy flax
reminder.
timber leaf
it’s like the opposite of something right
Ok, if a^b=a^c, then b=c
So we see a=5 here
What is b and c?
frail parcel
2(x-1) and 3(4x)
timber leaf
We dont mind them anymore
Since we need to find x
if a^b=a^c, then b=c. We are comparing b and c
frail parcel
so -0.2
so the whole thing is just finding common base
in this case 5
okay thank you
if your free can you help me with 42 too?
sturdy flax
what about 42 is confusing you?
frail parcel
this is what i have so far, it’s 42 not 41 i just needed extra space for work
sturdy flax
**Yukari**
reminder to this...
frail parcel
however there’s two Xs
sturdy flax
so if its addition its multipy right
and this.
sturdy flax
however there’s two Xs
have you ever solved a quadratic before?
frail parcel
have you ever solved a quadratic before?
yes i feel like i should know this but my mind is blanking for some reason
sorry
ik it’s + 5x on both sides
sturdy flax
yes.
I will give you a heads-up, however. one of the answers you will get is not a valid solution to the original problem.
sturdy flax
not necessarily because it itself is negative, but rather when used as the argument to the log functions, it will make the argument negative.
if, for instance, you have log_5(x + 6), then x = -3 is valid, because then the argument is still positive.
frail parcel
cuz -3-5 is is -8 so it makes a negative log
sturdy flax
negative log argument, yes, which is not defined.
frail parcel
okay and i got 150 for 43 which is also right
alright thank you sm i understand it now
2 times in a row
timber leaf
okay thank you
Yukari can do it
frail parcel
.close
final saddleBOT
final saddleBOT
Available help channel!
oak mica
I managed to prove a|bc but I can't get any further. I attached hints for the question
final saddleBOT
I managed to prove a|bc but I can't get any further. I attached hints for the qu...
rancid idol
tell us what youve tried / ideas
oak mica
I said c <= b <= a and use the bounds a <= LCM(a,b,c) <= abc. Then I tried simplifying by dividing by abc but then I got stuck.
scarlet sequoia
I said c <= b <= a and use the bounds a <= LCM(a,b,c) <= abc. Then I tried simpl...
you know that a | bc
how could this give you a better LCM upper bound than abc
oak mica
Maybe write it as bc = ka
scarlet sequoia
uh, sure
but remember the definition of lmc(a,b,c)
it's the smallest common multiple of the three
see how it links to bc?
oak mica
Can you write the upper bounds as bc
scarlet sequoia
Can you write the upper bounds as bc
yes
bc is a common multiple of a,b,c
so lcm(a,b,c) <= bc
(more specifically lcm(a,b,c) | bc, so bc = k * lcm(a,b,c)
Now, use the equality to get a lower bound
oak mica
Is the best lower bound not a
scarlet sequoia
Is the best lower bound not a
definitely not
you know what lcm(a,b,c) is equal to
try to lower bound that quantity
oak mica
When I say it's equal to BC/k I don't get k(ab+ac) = 4. Am I doing it wrong
I kinda have to go so I need to close
.close
final saddleBOT
final saddleBOT
Available help channel!
spring abyss
final saddleBOT
spring abyss
- Does this work?
- Any smart way to calculate this other than pressing my calculator from start to finish?
also unrelated to what i sent but is homogenous difference equation different from the one with lambdas? is that called second order difference equation?
ebon agate
1. Does this work?
2. Any smart way to calculate this other than pressing my cal...
You can notice this is every 2nd term of a binomial expansion
spring abyss
yeah
i think the entire thing is one binomial expansion but without the odd powered terms
ebon agate
i think the entire thing is one binomial expansion but without the odd powered t...
How do you find only the even terms of (1+x)^n
spring abyss
no clue yet
ebon agate
(7/8+1/8)^20 even terms
Can you apply the same here?
spring abyss
why does this look like the pgf(-1) trick to find Pr(X=even) are they related
ebon agate
why does this look like the pgf(-1) trick to find Pr(X=even) are they related
Yes
Same thing
spring abyss
what kind of other summations is this trick useful on
ebon agate
Basically you've got smth like
(P(x))^n
Where P(x) is a polynomial in x
Suppose you need to find the sum of every third coefficient
You'll find P(w)
P(w^2)
P(1)
Where w,w^2 and 1 are the cube roots of unity
And add them and then divide by 3
Basically if you need to find the sum of every kth coefficient in such expansion
You compute the polynomial at every unique kth root of unity
Add them and then divide by k
ebon agate
Add them and then divide by k
@spring abyss
ebon agate
(7/8+1/8)^20 even terms
Here you just need to find the even terms for these
So
((7/8+1/8)^20 + (7/8-1/8)^20)/2
Should suffice
spring abyss
i dont think the answers want 1+0.25^20, i think the 2nd part should be larger
ebon agate
1/8 is 0.125
spring abyss
bruh its 0.75^20 not 0.25 wait ima redo
ebon agate
Oh right
ebon agate
i think it worked, 0.5016 final
Nice
spring abyss
Where w,w^2 and 1 are the cube roots of unity
this some very advanced stuff tho ive never seen anything like this introduced to cancelling summations
ebon agate
this some very advanced stuff tho ive never seen anything like this introduced t...
I think you'll see this in generating functions
It's a commonly deployed trick
spring abyss
what counts as a generating function? im learning mgf and pgf, and previously multiplying barebone power series to do counting
ebon agate
I think you'll see this in generating functions
Suppose you roll three 6 sided, find the probability that their sum is a multiple of 3
ebon agate
what counts as a generating function? im learning mgf and pgf, and previously mu...
I think you'll encounter it later on then
spring abyss
then id write (x + x^2 + ... + x^6) ^3
ebon agate
There's a good 3b1b video on it
ebon agate
then id write (x + x^2 + ... + x^6) ^3
Yeah
You need to find every 3rd coefficient in this expansion
From 3 to 18
Then divide by 216
You can use the trick I mentioned
It will take a really long time to find coefficient through multinomial method
spring abyss
also unrelated to what i sent but is homogenous difference equation different fr...
do you also have any idea on what this is?
ebon agate
also unrelated to what i sent but is homogenous difference equation different fr...
What's the one with lambda
spring abyss
prolly this
ebon agate
also unrelated to what i sent but is homogenous difference equation different fr...
This in itself is a 2 degree homogeneous recurrence relation
pn + 1/3*p(n-1) = 1/3
p(n+1)+1/3*pn=1/3
ebon agate
isnt it about power
Yes
You can make it to be a 2 degree equation through manipulation
pn + 1/3p(n-1) = 1/3
p(n+1) + 1/3pn = 1/3
Subtract both these equations
You get a 2 degree recurrence
I assume one of the roots of the characteristic equation will always be 1
And it's respective coefficient will be 1/4
ebon agate
also unrelated to what i sent but is homogenous difference equation different fr...
But go with this method because the manipulation method gives redundant roots
ebon agate
why 1/3 as coeff means its degre 3
Sorry it's 2 degree my bad
peak abyss
final saddleBOT
oak nimbus
What idea u got ?
oak nimbus
Can u try to make rough graph what he has to say ?
oak nimbus
bcz both the y coordinates will be on the same size of the x axis ryt ?
Make rough sketch what questions is saying
peak abyss
wait
peak abyss
Make rough sketch what questions is saying
its has to be either the red one or the blue one
oak nimbus
its has to be either the red one or the blue one
Hmm right
oak nimbus
its has to be either the red one or the blue one
What u think from this graph ?
oak nimbus
yes k<24
There is another rule for root to be less than each other
Smthing like coeff of x^2 × f(1) <0 or smthing ?
Do you know anything like that ?
peak abyss
Do you know anything like that ?
i think i've heard of it
but rn i do not remember it
oak nimbus
i think i've heard of it
You can also check it from graph
peak abyss
why should the coeff of x^2 be +ve ?
oak nimbus
See when you take x^2 coefficient greater than 0 your f(1) is negative and vice versa
oak nimbus
why should the coeff of x^2 be +ve ?
Just see from graph for both case conditions is same
When coefficient + f(1) become - when , coefficient - f(1) become positive
peak abyss
i am sorry..what?😭
oak nimbus
i am sorry..what?😭
Umm make 2 diff graph when coeff of x^2 is + and -
Then find similarity that will help in finding k
oak nimbus
its has to be either the red one or the blue one
Make this 2 diff graph with condition
Or just check notes where u right this condition when u learn that
peak abyss
peak abyss
See at k-5<0 what f(1) and f(2) giving ?
both +Ve
peak abyss
wait
peak abyss
Put value in equation and check u will get inequality in k
what value 😭
oak nimbus
X=1 and 2
oak nimbus
what value 😭
??
peak abyss
writin
oak nimbus
Okok
peak abyss
oak nimbus
Ye can you take case k-5<0 here ?
oak nimbus
See this and say
oak nimbus
Noo man see if u take k-5<0 so for that f(1) shoud be positive right ?
oak nimbus
What your f(1) ?
peak abyss
oooohhh
peak abyss
ya but why tf is it negative
oak nimbus
ya but why tf is it negative
Bec it is contradicting case that why k-5>0 shoud be there
peak abyss
oohk
oak nimbus
So use case - 2 for ans you were asking why k-5>0 for that we check value if it contradict then we delete that case
So now what ans ?
oak nimbus
and we want it to be -ve ryt
?? For case 1 we need f(1) positive but that is contradicting our situation so we cancel that case and go to next case
oak nimbus
and we want it to be -ve ryt
We want to be + in that case but it came -
peak abyss
?? For case 1 we need f(1) positive but that is contradicting our situation so w...
ya i mean next case
oak nimbus
ya i mean next case
Yep
peak abyss
we want it to be -ve
oak nimbus
do i check this every time
Yep some practice will make it fast
peak abyss
Yep some practice will make it fast
what if i make the coeff of x^2 constant ?
oak nimbus
You need to find value of point given and make rough sketch u can watch from there also
peak abyss
for example by taking k-5 common
oak nimbus
what if i make the coeff of x^2 constant ?
You can't decide that question will provide
oak nimbus
for example by taking k-5 common
You can't do that in quadratic
Ig
Umm just check for yourself what u get
peak abyss
oak nimbus
Now evaluate? What u getting ? It will make hard for finding f(1).f(2)>0
U making it complex
peak abyss
peak abyss
U making it complex
yea you are right
oak nimbus
This is way to do faster
First check given point value and check from graph which case is correct
I hope u got ans what i looking for ?
@peak abyss done ?
oak nimbus
That what i told u do it my way that easy
oak nimbus
This is way to do faster
See this
peak abyss
That what i told u do it my way that easy
yesss
oak nimbus
So what you preparing for
peak abyss
.close
final saddleBOT
final saddleBOT
Available help channel!
gentle zephyr
final saddleBOT
onyx peak
so where're you stuck at?
soft zealotBOT
Renato
gentle zephyr
so where're you stuck at?
b)
onyx peak
here are the axioms of ordered field, + the field axioms ofc
but these 2 are about order
which one do you think we're gonna use in b) (mainly)?
onyx peak
here are the axioms of ordered field, + the field axioms ofc
notice that the first one is about addition and the second one is more about multiplication
granite sage
b)
If you multiply an inequality with positive number the equality sign doesn't changes
onyx peak
If you multiply an inequality with positive number the equality sign doesn't cha...
the point of the question is to prove that using ordered field axioms
gentle zephyr
jaja
gentle zephyr
notice that the first one is about addition and the second one is more about mul...
the second ordered field axiom is almost ot what we need to prove
granite sage
the point of the question is to prove that using ordered field axioms
Oh I didn't noticed my bad
onyx peak
the second ordered field axiom is almost ot what we need to prove
yeah, so we will try to apply that one
gentle zephyr
no worries
onyx peak
there is one issue though, we need to have 0 on one side to apply it
but we have a < b
how could we make it have 0 on the left hand side?
gentle zephyr
0 < a < b
onyx peak
the question doesnt mention that 0 < a
you cant just assume that
try instead rearranging the equation a < b somehow (you can again use the axioms of ordered field)
gentle zephyr
0 < b - a
onyx peak
here are the axioms of ordered field, + the field axioms ofc
perfect, so we use the first axiom of ordered field and then then the additive inverse axiom of normal field
okay now we have 0 < b-a and 0 < c
try applying the second axiom of ordered field now
onyx peak
here are the axioms of ordered field, + the field axioms ofc
oh and we should also distinguish between <= and < ig, does your book use < or <= when stating the axioms?
gentle zephyr
oh and we should also distinguish between <= and < ig, does your book use < or <...
it doesnt matter for the axioms if its <= and <
both are total order relations
onyx peak
it doesnt matter for the axioms if its <= and <
Okay, fair enough then
so now we have 0 < b-a and 0 < c. In this form, you can finally apply the 2nd axiom of ordered fields
gentle zephyr
so now we have 0 < b-a and 0 < c. In this form, you can finally apply the 2nd ax...
0 <= c(b-a)
right?
we are not getting anywhere with this though?
plucky rover
0 <= c(b-a)
Now break this into cases
Case 1, 0 < c(b-a) which would mean you're done
Case 2, 0 = c(b-a)
You want to show that this cannot occur
You can do this using the field axioms
onyx peak
Now break this into cases
it was said that it doesnt matter whether its <= or <, so ig we can keep it as 0 < c(b-a)
onyx peak
we are not getting anywhere with this though?
you can now use the field axioms to distribute the rhs
and rearrange it to ac < bc
final saddleBOT
@gentle zephyr Has your question been resolved?
onyx peak
that <= shouldnt even be there if we're working with < from the start. We can just go from 0 < b-a and 0 < c straight to 0 < c(b-a), if that's one of the axioms in @gentle zephyr 's book
Are you here btw? Do you need help with anything?
gentle zephyr
it was said that it doesnt matter whether its <= or <, so ig we can keep it as 0...
it was said that it doesnt matter whether its <= or <, so ig we can keep it as 0 < c(b-a)
yes
onyx peak
okay cool so now we have
0 < c(b-a)
can you find a field axiom which allows you to distribute this, to rewrite c(b-a) differently?
final saddleBOT
@gentle zephyr Has your question been resolved?
final saddleBOT
gentle zephyr
.reopen
final saddleBOT
✅ Original question: #help-36 message
final saddleBOT
@gentle zephyr Has your question been resolved?
full compass
at what point is this in an indeterminate form that can have l'hopital's rule applied to?
final saddleBOT
at what point is this in an indeterminate form that can have l'hopital's rule ap...
deep condor
direct substitution yields $1^\infty$, which is an indeterminate form
soft zealotBOT
haseeb ♥
full compass
direct substitution yields $1^\infty$, which is an indeterminate form
wouldnt direct subsitution yield 1^(1/0) so like undefined?
deep condor
informally, we define $\frac 10 \to \infty$. the motivation is that dividing 1 by a really small number (0.001) yields a very big number (1000)
soft zealotBOT
haseeb ♥
deep condor
indeterminate forms are undefined, because 0/0 is not defined explicitly
full compass
ok i got that, but lhopital is only compatible with the form indeterminate form 0/0 or inf/inf right
when did i get it in that form?
deep condor
but all undefined objects are not indeterminate: we know $\lim_{x \to 0} \frac 1x = \infty$, so a limit of the form $\frac 10$ does not exist for sure
soft zealotBOT
haseeb ♥
full compass
ok
deep condor
ok i got that, but lhopital is only compatible with the form indeterminate form ...
it is compatible with indeterminate forms that look like $1^\infty$ as well
soft zealotBOT
haseeb ♥
drowsy epoch
i just need to take logarithms first right
You can also tackle this limit differently
full compass
oh how
full compass
i just need to take logarithms first right
out of curiosity, is there an intuitive reason for why this works or should i just take this as given
that is is it a long proof or straightforward reason
drowsy epoch
,,\lim_{x\to0} (1-2x)^{1/x} = \lim_{x\to0} (1-2/(1/x))^{1/x} = \lim_{u\to\pm\oo} (1-2/u)^u = e^{-2}
soft zealotBOT
full compass
hmm ok interesting
final saddleBOT
@full compass Has your question been resolved?
full compass
.close
final saddleBOT
gentle zephyr
okay cool so now we have
0 < c(b-a)
can you find a field axiom which allows you...
you mean a field axiom and not an ordered field axiom?
maybe the distributivity axiom, i believe is called ?
distributivity addition axiom?
gentle zephyr
okay cool so now we have
0 < c(b-a)
can you find a field axiom which allows you...
sorrh for the late response, my landlord dude is asking me for money for something he should be responsible of
final saddleBOT
Available help channel!
fallen valve
Other than the omissions that A must be real, is there anythign else missing from this problem? Because I am having a bit of trouble concluding this
final saddleBOT
Other than the omissions that A must be real, is there anythign else missing fro...
fallen valve
and this was one of the exercsies in the book (FIS)
vital crag
What's the definition of ||.||_E
final saddleBOT
@fallen valve Has your question been resolved?
fallen valve
What's the definition of `||.||_E`
I beliveve it is the matrix norm (assuming both spaces use the euclidean norm)
vital crag
I beliveve it is the matrix norm (assuming both spaces use the euclidean norm)
Try diagonalization on B
final saddleBOT
@fallen valve Has your question been resolved?
knotty rampart
I had recently taken the PSAT, and I think I preformed well; but, I would appreciate further tutoring. I would like to prepare for the actual SAT, therefore if someone was graceful enough, it would be greatly appreciated.
final saddleBOT
I had recently taken the PSAT, and I think I preformed well; but, I would apprec...
icy current
I had recently taken the PSAT, and I think I preformed well; but, I would apprec...
I tutor students would you like to discuss in DMs?
knotty rampart
I tutor students would you like to discuss in DMs?
Do you plan on discussing it through the entirety? (Should I close the 'help' if so?)
versed crater
We’re not supposed to offer tutoring services through this server
knotty rampart
We’re not supposed to offer tutoring services through this server
I apologize, I had not known that. I assure you it won't happen again.😅
.close
final saddleBOT
final saddleBOT
Available help channel!
fallen valve
Try diagonalization on B
I don't think, I end up with the exact claim to prove though. I.e. (assuming $B$ is real), $A = B^t B$. Also $B$ is self adjoint (under standard euclidean norm). So there exists orthonormal basis of eigenvectors and we can write $B = Q^t D Q$ where Q is the basis of eigenvectors and $D$ is the diagoanl matrix of eigenvalues. Also $Q$ is unitary. So $Q^t Q= I$. Defining $A = B^tB = Q^t D^2 Q$ have that $|B |_E$ is the sqaure root of the largest eigenvalue of $A$.
final saddleBOT
I don't think, I end up with the exact claim to prove though. I.e. (assuming $B...
soft zealotBOT
LXDL
fallen valve
But we do not know if the eigenvalues are all positive, so we have to take the absolute value
vital crag
You should then first prove that eigenvalues of symmetric matrices are positive
final saddleBOT
@fallen valve Has your question been resolved?
fallen valve
You should then first prove that eigenvalues of symmetric matrices are positive
Im not too sure if this is true
I thought this only holds for positive semidefinite
vital crag
Oh yes -I of course
fallen valve
Other than the omissions that A must be real, is there anythign else missing fro...
yeah i think this probelm missing PSD at least
but not sure what else
at least i can show jsut using PSD
.close
hardy jackal
.
hardy jackal
is this correct
final saddleBOT
@hardy jackal Has your question been resolved?
muted prairie
,w derivative of 50000(1-t/30)^2 at t=10
soft zealotBOT
muted prairie
,calc -20000/9
soft zealotBOT
Result:
-2222.2222222222
yes
mossy inlet
is this correct
work looks right, seems simple enough, chain rule and algebraic work
south dirge
is this correct
Yup!!
Dont forget to remove the minus sign as in the question it simply asks the rate of flow of water!!
south dirge
Its 0.3t^2
The answer? yeah!
hardy jackal
south dirge
hm still not right!
south dirge
The answer is correct but step might is wrong?
hardy jackal
south dirge
Spot on!!
mossy inlet
yeah should be 0.1 original coeff
hardy jackal
thank u for catching that
south dirge
Yeah all the differetiation steps are correct just let me cross check your calculations!
Everything correct!
Good job!!
hardy jackal
Everything correct!
Good job!!
Thank u sm u are too kind!
south dirge
Anymore?
south dirge
Okay go ahead!!
final saddleBOT
hardy jackal
.reopen
final saddleBOT
✅ Original question: #help-36 message
south dirge
wait how
?
Oh
You could just take the denominator in the numerator
just make the exponent -4
but your method isnt looong either!!
Both work just fine!
hardy jackal
just make the exponent -4
ur right omg
i remember doing that for another question
let me do that rq
south dirge
correct me if i am wrong, but in the yellow step of part e, the denominator isnt (_)^3/4
hardy jackal
correct me if i am wrong, but in the yellow step of part e, the denominator isnt...
what do u mean
south dirge
when you multiplied by (3-4x)^1/2 in numerator and denominator, the denominator would become (3-4x)^3/4
oh wait is that a three and not a five?
hardy jackal
yeah its a three
south dirge
Well either way its still little off
hardy jackal
oh really
south dirge
Tell me smth,
a * (a)^1/2 =?
south dirge
This is all fine!
hardy jackal
,w determine the derivative of f(x) = (3x^2)/(3-4x)^1/2
soft zealotBOT
south dirge
Yeah its the same w just didnt simplify the denominator
you can take multiply numtr and denomtr with -1 and you would end up with the same result!
south dirge
In this one your result was correct for du/dt but step was a little wrong!
Sorry if i am being a little annoying pointing step mistakes do lmk if you dont want that!
hardy jackal
Sorry if i am being a little annoying pointing step mistakes do lmk if you dont ...
no please please do that
i appreaciate it alot
hardy jackal
In this one your result was correct for du/dt but step was a little wrong!
where im a little lost
south dirge
The part you wrote in red
hardy jackal
The part you wrote in red
you might have to point it out im ngl
i dont see it lmaooo
what did i do
that was wrong
south dirge
This seems right!!!
