And the second digit?
#help-0
peak bough
steep moon
5
peak bough
Yes
And we know that for the 3-digit number to be even, the last digit can only be either 0, 2 or 6
So how many ways can the third digit be chosen?
steep moon
3
peak bough
Exactly
steep moon
4•5•3 or we doing sum diff?
ocean sealBOT
Mikkel
steep moon
yes so it is 60
peak bough
Yes
peak bough
So we now have the following information:
-there can be made a total of 100 3-digit numbers
-60 of them are even
-whole numbers that are not even are odd
peak bough
if 125 has 60 even numbers then it has 65 un even numbers?
125 was if we counted 0 as a possible first digit
steep moon
yes
ocean sealBOT
Mikkel
peak bough
Yes
steep moon
so 60 odd for it
peak bough
?
steep moon
65
peak bough
Yes, if 0 is counted as a possible first digit
steep moon
and if not just 60
peak bough
Wait no
peak bough
One moment
steep moon
oki
peak bough
If we count 0 as a possible first digit:
-125 total 3-digit numbers
-75 even numbers
If we don't count 0 as a possible first digit:
-100 total 3-digit numbers
-60 even numbers
There we go
peak bough
No it's alright. I got confused myself for a minute there
floral snow
Let f : [a, b] → R be a strictly increasing function. Suppose that im(f ) = [c, d] for some c, d (in other words, its image is the whole interval [c, d], without any gaps, and nothing else). Prove that f is continuous.
floral snow
Is this correct?
mortal trellis
f(x0) makes no sense if x0 is in [c,d]
presumably you mean it the other way around
why is f(x0) not an endpoint if x0 is not
you have to show that x1<x0<x2, you cant just say it
it is because of the next inequality you are writing down. you are mixing up the order
you havent shown that in the end if you pick an x with |x-x0|<delta for the delta you have chosen, then you need to have |f(x)-f(x0)|<eps
floral snow
.close
naive valley
the - hasn't been distributed inside the (x-5)^2
all they've done is:
-[a - b] = -a + b
where a = (x-5)^2 and b = 22
echo socket
a = (x - 5)^2 you mean?
naive valley
yep sorry fixed
pliant shale
Could someone explain the steps to rewrite this equation? Haven't had to do this yet as it wasn't covered in lecture so I'm a little lost here.
pliant shale
Assuming the equation is all that's needed but here's the graph as well if it helps.
<@&286206848099549185>
oh sorry, my pc time is not the same as discord so I thought it's been 15.
lone heartBOT
@pliant shale Has your question been resolved?
lone heartBOT
@pliant shale Has your question been resolved?
alpine sable
So far this is what you have?
Consider multiplying 4 by 1 in this way to get your desired form.
pliant shale
Consider multiplying 4 by 1 in this way to get your desired form.
Ah, that's simple enough. I think I was just confused about the wording. Thanks
.close
alpine sable
dusky kraken
Hey, I can’t exactly see where i went wrong here
lone heartBOT
Hey, I can’t exactly see where i went wrong here
limpid spade
u didnt divide by two in both sides
at left side u divided by two but at right side u mutliplied by two
instead of dividing two
hence disturbing the balance and invoking mathematical chaos pandemonium
dusky kraken
I don’t know how my brain managed that @limpid spade
Let me fix that
Ok I still face a problem of it being incorrect @limpid spade
limpid spade
you didnt log the conatants
dusky kraken
What do you mean?
limpid spade
log(2 3^x)!= 2(xlog(3))
limpid spade
second
limpid spade
no
lone heartBOT
@dusky kraken Has your question been resolved?
soft saffron
lone heartBOT
soft saffron
Hi, I think I have this down but I just wanted to make sure I'm interpreting this correctly: They mean plug in the numbers and evaluate, no?
alpine sable
yes, since they are continuious, you can do that
lone heartBOT
@soft saffron Has your question been resolved?
soft saffron
yes, since they are continuious, you can do that
Oh, I didn't realize that's what they were implying. Thanks Toby, I appreciate you alleviating my ignorance.
.close
true jay
I really need help on this
lone heartBOT
I really need help on this
true jay
I've been stuck on it for like 30 mins
true jay
Yes 💀 I tried searching ways to solve it but I don't even get what I'm being asked to do
it's very difficult
finite flax
Yes 💀 I tried searching ways to solve it but I don't even get what I'm being as...
what does the bottom line say?
what do you think that means?
lone heartBOT
@true jay Has your question been resolved?
wise forge
pls help
wise forge
can you do (a) as an example? not really understanding
wise forge
y=2g?
queen narwhal
Erm
So
g isn’t a variable
So u don’t multiple it by 2
Instead it’s a function
So
Ok
If we look at the graph
Let’s start out at
X=2
Then we walk up the y axis until we hit
The graph y=g(x)
It hits at 1
wise forge
graphing y=g(2)
queen narwhal
How’d you get 1,2
We are inputing 2 into the function
And we can see that
Since the function is g(x)
We have x=2
So when x=2
What would y be
On the pink line
wise forge
1?
queen narwhal
Yesh
wise forge
uhuh
queen narwhal
Add them nowwwww :3
wise forge
5?
queen narwhal
Yesssss
wise forge
wtf
queen narwhal
At least I think
wise forge
just llike that?
wise forge
yeah (a) was right
queen narwhal
Nice
wise forge
ugh ty so much
queen narwhal
Np I’m just waiting for someone to help me now 😭
wise forge
but this time were (-)
queen narwhal
F-g
wise forge
okok
queen narwhal
Yep
wise forge
so then (c) would be multiplying
queen narwhal
Yeth
wise forge
and (d) would be dividing
wise forge
shuttle? :o
queen narwhal
…
wise forge
how far are you away from home?
queen narwhal
Thank u
queen narwhal
glad you made it home safe!
Go with ur hw 
wise forge
go where..?
queen narwhal
Gl*
queen narwhal
No type “.close”
queen narwhal
does this close on its own?
^
deft lagoon
What are the next two terms in the sequence 2, 3, 5, 7, 11, 13, 17
wary stream
What are the next two terms in the sequence 2, 3, 5, 7, 11, 13, 17
Determine the pattern
naive valley
what do those numbers all have in common?
deft lagoon
well they are integers
naive valley
more specifically?
deft lagoon
uh
deft lynx
Yup, all of those numbers share a trait
deft lynx
Yup!
lone heartBOT
lone heartBOT
✅
deft lagoon
Ok what about 2, 4, 7, 11
.
deft lagoon
3, 3 and 4
main ridge
3, 3 and 4
First one is wrong
main ridge
so, 16 and 22?
Yeah
pale flame
I do not understand how to solve problems 5 and 6. I would really appreciate any help on how to get started.
wary stream
I do not understand how to solve problems 5 and 6. I would really appreciate any...
For number 5, try to manipulate those numbers to have the same base
pale flame
Sorry, how do I do that with a fraction?
wary stream
Recall your exponent rules, like $\frac{1}{a^n} = a^{-n}$
ocean sealBOT
dldh06
pale flame
I’m sorry I’m SO confused. I definitely wasn’t taught any of this so I’m kinda struggling lol
wary stream
Take 25, that's the same as 5^2, correct?
How can you rewrite 36, with some base and some exponent?
pale flame
6^2 right?
pale flame
Yes, it would be 6 and would it be raised to the power of -36?
pale flame
Oh, so it would be 6^3
wary stream
Closer
pale flame
but would the cube be negative because it’s 1/216
wary stream
but would the cube be negative because it’s 1/216
Yes
When you manipulated that equation to have the same base, what's the current equation you have now?
pale flame
I have 6^2X = 6^-3
wary stream
Now recall that when you have an equation with the same bases, you can just set the exponents equal to each other
pale flame
Oh I didn’t know that! So I will cancels out the 6s which leaves me with 2x = -3 and then arrive at -3/2?
pale flame
Okay, sorry I’m kind of confused on how to set up this one as well. I’ll send what I have now
wary stream
It's going to be a quadratic equation
pale flame
Oh! That makes a lot of sense, I don’t know why I didn’t just recognize it.
I hope I am not wasting your time I just am very confused on how to solve this problem set.
pale flame
I thought that since they both included ln(x) I could cancel those features out
wary stream
No
That's not a legal step
If you have something like $log_{10}(x) = 2$, how would you solve for x?
ocean sealBOT
dldh06
lone heartBOT
@pale flame Has your question been resolved?
pale flame
I truthfully have no idea, I’m sorry.
wary stream
I truthfully have no idea, I’m sorry.
That's useful
You need to apply the last rule to your problem
pale flame
How do I apply that?
wary stream
What's the "opposite" of ln?
pale flame
e^x
pale flame
okay, so I get e^x+7 = e^x + 2
wary stream
You're skipping a bit there. The base needs to be e, so you apply rule 7 from this
Putting all that to e, it makes it $e^{ln(x+7)} = e^{ln(x) + 2}$
ocean sealBOT
dldh06
pale flame
okay, once I do that, do I not end up getting ^7= e^2 + 2
wary stream
No
Look at rule 7, notice how the base cancels out with the log base and you're left with the stuff on the inside?
pale flame
I don’t really understand what it’s saying.
wary stream
Do you agree on this, $10^{log_{10}(2)} = 2$?
ocean sealBOT
dldh06
pale flame
I don’t know, because I’m not sure how to read that
wary stream
https://www.youtube.com/watch?v=fnhFneOz6n8&ab_channel=TheOrganicChemistryTutor
Maybe that will help
lone heartBOT
@pale flame Has your question been resolved?
vapid shuttle
lone heartBOT
vapid shuttle
Okay so this is the question
I am unsure with where to begin. Normally least squares I know is for when Ax=b has no solution
but I can't tell what Ax=b is in this case
and we want to make it the best solution possible by doing some sort of projection
is it going to be projecting <0,1> onto <c, 0>
that's all that I can spit out of my mind about this one
naive valley
lemme take a moment to read
oh this looks familiar
i think i saw this screenshot the other day
but didn't have time to comment or think about it much
vapid shuttle
yeah I opened it up but after a few hours went offline and nothing came of it
naive valley
do you have general formulas for C-hat and D-hat for a linear fit?
given some data points
naive valley
they're not too hard to derive but my god doing it on 30 mins of sleep, that's asking a lot of my brain
vapid shuttle
I can wait for other helpers if you should get to sleep XD
vapid shuttle
do I put all the
<0, 0>
and the one
<0, 1>
into a matrix
call it A
and then multiply by column vector <c ,d>
naive valley
see for example here https://mathworld.wolfram.com/LeastSquaresFitting.html
naive valley
mm
for context, how do you typically approach this kind of problem
it's probably equivalent to what i just pasted
vapid shuttle
With a projection
vapid shuttle
No, normally I would set up the Ax=b that I cannot solve and then transform that into A^T A x-hat = A^T b and solve for x-hat
wary stream
No, normally I would set up the Ax=b that I cannot solve and then transform that...
You mean something like this? https://www.youtube.com/watch?v=RlQBEhLhM8Y&ab_channel=JeffreyChasnov
vapid shuttle
yes!
naive valley
No, normally I would set up the Ax=b that I cannot solve and then transform that...
ok so far so good
so your x would be [C D]
the two unknown parameters of the linear fit
now C is the constant term
so the corresponding column in the A matrix would be all 1's
and D is the slope term
so the corresponding column in the A matrix would be the 21 t values
to wit:
$$\begin{pmatrix}1 & -10 \ 1 & -9 \ 1 & -8 \ \cdots & \cdots \ 1 & 10 \end{pmatrix} \begin{pmatrix}C \ D\end{pmatrix} = B$$
ocean sealBOT
Bungo
naive valley
where $B$ is a column vector which is all zeros except at the middle entry (corresponding to t=0) where it is 1
ocean sealBOT
Bungo
vapid shuttle
1 moment let me read this
vapid shuttle
So, why is the left hand side 1's instead of 0's of the large column vector on the left
naive valley
read the system of equations row by row
the first row says:
C -10D = 0
the second row says:
C - 9D = 0
the third row says:
C - 8D = 0
...
then you get to the one with t=0
that one says:
C -0D = 1
(not 0 on the RHS)
then
C + 1D = 0
C + 2D =0
...
C + 10D = 0
in all cases:
C is the constant term
and D is the "slope" term
vapid shuttle
Yes okay no I do get that now!
naive valley
cool
now that's 21 equations and 2 unknowns
probably doesn't have a solution
so we do least squares to find the closest solution
vapid shuttle
yes
naive valley
that's where your
A^T A x = A^T B
x = (A^T A)^-1 A^T B
comes in
vapid shuttle
and B is all 0's except for the 11th entry?
naive valley
yep
vapid shuttle
okay so we have our Ax=B now
vapid shuttle
that's where your
A^T A x = A^T B
x = (A^T A)^-1 A^T B
comes in
This x-hat will be the solution for C and D right?
naive valley
yep
here x represents the column vector $$\begin{pmatrix}C \ D\end{pmatrix}$ which is annoying to typeset outside of tex
ocean sealBOT
Bungo
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
naive valley
and C and D are just numbers (scalars)
naive valley
cool
vapid shuttle
thank you for your help
vapid shuttle
.close
wise jewel
lone heartBOT
wise jewel
wise jewel
i dont understand how my work is wrong/what i even did wrong
merry depot
you need to do long division first
the degree of the top is larger than the degree of the bottom
wise jewel
ohhhh 
wise jewel
the degree of the top is larger than the degree of the bottom
but i dont need to do long division if the degrees are equal or if the bottom degree is larger than the top right
merry depot
you do if they're equal too
celest storm
lone heartBOT
celest storm
Any french here?
DSE = Developpable en série entière
Ma question est la suivante: comment pouvons-nous appliquer cette proposition à la fonction exp
La dérivée n-ième de exp(x) est exp(x) et exp n’est majorée par aucun nombre réel
<@&286206848099549185>
In english:
Prof gave us a theorem stating that if a function's nth derivative is < a real number M then this function is expandable in a form of a power series
And he applied this theorem to exp(x)
My question is: how can we apply this proposal to the exp function?
The n-th derivative of exp(x) is exp(x) and exp is not smaller then any real number
.close
lone heartBOT
soft pagoda
and this is the answer from my teachers answer key:
ive done everything i could to try to solve this
and get that answer
but i havent been able to
long axle
Show ur work
soft pagoda
long axle
,rotate
soft pagoda
oh uh
ocean sealBOT
soft pagoda
oh there it is
long axle
Hmmmm
soft pagoda
there is a chance my teachers key might be wrong though cause the last question also had the wrong answer on the key
long axle
Ur answer is correct
soft pagoda
oh that’s good
long axle
The teacher is wrong
long axle
Np
soft pagoda
.close
tall path
lone heartBOT
tall path
I originally thought that my answer was 7 but it says Im wrong
The only other number I can think of is 2
stuck geyser
mb
pseudo ice
$(x + 18)^{1/2}$ sure
ocean sealBOT
@pseudo ice
tall path
then I can derive it as 1/2(x+18)^1/2 right?
So where my 7 is would instead be 2 im guessing
pseudo ice
$\frac{1}{2} (x + 18)^{-\frac{1}{2}}$
ocean sealBOT
@pseudo ice
tall path
Ohhh gotcha so it moves back to the numerator which is why it's up there
pseudo ice
(actually it should be 1/2 rather than 2, my bad there)
tall path
No worries!
pseudo ice
Actually nooooo I'm being slow lol
pseudo ice
Ohhh gotcha so it moves back to the numerator which is why it's up there
it would be 2 there, you then put it in fraction form so you'd move the (x + 18)^{-1/2} to the denominator and convert back to square root
I ended up thinking they left it in power form for some reason without checking it 
tall path
Ohhhhhh okay that makes more sense haha. I was a little throw off for a second 
pseudo ice
lol even me 
sacred flax
sorry :-(
supple pecan
So in my math unit I’m given this question and my fiends and I are stumped, any thoughts on the steps and answer?
supple pecan
It’s just multiplying/dividing and adding/subtracting rational expressions
marble iron
Where are you getting stuck?
lone heartBOT
@supple pecan Has your question been resolved?
gloomy hatch
How do I find the equation for a normal line that intersects with a given point? So like y = x^2 and the point is (-2,3)
gloomy hatch
So in my math unit I’m given this question and my fiends and I are stumped, any ...
just multiply them then subtract with a common denom
slate monolith
How do I find the equation for a normal line that intersects with a given point?...
system of equations
lone heartBOT
@gloomy hatch Has your question been resolved?
tepid crane
lone heartBOT
tepid crane
I'm self studying and I'm kind of confused as to why it's a minus on the answer. When I was solving it I got 19b^7 + 12b^4 I was so sure it was correct that I never noticed it was 19b^7 - 12b^4 instead. Why is it minus? I thought the whole 12b^4 with the plus gets brought down.
slate monolith
thats just them messing up lol
tepid crane
Oh! I was confused for days. Had to put the book down to somehow rationalize this. I'm glad I asked. Thank you!
.close
steep osprey
hi
steep osprey
can someone help me figure out how to solve this? i’m confused how she put in 20x as theta from the equation a=20x^2
the work is all there i just need help understanding
what she did
<@&286206848099549185>
lone heartBOT
@steep osprey Has your question been resolved?
sonic falcon
Hello, I need help with Multivariable Limits. In normal limits if we got a limit in 0/0 form we just used L'hopitals rule, is there something similar for multivariable limits? I can't find anything on google.
I need help with the question in the image, the correct answer is 0.5 but how do I get it? Also I'm very new to the topic so I'm sorry if the question is stupid lol.
fallen verge
use difference of squares
sonic falcon
on the denominator? how would that help?
chrome salmon
on the denominator? how would that help?
it will
paper bloom
(a/b)^-3
paper bloom
what that be b^3/a^3
chrome salmon
yes
quartz panther
lone heartBOT
spiral dragon
Pls help on this question
chrome salmon
what did you try
quartz panther
Pls help on this question
please move to another thread
quartz panther
what did you try
making two simontaneuous equations and solving
chrome salmon
so what did you egt
chrome salmon
3 = 5a+b , -3 = 3a+b so we get a=3 , b=-12
keen plinth
!nosols
lone heartBOT
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.
chrome salmon
don't ban me 
quartz panther
.close
lone heartBOT
lone heartBOT
Available help channel!
mighty knoll
vale wigeon
,rccw
ocean sealBOT
lone heartBOT
Show your work, and if possible, explain where you are stuck.
vale wigeon
@mighty knoll
amber spire
a^2 + b^2 = (a+b)^2-2ab how?
lone heartBOT
a^2 + b^2 = (a+b)^2-2ab how?
vale wigeon
(a+b)^2 = a^2 + 2ab + b^2
amber spire
If i calculate a^2 + b^b then how can i get this?
amber spire
(a+b)^2 = a^2 + 2ab + b^2
Ok, got it, next?
vale wigeon
subtract 2ab from both sides of that, and you get the equality you asked about.
amber spire
a^2 + b^2
We can write like ( a+b )^2 and after calculate, we get a^2 + 2ab + b^2.
vale wigeon
no, a^2 + b^2 is not equal to (a+b)^2.
amber spire
no, a^2 + b^2 is not equal to (a+b)^2.
Then how you write this? (a+b)^2 = a^2 + 2ab + b^2
amber spire
Ok, got it, but where we get -2ab?
amber spire
subtract 2ab from both sides of that, and you get the equality you asked about.
This one, i didn't understand.
vale wigeon
(a+b)^2 = a^2 + 2ab + b^2
subtract 2ab from both sides of this
(a+b)^2 - 2ab = a^2 + 2ab + b^2 - 2ab
the +2ab and -2ab on the right cancel out
amber spire
My question is how we get 2ab?
vale wigeon
i thought you were asking how the negative 2ab comes about.
amber spire
after calculate a^2 + b^2 = a^2 +2ab+b^2
amber spire
i thought you were asking how the *negative* 2ab comes about.
yes
vale wigeon
and i am trying to answer the questions you ask
amber spire
(a+b)^2 = (a+b)(a+b) = a\*a + a\*b + b\*a + b\*b
I have understood this. but i didn't understand how we can get ( -2ab )
vale wigeon
do you understand why (a+b)^2 = a^2 + 2ab + b^2? yes or no
vale wigeon
so then are you asking why i chose to subtract 2ab from both sides, as opposed to doing something else?
amber spire
(a+b)^2 = (a+b)(a+b) = a\*a + a\*b + b\*a + b\*b
isn't done here? why -2ab again with this?
amber spire
I am sorry, i understand a little less
wild trail
Do you know how to calculate (a + b)^2 ?
amber spire
do you understand why (a+b)^2 = a^2 + 2ab + b^2? yes or no
(a+b)^2 = (a+b)(a+b) = aa + ab + ba + bb
= a^2+2ab+b^
I have undestood this clearly.
amber spire
Do you know how to calculate (a + b)^2 ?
yes
amber spire
- 2ab where you got this?
pale kestrel
from the air.
But u subtract it from both sides
understood?
2ab
(a+b)^2 = a^2 + 2ab + b^2
(a+b)^2 = a^2 + b^2 + 2ab
(a+b)^2 - 2ab = a^2 + b^2 + 2ab - 2ab
(a+b)^2 - 2ab = a^2 + b^2
amber spire
(a+b)^2 = a^2 + 2ab + b^2
(a+b)^2 = a^2 + b^2 + 2ab
(a+b)^2 - 2ab = a^2 + b^2 ...
You are saying (a+b)^2 - 2ab = a^2 + b^2.
My question is a^2 + b^2 = ?
You are telling me (a+b)^2 - 2ab = a^2 + b^2.
pale kestrel
notice B = A
winged sentinel
You are saying (a+b)^2 - 2ab = a^2 + b^2.
My question is a^2 + b^2 = ?
You ar...
its just an easier way of figuring out a^2 + b^2
subtracting 2ab from both sides makes it so that you only get a^2 + b^ on both sides
amber spire
notice B = A
OK, So there is no need to prof to calculate, this = this, right?
thorn monolith
it should have been easier to notice you can isolate (a+b)² instead of typing that entire comment
pale kestrel
(a+b)^2 = a^2 + 2ab + b^2
(a+b)^2 = a^2 + b^2 + 2ab
(a+b)^2 - 2ab = a^2 + b^2 + 2ab - 2ab
(a+b)^2 - 2ab = a^2 + b^2
a^2 + b^2 = (a+b)^2 - 2ab
amber spire
I dont understand your query
How you can prof ? b = a?
thorn monolith
equation isnt only one way bruh
pale kestrel
Im saying if <something> = <something else>, then <something else> = <something>
if apple = banana, then banana = apple
amber spire
I can memorize it. a= b | b=a.
I am trying to prof it, how?
vale wigeon
you are trying to prove that equality is symmetric...?
waxen rose
whats central tendency
vale wigeon
@waxen rose wrong channel. #❓how-to-get-help
pale kestrel
So you understand what = is representing in math
thorn monolith
I can memorize it. a= b | b=a.
I am trying to prof it, how?
for starters you can type "b" where you see "a"
thorn monolith
not only way, you can also type "a" where you see "b"
amber spire
bruh what
I can memorize it. a^2 + b^2 = (a+b)^2-2ab.
I want to prof this. to calculate a^2 + b^2. that is my goal.
pale kestrel
(a+b)^2 = a^2 + 2ab + b^2
(a+b)^2 = a^2 + b^2 + 2ab
(a+b)^2 - 2ab = a^2 + b^2 ...
this is a proof...
memorize what
pale kestrel
I can memorize it. a= b | b=a.
I am trying to prof it, how?
dont use the same letters otherwise everything will be mixed up
i use A and B earlier which is different
amber spire
(a+b)^2 = a^2 + 2ab + b^2
(a+b)^2 = a^2 + b^2 + 2ab
(a+b)^2 - 2ab = a^2 + b^2 ...
The confutation is that. last -2ab.
pale kestrel
,tex \begin{align*}
(a+b)^2 &= a^2 + 2ab + b^2
\
(a+b)^2 &= a^2 + b^2 + 2ab
\
(a+b)^2 - 2ab &= a^2 + b^2 + 2ab - 2ab
\
(a+b)^2 - 2ab &= a^2 + b^2
\
a^2 + b^2 &= (a+b)^2 - 2ab
\end{align*}
ocean sealBOT
pale kestrel
Have you got an image editor
can u circle where the confusion is
or call them lines 1-5
and say which one to which one
amber spire
pale kestrel
yes...
amber spire
Now we are align, like (a + b)^2 = a^2 + b^2 + 2ab, right?
pale kestrel
?????
oh
I am using the fact that the order of addition doesnt matter
X + Y + Z = X + Z + Y
amber spire
Ok, So here we are change the order just. nothing elase, right?
amber spire
Ok,
The next line ( a+b)^2 - 2ab , how?
amber spire
> If X = Y, then X - Z = Y - Z
I didn't understand this example.
amber spire
Ok,
The next line ( a+b)^2 - 2ab , how?
Here is my confusion
amber spire
yes
pale kestrel
I added 1 to both sides
and got something thats still true
You can add or subtract to do this
3 - 3 = (5 - 2) - 3
If X = Y, then X - Z = Y - Z
This is what im using
amber spire
Ok, let me understand.
lone heartBOT
@amber spire Has your question been resolved?
sturdy zenith
Hi ! I need help starting this question
lone heartBOT
Hi ! I need help starting this question
sturdy zenith
thorn monolith
use Fundamental theorem of calculus
thorn monolith
chen lu (chain rule) too
sturdy zenith
would i differentiate e^t^2 then?
thorn monolith
e^(0.5)^2-e^0
also e^0 wouldnt be a thing, you are thinking derivate with respect to x
thorn monolith
would i differentiate e^t^2 then?
no
reminder: dont forget the chain rule (its 3x instead of x here)
alpine sable
lone heartBOT
alpine sable
How do I do i?
lone heartBOT
@alpine sable Has your question been resolved?
cosmic roost
hi
cosmic roost
if x=loga(b) and y=logb(a). show that loga(b)=1/logb(a)
the letter outside the bracket is the base
how do I prove this?
<@&286206848099549185>
normal flower
lone heartBOT
What step are you on?
1. I don't know where to begin
2. I have begun but got stuck midway
3. I got an answer but I'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
@cosmic roost Has your question been resolved?
paper marsh
lone heartBOT
thorn monolith
not enough info
gray isle
the best you can do is express that in terms of d
lone heartBOT
heady cargo
Hello?
lone heartBOT
Available help channel!
hidden pike
need help in this question.
lone heartBOT
need help in this question.
hidden pike
this is what basically i've did till now, idk how will i reach to the options with such method.
oh the question is wrong in that picture :weird_kid:
thanks anyways
.close
alpine sable
I need help on question 5. I'm not sure what equation to use to find how fast the rope is stretching
steel spire
I need help on question 5. I'm not sure what equation to use to find how fast th...
Im not sure if Im correct but this is how I did it
alpine sable
I have inf tries on it so don't worry
steel spire
So initially, we can use the pythagorean theorem to determine the length (hypotenuse of the imaginary right angle triangle"
alpine sable
yeah
steel spire
If your position (the runner) is at 7.4m and you were going 2m/s how many seconds did it take for you to reach 7.4m?
alpine sable
3.7
steel spire
yup
alpine sable
my friend is 11.1
steel spire
assuming you and the air ballon started moving at the same time
yup exactly
and if you use the pythagorean theorem whats the length of the triangle?
alpine sable
13.34
steel spire
nice
so we got a length of 13.34m in 3.7 seconds
is there anyway to manipulate this so that in turns into x m/s
alpine sable
so just into per second?
steel spire
hint: 13.34 | 3.7
alpine sable
13.34/3.7=x/1?
steel spire
well it would be better if you divide both sides by 3.7
if you divide 3.7 by 3.7 that is 1
13.34/3.7 = 3.61
steel spire
3.61 m/s
steel spire
if im wrong im sorry
steel spire
np. You too!
alpine sable
.close
lone heartBOT
lone heartBOT
Available help channel!
simple swan
I want help with this limit pls
thorn monolith
I want help with this limit pls
l'hopital allowed im assuming?
vale portal
lone heartBOT
vale portal
Hello
Unsure as to how lecturer got from A:( 1/√2, 1/√2) D: (-1/√2, -1/√2) to At A and D, f=1/2
lone heartBOT
@vale portal Has your question been resolved?
alpine sable
$\frac{1}{\sqrt2}\times\frac{1}{\sqrt2}=1/\left(\sqrt2\times\sqrt2\right)=1/\sqrt2^2=1/2$
ocean sealBOT
0†Deathdoll∞Enchant†0
alpine sable
$\frac{1}{\sqrt2}\times\frac{1}{\sqrt2}=\frac{1}{\sqrt2\times\sqrt2}=\frac{1}{\sqrt2^2}=\frac{1}{2}$
ocean sealBOT
0†Deathdoll∞Enchant†0
vale portal
.close
feral sonnet
Hello I have a few questions about my geometry work
lone heartBOT
Hello I have a few questions about my geometry work
feral sonnet
First i’m trying to find the area of a trapezoid without the height
neon thicket
post the question or image
feral sonnet
I sent it but it wouldn’t send
one moment
So the question is “ what is the area of a trapezoid in square units
All I know is Top base is 27 Bottom base is 37 and the Left side is 13
lone heartBOT
@feral sonnet Has your question been resolved?
lone heartBOT
@feral sonnet Has your question been resolved?
hazy ferry
All I know is Top base is 27 Bottom base is 37 and the Left side is 13
you need to find out the green length first, then you can calculate the height with pythagorean theorem
and then you can use this
teal dust
narrow plank
feral sonnet
you need to find out the green length first, then you can calculate the height w...
that is the part i need help with
because I don’t know any other variables
all i know is it’s a right triangle with hyp that is 13
also it doesn’t say that they are congruent to eachother
vital wedge
hazy ferry
that is the part i need help with
the green lengths are the same
and the red one is 27
lone heartBOT
@feral sonnet Has your question been resolved?
north hemlock
How do I use Euler's Totient Theorem to calculate 3^103 mod 1000?
north hemlock
oh god
vapid shuttle
north hemlock
fr
the worst part is that 103 is prime :)
wait so like $3^{103} \text{ mod } 1000 \equiv a^{103 \text{ mod } \phi(1000)} \text{ mod } 1000$
and uh
ocean sealBOT
Juke | ping me if no response
north hemlock
from here....
103 mod 400??
oh
1000 = 8 * 125 so
hm
yeah not sure
,calc 400(103)
ocean sealBOT
Result:
41200
So Euler's totient theorem states that $a^{\phi(n)}\equiv 1\left(\mod{n}\right)$
ocean sealBOT
Labyrinth
alpine sable
Meaning that $a^{103}a^{\phi(1000)}\equiv a^{103}\left(\mod{1000}\right)$
ocean sealBOT
Labyrinth
alpine sable
I hate that space after the left parentheses askjdaklsdja
$\left(!\mod{1000}\right)$
Cruel world
north hemlock
and then $\phi(1000)$ is $\phi(4)\phi(125) = \phi(4)\phi(25)\phi(5) = \phi(5)^3 \phi(4) = 2(4)^3$
ocean sealBOT
Juke | ping me if no response
north hemlock
,calc 4^3
ocean sealBOT
Result:
64
128?
wait a sec
that's not right
oh wait they need to be coprime for me to just keep doing tht
so just $\phi(4) * \phi(125)$
ocean sealBOT
Juke | ping me if no response
ocean sealBOT
Juke | ping me if no response
alpine sable
Alright so I calculated the result using another method just so we can double check
alpine sable
so I just need to do 3^503
Hmmm what okay where did the 503 come from?
north hemlock
we get that $3^{103}a^{\phi(1000)} = 3^{103}a^{\phi(8)\phi(125)} \equiv 3^{103} \text{ mod } 1000$
ocean sealBOT
Juke | ping me if no response
north hemlock
so then $3^{103}a^{\phi(8)\phi(125)} = 3^{103}3^{400} = 3^{503} \equiv 3^{103}\mod 1000$
ocean sealBOT
Juke | ping me if no response
alpine sable
Yeah I'm starting to feel stupid
north hemlock
huh?
alpine sable
...Does that look like progress to you?
I feel like I just found a way to get further from the answer
north hemlock
well we have that our problem simplifies to 3^503
as for the value of that.... no clue
alpine sable
Yes
north hemlock
on the LHS we get a^503
alpine sable
Are you trying to do it the other way around like I was thinking of doing?
alpine sable
I meant to write 3^103 but yea
alpine sable
Actually it's anything*(anything else)^phi(1000)==anything(mod 1000)
Which is even more powerful
alpine sable
Yeah
north hemlock
ok ye
is that the solution u got doing it another way?
also is the other way CRT?
alpine sable
also is the other way CRT?
No idea what that is
north hemlock
chinese remainder theorem
alpine sable
Eh, probably, I embarassingly don't remembeer the CRT
I implemented a binary recursive algorithm if that makes sense
alpine sable
No, I implemented a binary recursive algorithm...
Well bruteforce is subjective but that doesn't seem like bruteforce to me
north hemlock
to calculate $3^{103} \mod 1000$
ocean sealBOT
Juke | ping me if no response
north hemlock
wait
alpine sable
def calculate(pow):
if pow == 1:
return 3
else:
return (calculate(pow//2)*calculate(pow-pow//2))%1000
did u end up with a nice number?
alpine sable
Yep
alpine sable
3^503 is bigger than 1000
fun fact
Literally could not have been it
Although, as expected, it has the same remainder mod 1000
alpine sable
Okay so there's a fact I learned from looking at the solution