no there isnt context
#help-43
rose goblet
book splits in cases
in some it uses trig subs but in many it suggests this
and not only on 1 book
native shard
rose goblet
i have seen this sub a lot and was wondering if it works on all cases where a is positive
native shard
šš»
rose goblet
https://www.integral-calculator.com/
?
does this work in all cases where a,b are real and a isnt negative
rose goblet
so only avoiding these types sqrt(-x^2 + 1)
rose goblet
type in sqrt(ax^2 + b) it gives a full solution with steps
how do i even put an indefinite integral there
rose goblet
fire brother
so it just took 1 case
z-x root(a)= root(ax^2+b) does this sub work for any b and positive a?
native shard
so it just took 1 case
then just rewrite it to be -b and -a
rose goblet
-a makes complex integration
and its gonna use different steps for each case
the question stands
z-x root(a)= root(ax^2+b) does this sub work for any b and positive a?
compact pewterBOT
@rose goblet Has your question been resolved?
rotund nimbus
Consider an isosceles right triangle ABC, with the right angle at A. There's a point M inside the triangle such that MA = 1, MB = 2, and MC = 3. Calculate the angle ā AMB
compact pewterBOT
Consider an isosceles right triangle ABC, with the right angle at A. There's a p...
open plaza
show your diagram
mortal nimbus
This could probably be done with vectors
sturdy forge
Could you help me, please?
what's confusing you about the diagram?
do you know what isosceles means and what a right triangle is?
signal willow
If I could draw diagram, I wouldn't need your help
draw a rough diagram?
check that you understand the question
sturdy forge
there's no need to construct an exact diagram btw. I think it'd be most helpful to construct a rough sketch with the most important features listed
signal willow
[hmm, wait is the situation possible...? lemme check]
sturdy forge
if it's a question, then probably....
it looks like an AMC question
so it's likely been thoroughly checked
signal willow
yeah, it's possible, i convinced myself with intermediate value theorem
rotund nimbus
it's a math problem for a 10th-grade student
signal willow
in these sorts of questions you usually want to construct congruent triangles to move your lengths about
rotund nimbus
What do you mean?
signal willow
you usually need to construct additional points to build useful shapes
sturdy forge
uh
your responses indicate that you may not be particularly experienced with geometry
this question might be a little difficult...
sturdy forge
What do you mean?
when you can show that different triangles are congruent, you can relate the lengths and measures of their corresponding sides and angles
it's very useful for these sorts of questions
rotund nimbus
yes,I'm not good at math
sturdy forge
are you aware of your triangle congruence theorems? (ASA, SSS, AAS, etc..)?
signal willow
okay i have an idea of how to get a solution... but so far i need to use trigonometry
sturdy forge
yes,I'm not good at math
that's perfectly alright. Unless this is a problem required for school, however, it could be better to first read up on some geometry and work on some easier problems to build up to it
signal willow
like it's some construction + a simple ending trigonometry step
it might be a nice angle, but I'll have to check
sturdy forge
like it's some construction + a simple ending trigonometry step
I haven't tried it yet but it's an AMC problem so I certainly expect some difficulty to arise
I wonder if it's amc 8 or 10
that will give hints to what tools are necessary I think
open plaza
I don't think this is an AMC
sturdy forge
I don't think this is an AMC
am i just getting baited by the angle in question being called "AMC" š
open plaza
sturdy forge
it's soooo sus
ain't no way it's just a coincidence
wait damn I actually cannot find it in the aops wiki on google
rotund nimbus
So, how do you solve this problem?
open plaza
toy around with stuff
open plaza
AMC looks like 90
sturdy forge
toy around with stuff
^
signal willow
but there's a way to make the trigonometry easy
open plaza
AMC looks like 90
from what I drew
signal willow
AMC looks like 90
definitely less than 90 deg from my analysis
rotund nimbus
what is amc
sturdy forge
what is amc
in this case, he's referring to the angle in your problem
earlier, we were talking about the american math olympiad called the AMC
signal willow
I'm getting about 80 degrees
open plaza
hmmm
signal willow
i'll clean up and try to get a diagram
open plaza
I mean I have saw this problem a couple of times before
these kinds of problems are designed so that the end result is somewhat pretty
so I still believe it is 90
signal willow
I'm completing the isosceles triangle into a square since squares are nice
open plaza
lemme try to prove it tho
signal willow
this should be roughly the sort of diagram you should be getting, and then what we want to do is basically construct new points to shift the angles about
this may be a very big hint
lavish rock
AMC = 135Ā°?
rotund nimbus
can you use trigonometry to solve
signal willow
can you use trigonometry to solve
yes, after constructing the diagram, and chasing down some simple angles, it's pretty basic trigonometry
rotund nimbus
It's more like 150.
signal willow
if my calculations are right, it's about 80 degrees
rotund nimbus
AMC
signal willow
it says around 79.4 degrees
yeah it's about there
not too hard to calculate with trigonometry, @rotund nimbus are you interested to try to go through that solution?
open plaza
my initial assumption is dead wrong
rotund nimbus
i use ruler and it's about 150
signal willow
i use ruler and it's about 150
hmm, are you measuring AMB instead of AMC?
AMB is about 167 degrees
rotund nimbus
actually,the question require calculate AMB,I am wrong
open plaza
show us the original problem
in original language
take a photo or screenshot of it if you can
rotund nimbus
the message above,i edited the message
compact pewterBOT
Available help channel!
deft tangle
compact pewterBOT
deft tangle
B option is RUR^-1 subset of and equal of I_A
Opps it was very easy
C is correct
A is reflexive
.close
young tinsel
please help me
compact pewterBOT
please help me
young tinsel
i tried finding a pattern for the tens digit
subtle helm
what do u notice
subtle helm
?
young tinsel
8^3 has 1 in tens?
young tinsel
idk cause this problem was under the binomial theorem category
so maybe try to use that instead?
young tinsel
ok
viscid canopy
Also if binomial is used then u can rewrite it 8 using 10-2 and 12 using 10+2
civic quartz
hm we could use binomial here but its a bit lenghty tbh
young tinsel
do you know eulers theorem
is it the same thing as fermat's little theorem
native shard
donāt
civic quartz
Also if binomial is used then u can rewrite it 8 using 10-2 and 12 using 10+2
yeah^
civic quartz
donāt
i mean we can use both concepts here
first the binomial to study the terms , then using flt for the relatively easy term's last digit
compact pewterBOT
compact pewterBOT
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wind spindle
compact pewterBOT
wind spindle
Suppose T is injective
Then my thought was basically that there exists S because
S is essentially just the inverse
It takes Tv to v
and any excess elements in W to 0
Not sure how to formally say this, but it seems to be essentially the right idea
grim arch
You've got the right idea. Use the linear map lemma and choose a basis for V and W to make it more formal
wind spindle
Alright
I'll get back to u with that and then I'll need to consider the other direction
grim arch
alright - feel free to ping me!
wind spindle
Let v1,...,vn be a basis of V. Our injective linear map T is then defined as follows: T(a1v1 + ... + anvn) = a1z1 + ... + anzn, where the z_j's are some list of linearly independent vectors in W. Thus, we can extend our list of z's to z1,...,zn,w1,...,wj to be a basis of w. Then there exists S defined by: S(a1z1,...,anzn,b1w1,...,bjwj) = a1v1 + ... + anvn, and it's easy to see that ST is the identity operator on V.
@grim arch
grim arch
Looks mostly good - might want to justify the fact that the z_js are linearly independent
wind spindle
okay now for the other way around
Suppose there exists S s.t. ST is the identity operator on V. Show that this implies T is injective
ST(v) = v
If T(v1) = T(v2)
then ST(v1) = ST(v2)
then v1 = v2
done
yeah that was easy
grim arch
yup
compact pewterBOT
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feral fog
If the definition of sine is opposite/hypotenuse (both always positive values), how can it have negative outputs?
compact pewterBOT
If the definition of sine is opposite/hypotenuse (both always positive values), ...
subtle helm
If the definition of sine is opposite/hypotenuse (both always positive values), ...
This definition of sin is pretty restrictive
feral fog
I know about the unit circle definition, but arent both of them correct?
keen granite
|sin| = opp/hyp
subtle helm
I know about the unit circle definition, but arent both of them correct?
How u do make a triangle of angle 3pi/2
feral fog
|sin| = opp/hyp
why is it absolute value?
subtle helm
|sin| = opp/hyp
I think this is formed using similar triangles with the definition using the unit circle (how I was taught)
subtle helm
why is it absolute value?
Because sin can be negative
But opp/hyp must always be positive
rose goblet
proof for this?
compact pewterBOT
proof for this?
rose goblet
anyone can help me figure it out
upbeat gorge
anyone can help me figure it out
what is the formula for area of a 2D region, with a double integral?
rose goblet
what is the formula for area of a 2D region, with a double integral?
double integral?
1 integral it is
double is volume
upbeat gorge
generally, yes. but if you doubly integrate over a certain function, you get the area of D, the domain of integration
do you know/recall that integral?
upbeat gorge
$\mathrm{Area}_D = \iint_D 1 ; \dd y\dd x$
boreal girderBOT
haseeb
upbeat gorge
yes, but because we are integrating over 1, you get the original region back, with no height
rose goblet
when is that
grim arch
1 integral it is
single integral of a function gives area under the curve. The formula haseeb posted is more general - it will simplify to an integral of a function if you use it for that
have you seen this before?
rose goblet
send link
upbeat gorge
it's paul's online notes
rose goblet
it's paul's online notes
yes what page of what course
rose goblet
havent done this yet
upbeat gorge
i thought you finished calc 3?
rose goblet
i did quickly calc 2 and part of 3 not all
upbeat gorge
ah
rose goblet
i have done a lot on calc 3 but not all
i came back to calc 2
to do excersices too cause i didnt
and do all proofs
tbf i didnt do calc 3 in correct order i think
rose goblet
ah
i might have done first part of it
ik how to calculate multiple integrals etc and stuff but i didnt know the area interpertation
i see why the top A= is correct
but how is that translateing to double int
oh i see
g2-g1 is int of dy
upbeat gorge
well, that would be where the formula stems from, ish, because $\frac{1}{2}r^2 = \int_0^{r=f(\theta)} t ;\dd t$
rose goblet
yes i see why
boreal girderBOT
haseeb
upbeat gorge
and you are taking the integral $\dd t \dd\theta$ (normally $\dd r \dd\theta$, but $r$ was in use)
boreal girderBOT
haseeb
upbeat gorge
g2-g1 is int of dy
so it "collapses" to a single integral
upbeat gorge
ye just the inner integral
upbeat gorge
uhh yeah but you can stay in (r, theta) because polar coords anyways
unless its easier to think about dydx
rose goblet
unless its easier to think about dydx
easier cause i just proved thats the area
how would we show it in polar
upbeat gorge
yeah, we get the integral from g2 to g1, and in this case, g2 and g1 are the lines
upbeat gorge
how would we show it in polar
convert to polar with the jacobian, then it's understood the area formula is $$\iint_R r ; \dd r \dd\theta$$, where $R$ is the transformation of $D$. then i think you can see it
boreal girderBOT
haseeb
upbeat gorge
oh ok
rose goblet
**haseeb**
show this is area same way we did for the other one
but we cant really type g2-g1
if im being honest i dont have an image in mind of how polar integration works
geometrically
upbeat gorge
once you convert the area integral to polar coordinates, you can stay inside polar and just show it there
the integrals are equivalent because we did the jacobian
rose goblet
yes ik that
i get that
but if we never had xy coordinates
and we started from polar and only show the problem in polar
how can we do some like this
where r dr dĪø will convert to something obvious
g2-g1
but as i said i dont haave an image of polar integration so idk what that obvious would be
in xy its obviously g2-g1 dx but in polar i have no idea
or we cant
idk i might be doing circular reasoning like this
upbeat gorge
in polar, the g2 - g1 will likely become circular rays, like theta = beta or theta = alpha in the original picture
you could probably show the area formula independently in polar coordinates by approximating an area with infinitesimal shapes that look like this
weeh transparent D:
upbeat gorge
and this is the idea of polar integration: if you vary y, you are varying r, and if you vary x, you are varying theta
rose goblet
and this is the idea of polar integration: if you vary y, you are varying r, and...
geometrically i meant
upbeat gorge
you can define equations for the lines at these endpoints
rose goblet
oh wait but this difference is curved
upbeat gorge
take a look at your original picture, and you will see them
rose goblet
in polar
and from what i know there isnt such thing as curved vector
so i dont think we can show it like that
upbeat gorge
oh wait but this difference is curved
which is why we vary an angle
rose goblet
no but that doesnt find that curved arc
it finds the line that connects the 2 points
anyways
jacobian is good enough
upbeat gorge
think of walking on a sphere: if you go straight, from the outside it looks like a curved path
rose goblet
i think im just going backwards like this
upbeat gorge
but it's a straight path "in the sphere"
yeah hopefully you understand where it came from
rose goblet
ight i mean ill just use jacobian i fully see it that way
upbeat gorge
generally, if i see (1/2)x^2, i immediately think integral of x
upbeat gorge
ight i mean ill just use jacobian i fully see it that way
thats fair. non-euclidean geometry can be unintuitive, but imaigning yourself "on that shape" will help
rose goblet
thats fair. non-euclidean geometry can be unintuitive, but imaigning yourself "o...
alr
if im being honest i stopped using visualization
it sometimes helps but in maths you should go of of proven theorems that are built on axioms not just your imagination and it also puts a lot of limitations on more complex stuff
upbeat gorge
i wouldnt recommend that tbh. imo visualization of complex structures can happen, it just takes time because it is a skill to practice. it is helpful to fall back on definitions, but geometric thinking is a big part of something like integration
complex integration relies pretty heavily on what path you choose, in relation to "where the function is"
rose goblet
i wouldnt recommend that tbh. imo visualization of complex structures can happen...
ik it is possible but i think in math you should rather be able to think in theorems and more algebraicly and rigorously/formaly
rather than constracting an image and using information from it
idk atleast for me this way i dont use stuff as obvious and i rather prove majority
anyways man thanks for the help
upbeat gorge
it depends on what you're doing. analysis is pretty heavily geometric, but in algebra you probably dont want to spend time visualizing stuff
hopefully i clarified some stuff tho 
rose goblet
hopefully i clarified some stuff tho <:catthumbsup:614540188747563008>
yes ofc i understand everything where it comes from in the picture i posted now
thanks
.close
compact pewterBOT
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severe solar
help
chilly basalt
Hi, question?
chilly basalt
Alr, youāre good with graphing it right?
severe solar
yea no need to graph
chilly basalt
Ok so, hereās the first question
Do you know what we need in polar form?
Two elements
severe solar
If you're talking about the numbers, 8 and 6
chilly basalt
If you're talking about the numbers, 8 and 6
Nope
severe solar
trignometry?
chilly basalt
We need to know two elements to form a polar coordinate
severe solar
cos and sin
chilly basalt
One more guess and Iāll reveal the answer
severe solar
arctan
chilly basalt
We need length and angle
severe solar
x ( cosx + jsinx)
chilly basalt
x ( cosx + jsinx)
Nah, replace the outside X with k or something else
severe solar
ya sure
chilly basalt
x plays an important role in this form
severe solar
it can also be Kangle (degree)
chilly basalt
z = k ( cosx + jsinx)
severe solar
yes
chilly basalt
Quick question number 2, can you find the exact value of theta in this case
We are now looking for the angle
chilly basalt
I can find the length
Decent, whatās the length?
severe solar
isnt it sqr a^2+b^2
sqr8^2+6^2=100
sqr100=10?
chilly basalt
isnt it sqr a^2+b^2
sqr8^2+6^2=100
sqr100=10?
Correct
chilly basalt
Because the triangle is not a special one. Therefore, we cannot find the exact value of theta, do you agree?
Special triangles are like 30-60-90 45-45-90 etc
severe solar
yea if it was special it would be on the left instead of it being on the right
its not touching 90
chilly basalt
its not touching 90
Itās a right triangle actually
But we have no idea about its exact value
severe solar
correct
I do watn to tell you tho I have the answer, but the point of this is for me to understand how that number came to be
but my question is tho
if a special triangle assuming you're saying left has ''30 degrees''
how come this one's exact value is 36
36.9 to be exact
does 6.9 make a difference
chilly basalt
I do watn to tell you tho I have the answer, but the point of this is for me to ...
I have a question personally. Does your teacher regulate the form to be presented in such case? Like you need to apply inverse trig or something.
severe solar
I don't think so
we are doing complex numbers
I think we just need to find exact value
not really anything too deep with trig
since its not a trig unit
chilly basalt
Allowed to use calculator?
severe solar
yes
chilly basalt
Ah I see
chilly basalt
In this case, you can first find out the value of sine theta
severe solar
and thats b/a right
chilly basalt
or tangent theta, which is easier and much more direct
chilly basalt
and thats b/a right
What do a and b represent?
chilly basalt
Thatās tangent theta
severe solar
ohh
chilly basalt
and thats b/a right
Tangent
severe solar
you're right my fault
I forgot this doesnt have a hypotenuse
ok tangent
so I'd do tan(6/8)
severe solar
tan^-1?
chilly basalt
yes
chilly basalt
tangent(theta) = 6/8
tan^-1(6/8) = theta
severe solar
I thought I was getting my values for sin and cos using sin and cos
but its actually tan
thanks man
chilly basalt
I thought I was getting my values for sin and cos using sin and cos
No, you just need to find the angle and the length
Thatās everything
severe solar
yep
chilly basalt
thanks man
Alr, have a good one
severe solar
you too man
chilly basalt
Donāt forget to close the channel before leaving
severe solar
.close
compact pewterBOT
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carmine garden
If $G$ is a finite group,show there exists a positive integer $N$ , such that $a^N=e \forall a \in G$
compact pewterBOT
If $G$ is a finite group,show there exists a positive integer $N$ , such that $...
boreal girderBOT
wai
carmine garden
sorry, wrong react
kind viper
... such that $a^N = e$ for all $a \in G$
boreal girderBOT
Ann
kind viper
$\forall$ is basically only for the logicians or when you're writing shorthand, and even then it has to go on the front of the statement.
boreal girderBOT
Ann
carmine garden
got it
kind viper
anyway yes share progress.
carmine garden
one minute, trying to find it in my notes
worthy pendant
Hello
carmine garden
Let $\abs{G}=M$
\
Let $N>M$
\
We then have $a^N= a^k; 0ā¤k<M$
\
This then gives us
$a^{N-k}=e$
compact pewterBOT
Hello
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #āhow-to-get-help for instructions).
boreal girderBOT
wai
carmine garden
I suppose this is wrong
kind viper
Let $\abs{G}=M$
\\
Let $N>M$
\\
We then have $a^N= a^k; 0ā¤k<M$
\\
This then give...
this is imprecise.
but let me try to steer you on the right track
you have been introduced to the concept of order of an element, yes?
kind viper
of an element.
so you know that for each element a there exists a natural ord(a) satisfying a^ord(a) = e
(and is the smallest one that does the job)
yes?
carmine garden
yes
kind viper
ok
so far, though, this depends on a.
let me also ask you this, and to be clear i will temporarily forbid you from running off and finishing the proof on your own:
if it is known for some $a \in G$ and $n \in \bZ$ that $a^n = e$, what can you say about $n$ and $\mathrm{ord}(a)$?
boreal girderBOT
Ann
carmine garden
nā„ord(a)
kind viper
incorrect
i specifically said n ā Z so you couldn't do this
ord(a) = 7 implies a^(-49) = e but -49 ā„ 7 is false
think again
kind viper
ok that's now technically correct but not morally correct
well like
i say not morally correct
i mean it's not what i am expecting of you
do you want to try again or do you want me to just tell you what i am looking for
carmine garden
for some $n \in \Z a^n=a^{\ord{a}}$
boreal girderBOT
wai
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
dull raft
i mean it's not what i am expecting of you
Am I allowed to say that you're looking for something number-theoretic?
kind viper
for some $n \in \Z a^n=a^{\ord{a}}$
mmmmmnobad
kind viper
Am I allowed to say that you're looking for something number-theoretic?
yes you are and i will agree
kind viper
for some $n \in \Z a^n=a^{\ord{a}}$
this says nothing
carmine garden
ord(a)|n
kind viper
ok there we go yes
so in other words, the solutions to a^n = e are precisely the integer multiples of ord(a).
now here's a thing
say you're looking at two elements of your group $a, b \in G$. you want to find an exponent $m$ such that $a^m = b^m = e$. can you give the \textbf{lowest} such exponent?
boreal girderBOT
Ann
kind viper
in terms of ord(a) and ord(b) obviously.
carmine garden
it would be the lcm of the order of a and order of b
kind viper
ok
yes it would
now we can go back to the problem
of finding an exponent N that "works" for every single element of G at once
it will be crucial that G is a finite group for this
carmine garden
I would think it would be the LCM of orders of all elements of G
carmine garden
is that it?
kind viper
I would think it would be the LCM of orders of all elements of G
indeed
compact pewterBOT
odd shale
9+10
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Available help channel!
boreal aurora
Idk what Iām doing
compact pewterBOT
Idk what Iām doing
kind viper
factor out an (x-2)
open plaza
set condition for x as well
kind viper
(x-2)(2-(x-2))=0
boreal aurora
Iām still confused
wooden bear
Idk what Iām doing
Hi, here you are workign in circles
yk how to expand (x-2)^2?
do that, then move the 2x-4 from the right to the left
and set the entire left equation = 0
so it would be x^2 -4x + 4 = 2x-4
and x^2 -6x + 8 = 0
which would give you x = 4 or 2
however you need to check both answers
in this case both are correct cus there is no negative value for x
in square root questions like these sometimes x cannot be negative
chilly basalt
yk how to expand (x-2)^2?
why?
wooden bear
why?
to solve it
?
idk the original question but its likely asking for values of x
chilly basalt
no need to expand it tho
wooden bear
how would you do it
can u like write it in chat
cus theres def more than one way of doing it
but expanding is the most simple for me
(a+b)^2 = a^2 + ab + ab + b^2
chilly basalt
||(x-2)^2 = 2(x-2). >> (x-2) = 0 or (x-2) = 2||
boreal aurora
This is what I got, what did I do wrong
grizzled geyser
This is what I got, what did I do wrong
The one I underlined in red is wrong
Check again when you open the bracket.
Or you can just don't and unite the (x-2)'s
Also, pro tip, you can split cases depending on whether (x-2) is zero or not. If (x-2) is zero, you can find the solution and check if it works on the original equation. And if (x-2) is not zero, you can now divide by (x-2)
grizzled geyser
Try expanding (x-2)(x-2)
compact pewterBOT
@boreal aurora Has your question been resolved?
boreal aurora
Try expanding (x-2)(x-2)
I get xĀ²-4x+4
boreal aurora
The one I underlined in red is wrong
How is it wrong then?
boreal aurora
is that kumon?
No
boreal aurora
<@&286206848099549185> I still canāt figure this shit out and Iāve been looking at it for over an hour
tribal nacelle
<@&286206848099549185> I still canāt figure this shit out and Iāve been looking ...
yes?
boreal aurora
This is what I got, what did I do wrong
.
quasi mortar
Check your expansion again - it may help to put brackets around the expanded term first.
grizzled geyser
I get xĀ²-4x+4
How do you negate that expression?
boreal aurora
How do you negate that expression?
What?
grizzled geyser
How do you negate it?
boreal aurora
What does negate mean
quasi mortar
@boreal aurora it might help to look at it this way:
When we expand -(x-2)(x-2), it is the same as expanding -1( (x-2)(x-2) ). As you have noticed, (x-2)(x-2)=x^2-4x+4 right? So what happens if we substitute this into the expression?
boreal aurora
Isnāt that literally what I did though
kind viper
uh oh stinky
you got a sign error right here
this should've been either
2(x-2) - (x^2 - 4x + 4) = 0
or
2(x-2) - x^2 + 4x - 4 = 0
oh bruh i am repeating after xwtek
vale moth
nah bro keep your head up mistakes happen
lavish gull
it happens to the best of us
vale moth
and the worst (me)
lavish gull
maybe make more use of brackets
grizzled geyser
Instead of mocking yourself, do more exercises and watch yourself become smarter.
lavish gull
i mean, that's what practice is for, isn't it
i crash and burn on simple +/- on calculus too, and it just happens
sometimes you are so tunnel-visioned on the hard stuff
you just miss the low hanging fruit
boreal aurora
My unit test is tomorrow (today in 6 hours) Iām definitely gonna do terrible
boreal aurora
sometimes you are so tunnel-visioned on the hard stuff
Yeah it could also be that itās 2:40am
Or maybe Iām just stupid
Idk
boreal aurora
I have to get an A on this test or else I might not get an A in the class
Iām only at an 83% average rn and I donāt want it to drop because of another B
lavish gull
remember your groupings, abuse brackets if you have to
grizzled geyser
Finally got it
Also, you need to check if the answer you got actually a part of the solution because you performed a squaring, which is not a one-to-one function
Both of the solution happens to be correct, but you have to make sure.
compact pewterBOT
vale moth
Imma go to bed now
goodluck on your test
compact pewterBOT
Available help channel!
near hound
is this channel free?
compact pewterBOT
is this channel free?
near hound
How do i solve this
kind viper
is this channel free?
if it doesn't have somebody's name on it then it's free
near hound
sweet
kind viper
anyway, progress so far?
near hound
i did a little but idk if im going to right steps i have a feeling im just over complexing working out
kind viper
well let's see what you got
near hound
so far
kind viper
1.5 = 159.15 + 1.5
what's that equals sign doing there...
this looks a bit disorganized even if vaguely on the right track
near hound
the inter circle + the outer circle = 1.5m
kind viper
?
kind viper
your roadmap will be roughly as follows:
- find the inner radius (based on inner circumference 1km)
- find the outer radius (inner+1.5m)
- find the difference between the outer and inner circumferences (answer to a)
- find the difference between the outer and inner areas (answer to b)
cerulean bramble
this looks a bit disorganized even if vaguely on the right track
eyyyyy
But also, not quite?
The question's implying that, if you ran the inner length of the track, the length is 1 km
We can probably assume that to be the length if you ran straight down the middle
- find the radius of a circle with such a circumference
call this R (best to calculate things in metres);
- find the circumference C1 of the circle with radius (R - 0.75) metres
- find the circumference C2 of the circle with radius (R + 0.75) metres
- Evaluate C2 - C1
compact pewterBOT
@near hound Has your question been resolved?
compact pewterBOT
compact pewterBOT
Available help channel!
final eagle
So, i took a test a while a ago and there's a question that has been bugging me since i cant solve it.
here's the question ( I might remember some part of it wrong so if it is unsolveable maybe i remember it wrong)
Jimmy wants to start a rabbit farm, so he bought a pair (M / F) of a certain type of rabbit which has the following traits:
-
This type of rabbit will give birth after 2 months since it's birth.
-
After giving a birth once, this type of rabbit will give birth again every month.
-
Each time, the rabbit gives birth to one male and one female.
4.This type of rabbit is immortal
compact pewterBOT
So, i took a test a while a ago and there's a question that has been bugging me ...
final eagle
So i think i can solve it now but im wondering if theres a way to write a general form for this sequence
like this sequence is basically adding the current number with the number behind it to find the next number
final eagle
i might be wrong tho
winged dock
Hmmm does it match some type of well known sequence??
solemn matrix
So, i took a test a while a ago and there's a question that has been bugging me ...
This type of rabbit is immortal š
final eagle
Hmmm does it match some type of well known sequence??
As far as i can think of, no
solemn matrix
wait wait wait
final eagle
`This type of rabbit is immortal` š
The in-breeding will goes crazy tho
solemn matrix
wait
im confusing about the number 1
this type of rabbit will give birth on its own?
š
oh i see
winged dock
@final eagle what did you get
solemn matrix
idk if im right
winged dock
Yes this is a pretty popular recuraion
final eagle
like 1, 2, 4, 8, 16,...
If im not wrong then prob not
subtle helm
There is an explicit formula for the fibonacci sequence.
Found by matrix diagonalisation and friends.
arctic portal
Is it every possible pairing of male and female give birth (so like n/2 choose 2 or you pair the rabbits off (so just n/2 pairs)?
final eagle
Is it every possible pairing of male and female give birth (so like n/2 choose 2...
i think its just n/2
solemn matrix
so at first
you have 2 rabbits
these 2 will give birth to a male and female
which gives you 4
winged dock
We can simplify the sequence by taking just one rabbit and letting it breed
winged dock
More like a single parent
solemn matrix
we keep track of variables
winged dock
Instead of a pair giving birth to pair
winged dock
Just one rabbit giving birth to one
final eagle
Just one rabbit giving birth to one
isnt that just dividing by 2?
solemn matrix
it would be like this:
start: 2
gave birth: x = 0, y = 2
month 1: x = 0, y = 2
month 2: x = 2, y = 2
month 3: x = 2, y = 4
month 4: x = 4, y = 6
month 5: x = 6, y = 8
final eagle
There is an explicit formula for the fibonacci sequence.
ohh ye its fibonacci sequence
thx
solemn matrix
where x is the amount of rabbits that gave birth
and y is the amount of rabbits that was just born
subtle helm
I like skibidi toilet.
solemn matrix
it would be like this:
start: 2
gave birth: x = 0, y = 2
month 1: x = 0, y = 2
m...
yeah this one is wrong
final eagle
alr i think this is prob just it ig
for the general form
i thought it needs to be like (N+1)3/2 or smth like that
ig thats it
anyways thx for the help yall
.close
compact pewterBOT
compact pewterBOT
Available help channel!
quartz yoke
,,\int \frac{\sqrt{x^2+1}}{x}, dx
compact pewterBOT
,,\int \frac{\sqrt{x^2+1}}{x}\, dx
boreal girderBOT
!
compact pewterBOT
@quartz yoke Has your question been resolved?
quartz yoke
<@&286206848099549185>
vale star
Try substituting x with tan(theta)
strange bolt
so when you set x=tan(theta) you can use the formula tan(theta)^2 + = sec(theta)^2
Also dx = sec(theta)^2*dt
Then you can do some manipulations and get to the integral: 1/(sin(theta)*cos(theta)^2)
From this point maybe you could do some integration by parts but I don't know for sure
(sorry for not doing latex, i'm not so good with it)
compact pewterBOT
compact pewterBOT
Available help channel!
fossil sorrel
HEEELPPP
fossil sorrel
people answered c.
but it just doesnt make sense
can someone help me with this part again
C.
all i did
is expanded the left parenthesis
then foiled the right parenthesis
combine like terms
and got a standard form of
4x2 - 16x -1 = 0
BUT since i cant factor
lavish gull
!show
compact pewterBOT
Show your work, and if possible, explain where you are stuck.
lavish gull
oh ok sorry
civic quartz
then foiled the right parenthesis
instead of that if you would have considered (2x-1) as t , dont you think it would have been faster to first solve for t and then subing 2x-1 back for t again
civic quartz
i asked chtgpt for this
tho its nice you are keen to learn , its suggested not to use ai sources as it hinders with learning and potential missinformation
lavish gull
actually yeah. this is a quadratic in (2x - 1)
strange ermine
i asked chtgpt for this
!nogpt
compact pewterBOT
i asked chtgpt for this
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).
fossil sorrel
okay yeah u guys are right
chtgpt is yapping nonsense
can u guys teach me this step by step?
ok so here
i got 4x2 - 16x - 1 = 0
standard form
so as provided in the picture
it says to fctor
but i cant do that since the constant is smaller than the linear term
mening no factors to add up to 16x
so i used quadratic equation
whinc i got like
16 square root of 272 over 8
at the end
idk wht to do next
oop
wrong one
molten badger
This equation doesn't have nice roots so I don't think we can factor it
fossil sorrel
fossil sorrel
This equation doesn't have nice roots so I don't think we can factor it
bro exctly š
thats why i used quadratic equation
molten badger
how about complete the square or use formula?
fossil sorrel
how about complete the square or use formula?
can u do squaring
i kinda forgot about that
molten badger
Well if you're allowed to ,just use it
strange ermine
,tex .cts
boreal girderBOT
Alberto Z.
fossil sorrel
i have no idea what im looking rn
sorry
my brin is so fried up
staying up late
for this ONE
C.
QUESTION
im blank
brain fog
PLLLLLLLLLLLLLLLLLEEEEEEEEEEEEEEEEEEEEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAASEEEEEEEEEEEEE
i just wanna go to sleep
but i cant
school start tomorrow
nd it paass tomorrow
and i dont want to copy others at school
š
kind crane
Wut
molten badger
kind crane
Why do you need to cheat by copying
molten badger
okay let start with $4x^2 - 16x - 1 = 0$
boreal girderBOT
Alexis_Fx
fossil sorrel
yeh yeah
molten badger
**Alexis\_Fx**
how can we get from this to $x^2+bx+c=0$
boreal girderBOT
Alexis_Fx
fossil sorrel
thats already a quadratic equation what do u mean
fossil sorrel
**Alexis\_Fx**
this
strange ermine
**Alexis\_Fx**
Is it of this form?
frail bane
look in front of the x^2
fossil sorrel
what
frail bane
you have a 4 in front of x^2
fossil sorrel
yeh
quadratic equation
i mean
quadratic term
andddd im alone
honeestly
i should just leave c. blank and let the teacher cook
and learn from it
kind crane
you have a 4 in front of x^2
The point of this is you need to divide both sides by 4
strange ermine
...
kind crane
,calc 0/4
boreal girderBOT
Result:
0
,calc -16/4
boreal girderBOT
Result:
-4
,calc -1/4
molten badger
x2 -16x - 1 = 4
wut
boreal girderBOT
Result:
-0.25
Yes and you did only one term
fossil sorrel
LET THE TEACHER COOK š„
strange ermine
u guys said bothh sideess
YES and you didn't do it
kind crane
LET THE TEACHER COOK š„
!done
compact pewterBOT
LET THE TEACHER COOK š„
If you are done with this channel, please mark your problem as solved by typing .close
molten badger
bro 0/4 is 0 and not 4
fossil sorrel
!done
compact pewterBOT
If you are done with this channel, please mark your problem as solved by typing .close
fossil sorrel
.close
compact pewterBOT
compact pewterBOT
ā
quiet sentinel
Then shouldnt you have the answers
quiet sentinel
4x2-16x-1 right?
fossil sorrel
yeah
quiet sentinel
So a is 4, b is -16, c is -1
strange ermine
what do u mean
You said you've solved it with the quadratic formula and you've found the two answers. So, where's the problem?
quiet sentinel
No plus or minus sign?
strange ermine
Are you sure you aren't missing anything?
fossil sorrel
No plus or minus sign?
i skip that prt
its the root im focusing
its a big number
quiet sentinel
i skip that prt
Why
fossil sorrel
i put plus or minus later
strange ermine
i skip that prt
If you skip it, your results are wrong!!
strange ermine
i put plus or minus later
But you can't write it like that!
fossil sorrel
plz dont shout
strange ermine
plz dont shout
You're the one who's started shouting when writing all on capslock, you know?
fossil sorrel
fossil sorrel
You're the one who's started shouting when writing all on capslock, you know?
oh, fair
quiet sentinel
Just calculate root272
fossil sorrel
how do i do that
quiet sentinel
And find the two roots
fossil sorrel
is there any method
quiet sentinel
CALCULATOR
quiet sentinel
Oh brub
quiet sentinel
Do you know laws of indicies/surds?
quiet sentinel
Ok
civic quartz
are you aware of divisibility rules
fossil sorrel
forgot
quiet sentinel
Well if you split 272 into factors you can take square roots of those factors
quiet sentinel
$\sqrt{4}\sqrt{68}=\sqrt{272}$
boreal girderBOT
ImOakley
civic quartz
so i just divide 272 to 1 then so on and so far
does 2 divide 272 (with remainder being 0 )
quiet sentinel
The point is you should find the factors of 272 that are square numbers
quiet sentinel
Take the largest one and use the rule $\sqrt{a}\sqrt{b}=\sqrt{ab}$
boreal girderBOT
ImOakley
civic quartz
then you can group repeating numbers in groups of 2 to find the square root
(similarly if it were to be cube root you will be grouping repeating numbers in groups of 3 )
compact pewterBOT
@fossil sorrel Has your question been resolved?
raw portal
compact pewterBOT
raw portal
I sort of guessed a part like you would multiply the left first column of identity matrix by 2 and that would be the matrix for first transformations as it's just basically scaling the first basis vector by factor of 2
is there a method to find out the matrix of corresponding transformations??
cuz idk how I would think about the b part
kind crane
you can make 3x3 variables for the entries in T and make 3 equations and the assumption that v1, v2, v3 and independent variables
e.g. first row could be [t11, t12, t13] so after multiplying by the column vector [v1, v2, v3] you get t11 * v1 + t12 * v2 + t13 * v3 = 2v1
repeat for the other two rows of T
raw portal
damn
we will actually have to solve the system of equations?
because that's like the second chapter in this book
kind crane
did you read this
raw portal
i sort of know it from highschool but not that intuitively
hushed magnet
with a tiny bit of practice you can literally just read the matrix off
kind crane
both a. and b. each takes < 10 seconds once you find the pattern
hushed magnet
I'm gonna write b as $T\begin{bmatrix} v_1 \ v_2 \ v_3 \end{bmatrix} = \begin{bmatrix} 0v_1+v_2+0v_3 \ v_1 + 2v_2+0v_3 \ v1+0v_2+v_3 \end{bmatrix}$
boreal girderBOT
Denascite
hushed magnet
now take a guess what the matrix could be
raw portal
Oh since the book didn't have any example they just stated definitions of linear transformations so the purpose of exercise was probably figuring out what you just told me it's really cool
.close
compact pewterBOT
raw portal
Hey guys I don't have solutions can someone cross check my answer
what I did is I multiplied it with a 12 Ć 1 matrix
Whose each rows entry is basically factor of contribution of that particular sem of assignment
Is it right ?
compact pewterBOT
Available help channel!
rigid jungle
compact pewterBOT
rigid jungle
i know how to do the problem just answer checking here
1,3,5?
def not 2 or 4
so for 1 would the HA be 2?
or just 1
slate bolt
def not 2 or 4
Right
rigid jungle
if exponents are the same i divide coefficients to get y?
slate bolt
so for 1 would the HA be 2?
Two
icy nymph
-# desmos
slate bolt
if exponents are the same i divide coefficients to get y?
Yeah i donāt understand that question sorry, anyone English native can answer that
slate bolt
Sorry and yw
rigid jungle
the 0 thing is stil tripping me up
junior vault
the 0 thing is stil tripping me up
for option 3?
rigid jungle
new problem
look for x and y
but i think i got it
bc for x im only solving for numerator
and for y x=0 so x is always 0
why tf are there 4 answers?
scenic forge
there are three x-intercepts and one y-intercept
rigid jungle
i figured it out\
thanks
how do i determine the end behavior
the other end behavior looks like this for reference
scenic forge
three cases:
if degree of numerator > degree of denominator: end behavior is infinity (sign depends on numerator - denominator)
if degree of numerator < degree of denominator: end behavior is 0
if degree of numerator = degree of denominator: end behavior is (leading coefficient of num) divided by (leading coefficient of denom)
so if your function were $\frac{2x+9}{1+3x}$
boreal girderBOT
DaveyLovesSocks
scenic forge
the end behavior would be 2/3
slate sand
The constants ādonāt matterā in the limit
Or rather, the lower order terms ādonāt matterā
If your function was $\frac{2x^2 +9x + 3}{2 + 1x + 3x^2}$ the answer is still 2/3
boreal girderBOT
frosst
scenic forge
essentially as x grows larger the x^2 term is much much larger than the x term
so it stops mattering
rigid jungle
yea i just mean ignore constants for the end behavior