B_w is many linear maps for each w ?
#help-10
agile oriole
No, it signifies that v is mapped to that
junior zephyr
aha
can I write B( v1 + v2,w) = B( v1,w ) + B(v2,w) and B(kvw) = k B(vw) ?
for all w E W
B( v , w1 + w2) = B(v, w1 ) + B (v,w2) and B(kvw) = k B(vw)
for all v E V
agile oriole
i mean can it be inferred from the definition
It is equivalent
I think that is the definition
warm shaleBOT
rti
junior zephyr
the definition doesnt say it explicitly so im curious from where to infer
agile oriole
the definition doesnt say it explicitly so im curious from where to infer
it should be exactly the definition
junior zephyr
https://en.wikipedia.org/wiki/Bilinear_map in this i couldnt find that
agile oriole
junior zephyr
In this page it shows explicitly https://en.wikipedia.org/wiki/Bilinear_form
agile oriole
can I write B( v1 + v2,w) = B( v1,w ) + B(v2,w) and B(kvw) = k B(vw) ?
this is the same
junior zephyr
yes
so this isnt a corollary but a definition?
obtuse pebbleBOT
@junior zephyr Has your question been resolved?
modern geode
how do you find gradient value and y-intercept coordinate if y=2x+3
obtuse pebbleBOT
how do you find gradient value and y-intercept coordinate if y=2x+3
spring steeple
Do you know about the slope-intercept form of a line?
modern geode
no
spring steeple
I'd recommend going through this video
https://youtu.be/IL3UCuXrUzE?feature=shared
obtuse pebbleBOT
@modern geode Has your question been resolved?
modern geode
no not yet i am a bit busy
oblique nebula
Just a quick question,
How to denote that a set is finite in math notation?
sage geode
Just write that its cardinality is equal to some natural number
oblique nebula
<:hmmCat:831508801646755850> Just write that its cardinality is equal to some na...
I want to define natural numbers with sets so...
sage geode
Ah
oblique nebula
Any other ideas?
sage geode
You can construct it using equivalence classes as far as I remember
Consider equivalence classes of N^2 under the equivalence relation (a, b) ~ (c, d) iff a + d = b + c
Are you using the set-theoretic definition of natural numbers?
In that case you can say a set is finite if there is a bijection between that set and some natural n
oblique nebula
You can construct it using equivalence classes as far as I remember
You just gave me an idea. Can I say that Z is the set of all numbers that have 1 as a divisor?
fossil crag
I made a whole class on set theory
Finite cardinality is bijection with a set of the form {1,...,n}
Requires natural numbers to be defined beforehand
Either through Peano's axiomatic or by construction with ZF axioms
obtuse pebbleBOT
@oblique nebula Has your question been resolved?
spring raptor
Is their something I can search for to find information about a function that maps all functions to reals such that some function, a's, mapped value is greater than an other function, b, if the limit of a/b as you approach infinity is infinite?
obtuse pebbleBOT
Is their something I can search for to find information about a function that ma...
warm canopy
"all functions"? what functions
obtuse pebbleBOT
@spring raptor Has your question been resolved?
spring raptor
"all functions"? what functions
I guess real -> real functions
obtuse pebbleBOT
@spring raptor Has your question been resolved?
timber shell
how do i solve for theta when Cos(Theta - 45 deg) = 1 ?
obtuse pebbleBOT
how do i solve for theta when Cos(Theta - 45 deg) = 1 ?
timber shell
is it just 45 deg?
spring steeple
for what values of theta cos(theta) = 1?
timber shell
for what values of theta cos(theta) = 1?
270,180,90
but in this case theta greater than or equals to 90
so 90-45=45 and theta = 45 right
spring steeple
yes but what's the general form?
timber shell
yes but what's the general form?
? wdym
spring steeple
lmao no wait
spring steeple
270,180,90
that's incorrect
spring steeple
not even 90 and 270
silk ledge
for what values of theta cos(theta) = 1?
cos(theta - 45) = cos(360n), so theta - 45 = n360, theta = 360*n + 45 degrees where n can be any integer
spring steeple
cos 90 = 0
yes, I asked when is cos(theta) = 1
obtuse pebbleBOT
cos(theta - 45) = cos(360*n), so theta - 45 = n*360, theta = 360*n + 45 degrees ...
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.
timber shell
360 and 0
timber shell
like cos 360 = 0
spring steeple
No I mean the general form of the solutions to the equation cos(theta) = 1 , you answered two particular solutions 0 and 360
spring steeple
like cos 360 = 0
that's not correct , cos(360°) = 1
timber shell
i keep doing that when typing
what the hellll
cos (theta -45) = 1 , theta = 360+45 and 0+45
right?
and cuz theta is smaller or eq to 90
theta = 45
indigo pendant
how do I find the exact value of
obtuse pebbleBOT
how do I find the exact value of
indigo pendant
hey how to use latex
cyan kite
send photo
light hedge
It asks you to find inverse csc of sqrt(2), basically finding x such that csc(x) = sqrt(2)
Do you know how to express csc using sin, cos, tan trigonometric functions?
indigo pendant
so is csc = hyp/op?
light hedge
That's correct in a right triangle, but using sine it's 1/sin(x)
Can you solve 1/sin(x) = sqrt(2) ?
light hedge
No.
indigo pendant
how do i show my work for this i am kind of confused
light hedge
Usually, f^-1(x) denotes the inverse function
indigo pendant
Yes
light hedge
If you're solving f^-1(a) = x you're looking for an x such that f(x) = a
In your case, f is the cosecant, a = sqrt(2)
indigo pendant
Um
light hedge
light hedge
Do you know the definition of domain and range?
indigo pendant
yes
obtuse pebbleBOT
@indigo pendant Has your question been resolved?
tacit pier
Help my algebra is messed up for the integration
obtuse pebbleBOT
Help my algebra is messed up for the integration
obtuse pebbleBOT
@tacit pier Has your question been resolved?
tacit pier
No
feral sedge
@tacit pier
Did you change your upper limit and lower limit?
feral sedge
You should change the upper limit and lower limit immediately since you have already changed from dx to du.
Anyways, let me have a further check
(BTW, remember to tag me if you reply me, so that I won't miss your message)
tacit pier
@feral sedge I know it’s right I’m just having trouble reducing the a section of the integral without a online calculator
feral sedge
Do you mean this transition?
tacit pier
@feral sedge yes sorry I had this question a few places and forgot to add that note
@feral sedge also what software was that pic u sent seems really helpful I’ve just been using Symbolab
feral sedge
<@683299579307098167> yes sorry I had this question a few places and forgot to a...
Does it help?
tacit pier
@feral sedge yes thank you!!! I need to try writing it out with that help
feral sedge
<@683299579307098167> yes thank you!!! I need to try writing it out with that he...
Simplification and solving equations are tedious and boring job so actually we could use CAS like Mathematica to simplify LOL
tacit pier
@feral sedge lol tell my calc 2 teacher 😭😭 thank u though
obtuse pebbleBOT
@tacit pier Has your question been resolved?
median palm
what is sin
obtuse pebbleBOT
what is sin
warm shaleBOT
MrFancy
drifting wraith
obtuse pebbleBOT
@median palm Has your question been resolved?
dense stirrup
A ratio of the sides of a triangle
right triangle... to be more specific
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
lapis bolt
obtuse pebbleBOT
lapis bolt
How should I formulate in a)? Im having mass of 1xH & 1xHe at my disposal.
obtuse pebbleBOT
@lapis bolt Has your question been resolved?
tardy epoch
How should I formulate in a)? Im having mass of 1xH & 1xHe at my disposal.
!show
obtuse pebbleBOT
How should I formulate in a)? Im having mass of 1xH & 1xHe at my disposal.
Show your work, and if possible, explain where you are stuck.
tardy epoch
Find the mass of the sun to be converted
tardy epoch
Yea use that
lapis bolt
For m you mean?
tardy epoch
Read the problem. What's being converted
lapis bolt
Read the problem. What's being converted
The mass is converted from 100% H to 90%H & 10% He
And they are asking me to find out the energy difference
obtuse pebbleBOT
lapis bolt
.reopen
obtuse pebbleBOT
✅
lapis bolt
Read the problem. What's being converted
So should I multiply the mass of 1xH atom to the Mass of sun and then the same for He. And then take 0.9 x hydrogen mass energy and 0.1 x helium mass energy and compare the difference?
tardy epoch
The mass is converted from 100% H to 90%H & 10% He
No that's not what it's saying
lapis bolt
No that's not what it's saying
Why not? I dont understand
tardy epoch
"10% of its mass" refers to the mass of the sun
lapis bolt
Yes. So first we have 100% H mass of the sun. Therafter we have 90% H & 10% He
lapis bolt
"10% of its mass" refers to the mass of the sun
Right?
tardy epoch
You're using the word converted wrong then
100% of the original hydrogen doesn't get converted to 90% hydrogen. It just stays hydrogen
Convert means to change
tardy epoch
So is it a correct interpretation? <@261933205387477002>
Is what
Did you correct your statement
lapis bolt
If yes, can I not then Calculate E1 of 100% H = (1.0079 x 1.989x10^30) x C^2
And then E2 of 90% H & 10% He = ((0.9)x1.0079x1.988x10^30)+((0.1)x4.0026x1.989x10^30) x c^2
Will this be accurate calculation for the problem? @tardy epoch
Calculating the difference between E1 & E2?
obtuse pebbleBOT
@lapis bolt Has your question been resolved?
lapis bolt
Why are you still converting the energy of the entire sun
Because thats how I interpret the problem
tardy epoch
Find the mass of the sun to be converted
No. Do this
lapis bolt
No. Do this
Can you elaborate? The mass of the sun I have already in the sheet
tardy epoch
"10% of its mass" refers to the mass of the sun
.
10% of the mass of the sun is what's being converted
lapis bolt
10% of the mass of the sun is what's being converted
So 10% of 100% H converted to He?
tardy epoch
Yes
warm shaleBOT
tardy epoch
Either 2 or 4
lapis bolt
Yeah this is not helping sorry
tardy epoch
Yeah this is not helping sorry
Do you understand the word convert
lapis bolt
Do you understand the word convert
I have converted units and real life money for ages. Yes I Do understand that Word lol 😂
tardy epoch
So I should take 0.1 x Mass_sun x Mass_he ?
Then why do you multiply the masses when you're supposed to be converting hydrogen to helium
When you convert currency, you don't multiply them do you
lapis bolt
When you convert currency, you don't multiply them do you
I mean technically I do multiply since there is a conversion number
tardy epoch
So I should take 0.1 x Mass_sun x Mass_he ?
Where's the conversion here?
lapis bolt
Where's the conversion here?
I dont know. Thats where Im failing to connect I think
Is it 4H—>1xHe number?
tardy epoch
Yes try it
lapis bolt
Yes try it
So E1 = (4xmass H) x C^2 and E2=(mass He)xc^2?
obtuse pebbleBOT
@lapis bolt Has your question been resolved?
lapis bolt
I forgot to square the C but besides that
obtuse pebbleBOT
@lapis bolt Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
feral kraken
are we supposed to divide ratios
obtuse pebbleBOT
are we supposed to divide ratios
feral kraken
like 8/10 could be a ratio
but is 0.8
i kinda dont get the difference betwee ratios and fractions
daring rock
either one represents the ratio of 8 to 10
do you have a specific problem you're working on? some context might help us to understand what you're confused about
violet sentinel
i kinda dont get the difference betwee ratios and fractions
Kahn academy to the rescue!
obtuse pebbleBOT
@feral kraken Has your question been resolved?
feral kraken
do you have a specific problem you're working on? some context might help us to ...
no im just having trouble understanding them generically
feral kraken
Kahn academy to the rescue!
https://youtu.be/UK-_qEDtvYo?feature=shared
ok ill watch
ok i understand it using for every
like for every 2 apples there are 5 total fruit
but i have trouble using that for the trigonometric functions
like for every opposite there are hypotenuse
wat
severe plover
I know I probably have to somehow express that sum in terms of 2^3n but im not sure on the details
obtuse pebbleBOT
I know I probably have to somehow express that sum in terms of 2^3n but im not s...
@severe plover Has your question been resolved?
severe plover
<@&286206848099549185> ?
obtuse pebbleBOT
@severe plover Has your question been resolved?
delicate kettle
whats the diff between cube root and square root
obtuse pebbleBOT
whats the diff between cube root and square root
delicate kettle
is it just Cube root: 2'3 then square root is 1'2?
tardy epoch
Cube root of x is the number whose cube is x
high lily
wdym with that ' notation
high lily
still not sure waht you mean as the ' is between two numbers
delicate kettle
still not sure waht you mean as the ' is between two numbers
the 3 on topo
delicate kettle
Cube root of x is the number whose cube is x
wdym by that
high lily
then where's the 2
agile oriole
whats the diff between cube root and square root
4 is the cube root of 64, and 8 is the square root of 64
high lily
$\cbrt{k} \times \cbrt{k} \times \cbrt{k} = k$
warm shaleBOT
ℝam()n()v
high lily
$\sqrt{k} \times \sqrt{k} = k$
warm shaleBOT
ℝam()n()v
delicate kettle
4 is the cube root of 64, and 8 is the square root of 64
oh so whats the diff
high lily
$n$th root of $k: \sqrt[n]{k}$
warm shaleBOT
ℝam()n()v
agile oriole
oh so whats the diff
4 and 8 are different numbers, and 4 x 4 x 4 = 64, while 8 x 8 = 64
as ramonov is pointing out
high lily
horrid wording
delicate kettle
horrid?
high lily
horrible
agile oriole
so cube root is 4 x three times
can you try finding the square root and cube root of 729
delicate kettle
cube root of 729 is
high lily
the cube root of 64 is 4
because 4^3 (4 cubed) is 64
delicate kettle
729/2
high lily
the square root of 64 is 8
because 8^2 (8 squared) is 64
high lily
no
high lily
yes, that's usually what you'd do
delicate kettle
ok
obtuse pebbleBOT
@delicate kettle Has your question been resolved?
rotund vector
Could I get some clarification on this question?
obtuse pebbleBOT
Could I get some clarification on this question?
rotund vector
In the (Merit) section why is the four being distributed to both parantheses?
Shouldn't it be like this $$4(x-10)^2+(y+2)^2$$
warm shaleBOT
Lex1729
hot coral
It wants you to work out what shape |z + 2 - 7i| = 2|z - 10 + 2i| makes on the complex plane, and then rewrite it in terms of x and y in that form
hot coral
Shouldn't it be like this $$4(x-10)^2+(y+2)^2$$
no because 2 is multiplying the whole square root, not just the first term inside it
rotund vector
Oh I see
obtuse pebbleBOT
@rotund vector Has your question been resolved?
placid forge
how do i get q1 and q2 ?
obtuse pebbleBOT
how do i get q1 and q2 ?
spring steeple
the charge on each ball is doubled
spring steeple
If q_1 and q_2 were the charges earlier their product was q_1q_2 what it should after each doubled ?
Also no need to write the value of epsilon_0 , you are only gonna compare the magnitude of the forces
spring steeple
also just noticed i forgot to square the r lol
That formula works in SI units , 50cm = 0.5 m
spring steeple
What's confusing exactly?
placid forge
all of it
spring steeple
Alright
spring steeple
That formula works in SI units , 50cm = 0.5 m
You understand this?
placid forge
sorta
i don’t remember which formula works in which units is all
sometimes i do the wrong one
spring steeple
Fine usually all the formulas are given wrt SI units
placid forge
oh okay
spring steeple
If q_1 and q_2 were the charges earlier their product was q_1q_2 what it should ...
This?
placid forge
i don’t get that
spring steeple
hmm what's confusing you in that sentence?
placid forge
all of it
spring steeple
alright
placid forge
i get what. you mean by doubling because the question says it, that’s about it
spring steeple
Yea fine that works
I was just asking the new magnitude of charges will be 2q_1 and 2q_2?
Do you understand this?
spring steeple
Nice
spring steeple
Also no need to write the value of epsilon_0 , you are only gonna compare the ma...
Also this?
placid forge
i don’t get that
spring steeple
hmm so you're asked to write the magnitude of the new force in terms of the magnitude of F, you're going to comparing the magnitudes no?
spring steeple
No , New force magnitude = (1/4πε₀)(2q_12q_2)/(r/2)^2
and F = (1/4πε₀)(q_1q_2)/(r)^2 you can substitute F in the New force magnitude
placid forge
shouldn’t r be halved?
spring steeple
so what do i do with it then? get rid of it?
Just no need to write down the value of epsilon_0 and even r
spring steeple
shouldn’t r be halved?
Yeah sorry mb
placid forge
so how do i work it out then?
spring steeple
Factor out (1/4πε₀)(q_1q_2)/(r)^2 from (1/4πε₀)(2q_12q_2)/(r/2)^2
placid forge
like this?
spring steeple
1/2 squared
placid forge
why doesn’t that get factored?
spring steeple
$\frac{1}{4\pi\epsilon_0} \frac{2q_12q_2}{(r/2)^2}$ =
$\frac{1}{4\pi\epsilon_0} \frac{4q_1q_2}{\frac{r^2}{4}}$
warm shaleBOT
Normed
spring steeple
Yea
royal basin
watch the units
spring steeple
Temperature should be in Kelvin
royal basin
aside from B, all of the answer options are given in joules per gram per kelvin
you calculated it in joules per kilogram per kelvin
royal basin
Temperature should be in Kelvin
makes no difference in our case since all that's described is a temperature rise but yes
spring steeple
Ohh yeah right
placid forge
yeah i see
royal basin
but also yes it's rounded
random gazelle
So Someone Already helped me with this one but I forgot to ask one more question
random gazelle
We got the number 70.35
But idk if I need to put it as 70.35° or with the “<“ symbol which is angle
royal basin
well given you got your angle as 70.35 degrees, most likely...
royal basin
But idk if I need to put it as 70.35° or with the “<“ symbol which is angle
where do you see a < symbol on your toolbar tho
cinder lintel
Or is that a calculator
random gazelle
Leave it in arctan ig
Arctan?
cinder lintel
Do you know how you get the angle
Tan theta =m1-m2/1 +m1m2
So you get theta as its arctan
Now idk if the arctan shown there is used for input or not
random gazelle
I did it with someone else and they said the answer was 70.35 but I forgot to ask if I needed to put it as that or with a symbol
Hold on
cinder lintel
Thats basically what i am doing
Tan theta =m1-m2/1+m1m2
Here m1 is 2/3
M2 is -4
10/3/5/3
2
Wierd
Im getting arctan 2
random gazelle
So the answer isn’t correct or it is ?
cinder lintel
Im p sure i did ir correctly
random gazelle
So 70.35 is incorrect?
Im not good with angles which is the problem so I’m not sure what’s correct or not 😅
cinder lintel
Arctan 2 is like 64
obtuse pebbleBOT
Available help channel!
brisk arrow
how to get the value of log 11
obtuse pebbleBOT
how to get the value of log 11
brisk arrow
an approximate value
royal basin
log_10(11)?
warm shaleBOT
Result:
11.22018454302
You could raise 11 to the power 1000, and then see how many digits it has
royal basin
ok yeah you're right
agile oriole
You could raise 11 to the power 1000, and then see how many digits it has
Not that bad
You can use the method of repeated squares
And keep only 3 sf
royal basin
ok yeah might as well do scientific notation lol
brisk arrow
does math instinct really exists
brisk arrow
i can tell it is between 1 and 1.05
how do i know it is between 1 and 1.05
am i being gifted by the god of math to be able to gauge the value of a unknown number
.close
boreal mantle
How do I even start to find the domain?
obtuse pebbleBOT
How do I even start to find the domain?
agile oriole
How do I even start to find the domain?
Domain?
boreal mantle
Limit
sage geode
why the upper-case
English grammar, no?
agile oriole
why the upper-case
Because it is the first word of a sentence
brisk arrow
ohh
boreal mantle
Yeah my bad, im just unsure where to start
sage geode
First combine the logarithms into one
boreal mantle
you can combite ln?
sage geode
Are you familiar with logarithm properties?
boreal mantle
I havent used log in a while, I could use a refresher if possible
sage geode
$\ln(b) - \ln(a) = \ln\mathopen{}\left(\frac{b}a\right)$
warm shaleBOT
A Lonely Bean
boreal mantle
Oh that looks fimiliar for sure
sage geode
How do I even start to find the domain?
Right, so, what do you get when you combine the logarithms here?
boreal mantle
so then we get ln (1+x^2)/(1+x)
sage geode
No actually ln can be replaced by any other logarithm here and the answer will be the same, e is irrelevant
boreal mantle
ohh okay
sage geode
Or what is "continuous function"
The type of function such that f(x) -> f(a) as x -> a
Okay, if you don't know continuitous functions, we can argue without them
boreal mantle
With this, we get the same thing over one another, so would it just go to ln(1)? making the answer 0?
rustic veldt
help me
sage geode
No
sage geode
With this, we get the same thing over one another, so would it just go to ln(1)?...
What does (1 + x^2)/(1 + x) approach as x goes to infinity?
boreal mantle
would they both be approching positive infinity?
becuase technically the numberator is growing at a faster rate. and can never be negative?
but the denominator could be negative
sage geode
Yes
(1 + x^2)/(1 + x) diverges and then so does ln when its input grows arbitrarily large
boreal mantle
so would we be able to rewrite it as lm(1+ |x|)/ (1+x)?
sage geode
No, (1 + x^2)/(1 + x) and that are not the same
sage geode
but the denominator could be negative
Wdym, it can't
x is positive and then so is 1+x
boreal mantle
But wouldnt be technically be able to plug in a negative number? with the limit being As X approches infinity, that would still include negatives
sage geode
You can restrict the values of x
And like
Why would you look at what happens when x < 1 if in your limit x -> 2
The point of taking the limit is to look at what happens on the neighbourhood around a certain point
Or what happens as x grows arbitrarily large in the case of x -> infinity
boreal mantle
So when it says "as x approches infinity" we shouldn't even look at the negatives?
sage geode
You can, but, that will not be useful
In fact it will make the whole process more complicated
While assuming x > 0 doesn't change anything
boreal mantle
Got it! that helps a lot
so then with where we are, would they just continue to grow? making the limit infinity?
sage geode
Yes, (1 + x^2)/(1 + x) goes to infinity as x -> inf and so does ln
boreal mantle
got it THANK YOU!
Also thank you for being patient with me, really helps.
.close
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
long hedge
obtuse pebbleBOT
spring steeple
When is tan(θ) = 0 (that gives you all the points of discontinuity)
Also when is sin(x)-cos(x) = 0 ?
spring steeple
Yes , but what's the general form?
long hedge
Yes , but what's the general form?
i dont know
spring steeple
nπ ; n is a integer
long hedge
nπ ; n is a integer
ok
spring steeple
yea so those are the points of discontinuity
ohh lmao
I didn't read it says at θ = π
Yea so you just need to check at θ = π
long hedge
so just plug in pi for theta
spring steeple
yea
obtuse pebbleBOT
@long hedge Has your question been resolved?
covert lion
Can someone PLEASE help me with this (last part)
obtuse pebbleBOT
Can someone PLEASE help me with this (last part)
agile oriole
Can someone PLEASE help me with this (last part)
Where are you stuck
agile oriole
You’re supposed to express v_5 as a linear combination of the basis vectors in B
covert lion
yeah idok how to do that
agile oriole
you know what the basis vectors of B are?
obtuse pebbleBOT
@covert lion Has your question been resolved?
covert lion
you know what the basis vectors of B are?
?
agile oriole
:/
covert lion
v1, v3
agile oriole
thats the basis vectors of B
and then
You’re supposed to express v_5 as a linear combination of the basis vectors in B
agile oriole
you have v_5
and you have v_1 and v_3
multiply v_1 and v_3 by the right constants and then add to get v_5
agile oriole
let the coefficient of v_1 be a
covert lion
how do i have v5
agile oriole
Can someone PLEASE help me with this (last part)
the rightmost column of U
just like how you have v_1 and v_3
covert lion
<3,-3,0>?
agile oriole
this is v_5
covert lion
okie
omg i get it
6v_1-3v_3
bruh its so easy
rti your are the goat i love you
I hope both sides of your pillow are cold at night
agile oriole
thanks
agile oriole
I hope both sides of your pillow are cold at night
You too
covert lion
it equals v5?
covert lion
oh mb
agile oriole
It does
covert lion
not underscore
the math quiz practice doesnt use _ for subscripts for some reason lol
agile oriole
And then you can follow the format given in red letters to get to the answer
ripe knoll
obtuse pebbleBOT
ripe knoll
can anyone help me with this
royal basin
transcribing the problem for my own sake since the image is hard to read:
Prove that the greatest term of the longest geometric progression in {100, 101, ..., 10000} is 6561.
or to put it another way:
Prove that the longest geometric progression consisting of integers between 100 and 10000 inclusive ends in 6561.
agile oriole
,w 9^4
warm shaleBOT
obtuse pebbleBOT
@ripe knoll Has your question been resolved?
tender stratus
I suppose this must be the longest GP then 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561
with ratio 1.5
or is it something else
ripe knoll
yea but how can we get the first term
kind hawk
in each step you need to divide by at least one prime factor of the starting number. to get a GP which is longer that means the first number needs to contain at least 9 times the same prime. that limits it quite a bit
tough hull
And this
obtuse pebbleBOT
@tough hull Has your question been resolved?
royal basin
don't ping helpers like this individually
tough hull
Okk
obtuse pebbleBOT
warm shaleBOT
Şêro
robust sleet
try setting common denominaytor
timid silo
Trig substitution in the first step will not really simplify the expression. And I don't really see why you would need implicit differentiation here
I see. Say you are doing x = sin(a) substitution. Find dx/da and then "multiply" by da on both sides. Note that this is just a shortcut trick and there is a proof why this works.
For substitutions like u = f(x), find du/dx and then "multiply" dx on both sides
warm shaleBOT
Şêro
frank monolith
what the-
timid silo
I was just giving you an example of how to do the process of substitution in general but this is not even needed in this question
Always try to simplify the integrand. Substitution may not always work
frank monolith
this sub will make the problem worse
timid silo
In fact you would have to solve a 4th degree polynomial to get x in terms of u.
And then find the dx
But.... why lol
frank monolith
yeah
and for the left, use quotient rule and chain rule
but still this sub is not recommended
just like du appeared
xD
you just said you want to differentiate implicitly
if you are differentiating with respect to x, then the right side should have du/dx instead of du
wrong
warm shaleBOT
Dyssrupt
warm shaleBOT
frank monolith
yep correct
wolfie's high
nope
its correct
the integral you solved is correct
oh
lemme check
yeh this is correct acc to your answer
,w integral 0 to 1/2 sqrt(1 + (4x^2)/(x^2 - 1)^2)
warm shaleBOT
frank monolith
,w ln(3) - 1/2
warm shaleBOT
frank monolith
hmm
yup yup
that one is negative for 0 to 1/2
nice one
just remove the abs
with a negative sign
to easily see this, take the sqrt of ((x^2+1)/(x^2-1))^2
the numerator is always positive
and the denominator is negative for x belonging to (-1,1)
so abs opens with negative sign
and then split the fraction
haha
congrats
if you dont need further help, type .close
obtuse pebbleBOT
@timid silo Has your question been resolved?
half otter
Is it possible to find the diatance between A and segment PG?
obtuse pebbleBOT
Is it possible to find the diatance between A and segment PG?
agile oriole
Is it possible to find the diatance between A and segment PG?
Yes, if you have the equations of both
half otter
Yes, if you have the equations of both
Let say the cubic edges is 6, then what's the distance?
agile oriole
6 sqrt(2)
half otter
Should i extend PG to GC?
obtuse pebbleBOT
@half otter Has your question been resolved?
undone sluice
Is this true? \ Because the text book isn't giving me an answer \ Evaluate the limit (if possible) of the sequence ${a_n}$ \ $a_n = \frac{n^{2}-2\sqrt{n}+1}{1-n-3n^{2}} \Rightarrow \frac{2\infty}{\infty^{2}} \approx 0$
obtuse pebbleBOT
Is this true? \\ Because the text book isn't giving me an answer \\ Evaluate the...
warm shaleBOT
HqppyFeet
warm shaleBOT
undone sluice
well if n approaches infinity... xD
perhaps I should just stick with 2n/n^2 then 2/n. my bad
agile oriole
Is this true? \\ Because the text book isn't giving me an answer \\ Evaluate the...
also, you can divide the top and bottom by n^2
to evaluate the limit
undone sluice
also, you can divide the top and bottom by n^2
yeah and it ends up with -2\sqrt(n) / n. Same said, it would approach 0.
thank you ❤️
agile oriole
yeah and it ends up with -2\sqrt(n) / n. Same said, it would approach 0.
Are you aware the numerator and denominator have the same degree
undone sluice
at what point? Here? -2\sqrt(n) / n
they aren't? perhaps my understanding of "same degree" is different, you mean the index right?
sqrt(n) would be n^(1/2)... n would be n^1. Sorry im not understanding
agile oriole
Is this true? \\ Because the text book isn't giving me an answer \\ Evaluate the...
there’s a n^2 in the top and a -3n^2 in the bottom
undone sluice
--oh yeah I can't cancel them xD wait
Found the limit. thanks for the heads up @agile oriole ❤️
.close
timid silo
obtuse pebbleBOT
obtuse pebbleBOT
@timid silo Has your question been resolved?
arctic merlin
Yo yo yo gangster mathematician is here
obtuse pebbleBOT
Yo yo yo gangster mathematician is here
royal basin
Yo yo yo gangster mathematician is here
do you have a math question to ask or did you just want to chat
high lily
does the gangster mathematician have a math question they need help with?
arctic merlin
Im good thx
high lily
.close
obtuse pebbleBOT
shrewd scroll
sup
obtuse pebbleBOT
Available help channel!
tropic turtle
I need help
obtuse pebbleBOT
I need help
primal steeple
need help understanding some ODE stuff
tropic turtle
the lesson is factoring but I've used trial and error and all I know, it seems unsolvable or am I wrong
frank monolith
need help understanding some ODE stuff
obtuse pebbleBOT
@tropic turtle Has your question been resolved?
violet sentinel
the lesson is factoring but I've used trial and error and all I know, it seems u...
yeah I'm not really seeing how this can work. Is that an option to put that it's not factorable? From what I can tell (I can be wrong) but in order to end with -y^2, your Blank 1 would have to be -1/5y, But then that creates problems with other parts of the FOIL
tropic turtle
i tried factoring by grouping
(3-5x)(3+5x) -y(50x+y) and is equal to 9-25x^2-50xy-y^2
but how do i make (3-5x)(3+5x) -y(50x+y) into (3-5x-BLANK 1)(BLANK 2+BLANK 3+5y)?
violet sentinel
yeah I'm not see how unfortunately.
violet sentinel
i tried factoring by grouping
(3-5x)(3+5x) -y(50x+y) and is equal to 9-25x^2-50x...
nice work on this tho 🙂
tropic turtle
yeah it's really tricky
ruby path
I guess you should first check it's determinant before trying to factor it any further
violet sentinel
If it was 9 - 25x^2 - 50xy - 25y^2 it would factor really nicely to (3 - 5x - 5y)(3 + 5x + 5y)
but it's not 😦
tropic turtle
I tried (3-5x-5y)(3+5x+5y) and I got 9-25x^2-50xy-25y^2 instead of 9-23x^2-50xy-y^2. What do I do?
violet sentinel
I would maybe talk with your teacher at this point. Either it's not factorable or there's a typo and one of the constants are incorrect.
tropic turtle
that's true it's almost been an hour and this is the only question I can't solve
violet sentinel
yeah sorry
best of luck, hopefully the teacher is forgiving and we're not just missing something vital
but WA even can't come up with a solution
,w factor 9 - 25x^2 - 50xy - y^2
warm shaleBOT
obtuse pebbleBOT
@tropic turtle Has your question been resolved?
tropic turtle
.close
meager pumice
Idk where to start
obtuse pebbleBOT
Idk where to start
plucky burrow
do u know how to divide fractions?
meager pumice
I do know kcf just the algebra made me confused
plucky burrow
kcf?
meager pumice
Keep change flip
meager pumice
meager pumice
(p+3)?
plucky burrow
yup
meager pumice
plucky burrow
can u cancel anything else?
meager pumice
Can I cancel 2
plucky burrow
yes u can
meager pumice
Can I do this
plucky burrow
no u cant
u cant cancel p
there is adsition in the fraction
if it was multiplication, then u cud cancel
meager pumice
Oh ok
plucky burrow
the cancellqtion of 2 is correct tho
meager pumice
meager pumice
Could I swap the p on top with the 9
plucky burrow
what?
meager pumice
plucky burrow
u cant just swap them like that
it makes no sense to do that
instead, u can write 3p + 9 as 3(p + 3)
plucky burrow
yes
meager pumice
What about the times sign
plucky burrow
yeah so (1 × 9p) / 3
meager pumice
Tysm
plucky burrow
np
obtuse pebbleBOT
@meager pumice Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
warped dune
Hi I have a math problem I can't solve
obtuse pebbleBOT
Hi I have a math problem I can't solve
warped dune
Am I supposed to send what's the problem
dense stirrup
yes
warped dune
All right wait— gotta translate the whole thing
four circles with the same radius are both tangent to each other and tangent to two concentric circles
Here's a scheme
What we know
Is that little circle has a radius of 1cm
dense stirrup
what u have to find?
wooden cipher
The one in the top right is the most useful
Use the pythagorean theorem to find the radius of the medium circles
warped dune
But with what measures
I tried that but I would need one more information
Such as an angle or something
timid silo
There centres are uhhh in line right?
warped dune
They're all tangents
timid silo
Like a plus sign
warped dune
So yes
timid silo
Yeah
wooden cipher
Would you agree OBC is a right triangle?
wooden cipher
Call the medium radius r, can you make an equation using pythagoras?
timid silo
What would be BO ??
wooden cipher
Because its not right
warped dune
Why is it not
wooden cipher
Hint: OACB is a square
wooden cipher
Doesnt it?
Excuse the bad drawing, but all these are squares
I have to go to, im calling for backup rn
Hopefilly someone else can help
timid silo
And you get half the side of the square from this
And also
- half the side of this square = radius of that medium sized circle
Now radius of big circle is easy
@warped dune
obtuse pebbleBOT
@warped dune Has your question been resolved?
tepid urchin
what.
still flower
obtuse pebbleBOT
tall summit
you can start trying to do some substitutions
bold bane
Whenever you see a x/(x+c) situation, try adding +c -c to the top.
still flower
u = sqrt(...) is smth you can try
ima try that
tall summit
or you try to integrate dy parts
still flower
is it better to seperate it
tall summit
imo its easier
tall summit
i havent seen partial yet
its easy
tall summit
still flower
ye that i seen
still flower
u = sqrt(...) is smth you can try
is u = sqrt (x / x+1)
still flower
this is not a theorem
like this
solar trellis
cool
tardy epoch
Try integration by parts
if your form is int udv, then do u = entire integrand and dv=dx
tardy epoch
is u = sqrt (x / x+1)
Same u as here
still flower
idk what to do after this
Well integration by parts here then right
tardy epoch
Looks like you're doing u sub there
obtuse pebbleBOT
@still flower Has your question been resolved?
tardy epoch
Yea don't do u sub
obtuse pebbleBOT
@still flower Has your question been resolved?
still flower
.close
digital remnant
$let a \in \mathbb{R}$
Depending on the value of $a$, what is the nature of $\sum_{n\geq1}\frac{ln(n)}{n^a}$
obtuse pebbleBOT
$let a \in \mathbb{R}$
Depending on the value of $a$, what is the nature of $\s...
digital remnant
i need help please, i dont know how to present it
if know if a <= 1 then diverges
and a > 1 converges
but i dont know how to prove it
prove it properly for the least
warm shaleBOT
demiryolu mühendisi
slate zephyr
if know if a <= 1 then diverges
if a != 1 you can just do one of the standard comparison tests.
For example you can simply do root test or ratio test.
Let me know if there is more trouble.
slate zephyr
i dont know that
Haha funny that you say that, I am from Germany
slate zephyr
in german we call them "Kriterium", I guess "criteria".
slate zephyr
i dont learn american maths
What have you learnt about infinite series so far?
digital remnant
What have you learnt about infinite series so far?
supposedly all of L1
but yeah i suck
slate zephyr
What does L1 mean here?
slate zephyr
I think that varies greatly between different countries and universities
digital remnant
in france that is
slate zephyr
Ah...
Well I think they each it more similarly to us.
Anyway, we need something specific about series.
digital remnant
with the integrals ig i get equivalent
slate zephyr
You are talking about itnegrals but there are no integrals involved here.
digital remnant
yeah
$\int^{n+1}2 \frac{ln(t)}{t^a}dt \leq \sum^n{k=1} \frac{ln(k)}{k^a} \leq \int^{n}_1 \frac{ln(t)}{t^a}dt$
warm shaleBOT
demiryolu mühendisi
slate zephyr
I think this is related to ingral test, but there are futher preconditions and it is also not the easiest route.
What have you learnt about what convergence of a series even means?
digital remnant
a <= 1 i did $1 \leq ln(n)$ so $\frac{1}{n^a} \leq \frac{ln(n)}{n^a}$ so $\Sigma_n \frac{1}{n^a} \leq \Sigma_n \frac{ln(n)}{n}$ and we know $\Sigma_n \frac{1}{n^a}$ diverges because a <= 1 so $\Sigma_n \frac{ln(n}{n^a}$ diverges also
warm shaleBOT
demiryolu mühendisi
digital remnant
What have you learnt about what convergence of a series even means?
means the sequeces inside converges to 0
ig