there are, well maybe the helpers who know multivar calc are sleeping
#help-0
lavish cave
for ratio and proportion, here
dilation should be in here
Khan Academy has both videos and quizzes
lone heartBOT
Available help channel!
grand falcon
if 2+5 is 3, what is 4+7
lone heartBOT
if 2+5 is 3, what is 4+7
alpine sable
if 2+5 is 3, what is 4+7
is that the entire question?
grand falcon
is that the entire question?
yes
zinc bolt
if 2+5 is 3, what is 4+7
I'd say undefined because 2 + 5 = 3 doesn't exactly define the operation + is doing
Yea, maybe mod 4, maybe a + b is defined to be b - a
alpine sable
idk if mod 4 makes sense cause they use 5
idk
can you do that? i have no experience with mod
zinc bolt
can you do that? i have no experience with mod
You could make it a property of + so yea
zinc bolt
or + could just be the constant function 3 lol
Exactly
zinc bolt
Actually in all the relations we defined, it seems to be coming to 3, what a crazy coincidence lol
alpine sable
lmao
zinc bolt
Yea but still this doesn't mean that it'll always be 3 tho
ruby current
a + b - 4
ruby current
a + b = 3 cos^2(a - 2) sin^2(b - 5) + 3
alpine sable
a + b - 2(a!)
glacial sundial
wait hi
mossy reef
can someone help me with this please
Have you heard of a MacLauren Series?
glacial sundial
yeah
glacial sundial
ive done this problem like a handful of times and i am getting it wrong
mossy reef
ive done this problem like a handful of times and i am getting it wrong
Can you show what you have done previously?
glacial sundial
yeah leme type it out my phoen is broken
I first did (0,0), which resulted in 3
then I did the first partial derivs (fx and fy)
which were 2/sqrt(9+4x+5y)
and 5/2sqrt(9+4x+5y)
which when you plug in (0,0) you get 2/3 and 5/6
then I solved fxx, fxy, and fyy
mossy reef
,w partial z = sqrt(9 + 4x + 5y) at (x, y) = (0, 0)
glacial sundial
fxx was -8/(9+4x+5y)^3/2
ocean sealBOT
mossy reef
fxx was -8/(9+4x+5y)^3/2
This needs to be fixed.
mossy reef
Also, what did you get for $\diffp{f}{{y^2}}$?
glacial sundial
is that the error youre thinking of?
ocean sealBOT
@mossy reef
glacial sundial
fyy = -25/(4(9+4x+5y)^3/2
mossy reef
-# Let me check once more...
I checked, and it's definitely wrong.
mossy reef
fxx was -8/(9+4x+5y)^3/2
Actually, you're right and this should be negative. But that's not the only error.
glacial sundial
wait so fxx is correct, is fxy?
nevermind i just reread what you sent
do you know what i am doign wrong?
mossy reef
do you know what i am doign wrong?
I'm not sure entirely, but it's definitely just a silly error.
glacial sundial
thats the worst type lmao
mossy reef
I'm not sure entirely, but it's definitely just a silly error.
||Your partial derivatives differ from the true derivatives by just a constant multiple.||
glacial sundial
wait am i off by a factor of 2
okay so, I should multiply fxx, fyx, and fyy by 2?
or just fxx
i tried doing both and im still incorrect
mossy reef
okay so, I should multiply fxx, fyx, and fyy by 2?
Both fxx and fxy should be divided by 2.
glacial sundial
oh my god thank u so much
lone heartBOT
@glacial sundial Has your question been resolved?
subtle meadow
Im not too sure what to do for part b. I can take K to be 1/2 as a good choice probably, but I'm not sure how to lay it out. I think I can assume properties of familiar functions, even though its an Analysis 1 course P-Set 4, but how should I lay this out properly
lone heartBOT
Im not too sure what to do for part b. I can take K to be 1/2 as a good choice p...
@subtle meadow Has your question been resolved?
vale wigeon
oh ffs
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slate oracle
The flight of an airplane from Korea to Poland can be modelled by the relation h=-2.5t^2+200t where t is the time in minutes and h is the height in metres.
a. How long does it take to fly from Korea to Poland?
b. What is the maximum height of the plane? At what time does it reach this height?
slate oracle
im cooked for my math exam i gotta review and i forgot how to do this pls help 😭🙏
fervent knoll
im cooked for my math exam i gotta review and i forgot how to do this pls help �...
When the plane lands the height will be 0
slate oracle
ive set the height to 0
fervent knoll
Find t then
fervent knoll
Cancel one t. As we know t is not zero
fervent knoll
It's right
fervent knoll
U need (b)??
slate oracle
oh right
slate oracle
uhh not really..
fervent knoll
Do you know calculus??
slate oracle
i havent learnt calculus
uneven jewel
for quadratic optimization you don't need calculus
just complete the square and go from there
slate oracle
currently i have learnt this but forgot most of it..
fervent knoll
just complete the square and go from there
Yea do this
fervent knoll
i got max height of 4000meters
Yes
lone heartBOT
lone heartBOT
@dense wren Has your question been resolved?
raw jetty
if you asked there then why open here
dense wren
im getting n*2^(-n+1)
cause for an angle/point to be acute there should be n-1 points on the semicircle facing that point
which has 2^(-n+1) chance of happening
and since they are disjoint u add them up
lone heartBOT
@dense wren Has your question been resolved?
vale phoenix
Just a simple arithmetic question, how did he squared the whole fraction of complex numbers? Wouldn't that change the ratio?
Is there a property or identity I'm missing?
vale wigeon
he's working with |ratio|^2 with the intent of taking the square root at the very end
@vale phoenix
vale phoenix
but wouldn't that change the question itself?
i am little confused about that..
like (2/5)² is not equal to 2/5 in ratio
so what im finding it wrong?
vale wigeon
i am little confused about that..
like (2/5)² is not equal to 2/5 in ratio
so ...
it isn't and nobody is claiming it is
vale wigeon
but wouldn't that change the question itself?
not really, it is more of a change of approach
again the idea is
instead of working out x directly
we work out x^2 and then AT THE END we take the root
do you accept this as a valid approach yes or no
vale phoenix
do you accept this as a valid approach yes or no
it makes sense a bit to me, but ig this feels new to me lol
copper pond
Could somebody explain to me why it is negative 1 and not positive 1
vale phoenix
Could somebody explain to me why it is negative 1 and not positive 1
is it related to this question? you can create your own question forum #❓how-to-get-help
copper pond
vale phoenix
.close
lone heartBOT
tawdry lily
Oh here you go
tawdry lily
Why not use k= 2...?
From which line?
lucid surge
The last?
Like ig since I said to solve z⁴, we got 4 solutions (kinda confused why we don't have more tho)...so instead of using k= -2, why can't we also use k = 2
tawdry lily
Like ig since I said to solve z⁴, we got 4 solutions (kinda confused why we don'...
k = -1 and k = -2 correspond to the same roots as k = 2 and k = 3, just expressed with negative values
lucid surge
Wait so apparently negative values won't be correct answers?
Literally got my main fp2 exam in a couple of days, and I'm dying over dis–
tawdry lily
Wait so apparently negative values won't be correct answers?
It will
Since we're using complex numbers the 4th roots are periodic
Which means the roots should repeat every 4 values of k, correct?
lucid surge
Ohh
Yes yes
So is k= 2 bringing a repeated root here?
The theta it gives is 13π/12
<@&286206848099549185> ...
tawdry lily
So is k= 2 bringing a repeated root here?
Yep
Correct
So that's why k=2 is unnecessary since we already have k=-1
proper glade
i can’t figure out where i went wrong, because apparently the answer is 8pi/3
vast harbor
,rccw
ocean sealBOT
vast harbor
i can’t figure out where i went wrong, because apparently the answer is 8pi/3
you calculated the area not the volume
proper glade
i think i just forgot to add the pi, but i squared it so the answer should still be 8/3
but i didn’t get it for some reason
vast harbor
how did you get $\frac{4y^{\frac{3}{2}}}{3}$
shouldnt it be just $\frac{y^{\frac{3}{2}}}{3}$
nvm lol
proper glade
for which part?
glad fern
the second last part where you worked out to get 40/3
proper glade
ohhhhhh i tried it again but i still didn’t get the answer 😭
vast harbor
the solid is a cylindre ?
proper glade
it’s the shape i drew out i think, cuz it’s rotated about the y axis
glad fern
i think it should $4\sqrt{y}$
ocean sealBOT
MaxeulRaggedus
ocean sealBOT
MaxeulRaggedus
proper glade
OHHHHH
glad fern
you know when you expanded it?
glad fern
np 👍
proper glade
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proper glade
how do you get the volume btw ?
i just used the volume formula for volumes of a solid revolution
rich quiver
i just used the volume formula for volumes of a solid revolution
Can you send the formula?
vast harbor
i just used the volume formula for volumes of a solid revolution
interesting , thx
proper glade
Can you send the formula?
ocean sealBOT
tight pier
Your y-bounds are from 0 to 4 and x is supposed to vary within [0,2] so the negative solution is the right one.
@proper glade
proper glade
ohh ok thanks!
lone heartBOT
@proper glade Has your question been resolved?
hearty nest
watch me solve, if I get stuck I will ask for hints
lone heartBOT
watch me solve, if I get stuck I will ask for hints
hearty nest
gcd(a, b) = 3
a, b < 100
lcm(a, b) = 12 * a
max( a + b ) = ?
3 * 12 * a = a * b
b = 2^2 3^2
gcd(a, 3^2 2^2) = 3
lcm(a, 3^2 2^2) = 3 2^2 a
a = 3 ?
but that gives a + b = 39 which is wrong answer
how is this giving definite values of a, b when it is asking to maximize it
empty mason
b = 2^2 3^2
I’m pretty sure that once you find what b is from here the LCM condition is not needed anymore. You just focus on the GCD condition
empty mason
but that gives a + b = 39 which is wrong answer
Yes it’s wrong
empty mason
how is this giving definite values of a, b when it is asking to maximize it
Try maximise a now. You should try test some values
hearty nest
yeah
empty mason
gcd(a, 3^2 2^2) = 3
You actually know from this that || gcd(a/3, 2^2 * 3) = 1 || so you can limit your search
hearty nest
3 * 29
hearty nest
wro9ng
empty mason
You actually know from this that || gcd(a/3, 2^2 * 3) = 1 || so you can limit yo...
Say a = 3x. From here, we get that || gcd(x,12) = 1 || Now what’s the largest value of x less than or equal to 33 that satisfies this gcd condition?
hearty nest
3 31
hearty nest
Correct. That’s the value of our a, 3*31
thanks, I got it
common 3 will be taken out from gcd.
and since 3x <= 100
x <= 33
empty mason
thanks, I got it
common 3 will be taken out from gcd.
and since 3x <= 100
x <...
Great! And yes, exactly
hearty nest
.close
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crisp nova
how do i claim this>
lone heartBOT
how do i claim this>
vale crag
it is claimed now
crisp nova
oh
so i was working on a past math exam paper to prep for my math finals next week so i stumbled accros this question
and got very confused
i assigned x as the radius of the sphere
im not sure if im on the right track
similar triangles got me x/15 = (36-x)/36
but i just dont feel like its right
i got x = 180/17
i can technically go on and calculate the volume of the sphere but am i doing it right?
i clearly did went on the wrong track at first cuz i messed up the base radius and height
slate vortex
I don't think you need similar triangles for this problem
slate vortex
Use this diagram
crisp nova
im sorry but i still dont get it, i know that the hypotenuse is 39, the height is 36 and the radius is 15..
crisp nova
ohhhh
slate vortex
The key thing to notice is the radius divides the hypotenuse into 15 and 24
crisp nova
thank you!!
slate vortex
lone heartBOT
@crisp nova Has your question been resolved?
crisp nova
i clearly did went on the wrong track at first cuz i messed up the base radius a...
em may i as well get help on part b lol
how can i get the top part hypotenuse
i have to calculate the curved surface area of the bottom half cone
can i get some extra help?
lone heartBOT
@crisp nova Has your question been resolved?
pseudo aurora
lone heartBOT
pseudo aurora
hi anyone here??
pseudo aurora
I got 99%
spring dagger
correct
spring dagger
1.1 x 0.9 is 0.99
winter light
hi anyone here??
You don't need to write it every time you claim a help channel
lone heartBOT
pseudo aurora
.reopen
lone heartBOT
✅
elder forge
what did u try
pseudo aurora
why??
elder forge
consider intital length is l and breadth is b
winter light
Huh? 99% is correct 🤔
elder forge
so area is lb right
elder forge
Huh? 99% is correct 🤔
nah 21% is correct answer
pseudo aurora
working??
elder forge
ok so
winter light
nah 21% is correct answer
What the heck?
elder forge
initial area is lb
new length is l + 0.1l = 1.1l
new breadth is b + 0.1b = 1.1b
new area is 1.21lb
and initital area is lb
so 21% increase
or u can say new area = 1.21*old area
spring dagger
nah 21% is correct answer
by definition the area is lw. l increased 10%, meaning it mulitlies by 1.1. w is decreased by 10%, meaning it multiplies by 0.9. lw becomes (l)(1.1)(w)(0.9). by commutivity, is 0.99lw
crude urchin
new length is l + 0.1l = 1.1l
new breadth is b + 0.1b = 1.1b
It decrease isn't it
pseudo aurora
new length is l + 0.1l = 1.1l
new breadth is b + 0.1b = 1.1b
the breadth decreases by 0.1
spring dagger
1% decrease is 99% of the original
lone heartBOT
woeful zenith
A2 = l1.1 * w0.9 = lw*0.99 = 0.99A
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Available help channel!
burnt reef
Where does this conversion comes from?
lone heartBOT
Where does this conversion comes from?
burnt reef
Random variable theory and Bayesian network.
ocean sealBOT
Sherif Player
charred summit
Then
$$P(A\cap B) = P(A|B)P(B)$$
ocean sealBOT
Sherif Player
burnt reef
but there is a new variable on the right of the |
That's what confuses me the most.
.close
lone heartBOT
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Available help channel!
lone heartBOT
tacit arch
no
warped hedge
shadow pond
Hello.
shadow pond
So, I have a question regarding problems in introduction to proofs.
So, I'm confused to how epsilon is selected in each case and why n should be an even and odd integer in cases 2 and 3, respectively.
tacit arch
can you negate the statement "a_n is divergent" using epsilon-N?
tacit arch
negation of "a_n is divergent" is "a_n is convergent"
can you write convergence using epsilon-N?
misty hamlet
negation of "a_n is divergent" is "a_n is convergent"
hi
tacit arch
hi
read #❓how-to-get-help
shadow pond
I'm don't know if I can write convergence using epsilon - N.
tacit arch
then start by reading definitions of words you're taught
something like 2.1.2 https://connect.gonzaga.edu/asset/file/1514/ch02_1_convergence_v00.html
shadow pond
So, doing the example problems, N would equal a ceiling function in relation to epsilon.
I can't seem to find a N equal to a ceiling function in relation to epsilon.
tacit arch
what are you even doing
tacit arch
can you write convergence using epsilon-N?
did you do this yet for your sequence?
shadow pond
No.
shadow pond
Yeah, I'm reading it right now.
lone heartBOT
@shadow pond Has your question been resolved?
shadow pond
Could you close this problem? I think I'm too lost and maybe need more exposition.
tight pier
Could you close this problem? I think I'm too lost and maybe need more expositio...
You can close your channel with .close
sharp stag
what is Fundamental Theorem of Calculus?
lone heartBOT
what is Fundamental Theorem of Calculus?
sharp stag
in a nutshell, what I'm understanding is basically it says that 1. integration and differentiation are inverse operations 2. integral a to b f(x)dx = F(b) - F(a) where F is antiderivative
is that it?
tight pier
in a nutshell, what I'm understanding is basically it says that 1. integration a...
When you say "inverse operations" what do you mean?
sharp stag
When you say "inverse operations" what do you mean?
what addition is to subtraction and what multiplication is to division
tight pier
Well, you have to watch the bounds too.
Because then the integrand is not necessarily the derivative of the antiderivative.
safe tartan
that statement is wild
tight pier
sharp stag
that statement is wild
I just saw it in a vid
ocean sealBOT
tight pier
Your bounds have to be in the form like there so that the differential operator and integration act like an "inverse operation".
Where they "cancel out".
sharp stag
okay cool
yeah I think ik what ur talking about
ima look into it more then '
thanks sm
tight pier
what use is the FTC?
The FTC is pretty useful when it comes to computing the derivative of an antiderivative, that doesn't exist in terms of elementary functions. For example:
[ F(x) = \int_0^{g(x)} e^{t^2} , dt. ]
You can't evaluate the integral, yet you can still find $F'(x)$ using FTC.
ocean sealBOT
sharp stag
The FTC is pretty useful when it comes to computing the derivative of an antider...
okay bet, I'll make note of this
appreciate it!
.close
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zenith flax
lone heartBOT
zenith flax
Unsure of how to do this
Not understanding why we'd put the .8/.23 for part one
I read it as, given the probability that they read books is .23, what is the probability they read art
which means both
north rover
P(B)=0.07+0.03+0.05+0.08
north rover
which means both
No
We cannot deteremine if these events are independence, hence we use the formula
pseudo ice
P(B)=0.07+0.03+0.05+0.08
(this is given in the table already btw)
north rover
We didnt use a formula int his one though?
Yes
zenith flax
So what formula are you signaling to
Is it because thats the only thing thats not in A
north rover
So what formula are you signaling to
$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$
ocean sealBOT
;(
north rover
Cause we are considering something given something here
zenith flax
was just guessing
north rover
Not understanding why we'd put the .8/.23 for part one
I mean, this is the question you asked
zenith flax
Okay thats not really explaining much to me sadly
north rover
There isn't even worded context, that's the thing
north rover
Okay thats not really explaining much to me sadly
So what don't you understand
zenith flax
Why we chose .8. I understasnd that A n B is the intersection aka what a and b have in common
wait
.5+.3
my god
Im so slow lmfao
reef wren
Im so slow lmfao
at least you got it
zenith flax
im lost again
reef wren
why ?
zenith flax
north rover
zenith flax
Okay so the .04, and .05
north rover
Drumroll please
You're taking the common things between A and the region contained by both B and C
zenith flax
I get the A part
Not the B and C part
Why did we select .04 and .05 for the B and C?
north rover
It's these fuckers we wanna analyze
north rover
Why did we select .04 and .05 for the B and C?
Ok, get a piece of paper
Or drawing tool
Whatever
Draw this diagram
Label the circles A, B, C
Then first shade in the region $B\cup C$
ocean sealBOT
;(
north rover
And then $A$ afterwards, and find the common region between $A$ and $B\cup C$ (this is your region of $A\cap (B\cup C)$)
ocean sealBOT
;(
zenith flax
no like i totally get that .05 is shared amonst a b and c in that case
I guess im getting logic and alg mixed up with this
because
for a union, we'd include all of it when doing b u c
Im confused as to why .8 isnt included
as b and c share it
If were doing the intersection, B and C share .05, and .08
I cant tell if im over thinking this lmfao
north rover
Sorry, I went to get some food
north rover
Im confused as to why .8 isnt included
.8 isn't in A
We are finding the common area between A and B U C
zenith flax
Yes I get that part
can we try breaking down step by step
So were first combining B and C
Correct?
Or is it, we're finding where both B and C intersect eachother
I think im getting mixed up because Im just now figuring out union is different in stats versus logic & algorithms
north rover
So were first combining B and C
Yes
zenith flax
Yes
Is that .05 and .08 as the intersection of the two, or is it .04 + .05 + .03 + .20 + .08 + .07
Like the .04 + .05 + .03 + .20 + .08 + .07
is the combination of them both (the denominator)
Im having trouble getting the numerator
north rover
Is that .05 and .08 as the intersection of the two, or is it .04 + .05 + .03 + ....
Yeah, part of it
It'll get excluded from A
zenith flax
wait that was a "or" question 😢
So the .47 is excluded
aka the combination of b and c
Im just not getting how im getting .03 + .04 + .05 in the numerator
north rover
It's these fuckers we wanna analyze
^
zenith flax
im just not getting why its those, Im trying to understand with the intersection and union why we picked those
wait
i get it
i got it thank you
.close
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misty sleet
I was wondering if anyone has good resources (free if possible) to study for the ALEKS. I need to do well on this test next year and I want to study.
quiet birch
Hey @misty sleet , try to check out the book recomendations / study discussion rooms. They might help you better there.
lone heartBOT
@misty sleet Has your question been resolved?
lone heartBOT
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forest cargo
Hi
ocean sealBOT
forest cargo
My work coming
Here it go
I have no idea about the question
Anyone helping me????
If yes then type something
<@&286206848099549185>
If i don't get respond soon i might be afk as i have short time for my classes if u have any clue how to solve the question leave some hints for me ok
<@&286206848099549185>
@mossy reef
<@&286206848099549185>
lone heartBOT
@forest cargo Has your question been resolved?
nocturne veldt
you're pinging helpers a little frequently...
hmmm wow this is a pretty annoying problem
nocturne veldt
might be better to just represent this as a system of equations somehow
rather than working with all these pictures
lilac scroll
My work coming
Can you try using the formula instead of Venn diagram?
wait let me attempt it a bit. Not sure if I can do it.
eager swallow
.help
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Type .help <command name> for more info on a command.
eager swallow
Oh
A cylinder is cut from solid sphere of radius R. Find maximum volume of cylinder
eager swallow
Yes
I'm not even able to understand how to start it-
So i haven't made any progress yet
unique dune
Maybe let’s start by drawing a diagram of what it could look like
lilac scroll
Is there any diagram?
eager swallow
eager swallow
Let’s consider one cross section
Yes I did
unique dune
eager swallow
According to my teach this question includes the concept of maxima minima
unique dune
Then cylinder is this cross section rotated about the y axis
unique dune
.
eager swallow
Then cylinder is this cross section rotated about the y axis
Yes
unique dune
Find a rectangle that maximizes the area and u should be done
unique dune
,rccw
ocean sealBOT
unique dune
Maximize $2x\sqrt{r^2 - x^2}$
ocean sealBOT
k
unique dune
Do u know why I derived this?
eager swallow
No
unique dune
Consider a rectangle inscribed in this semicircle
unique dune
.
Its width is x
Its height is 2 times rhe function
Remember the height from the x axis at x=a is the same as the function non-negative f(a)
eager swallow
Yes
eager swallow
Gotcha but
unique dune
Consider a rectangle inscribed in this semicircle
.
It is the area of that rectangle
lone heartBOT
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south geode
Does anyone know how to create pictures in desmos
lone heartBOT
Does anyone know how to create pictures in desmos
south geode
It is like torture
lavish cave
I've seen this exact task asked by another student
south geode
Haha I gues all the teachers are using it
void bear
It is like torture
yes
lavish cave
should be self-explanatory how to type those in
the thing you need to learn is how to make the domain restrictions
south geode
Could you slide the tutorial
lavish cave
south geode
oh
south geode
okay thank you
lavish cave
no worries!
south geode
.close
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tulip helm
need help on q14 a pls
lone heartBOT
need help on q14 a pls
cold cargo
I'm guessing you mean 14b?
cold cargo
you need to think about what the relation in the picture between e^x and ln x is
this is true for all inverse functions
f^{-1} is the reflection of f in the line y=x
so given this, can you translate the area into something you can calculate?
tulip helm
I’m not sure how honestly, it’s not making sense
cold cargo
try rotating the picture by 90 degrees and looking at the shaded region
cold cargo
I'm not sure if you understand, do you know what the calculation is now?
tulip helm
tulip helm
-1 as upper bound
ahh i see now i didnt even think of rotating it
but the y=x thing makes sense
cold cargo
yeah all we are doing is swapping x and y essentially
in order to do that we must be able to invert the function
cold cargo
no problem
lone heartBOT
lone heartBOT
Available help channel!
pliant shale
lone heartBOT
pliant shale
pliant shale
Oh wtf that's not one of the answers?
yeah that's why I am asking am i correct?
pliant shale
Looks good to me, idk
Thank you for your help!(I will check what's wrong with the options)
alpine sable
Yeah your working seems fine
lone heartBOT
raw jetty
its already closed
alpine sable
@pliant shale could you please reopen the channel?
pliant shale
.reopen
lone heartBOT
✅
alpine sable
Thank you. Your working is incorrect unfortunately
Sure the substitutions seem correct but you didn't factor the nature of f into the picture
To see that,
Since f(x + f(y)) = f(x) + y for every Real x,y
Set y=0
This implies f(x + f(0)) = f(x)
This may imply that f(x) can be periodic with period f(0), but it's not the case (can you see why and hence prove that f(0) must be 0)?
pliant shale
what is the problem with the function being periodic?
alpine sable
what is the problem with the function being periodic?
Hint: ||Separately plug in x=0||
pliant shale
Hint: ||Separately plug in x=0||
Do you mean f(0+f(y))=f(0)+y?
alpine sable
Actually wait an easier way to see this is assume f is not injective, meaning, let a,b in R such that f(a) = f(b) => a ≠ b
So now in the functional equation, put y=a then y=b
You get f(x + f(a)) = f(x) + a
f(x + f(b)) = f(x) + b
Can you see the contradiction now?
If f(a) = f(b)
Then x + f(a) = x + f(b)
So....
pliant shale
oh yes i see
alpine sable
Yep
So by contradiction, f must be injective, and hence must have period 0
which implies f(0) = 0
pliant shale
So by contradiction, f must be injective, and hence must have period 0
how the period 0?
alpine sable
Because if f(x+T) = f(x) and f is injective, then it implies x+T = x which implies T=0
alpine sable
Do you mean f(0+f(y))=f(0)+y?
So we have f(f(y)) = y
Can you see how this will be useful in f(x + f(y)) = f(x) + y?
alpine sable
And the solution to that functional equation assuming f(x) is nice and all is...
alpine sable
does the period 0 mean aperiodic?
Yeah
pliant shale
How do we get f(0)=0?
alpine sable
To see that,
Since f(x + f(y)) = f(x) + y for every Real x,y
Set y=0
This imp...
Here
y=0 implies f(x + f(0)) = f(x) for every Real x
So f(x) could be periodic with period T=f(0)
Like sin(x + 2π) = sinx for every Real x
pliant shale
since the function is injective x+f(0)=x
alpine sable
Mhm
pliant shale
then do f(x+f(y))=f(x)+y invariably means f(x+y)=f(x)+f(y)?
alpine sable
Yes because f(0+f(y)) = y + f(0) = y
So putting y=f(y) in the original functional equation gives that
pliant shale
Can I use this result anywhere for x,y belongs to some R?
alpine sable
Yeah
I mean in this case, from the given functional equation only that holds for every x,y in R by assumption we deduced that f is injective and f(f(y)) = y, so it's valid here
pliant shale
Is there any reason why we can't directly use that functional rule for numbers? @alpine sable
lone heartBOT
@pliant shale Has your question been resolved?
alpine sable
You indeed can, and it's one of the ways to get the values of f(x) for certain x's
The problem is that the question is inconsistent
No such function exists such that
f(x+f(y)) = f(x) + y and f(1) = 2
The bold part is where the inconsistency lies
This is because solving f(x+y) = f(x) + f(y) gives f(x) = cx for some c in R
But f(f(y)) = y forces c² = 1
So the only possible solutions (remember we didn't even use the inconsistent f(1) = 2 condition yet) are f(x) = x or f(x) = -x
You can even substitute these functions in the given functional equation and check if they satisfy it or not for every Real x,y
With this, we get f(1) = ±1, which is not 2
alpine sable
No such function exists such that
f(x+f(y)) = f(x) + y and **f(1) = 2**
Because of this, in your working, you assumed f(1) = 2 right from the start. Starting with something inconsistent will yield inconsistent results
@pliant shale
So yeah the question is itself inconsistent
alpine sable
But f(f(y)) = y forces c² = 1
So the only possible solutions (remember we didn...
If you naively ignore the f(f(y)) = y condition we got from injectivity, then f(x) = 2x (mind you this is still not the correct answer for the original functional equation) and you get f(4) = 8. That would be one of my answers by guessing, even though it's wrong
The correct answer should be f(4) = 4, and the condition in the question should be f(1)=1 not f(1)=2
pliant shale
If you naively ignore the f(f(y)) = y condition we got from injectivity, then f(...
Do you mean the question itself is wrong for the options given?
alpine sable
The condition f(1)=2 is wrong yes
pliant shale
The condition f(1)=2 is wrong yes
for the functional rule give, right?
alpine sable
Welcome. Have a great day
pliant shale
.close
lone heartBOT
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Available help channel!
dull pilot
dull pilot
<@&286206848099549185>
gray isle
what issues are you having with this
dull pilot
Tbh everything
It’s like everytime I get it it changes and the answer key they provide is like
Idk a foreign language almost
😞
I also need help with geometry as well
gray isle
don't skip the work
write steps clearly
frigid niche
U need someone to help you solve or explanation
dull pilot
Little bit of both honestly
frigid niche
Little bit of both honestly
Which question
dull pilot
Its more or less pattern recognition knowing what to do for what problem
gray isle
e.g. in the first question, you jumped directly to an (incorrect) result
so noone knows where exactly it went wrong
dull pilot
But when I get to a pattern it just switches randomly so now it’s all wrong
gray isle
try make another attempt
gray isle
the same mathematical rules apply,
a slightly different question shouldn't matter
unfortunately vc is very limited on this server. (and people usually don't like doing this through dms)
try your best to comminicate what you're doing
frigid niche
Is the first questions answer -10x +16?
dull pilot
What do I even do here💔
dull pilot
Is the first questions answer -10x +16?
Yeah :3
gray isle
they also seemed to provide an explanation for it
frigid niche
dull pilot
unfortunately vc is very limited on this server. (and people usually don't like ...
I honestly don’t mind I have my own server for my fans and what not but honestly inviting a tutor wouldn’t even hurt
frigid niche
The - in the middle is important
frigid niche
-
- =+
-
- =-
-
- = +
dull pilot
I’m so confused how’re you doing this
frigid niche
Like
gray isle
distributing that - to the ()
dull pilot
VA pls can I vc you I need like real time feedback on what I’m supposed to be doing or one of you it doesn’t even matter I’m so determined to pass this test
dull pilot
distributing that - to the ()
But the thing is where’d you get a 10 from
gray isle
focus on the first step first
dull pilot
Is this urgent?
I have like 3 more days to study for this test :<
frigid niche
Oh
gray isle
did you have any issues with that?
frigid niche
I will be home after 40minutes
dull pilot
And I gotta jampack all of this I have a basic understanding it’s just getting a concrete foundation
dull pilot
did you have any issues with that?
Absolutely
gray isle
what issues did you have with the first step of distributing the -
frigid niche
It's 9x+30
dull pilot
Ignore that
frigid niche
3 × 3x = 9x
dull pilot
Pretend I accidentally wrote the 9 upside down
gray isle
so what's your issue with this?
dull pilot
dull pilot
so what's your issue with this?
No clue where to go from here
gray isle
simplify all of them
dull pilot
How do I do that for 39x
gray isle
and should be clear that its not the same as 9x + 30 due both the different coefficient of x and lack of constant term
modern geyser
Here’s one
do you know what x² represents?
dull pilot
Yeah :3
gray isle
for this you could solve the given equation
or plug in the options and see which one works
dull pilot
No :<
modern geyser
do you have an answer in mind?
gray isle
yeah or no, which one is it?
dull pilot
X=-8
dull pilot
yeah or no, which one is it?
Not for this one I was taught that if no value is given to a variable it’s equal to 1
modern geyser
have you learnt what exponents are?
gray isle
they've been auto muted
gray isle
.close
glad cradle
lone heartBOT
glad cradle
this is my diagram
(refer to my diagram)
i think of the point in the first quadrant as x being plugged into an func to give y as f(x)
however, the point in the 4th quadrant, x is being plugged into a func to give f(-x)
what is wrong in my reasoning
modest cradle
It should be first and second quadrant for first question
lone heartBOT
lone heartBOT
Available help channel!
worthy lichen
lone heartBOT
worthy lichen
I've solved it up to this point, give me a hint for the next step so I can quickly finish it in one go.
zinc bolt
$7^{\log_{70}(2)} = 2 \log_{7 \times 10}(7)$ 🤔
ocean sealBOT
@zinc bolt
zinc bolt
Even then, $7^{\log_{70}(2)} = 2 \log_{7 \times 10}(7) = \frac 1 5$
ocean sealBOT
@zinc bolt
zinc bolt
i ate steps
How many?
worthy lichen
two
7 log_70 (2) = 2 log_70 (7) = 2 log__(7*10) (7)
from here
__ is for subscript btw
otherwise it becomes itali
c
worthy lichen
do you get now
what i did
after that 2*(1/10) log_7 (7)
because
log_b (a) = 1/b log (a)
zinc bolt
Ok, first of all, this value is just verifiable not gonna be 1/5
worthy lichen
2/10?
zinc bolt
Also, you can't willy nilly seperate bases in factors
zinc bolt
7 log_70 (2) = 2 log_70 (7) = 2 log__(7*10) (7)
Also, pretty sure this is also supposed to be $2^{\log_{70}(7)}$ rather than $2\log_{70}(7)$
ocean sealBOT
@zinc bolt
zinc bolt
Did you try seperating the whole and logarithmic powers and using $a^k \cdot b^k \cdot c^k = (abc)^k$?
ocean sealBOT
@zinc bolt
vale wigeon
$N = 2^{\log_{70}(9800)} \cdot 5^{\log_{70}(140)} \cdot 7^{\log_{10}(2)}$
this is what it said, right?
ocean sealBOT
Ann
vale wigeon
honestly i think this approach will be difficult and also a breeding ground for errors.
wait i misread
$N = 2^{\log_{70}(9800)} \cdot 5^{\log_{70}(140)} \cdot 7^{\log_{70}(2)}$
ocean sealBOT
Ann
lone heartBOT
@worthy lichen Has your question been resolved?
worthy lichen
Tell me the solution
lone heartBOT
lone heartBOT
Available help channel!
floral cairn
Need verifying for problem E.
lone heartBOT
Need verifying for problem E.
slate vortex
You can't do that
unkempt wind
No, idek what is that rule
Differentiate the top and bottom then take the limiy again, but as you can't use it, don't worry about it
slate vortex
You can just notice that x grows toward -infinity, the +3 and -4 become basically irrelevant
floral cairn
Yes
slate vortex
that should help you the ans
floral cairn
Bottom is -infinite, top is positive infinite
slate vortex
No
floral cairn
Since there is 1 negative. The whole equation is negative?
slate vortex
it is negative but you can't use infinities
unique dune
Note: ${\sqrt{x^2} = |x|}$
ocean sealBOT
k
slate vortex
$\frac{\sqrt{2x^2 + 3}}{x-4} \approx \frac{\sqrt{2x^2}}{x} = \frac{\sqrt{2}|x|}{x}$
ocean sealBOT
Ari
slate vortex
No, you can't just plug in infinity for these types of questions
slate vortex
Which makes the answer sqrt2
Negative
unique dune
There is no such thing as limit ‘at’ infinity, just limit ‘approaching’ infinity
unique dune
What about D then?
Correct
slate vortex
It's correct
floral cairn
Can’t we use the same method for E?
unique dune
Because denominator is stopping the growth that the numerator is doing
floral cairn
Oh, so the two infinites cancel out, leaving me with just sqrt(2)?
unique dune
So, you can’t really say that this won’t outgrow this
unique dune
Oh, so the two infinites cancel out, leaving me with just sqrt(2)?
Infinities are not numbers.
They are notions of unboundedness.
slate vortex
Oh, so the two infinites cancel out, leaving me with just sqrt(2)?
You can't say infinities cancel out
floral cairn
Hmm
slate vortex
for example in x^2/x, the infinities would also "cancel out"
slate vortex
-sqrt2
slate vortex
floral cairn
.close
radiant badger
I have a doubt.
lone heartBOT
I have a doubt.
radiant badger
in the third step, why there's a + sign in the middle, why cant it be -
vale wigeon
why would the plus sign from step 2 become a minus
north rover
in the third step, why there's a + sign in the middle, why cant it be -
You're distributing a positive 3...?
The terms will not match anyways
radiant badger
I'm asking in step 2 what sign has to go in the middle
gray isle
the sign depends on what you have
median oar
If you stick the - onto the number and put a + in front, then you see that + is associative
a + (-b) + c + (-d) = ( a + (-b)) + ( c + (-d))
radiant badger
This is a factorisation question
gray isle
are you referring to the
4x(x-2) + 3(x-2)
lone heartBOT
@radiant badger Has your question been resolved?
tender shard
Can someone help me solve this?
lone heartBOT
Can someone help me solve this?
tender shard
btw |EF| = |EK|
unique dune
,calc (4*180)/6
ocean sealBOT
Result:
120
its a equally-sided hexagon, right?
tender shard
yep
tender shard
yes
unique dune
do u know trig?
unique dune
,calc 360 - 120 - 75 - 90
ocean sealBOT
Result:
75
,calc (180 - 75)/2
ocean sealBOT
Result:
52.5
are u sure that they are 'equally-sided'?
unique dune
,calc 180 - (75*2)
ocean sealBOT
Result:
30
tender shard
didn't get a single thing lol
echo socket
||Draw KD||
tender shard
echo socket
Determine the angle KDL
unique dune
,calc 120 - 45
ocean sealBOT
Result:
75
echo socket
It’s pretty natural to want to make a triangle with x as one of its sides, for that you could either do what k did, draw KD or draw LE
Drawing KD happens to not require trig
lone heartBOT
tender shard
It's asking about the bottom piece's area
lone heartBOT
It's asking about the bottom piece's area
tacit arch
what is this even saying
lone heartBOT
@tender shard Has your question been resolved?
unique dune
Do u know what the area of a trapezoid is