well 1/3^2 + 1/5^2 + 1/7^2... isnt exactly 1/3^2 + 1/4^2 + 1/5^2...
#help-27
real osprey
so you'll need (d) to relate them
try use (d) for 1/3^2 + 1/5^2 +… < 1/4
if a<b and b<c then a<c
swift valve
so you'll need (d) to relate them
Ok so
I did 1/3^2 + 1/5^2 … <1/4
Then
1/2(bottom sum)<1/4
bottom sum < 1/2
bottom sum +1 +1/4 < 6/4
Required sum <6/4
Required sum <7/4
Please tell me I’m rigt
real osprey
I did 1/3^2 + 1/5^2 … <1/4
where did the 1/8 come from?
swift valve
where did the 1/8 come from?
Oh shit I meant 1/4
real osprey
bottom sum +1 +1/4 < 6/4
if you use bottom sum<1/2 this will be
bottom sum +1 +1/4 < 7/4
so yes this is correct
swift valve
Yay thanks
real osprey
nice
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trail apex
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trail apex
Can anyone explain what I did wrong
cold bone
Can anyone explain what I did wrong
using the limit as t tends to 0
and in the end it will be -ln9 and not -ln3
I don't see any major issues with your work except the above two
trail apex
It’s ok I got it I just didn’t know that t tending to infinity meant it would become ln4 so I need to try find out why other than that I’m right @cold bone
brisk panther
using the limit as t tends to 0
hey unrelated but why are you named 77 squared?
brisk panther
oh ok
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restive river
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somber isle
hello can someone help me prove this
devout snowBOT
hello can someone help me prove this
somber isle
brisk panther
have you tried induction
somber isle
n+1 ?
fierce ember
recurrence
brisk panther
=4/4
fierce ember
in french
somber isle
recurrence
ah ok
fierce ember
Two ways to do it in that case
somber isle
and told me to write 1^3 + 2^3 + 3^3 + ...... + N^3 with n only
somber isle
Two ways to do it in that case
tell me
brisk panther
and told me to write 1^3 + 2^3 + 3^3 + ...... + N^3 with n only
hmm
ya i always wondered that oo
too
one sec ik a book that explains how to do this
somewhat
one second
fierce ember
tell me
It's in french but you should understand
The other way is specific to powers of 3
somber isle
im having trouble ngl
brisk panther
im having trouble ngl
you speak french?
somber isle
yes
brisk panther
attends stp
somber isle
okok
fierce ember
im having trouble ngl
You need to know the expressions for the precedent exponents to do this method
But if you remember them
brisk panther
@somber isle i never did this on my own but
see if you can do it
they gave a hint
fierce ember
The other way that's specifique to k^3 is to solve this but it's probably more difficult than just guessing the answer lol
somber isle
The other way that's specifique to k^3 is to solve this but it's probably more d...
this is easy
since we know the sum on the right side
it's a usual sum
no ?
fierce ember
it's only a usual sum if u_k is some sort of usual progression
but it could be anything
brisk panther
it's a usual sum
did you undersatnd what i sent
somber isle
what's U-k in our case here
somber isle
did you undersatnd what i sent
im still trying to
brisk panther
If you look at the long line at the bottom
what the did is add everything on top
like how you do
4
5
+6
15
like this @somber isle
then they combined like terms
ya it makes sense
2^3 - 1^3
3^3 - 2^3
4^3 - 3^3
5^3 - 4^3
and so on
everything cancels out except (n+1)^3 - 1^3
somber isle
this is to find 1^2 + 2^2 + 3^2 + 4^2.... + n^2 right ?
brisk panther
this is to find 1^2 + 2^2 + 3^2 + 4^2.... + n^2 right ?
nope n^3
oh ya n^2
ya
mb
somber isle
so in order to find 1^3 +2^3 +3^3 +.. +n^3 well need to use the difference of 2^4- 1^4 then 3^4-2^4 and so on
somber isle
yes and we're gonna need to already have 1^2 + 2^2 + 3^2 + 4^2.... + n^2 in possesion to use it
ok ok but one other thing i dindt understand is
brisk panther
yes and we're gonna need to already have 1^2 + 2^2 + 3^2 + 4^2.... + n^2 in poss...
no
for finding 1^2 + 2^2 + 3^2 + 4^2.... + n^2
you need to find 1 + 2 + 3 + 4 + 5 + 6...
for 1^3+ 2^3 + 3^3 + 4^3.... + n^3
you need to find 1^2 + 2^2 + 3^2 + 4^2.... + n^2
somber isle
why did we go from (n+1)^3-n^3 to (n+1)^3 -1
brisk panther
why did we go from (n+1)^3-n^3 to (n+1)^3 -1
remember what i said about adding everything on top?
fierce ember
We summed over all terms
brisk panther
2^3 - 1^3
3^3 - 2^3
4^3 - 3^3
5^3 - 4^3
if you add these
2^3 gets cancelled out
right
so does 3^3
and so on
until (n+1)^3
and -1^3
is remaining
fierce ember
Sum from 0 to n of f(i+1)-f(i) = f(n+1)-f(0) basically
so you dont need to rewrite everything everytime
somber isle
oooooh
brisk panther
oooooh
for practice you could try finding 1^2 + 2^2 + 3^2 + 4^2.... + n^2 first
and then 1^3+ 2^3 + 3^3 + 4^3.... + n^3
somber isle
i can't believe this highschool maths
somber isle
for practice you could try finding 1^2 + 2^2 + 3^2 + 4^2.... + n^2 first
yesss
brisk panther
usually we are just spoonfed that identity
instead of it being proved
i hope the method to find these identities is easy
fierce ember
np
somber isle
let me save this rq
brisk panther
thank you <@1016754862362476565> and <@332588758660808705>
brisk panther
let me save this rq
you want the book?
brisk panther
we're able to this because f(-x) = f(x) ?
yes
you can do this with even functions
even function: a function where f(-x) = f(x)
somber isle
thank you so much
brisk panther
np[
somber isle
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fathom flint
que 17
fathom flint
tried and got till here
junior chasm
thats after u made the correction
junior chasm
if its 3
fathom flint
but he didnt give me a solution
junior chasm
then the options are incorrect
fathom flint
what is g anyways
junior chasm
what is g anyways
gradian
fathom flint
i also have doubt in 31
queen hearth
What did you try?
fathom flint
What did you try?
tried n(n+1)/2 to sum of natural numbers
junior chasm
im pretty sure it was g before the correction
fathom flint
1+2....n
junior chasm
because that gives 2nd option as the answer
fathom flint
im pretty sure it was g before the correction
ye
junior chasm
well if u want to solve for 3
fathom flint
then let x^3 = t
which question are u talking abt
queen hearth
then let x^3 = t
They solved it using lhopital which works
queen hearth
i also have doubt in 31
Do you know the formula for summation of n^2?
fathom flint
Do you know the formula for summation of n^2?
yes
queen hearth
Use summation formula then take n^3 common
fathom flint
Use summation formula then take n^3 common
what to do of n?
well i got the answer ig
queen hearth
You'll have one constant term and all other terms with n in the denominator
fathom flint
take everything common ghat leaves 1/6
thanks @queen hearth and @junior chasm have a great day
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junior chasm
which question are u talking abt
the limit one, net issue so msg took time to send
junior chasm
They solved it using lhopital which works
alright
sturdy mango
how to think of this? practice? i know but anyone have more important ones like this?
devout snowBOT
how to think of this? practice? i know but anyone have more important ones like ...
crude wasp
how to think of this? practice? i know but anyone have more important ones like ...
context?
it depends on what u want to do basically
sturdy mango
like i see it a lot in questions like this, limits, this was in a arctangent series
crude wasp
like i see it a lot in questions like this, limits, this was in a arctangent ser...
i think it'd be helpful to send the original question
crude wasp
how to think of this? practice? i know but anyone have more important ones like ...
but basically i think to answer ur question, how u think of doing X matters a lot more on the original question rather than the expression
sturdy mango
fair enough
crude wasp
how to think of this? practice? i know but anyone have more important ones like ...
like if u gave me the LHS and asked me 'what are some interesting ways of expressing the LHS', it would be quite hard to find the final expression because that kinda comes out of nowhere
sturdy mango
i think it'd be helpful to send the original question
crude wasp
like if u gave me the LHS and asked me 'what are some interesting ways of expres...
but often u can kinda work out "it might be helpful to write X like Y in order to do the question"
sturdy mango
hmm
crude wasp
my best answer for this particular question is that it might've been useful to write the series as a telescoping sum so u might try writing the thing in that particular way
sturdy mango
fair enough
crude wasp
but i don't think there's like a good 'seeing X means Y will work' for what ur asking
i think here, u'd still have to play around quite a bit with the expression
like it's quite common to have a lot of good ideas, none of which work before u stumble on the right idea
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grave acorn
can anyone explain why a is in the range
devout snowBOT
can anyone explain why a is in the range
grave acorn
in [0,a**)**, a is excluded
oh i see? but why is it excluded
sullen island
cause that's what the notation [0, a) means
all the numbers x for which 0 <= x < a
your question was "why is a in the range ?", what I'm saying is that you're misinterpreting what they wrote
or maybe I misinterpreted your question
grave acorn
or maybe I misinterpreted your question
my main question is in the range we did [0, a) but im confused as to why we added a all together in the range. is it because [0, 1) was the range of the inner function (N/N+1) so, a * 1 = just a?
sullen island
essentially yes
grave acorn
😮
the one in the range is excluded, is there a reason why? why couldn't we have had square brackets?
sullen island
well N is always less than N+1
so you can't really expect N/N+1 to ever be 1
it can get damn close to 1 sure
but never literally be 1
that's why it's excluded
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mortal aspen
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mortal aspen
how to find those points?
shy kestrel
how to find those points?
do u know the definition of local max/min
mortal aspen
do u know the definition of local max/min
yes yes its like the most mininmum point on graph
i am thinking its local maxima mean 3,5 nd local minima is 1,3
shy kestrel
i dunno where those numbers are coming from
shy kestrel
yes yes its like the most mininmum point on graph
local = over an interval
its asking for max/min over the interval they showed u
so can u find the max/min value on the graph they showed
mortal aspen
nopee😭😭😭
shy kestrel
wheres it highest and lowest
mortal aspen
its asking for max/min over the interval they showed u
no no they r asking just for the point
shy kestrel
no no they r asking just for the point
thats the definition of a local max/min???
mortal aspen
wheres it highest and lowest
lowest 0,1 highest 3,5
mortal aspen
thats the definition of a local max/min???
nope
shy kestrel
u r clearly confused about its meaning
over the interval they showed u (ie the blue line) they r asking for which x values is y highest/lowest
mortal aspen
but when x=1 its lowest no
shy kestrel
when x=1 u get y=0...
shy kestrel
use ur eyes
ur telling me green dot is the highest value that this function can attain over this interval which is wrong
is there anything higher than the green dot
mortal aspen
is there anything higher than the green dot
yess
shy kestrel
where is it
mortal aspen
u mean x=4 right?
yesyes
shy kestrel
x=4 thats a local max
shy kestrel
u dont consider an entire segment
mortal aspen
u dont consider an entire segment
nope
shy kestrel
is there anywhere on the graph thats lower than f(2)
hmm
actually try putting in x=2 for the local min and seeing what happens
mortal aspen
actually try putting in x=2 for the local min and seeing what happens
it came as wrong
imma give up at this point
mortal aspen
is there anywhere on the graph thats lower than f(2)
nopee
shy kestrel
look again
shy kestrel
thats not how any of this works
mortal aspen
ohh
shy kestrel
when x=5 the value of y is y=0
the y-value at the orange dot (x=2) is smaller than the y-value at blue dot (x=5)
atomic dome
it came as wrong
Try x=1,5
atomic dome
For the maximum
mortal aspen
For the maximum
yesss nd shld i try 0,3 for minimum?
mortal aspen
ohhhh
atomic dome
What are the answers then?
atomic dome
5 is not an endpoint
mortal aspen
ohhh
atomic dome
It is still local
atomic dome
Any x value except for 0 and 6 is local
mortal aspen
Any x value except for 0 and 6 is local
ohh
atomic dome
Is that all you needed help with?
mortal aspen
Is that all you needed help with?
yes yessss only in graphs
atomic dome
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left oyster
I need help in the last question. the answere is 1.362 meters. i tried finding the 40% by dividing 1.4 (50%) by 5 and then subtracting the result from 1.4. I got 1.12 but the value is not the same as the answere.
devout snowBOT
I need help in the last question. the answere is 1.362 meters. i tried finding t...
@left oyster Has your question been resolved?
lunar harbor
I need help in the last question. the answere is 1.362 meters. i tried finding t...
Shortest 40% of students -> you want the z score where P(Z<z) is 0.4
which is around -0.25 (you can get this to be more accurate with a calculator, but I'm too lazy)
This means you need to find the height such that the z score is around -0.25
,w \frac{x-1.4}{0.15}=-0.25
woven radishBOT
lunar harbor
yeah the process is fine, just get a more accurate z score than I did and you'll be fine
left oyster
I am using the Z table, for 0.25 the Z score is 0.0987. it should be diffrent if it was negative? because i dont have negatives in the table
left oyster
Shortest 40% of students -> you want the z score where P(Z<z) is 0.4
also what made you sure that the value of 40 percent is 0.4? it might be something else. like in the second question the value of 50% is 1.4 not 0.5
lunar harbor
also what made you sure that the value of 40 percent is 0.4? it might be somethi...
the total area under the normal distribution curve is 1
left oyster
oh ok
lunar harbor
the total area under the normal distribution curve is 1
so the area corresponding to the lowest 40% of the data is the 40% of the area (aka 0.4) that is furthest to the left
low sleet
How would i go about solving this question
devout snowBOT
How would i go about solving this question
low sleet
I've managed to simplify one side but don't know where to go from there
quaint citrus
U might need to get some logarithm action going
low sleet
im assuming natural logs but i'm not sure if we are allowd to use it in the questions as i dont recall being shown it
low sleet
i honestly dont see a way to do it without it so im going to assume you can...
i think ur right
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knotty plinth
why does the integrand being an even function make the bounds 0 to 2 and give 2 as a constant?
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why does the integrand being an even function make the bounds 0 to 2 and give 2 ...
@knotty plinth Has your question been resolved?
knotty plinth
<@&286206848099549185> \
modest dune
Even function is symmetric about x axis
So the area from 0 to 2 and-2 to 0 is the same
devout snowBOT
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oblique quarry
knotty cave
I'm trying to understand the shortest path between a line and a point, how does that make sense. I know that for point Q on line, P point, PQ - projection of the PQ on the direction vector = the orthogonal vector component, and that it's magnitude is the shortest path. Why does the formula in the image make sense and is the orthogonal vector component magnitude
devout snowBOT
I'm trying to understand the shortest path between a line and a point, how does ...
knotty cave
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proud bluff
A binary number a = 11.
Considering 4 bits -a =
devout snowBOT
A binary number a = 11.
Considering 4 bits -a =
proud bluff
is it 1101
devout snowBOT
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proud bluff
<@&286206848099549185>
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proud bluff
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cedar ginkgo
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cedar ginkgo
does anyone know if this question is possible
misty crest
cedar ginkgo
how do u do a
@misty crest
also how do u write the equation for this question
<@&286206848099549185> these two questions above please ^^
minor kite
also how do u write the equation for this question
What have you tried for this? It cost some amount, X, and that amount is going to reduce by 0.15 every year. So, it went from X to 350. So maybe something like, $X - 0.15X$ would tell you how much it is still worth after a year?
woven radishBOT
drippdropp
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woeful wave
can someone help me solve this, I am completely stuck
devout snowBOT
can someone help me solve this, I am completely stuck
woeful wave
versed jetty
!status
devout snowBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
1
cedar ginkgo
**drippdropp**
I tried that it didn’t work
minor kite
@woeful wave You need to think about how the graph changes from $x^2$ to $4(x+2)^2 + 3$.
woven radishBOT
drippdropp
cedar ginkgo
None of the answers match up
minor kite
@cedar ginkgo You need the growth/decay formula. Figure out how much you lost from $X$ to $350$.
woven radishBOT
drippdropp
minor kite
@woeful wave The $(x+2)$ will make it shift to the left. And the +3 is going to make it shift UP, right?
woven radishBOT
drippdropp
woeful wave
i think I messed up, I tried to make my points and I took (-2,3) to begin my graph
then did
-4
-3
-2 3
-1
0
so then i put -3 for my x and got
4(-3+2)^2+3
(-12+8)^2 +3
-4^2+3
16+3
19
i think i messed up
if i plugged it in it would make my -1 and -3 y 19
minor kite
You don't need to plug in any points
Just look at what is happening to the graph.
The (x+2)^2 is like x^2
For each answer, its probably saying UP, DOWN, LEFT, RIGHT or something and then you tell it by how much
any term that is going to increase the total of the $y$ is going to move the graph UP. if it decreases $y$, then it moves the graph DOWN, right?
woven radishBOT
drippdropp
minor kite
and then the $(x+2)^2$ versus $(x^2)$ means you will get a little shift to the LEFT. if it was (x-2) it would get to the right
woven radishBOT
drippdropp
minor kite
Exactly! This is good information for what you're trying to accomplish.
woeful wave
so my graph shifts to the left 2 because it's a postive instead of a negative
woven radishBOT
drippdropp
woeful wave
so would my graph go up vertically 4 because its a positive
minor kite
If it was were $(x+3)$ it would shift to the LEFT by $3$
woven radishBOT
drippdropp
woeful wave
$g(x) = 4(x+2)^2+3$
woven radishBOT
Roach
minor kite
See the $4$, that's going to widen your whole graph horizontally. If the $4$ was like, say $0.9$, it would shrink it horizontally.
woven radishBOT
drippdropp
woeful wave
if it was a -4 would it shrink it horizontally?
woeful wave
like flip it over the x-axis
minor kite
over the x-axis sure, but you'd get a mirror from the lowest point
woeful wave
I think i got the right answer, can you check?
minor kite
What'd you do? I couldn't see the dropdowns in the original image
woeful wave
minor kite
The secondn and third answers look wrong to me.
woeful wave
- (Left, Right, None) by (2,3,4,None)
- Vertically (up,down, none) by (1/2,2,3,4,None)
3.Vertically (Stretches, Shrinks, Neither) by (none, 4,1/4, 2, 3 )
minor kite
up by 3
woeful wave
and that's because the 3 on the outside
minor kite
yeah
so like, think about it this way
whatever 4(x+2)^2 does
you're always adding another 3 to it
minor kite
positve $y$ means the whole thing goes up
woven radishBOT
drippdropp
woeful wave
$f(x) = a(x-h)^2+k$
minor kite
f(x) is your y
woven radishBOT
Roach
woeful wave
so the x in parathesis is my x and f(x) is my y
woven radishBOT
drippdropp
minor kite
positive would not matter, -1 to 1 will narrow it, $(-\infty, -1) \cup (1, \infty)$ will widen it. If it's positive it will be in the same position, but if negative it will flip the entire graph
woven radishBOT
drippdropp
minor kite
Honestly, this is a really important topic. Being able to have a general understanding of how a graph will change based on these small things is really good to know. I would strongly recommend not just driving through these without understanding what is really going on. A lot of these concepts are cumulative meaning if you have trouble with something now, it can cause you serious problems with math down the road.
woeful wave
so I should learn the little things that affect the graph
minor kite
YES!
Whenever I don't understand something about a function, I graph/plot it. I want to see visually what is happening. This is a very powerful tool
woeful wave
Like on desmos?
minor kite
Yup, exactly
So, now that we've identified how these different things will IMPACT the graph with shifting. Now you can pick a point
PLug it in to both
and look at the difference
That's how you verify/check your anawer
woeful wave
the - or + in my parathesis affects which side of the y-axis it's on
if my A is negative it flips the whole graph
minor kite
$4(0 + 2)^2 + 3 = 4(2^2) + 3 = 16 + 3 = 19$
woven radishBOT
drippdropp
woeful wave
uh
woeful wave
so
A- means downward facing parabola A+ means upward
+K means positive on y-axis -K means negative on y-axis
-H means negative on x-axis +H means postivie on x-axis
i think i did this right
No, cause why am I actually starting to understand
long plume
Explain me the ans of (b) "state the set of value of x for which the expansion is valid"
devout snowBOT
Explain me the ans of (b) "state the set of value of x for which the expansion i...
lunar harbor
Explain me the ans of (b) "state the set of value of x for which the expansion i...
In general, the expansion of $(1+x)^n$, where $n$ is not a non-negative integer, is valid for $|x|<1$
woven radishBOT
Civil Service Pigeon
long plume
In general, the expansion of $(1+x)^n$, where $n$ is not a non-negative integer,...
ok also how did we solve |6x| < 1 further
lunar harbor
Recall that $|ax|=|a| |x|$
woven radishBOT
Civil Service Pigeon
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warped sequoia
What is this problem called? Excuse my description:
Does there exist another shape/object in RR^3 which contains the origin (other than the sphere), where every cross-section of that object which contains the origin has an equivalent boundary to any other cross section that contains the origin?
faint zinc
$\bR^3$?
woven radishBOT
OmnipotentEntity
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vital blaze
No clue really where to go
devout snowBOT
No clue really where to go
vital blaze
from part a, g = 2.7m/s^2
I understand that Fc = Fw and found a velocity of sqrt7.4
but I have no clue where to really go
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vital blaze
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storm wharf
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broken stone
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broken stone
Hey there can somebody teach me this trig ...?Is this correct way?
dapper fable
i think the easier way would just be to divide both sides by cos(x + pi/3)
is there a range of x given?
broken stone
is there a range of x given?
Yep, 0 ≤ x ≤ 2π
broken stone
i think the easier way would just be to divide both sides by cos(x + pi/3)
Oh become tan (x +π/3) = 1 right?
broken stone
thining about it in my head you 'll get 2 values for x
Now I get x + π/3 = π/4 , 5π/4 is this right?
dapper fable
not quite
dapper fable
since 0 < x < 2π, then π/3 < x + π/3 < 2π + π/3
notice π/4 is not in this range
but there is another value whihc is
broken stone
since 0 < x < 2π, then π/3 < x + π/3 < 2π + π/3
ohh yes you're right so the only ans should be 11π/12
dapper fable
actually x + π/3 = 9π/4 is also included
broken stone
actually x + π/3 = 9π/4 is also included
understood thanks
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edgy scroll
Hello this is a stats question and I don't fully understand what it's asking of me
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Hello this is a stats question and I don't fully understand what it's asking of ...
edgy scroll
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<@616880914135253013> You need the growth/decay formula. Figure out how much you...
Can u please send working out
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vernal grail
$2tyy'=y^2 - t^2 +1$
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$2tyy'=y^2 - t^2 +1$
woven radishBOT
alee
vernal grail
At this point can I proceed with the integrals or not yet? Also for I have a variable t on the left
Maybe better to use Bernoulli?
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cyan laurel
At this point can I proceed with the integrals or not yet? Also for I have a var...
You can't proceed with integrals here
On left there's both y and t
vernal grail
@cyan laurel What can I do
vernal grail
Ok thanks
cyan laurel
I think I got it
@vernal grail
First we move y^2 to the other side
Then notice that the LHS is of form
t * f' - f * dt
Which looks like the quotient rule
cyan laurel
( try and Identify f in our equation)
vernal grail
But do we have to go back to Bernoulli's equation?
cyan laurel
But do we have to go back to Bernoulli's equation?
I don't think Bernoulli can be used
cyan laurel
cyan laurel
These steps
Then, if you notice, the LHS looks like the quotient rule, but without the g^2 in denominator
So, we identify g in our case and then divide by g^2 on both sides
Do you follow?
@vernal grail ?
vernal grail
But with a substitution we can go back to a first order linear equation
cyan laurel
But with a substitution we can go back to a first order linear equation
Yeah u = y^2
Will also work
It's just my preference
vernal grail
Ok then
vernal grail
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fresh moat
Hello, this is a somewhat open question so perhaps not appropriate - but I wanted to see if anyone had a good example come to mind.
I am attempting to construct a counterexample for something, and I have determined that I need something that fulfills the following conditions:
I want a second order, non-linear ODE of the form
y'' = f(y)
with f a differentiable function and f(0) = 0 and some solution satisfying the boundary conditions y(0) = y(1) = 0. Ideally this solution happens to have a nice(ish) closed form to work with as well (i.e. I'd like to avoid elliptic integrals and the like if possible)
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Hello, this is a somewhat open question so perhaps not appropriate - but I wante...
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supple knot
Hello, this is a somewhat open question so perhaps not appropriate - but I wante...
Start around equation 4.4.7
https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/04%3A_Applications_of_First_Order_Equations/4.04%3A_Autonomous_Second_Order_Equations
fresh moat
Thanks, but I ideally want a nice closed form, which the (nonlinear) cases of the above don't have. I suppose I'll have to bite the bullet and work with elliptic functions after all
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lament schooner
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lament schooner
for the equivalence theorem
heavy current
does that mean x has to be = 0
it's showing that we must have x = 0, yes
faint gorge
https://tenor.com/view/love-mochi-mocha-milk-and-gif-26062771
hi gher!
heavy current
if we assume that x is a solution to Ax = 0, then the invertibility of A gives us that x = 0
hi adonis
heavy current
if we assume that x is a solution to Ax = 0, then the invertibility of A gives u...
hence the equation has only the trivial sol
heavy current
lament schooner
it's showing that we must have x = 0, yes
so basically the solution will always be 0? whenever A is invertible
lament schooner
hello syrex <:happy:908558103992225822>
hiii
lament schooner
oh you are alive syrex
unfortunately 
fervent hornet
so basically the solution will always be 0? whenever A is invertible
yes
heavy current
so basically the solution will always be 0? whenever A is invertible
the only solution to Ax = 0 is x = 0 when A invertible, yes
lament schooner
oh okay tyyty
upper schooner
Hi everyone 
lament schooner
Hi everyone <:AntlerLove:676606491624603661>
hiiii
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misty meteor
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misty meteor
I’ve wrote down all the information I think I can, but I don’t really know where to go from here 😭
woven radishBOT
bacc
faint gorge
Given that f, g and h are pseudo increasing
Meaning you can use these as facts
So you wanna start with the left side
misty meteor
I’ve tried to manipulate it around but I just don’t see how I can prove that it’s less than the right side
faint gorge
Let's do it term by term
woven radishBOT
bacc
faint gorge
Let's start with f(x_1)
You know it's pseudo increasing
That means 2f(x_2) is greater
And you can go up
woven radishBOT
bacc
faint gorge
And we can do this now with the other two terms
misty meteor
But how would I know if 4g(x_2) is greater than 2g(x_1)?
faint gorge
because g(x) is pseudo increasing
that means
g(x_1) < 2g(x_2) holds
Now that inequality is equivalent to
2g(x_1) < 2(2g(x_2))
multiplying both sides by two
misty meteor
Sorry I’m just not sure how to phrase proofs 😭
faint gorge
Do you see
woven radishBOT
bacc
misty meteor
I think I get it now I just don’t know how I’d justify it when I write it down :’)
The visual is very helpful though
faint gorge
Your justification is that you are using your assumption
that f,g and h are pseudo increasing
This is what you use
your help
Then you ask yourself what does this literally mean
you write down the definition
These is what you will be using
misty meteor
should I establish that in the proof?
faint gorge
And then you wanna go up and up until you arrived at what you wanted to arrive
faint gorge
should I establish that in the proof?
yes, as just shown
misty meteor
Like “assume f(x_1)<f(x_2)”
faint gorge
Like “assume f(x_1)<f(x_2)”
assume f is pseudo increasing that means f(x_1) < 2f(x_2)
misty meteor
Then should I not reference it at all in my proof if the problem already establishes it?
woven radishBOT
bacc
faint gorge
Then should I not reference it at all in my proof if the problem already establi...
you can
before you start you can mention what it means for f,g and h to be pseudo increasing
this
and then you can us this in your proof step by step
as we did
term by term
then it's clear that you've used the assumption
misty meteor
So should I start with “we know that” and then “by definition of pseudo-increasing?”
faint gorge
you can yea
Assume f,g,h are pseudo increasing, that means for f,g,h this holds bla bla bla.
We wanna now show f+2g+3h is also pseudo increasing. (Ask then what that means by your definition of pseudo increasing, then you know what you wanna show)
Then you can start with the left side and include your assumptions about f,g and h, to get to the result you want.
vast violet
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vast violet
anyone know where i can find more word problems like these
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restive river
hello
restive river
I have a problem with Linear Relations in Grade 10 math. My teacher explains very poorly and it's hard to understand what she's trying to say. I have a quiz tomorrow and I don't understand writing equations and solving them from word problems
and example i have is
something like this
,rotate
woven radishBOT
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restive river
A garden store sells two types of fertilizer: brand A for $5 per bag and brand B for $8 per bag. If a customer buys a total of 10 bags of fertilizer for $65, how many bags of each type did they buy? Create an equation to represent this situation and solve it without using the substitution method.
<@&286206848099549185>
twin thunder
so did you make the equation?
restive river
so did you make the equation?
equation 1 is x+y=10 and the second one is 5x+8y=65
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minor kite
$5X + 8Y = 65$ and $X + Y = 10$. You got it.
Then $Y = 10 - X$ which is going to be $5X + 8(10 - X) = 65$. Solve for $X$ and $Y$ so you know what they are. But even better, since it looks like it says don't use substitution, is you just set up a matrix and row reduce.
$\begin{bmatrix} 5 & 8 & 65 \ 1 & 1 & 10\end{bmatrix}$
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drippdropp
minor kite
julia> A = [ 5 8 65 ; 1 1 10 ]
2×3 Matrix{Int64}:
5 8 65
1 1 10
julia> rref(A)
2×3 Matrix{Float64}:
1.0 0.0 5.0
0.0 1.0 5.0
(5)5 + (8)5 = 65
For some extra context, you know you will get an answer from a row reduction because you have two equations and two unknowns, so $A$ is linearly independent.
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drippdropp
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coral wraith
can someone help me with this?
devout snowBOT
can someone help me with this?
coral wraith
?
sullen island
yeah that's what it means to be a subspace
coral wraith
im pretty sure it would be a subspace then
sullen island
you tried anything yet ?
coral wraith
zero vector would still exist
void ruin
check these properties
coral wraith
and the way this subspace works, either x_1, x_2, or both are zero
sullen island
indeed
coral wraith
so i just go through all cases of addition?
that would be 6
i mean im pretty sure the sum of any two vectors in this space would be in R^3
but i dont know how to say it without tediously showing it
sullen island
well 9 cases tbh
acoustic leaf
i mean im pretty sure the sum of any two vectors in this space would be in R^3
we can't just have the sum of them be in R^3, we need it to be in the subspace
coral wraith
we can't just have the sum of them be in R^3, we need it to be in the subspace
oh right
well i think i can find a counterexample
[ 1 0 0 ] + [ 0 1 0 ] = [ 1 1 0 ]
the result isnt in the subspace
sullen island
gg
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gentle helm
How would I solve this equation?
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How would I solve this equation?
dense jay
roughly how long does it take the earth to do a full rotation
acoustic leaf
how fast does the earth rotate?
gentle helm
that's all the word problem explains. could we add that the earth does a full rotation every 24 hours?
acoustic leaf
yes, it doesn't specify because that's common knowledge
gentle helm
In order to solve the problem, I would need to convert the 24 hours to the rpm and that'll make it 1440 rpm right?
acoustic leaf
no, the earth does not complete a full rotation 1440 times per minute
gentle helm
so the rpm would be 24 hours?
gentle veldt
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cloud ermine
lets see
gentle veldt
hello
cloud ermine
hello!
gentle veldt
i dont know how to do this its algerbra 1
gentle veldt
but which ones theres so manny
cloud ermine
do any
cloud ermine
yes
cloud ermine
if u see on the top it says y = x ^2
gentle veldt
so how does that work?
cloud ermine
i think u will have to plug in all the x axis numbers with ^2
gentle veldt
so any?
cloud ermine
any ones
gentle veldt
but like why
cloud ermine
becuase it says in the top that there is a x axis and a y axis
cloud ermine
put that in ^2
cloud ermine
yea
gentle veldt
but how would i plot that
it only goes to 10
wait
so
i did some stuff but i got 4 dots only
so i can go outside the quadrants right
cloud ermine
for each pair of x and y values in the table, find the corresponding point on the graph.
for example, the point (-10, 100) is 10 units to the left of the origin and 100 units up.
plot this point on the graph.
repeat this process for at least five points from the table.
connect the points:
once you have plotted the points, connect them with a smooth curve.
this curve should be a parabola, which is a symmetrical u-shaped curve.
gentle veldt
so like i got theese
uh
how
did u get u
i was getting a straight line
u cant go 100 units up is the problem
cloud ermine
correct
thats why i have a ^ so u will need to find the square root
<@&286206848099549185> please help me
mental crown
what happening
cloud ermine
i dont understand this
mental crown
the graphyng?
cloud ermine
i know the base of this problem but i dont know how to explain it
mental crown
how
gentle veldt
by taking the x
mental crown
yes
gentle veldt
and doig it to the power of 12
2
and then i plot
for example -3,9 would be 9,-9
so ye
idk how u got curved
i got straight
i need like examples to understand
mental crown
gentle veldt
how did u get that
mental crown
when x is 1, y is -1
gentle veldt
ok
gentle veldt
wait huh
i tohught -2 squared was 4
not -4
cause u multiply -2 against itself
creating positive numbers
gentle veldt
oh
i didnt know that..
so
i need 5 tho
and all the other numbers are to big for the y
likje i cant ploit them'
mental crown
then let y stay under 10
mental crown
what about negatives
gentle veldt
so i would do
-1 times 1? which is -1,-1?
OHH THEN I DDO THE SAMEEE
I GOT IT
i think
mental crown
yessir
gentle veldt
lemme test it
gentle veldt
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tender veldt
when its asking for values for p does it mean a literal answer or like p>1
hexed turtle
can someone help
robust scaffold
can someone help
If you want help then get your own channel, you do that by typing your question in the ''help'' channels above.
tender veldt
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supple knot
when its asking for values for p does it mean a literal answer or like p>1
Show the previous question
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tender veldt
Show the previous question
supple knot
when its asking for values for p does it mean a literal answer or like p>1
Try a u sub
supple knot
It'll get the integral to look close to this one
tender veldt
It'll get the integral to look close to this one
would this be enough?
supple knot
Small technicality that I don't know if your teacher intended: ln(2) < 1, so maybe adjust your u sub so that the lower limit is exactly 1
tender veldt
Small technicality that I don't know if your teacher intended: ln(2) < 1, so may...
wouldnt I have to just do integration by parts since letting u= xln(x) cause issue
supple knot
You should not need integration by parts
tender veldt
I can let u just be xln(x)?
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limber kernel
how to use the bot
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how to use the bot
limber kernel
$\polylongdiv{5x^4-12xy^3-7x^3y}{x^2-2xy}$
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marito1704
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