alright
#help-17
wanton shuttle
hardy vector
Basically the largest side is opposite the largest angle. The second largest side is opposite the second largest angle. And the third largest side (i.e. is the smallest side) is opposite the smallest angle
wanton shuttle
right
hardy vector
Because of the sine rule needs to hold to compensate AE and AD are smaller.
That is the best way I can explain it unfortunately
This is the key takeaway
wanton shuttle
alright thank you very much @hardy vector
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thorny ether
I keep finding difficulty in diagonalizing this matrix, could anyone help me please?
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I keep finding difficulty in diagonalizing this matrix, could anyone help me ple...
thorny ether
what do you have at the moment ?
umm I didn't understand
cyan talon
didn't understand what
thorny ether
like have what? do you mean my answer?
cyan talon
yeah, what are your attempts at the question ?
cyan talon
there's other ways of getting that determinant
with row reduction it's doable but it's quite a pain just for a 3x3
thorny ether
there's other ways of getting that determinant
oh sure
thorny ether
the one with the X-1
yeah
cyan talon
if you expand the determinant along that column you're almost done
it's gonna be much quicker than row reducing
thorny ether
if you expand the determinant along that column you're almost done
ughh how? you mean doing linear operations on the 2nd row?
cyan talon
no
have you seen something like that to compute determinants?
computing a 3x3 as a weighted combination of smaller 2x2 dets
thorny ether
have you seen something like that to compute determinants?
oh yes, I usually try to go this way
but I think you can do the same with other columns?
cyan talon
yes
cyan talon
you just have to be aware of the signs
thorny ether
you just have to be aware of the signs
and the total detirminant is the sum of the 3 right?
cyan talon
yeah
but since 2 of the terms are straight up zero on the second col
you only have 1 term that's interesting
thorny ether
now I understand why the lecturer suggests to do the row reduction, I thought it's the only way to calculate that detirminant to get eigen factors
cyan talon
well if you have bigger matrices you definitely want to go the row reduction way
for 3x3 it's just not worth it
thorny ether
you only have 1 term that's interesting
yeah it's way easier than I thought, thank you so much 
thorny ether
for 3x3 it's just not worth it
true
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jagged cargo
Given the pyramid $S.ABC$, with SA, SB, SC all makes an angle of 60$^{\circ}$ with the base. Given BC = a, angle BAC = 45$^{\circ}$. Calculate the distance from the apex S to the plane (ABC)
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Given the pyramid $S.ABC$, with SA, SB, SC all makes an angle of 60$^{\circ}$ wi...
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worm desu
jagged cargo
essentially i need to find the height of the pyramid
to do that, i need to find the radius of the circumcircle of the base. to do that i need to use the formula area = abc/4R. the problem is i cant find the area when im given only 1 angle and 1 side
would appreciate any hint
i also thought of sine rule too, but again, not enough information
<@&286206848099549185>
god i fucking wish my maths sir taught me about 3d geometry for hours
gonna close this since im tired and wont think properly
.close
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edgy sapphire
it's just ma=T-mg
velvet cove
it's just ma=T-mg
i derived the expression $F_t=30+3a_{lin}$
twin meteorBOT
Jash
velvet cove
how to find the linear acceleration
edgy sapphire
all the angular stuff can be converted to linear by multiplying by r
velvet cove
all the angular stuff can be converted to linear by multiplying by r
so linear acceleration = r * angular acceleration?
edgy sapphire
right
velvet cove
so $F_t=30+3(0.5\cdot-1.57)?$
twin meteorBOT
Jash
edgy sapphire
oh yea
they should combine since tension should be more than the weight here
the signs make sense that way
velvet cove
but isnt the block accelerating downwards
velvet cove
tension should be less than weight
edgy sapphire
yea
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wraith stirrup
Out of 40 sweaters in one boutique, 10% of them have a defect. Tanja didn't bring her glasses and is buying two sweaters. Find the probability of event A, that both purchased sweaters are without defects, and event B, that exactly one sweater purchased has a defect.
no idea
sweet flower
```
Out of 40 sweaters in one boutique, 10% of them have a defect. Tanja didn't ...
!status
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```
Out of 40 sweaters in one boutique, 10% of them have a defect. Tanja didn't ...
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
I assume they are independent trials, and try using a complement
wraith stirrup
anyway this is translated to english
dark kiln
they aren't independent
wraith stirrup
they aren't independent
how?
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frozen bobcat
```
Out of 40 sweaters in one boutique, 10% of them have a defect. Tanja didn't ...
do you know hypergeometric distribution?
sweet flower
i don't want to help, i just wanted to correct that one dude
Ah yeah I missed the selecting 2 from 40 without replacement part
upper zodiac
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twin meteorBOT
puckmyseen
upper zodiac
do i bring the exponent on all them
wicked shard
$\ln(ab)= \ln(a) + \ln(b)$
twin meteorBOT
hired
upper zodiac
it says logarithmic differentiation
that's why
not sure if that counts
how i've been taught
heavy yoke
**puckmyseen**
the strategy here is to simplify the right side as much as possible with log rules before taking the derivative
heavy yoke
yes
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upper zodiac
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upper zodiac
i don't know where to even start
potent anchor
what is the derivative of e^2x
upper zodiac
e^2x * 2
wicked shard
oh yes
potent anchor
differentiate is the verb of derivative makes sense thanks english
upper zodiac
f'(x) = Me^2x * 2
potent anchor
now just do that 10 times
(2^10)
potent anchor
multipying by 2 ten times is the same as multiplying by (2^10)
so you know taking a derivative "multiples it by 2"
what is the tenth derivative then
upper zodiac
M * 2^10 * e^2x?
potent anchor
that is right
then you just set that equal to 96
when x is 0
come to think of it this wasn't necessary because x is 0
but you would need to do this if x was something else so
potent anchor
upper zodiac
how do i simplify that?
upper zodiac
mm ok thank you
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charred grotto
i belive C,D abd F are false bc the matrix
-1 & -1\\
1 & 1
\end{bmatrix}
has an inverse and when squarted gives zero.
as for A and F i think they are the same Q and are true by the properties of square matrices. as for B IIRC the properties of square matrices make behave like normal algebra
charred grotto
steady quartz
top one is wrong cus if the determinant is 0 then no inverse
hot ledge
Can you expand the expression in B?
steady quartz
yes
it is equivalent
isnt c true
and d
and f
im acc so confused with those
its all 0s
hot ledge
d is false e is false, f is false
steady quartz
tfff why
hot ledge
a is true (take 0) B is false (B and N don't necessarily commute), C is true
For D take [[0,1][0,0]]
hot ledge
No they aren't
E is for all N,B
A is there exists
For a, take B = 0 and N whatever it works
charred grotto
top one is wrong cus if the determinant is 0 then no inverse
A?
steady quartz
mb bro i thought it was inverse not just any matrix
hot ledge
Don't worry
charred grotto
but then a and e
yeah what i msaying
charred grotto
For D take [[0,1][0,0]]
$\begin{bmatrix}
-1 & -1\
1 & 1
\end{bmatrix}$ is better as it can do C D and F
twin meteorBOT
Mr.Bossman's Misses
charred grotto
a is true (take 0) B is false (B and N don't necessarily commute), C is true
anyway why is B false
hot ledge
It does not have an inverse so can't do C
charred grotto
It does not have an inverse so can't do C
it does
hot ledge
anyway why is B false
Because (N+B)^2 = N^2 + B^2 + NB +BN and BN is not necessarily equal to NB
For C just multiply both sides by the inverse
charred grotto
Because (N+B)^2 = N^2 + B^2 + NB +BN and BN is not necessarily equal to NB
but its square so isnt BN equal to NB
charred grotto
because it says both B and N are square of size nxn
hot ledge
Yes but when you multiply there is no commutativity
charred grotto
so they are all false?
hot ledge
No, just B,D,E and F
charred grotto
thx i still dont understand why C and B are what they are. and how A and E are different
why are A and E are different
cyan talon
A says you can find one matrix B that commutes with N (where B might depend on N)
E says all matrices commute with N
E is a much bigger ask than A
@charred grotto
cyan talon
yea
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molten bolt
Is there any special reason the beginning variable is t, or was it just randomly picked to show that the variable changes from t to x
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Is there any special reason the beginning variable is t, or was it just randomly...
hybrid flicker
Is there any special reason the beginning variable is t, or was it just randomly...
we just need a different variable than x
you could have chosen f(u)du or f(y)dy doesn't matter
but you can't integrate with respect to a variable that is already taken, such as x
molten bolt
makes sense
i was confused because i first thought t meant with time
but that didnt make sense
thank you
.close
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celest nacelle
Heyo, trying to work on my fundumental understanding for dynamics here.
Working on this problem.
Here is my work.
My main question is when you use arrows to represent direciton instead of signs, how do I infer final rotation direction of alpha? In this problem alpha = .5 clockwise.
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celest nacelle
.reopen
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โ
celest nacelle
@sacred flume yo got me? lol
sacred flume
nah bro sorry ๐ been a min since physics
celest nacelle
oof
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celest nacelle
<@&286206848099549185> Maybe im onto something here.
the right hand side of formulas, does a = r๐ถ not determine direction?
does it only determine magnitude?
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waxen hawk
can I divide 0 on both sides of an equation.
vocal sleetBOT
can I divide 0 on both sides of an equation.
waxen hawk
why
lime basin
because
in simple terms
0 + 0 + 0 + 0 + 0.....n times, is still 0
so you can add 0 as many times as you want (finitely), you'll get 0
4/2 means (in simple terms) getting how many times you can add 2 to itself to get 4
so 1/0 means getting how many times you can add 0 to get 1, which is basically not possible
barren river
Very nice explanation kavin ๐
lime basin
thank you ๐
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rough zenith
Ross likes every number which is equal to 13 times the sum of itโs digits. How many
positive integers does Ross like?
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Ross likes every number which is equal to 13 times the sum of itโs digits. How m...
rough zenith
Im confused because how are we suppose to know how many digits there are?
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merry python
you can know the upper bound
take the case of three digit number
maximum sum of digits would be 27
13*27=351
so there should be no number abive 351 that would satisfy the relation
@rough zenith
rough zenith
i still dont udnesratnd
@merry python
i just dont understand one thing
why isnt there more than 4 digit numbers not work
merry python
why isnt there more than 4 digit numbers not work
because number above 351 won't satisfy the relation
sum of dogits of any four digit number will be less than 36
and 36*13 is 468
which is no where near a four digit number
@rough zenith
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grizzled mango
A load of 1600 kW at 0ยท8 p.f. is shared by two 1000 kVA transformers having equal turn ratios and connected
in parallel on their primary and secondary side. The full load resistance drop is 1% and reactive drop is
6% in one of the transformers, the corresponding values in the other transformer being 1.5% and 5%.
Calculate the power and power factor at which each transformer is operating.
(Ans. 700 kW, 0.76 lagging, 900 kW, 0.84 lag.)
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A load of 1600 kW at 0ยท8 p.f. is shared by two 1000 kVA transformers having equa...
grizzled mango
A load of 1600 kW at 0ยท8 p.f. is shared by two 1000 kVA transformers having equa...
i wanna know how to get the answer
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indigo comet
Your answer is not correct
Hint:Do the fish pass through the start point after it begins swimming?
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lyric lava
is this right, if so explain please
lyric lava
<@&286206848099549185>
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edgy sapphire
looks like m is wrong, it should be -2 - (-6)
I try to draw the points near where they should be on a graph so I can check if the slope looks about right
trim wind
How would I formally argue for b) here, should have to use time inversion property but I'm not sure. Need help
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cyan marsh
What would be the chance of me not hitting a 1 in 92, 430 times in a row?
sly sierra
!original
vocal sleetBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
cyan marsh
I'm currently playing a game and getting what i want has a chance of 1 in 92. I've tried it 430 times and i wonder what the chance of me being so unlucky is.
What would be the correct rule to apply? My math is really bad..
sly sierra
probability of failing is 91/92
cunning plaza
Binomial distribution
sly sierra
probability of failing 430 times in a row is (91/92)^430
frozen bobcat
product rule for probability
assuming independence (your chances don't change between attempts) its (91/92)^430
sly sierra
so the probability of NOT failing 430 times in a row is...?
cyan marsh
Dumb question, but.. What is ^ ?
twin meteorBOT
Result:
0.0090998228849429
Oh! Thanks! Can i somehow use the bot later on if needed?
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warped osprey
how do you know when to restrict a function to find its inverse if you don't know the shape of the graph?
vocal sleetBOT
how do you know when to restrict a function to find its inverse if you don't kno...
edgy sapphire
it comes down to checking where the function outputs the same value, which depends on the situation, there's loads of different ways
for most functions just checking that it never turns around is fine
warped osprey
it comes down to checking where the function outputs the same value, which depen...
wait wdym checking where it outputs the same value
edgy sapphire
like x^2 outputs the same value for x and -x, so you restrict to one side
those are also the parts where it's always decreasing/increasing
edgy sapphire
flat at a point is fine
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deft flower
Question : A rectangular walled garden is being made. North wall costs 30$ per foot, ans the other 3 are 20$ per foot. The garden must be 200 square feet in area.
a) make function c(x) that gives the cost to create the walled garden were x is the north wall
b) what is the minimum cost
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Question : A rectangular walled garden is being made. North wall costs 30$ per f...
deft flower
this is my work so far
p sure im dead wrong
im so lost
this optimization stuff in calc is hard
i also dont know how i should derive this
ik ita in terms of L
but do i need chain rule for A or do i just plug in 200
and is my cost function even right
keen thicket
Question : A rectangular walled garden is being made. North wall costs 30$ per f...
it is rectangular,
let x be width and y be length
area = xy = 200
radiant wedge
Part A is correct
keen thicket
c(x) = total cost = 30x (north) + 20y + 20y + 30x (rest)
so try substituting y in terms of x
deft flower
so try substituting y in terms of x
wait but other person said my part a was fine
is it not
or are you suggesting i substitute everything in terms of x just to make deriving easier
keen thicket
wait but other person said my part a was fine
it is, correct
deft flower
sorry its been very hard for me to visualize these sorts of problems
keen thicket
i have used x,y instead of l , w
flint idol
you can change the A in part a to 200, since area is always 200
that should make it easier to differentiate
deft flower
so the derivative should be 30 + 20 - 8000/l^2
flint idol
yep
deft flower
so from there should i solve for l for the critical points
deft flower
would the domain be all real numbers? im inclined to say its everyrhing except 0 bc of l being in the denominator but
wait
no it is (-inf, 0)U(0, inf) right
i think
deft flower
thats so true ๐ญ
deft flower
one sec im gonna chart it out
i got a decimal value for the absolute min, not really sure how correct it is - i got (4sqrt10, 400sqrt10)
im worried tho cause we donโt typically use calculators for this prof
and idk if hed give us a problem like this where we get weird radicals
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prisma badger
what does the disciminant of the quadratic formula find
vocal sleetBOT
what does the disciminant of the quadratic formula find
brisk tinsel
It tells you about the roots of the quadratic equation
Remember the discriminant is inside a squareroot?
If the discriminant is negative, then there will be a squareroot of a negative number
Meaning, there will not be a real solution to the equation
prisma badger
ah ok
brisk tinsel
So if you know that $B^2-4AC<0$, then the quadratic equation will have 2 imaginary solutions
twin meteorBOT
Renz
brisk tinsel
are you trying to solve for the value of x?
prisma badger
because i know its x^2 - ___ +7
prisma badger
are you trying to solve for the value of x?
no value of a
we got a point
how do i find B
brisk tinsel
try expanding the right first
prisma badger
try expanding the right first
?
brisk tinsel
use the $(a-b)(a+b)=a^2-b^2$
twin meteorBOT
Renz
brisk tinsel
what do you get
prisma badger
because i know its x^2 - ___ +7
^
prisma badger
x^2 - 7
prisma badger
wait but like just imagine there is no a or point
how would i take it into standard form
dont i need a b term
brisk tinsel
you mean lets consider y=xยฒ-7 for now?
prisma badger
ye
prisma badger
so b is considered 0
prisma badger
is that for all difference of squares?
brisk tinsel
the only restriction on the standard form of quadratic equation is that a cannot be equal to 0
brisk tinsel
is that for all difference of squares?
yes, if you are using the difference of 2 squares, there will usually be no B term appearing
prisma badger
and for perfect squares
brisk tinsel
since they cancel out as $ab-ab$ when expanding $(a-b)(a+b)$
prisma badger
its a^2 + or - 2ab + b^2
twin meteorBOT
Renz
brisk tinsel
its a^2 + or - 2ab + b^2
yep, just follow the formula
prisma badger
yep, just follow the formula
ok
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brave orchid
I am supposed to pick which answers are correct, I got this question wrong and I am unsure why and/or how to solve this. This is discrete math Recurrence Relations & Generating Functions.
vocal sleetBOT
I am supposed to pick which answers are correct, I got this question wrong and I...
regal bane
Hopefully you know that
1/(1 - x) = 1 + x + xยฒ + xยณ...
Then,
1/(1 - 4x) = 1 + 4x + (4x)ยฒ + ...
brave orchid
4^3?
regal bane
You following me so far?
brave orchid
I think so? From what I see, you're increasing power?
regal bane
I am using this identity:
1/(1 - x) = 1 + x + xยฒ + xยณ + ...
Where the right side powers do increase forever, yeah
Sub 4x into that, you get the solution to the first box
brave orchid
oh so it would be 1 + 4x? for the right side of it?
regal bane
Then,
1/(1 - 4x) = 1 + 4x + (4x)ยฒ + (4x)ยณ + ...
If we rearrange,
1/(1 - 4x) - 1 - 4x = (4x)ยฒ + (4x)ยณ + ...
brave orchid
is the equal part supposed to be what the summation should look like if it was true? "= (4x)ยฒ + (4x)ยณ + .."
so (4x)^k is what is would be?
regal bane
I'm merely saying
1/(1 - 4x) - 1 - 4x
Is equal to
(4x)ยฒ + (4x)ยณ + (4x)โด + ...
Now, that right side is clearly just ฮฃ (4x)^k
For k โฅ 2
= ฮฃ 4^k x^k
Hey, that's the first box!
regal bane
so is (4x)^k= 4^k x^k?
This is an exponent law yes
brave orchid
oh thanks, I didn't know that. I thought it would be 2 different answers if it was written that way
so just for clarification, 1/(1 - 4x) - 1 - 4x is equal to 4^k x^k because of the property 1/(1-x) = 1 + x + x^2 because the 1/(1 - 4x) - 1 - 4x rewritten is equivalent to (4x)^2 + 4x^3....?
regal bane
That wording isn't great. Hopefully you can get clarification from our conversation above instead. If anything isn't quite making sense, feel free to ask about one of the steps
brave orchid
sorry, I think I wrote it too cluttered. So to begin again.
- since the property is 1/(1-x) = 1 + x + (x)^2..... we learned that the first box would look like 1/(1-4x) + 1 + 4x + (4x)^2 correct?
- you said that 1/(1-4x) - 1 - 4x re-written is equivalent to (4x)^2?
- since (4x)^k is equivalent to 4^k x^k, this means that the first box is correct? @regal bane
please let me know if I made an error or a misinterpretation somewhere
<@&286206848099549185>
regal bane
sorry, I think I wrote it too cluttered. So to begin again.
1. since the proper...
The equal sign disappears in the last line there. Without it, that makes no sense
regal bane
2. you said that 1/(1-4x) - 1 - 4x re-written is equivalent to (4x)^2?
This doesn't parse. Also, be careful not to leave out ...
I think you're seeing ... and thinking it can be ignored
brave orchid
I forgot about that
ohh, I just got what you meant for the equal sign
so the property is 1/(1-x) + 1 + x + (x)^2.... correct?
so because of this, this means the first box is incorrect because the equation should be 1/(1-4x) + 1 + 4x + (4x)^2 but instead it's negative? @regal bane
regal bane
so the property is 1/(1-x) + 1 + x + (x)^2.... correct?
You need an equal sign
regal bane
Hopefully you know that
1/(1 - x) = 1 + x + xยฒ + xยณ...
This is the identity
brave orchid
You need an equal sign
equal sign to what specifically?
regal bane
Equal
Like, for all x
(Where x can be between -1 and 1)
You may want to see Taylor Series before attempting this problem
brave orchid
I'll be learning that later in my calc 2 class later in the sem, we're barely touching on series in general
ay man, should I just close the case or do you think we could continue @regal bane
I'm trying to understand you but it's a little difficult when I can't hear back a response that often
regal bane
Mb. I'll let other helpers help.
brave orchid
no worries bro, I appreciate your time. So should I close the case and start a new one or just wait here? @regal bane
.close
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toxic knot
anyone plz help in this
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anyone plz help in this
toxic knot
anyone plz help in this
ans is x=4a
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tepid hound
\begin{equation*}
\begin{cases}
y' = xy^2 \
y(1) = -1
\end{cases}
\end{equation*}
vocal sleetBOT
\begin{equation*}
\begin{cases}
y' = xy^2 \\
y(1) = -1
\end{cases}
\end{equation...
twin meteorBOT
alee
tepid hound
how can i start to solve this ?
thin vale
identify what type of differential equation it is
tepid hound
identify what type of differential equation it is
it is a nonlinear differential equation due to the $xy^2$ term ?
twin meteorBOT
alee
thin vale
not exactly the classification I was looking for
tepid hound
can you help me ๐ฆ
desert hornet
Itโs a Bernoulli differential equation
tepid hound
$\frac{dy}{dx} = P(x) \cdot y + Q(x) \cdot y^n$
twin meteorBOT
alee
tepid hound
something like that ?
thin vale
Itโs a Bernoulli differential equation
it's seperable sir, is it not?
desert hornet
it's seperable sir, is it not?
Ah yeah it is
My bad
It still is a Bernoulli equation though, just with P(x)=0
Separating variables is definitely way better
tepid hound
how can i continue ? ๐ฆ
desert hornet
Okay
A separable DE is one where, as the name suggests, you can completely separate the two variables (x and y) on either side
For example, here you can get $$\frac{yโ}{y^2}=x$$
twin meteorBOT
kheerii
desert hornet
Now you can integrate both sides wrt x and (hopefully) be able to solve for a general solution of y
Donโt forget +C!
tepid hound
what is a DE ?
desert hornet
DE stands for differential equation
tepid hound
ah ok
\begin{equation}
\frac{dy}{dx} = xy^2
\end{equation}
\begin{equation}
\frac{1}{y^2} dy = x dx
\end{equation}
twin meteorBOT
alee
tepid hound
something like that ?
desert hornet
I don't understand what y(1) = -1 means
This is telling you that at x=1, y=-1
It will help you solve for the integration constant
tepid hound
\begin{align}
\int \frac{1}{y^2} dy &= \int x dx \
-\frac{1}{y} &= \frac{x^2}{2} + C
\end{align}
twin meteorBOT
alee
desert hornet
Yes
tepid hound
but do I have to solve for C? Then should this value be replaced only in x or also in y, or only in y?
desert hornet
You need to replace both values
desert hornet
Not quite
tepid hound
ah
desert hornet
C=1-1/2=1/2
tepid hound
ah yes
yes correct
so
\begin{equation}
-\frac{1}{y} = \frac{x^2}{2} + \frac{1}{2}
\end{equation}
twin meteorBOT
alee
tepid hound
i have this ?
\begin{equation}
\begin{aligned}
-\frac{1}{y} &= \frac{x^2}{2} + \frac{1}{2} \
\Rightarrow -1 &= -\frac{x^2y}{2} - \frac{y}{2} \
\Rightarrow -2 + y &= x^2y \
\Rightarrow y(1 - x^2) &= -2 \
\Rightarrow y &= \frac{-2}{1 - x^2}
\end{aligned}
\end{equation}
twin meteorBOT
alee
twin meteorBOT
alee
tepid hound
.close
vocal sleetBOT
vocal sleetBOT
Available help channel!
frail violet
vocal sleetBOT
frail violet
Right so I've done everything correctly but in Q7 (b) they have omitted the angles I have encircled in orange
Why??
They are in range
Hey give the correct value 1/2
Like I get the negative how they got the negative values but why did they not write these values circled in orange
vocal sleetBOT
@frail violet Has your question been resolved?
frail violet
<@&286206848099549185>
bronze osprey
Like I get the negative how they got the negative values but why did they not wr...
The marking guidance says "allow different values of q as long as all seven solutions are found"
Yeah your values are clearly correct
vocal sleetBOT
@frail violet Has your question been resolved?
vocal sleetBOT
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Available help channel!
vast shale
vocal sleetBOT
vast shale
how i can use the squeeze theorem in this
bronze osprey
how i can use the squeeze theorem in this
Substitute x = pi + u
bronze osprey
Yeah the idea is to use the limit of sin(x)/x
As x goes to 0
We already have x in the question so I used u
vast shale
Substitute x = pi + u
like -1/pi +u - x <= sin(x)/pi+u-x>=1/x-pi+u
?
๐คจ
oh
well
wait
bronze osprey
like -1/pi +u - x <= sin(x)/pi+u-x>=1/x-pi+u
Don't use the squeeze theorem
Like forget that idea
bronze osprey
sin(pi+u)/u
Can you recognise what sin(pi + u) is?
Shift the graph of sin(u) pi units to the left
Cause of the +pi in u + pi
vast shale
Shift the graph of sin(u) pi units to the left
what
vast shale
Shift the graph of sin(u) pi units to the left
can you explain it to me
bronze osprey
can you explain it to me
Have you not covered function transformations yet?
Okay watch this video: https://www.youtube.com/watch?v=Tmdrjs9xufc&pp=ygUYZnVuY3Rpb24gdHJhbnNmb3JtYXRpb25z
vast shale
Okay watch this video: https://www.youtube.com/watch?v=Tmdrjs9xufc&pp=ygUYZnVuY3...
i know what it means (i think), but idk how it is used in this case of limit
my mother language isnt english 
bronze osprey
i know what it means (i think), but idk how it is used in this case of limit
You clearly need to work on your trig
bronze osprey
i know what it means (i think), but idk how it is used in this case of limit
If you can't tell me that sin(pi + u) = -sin(u)
vocal sleetBOT
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Available help channel!
torn spoke
I thought we could do all kinds of operations with both sides of equations and it wouldn't affect the roots of the equation. However consider t=-2 and if we square both sides we will get t^2 = 4 but it means that t=+-2 thus we introduced an extra root. Can we not square both sides in equations? If no, then why do I remember so many equations being solved by squaring both sides?
vocal sleetBOT
I thought we could do all kinds of operations with both sides of equations and i...
graceful canopy
I thought we could do all kinds of operations with both sides of equations and i...
you can square both sides but it sometimes adds an extraneous solution
torn spoke
If it sometimes adds an extra solution then it is unacceptable. But many equations are solved by squaring, so I don't find your answer good
bronze osprey
So you need to check all your solutions by substituting them into the original equation
That's not too big of a deal
thin vale
If it sometimes adds an extra solution then it is unacceptable. But many equatio...
-3=3
Proof:
square both sides.
(-3)^2=(3)^2
9=9
Therefore, -3=3
bronze osprey
If it sometimes adds an extra solution then it is unacceptable. But many equatio...
But then you would be stuck if you didn't multiply or do something that would give you an extra solution
torn spoke
.close
vocal sleetBOT
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Available help channel!
vast shale
What do these double lines above the x and y mean
vocal sleetBOT
What do these double lines above the x and y mean
vast shale
What are they asking for
bronze osprey
What are they asking for
Probably the centre of mass, the x and y coordinates
vast shale
that makes sense
in the lecture notes he uses this
does the double line mean i use I x and I y instead
bronze osprey
does the double line mean i use I x and I y instead
I'm not sure cause that's not standard notation
vast shale
I'm not sure cause that's not standard notation
ooo i found it in my textbook, nm, thankyou
.close
vocal sleetBOT
bronze osprey
No worries
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Available help channel!
wind orchid
How do you answer the 13th question?
vocal sleetBOT
How do you answer the 13th question?
@wind orchid Has your question been resolved?
vocal sleetBOT
@wind orchid Has your question been resolved?
severe estuary
vocal sleetBOT
severe estuary
This has been fucking with my brain
heavy yoke
so by log rules ln(2x) = ln(x) + ln(2)
severe estuary
I asked friends in multivar they were confused too lol
heavy yoke
therefore they are off by some constant
vocal sleetBOT
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leaden crane
What's the best way of getting the value here?
vocal sleetBOT
What's the best way of getting the value here?
bitter pilot
What's the best way of getting the value here?
You basically got linear functions
sine the bounds are -8 and 8
you really need F(8) - F(-8)
leaden crane
(-3) - (0) then?
bitter pilot
You would have many antiderivatives defined for certain intervals
But you really only need these
severe estuary
couldnt you draw triangles
then find the areas of the triangles
since the integral from -8 to 8 is just the area
bitter pilot
You could but I think it's lees work to determine the two linear functions, integrate them and do F(8) - F(-8)
leaden crane
sooo we find the slope of those two lines?
bitter pilot
yea first
leaden crane
first one is 3/1 and second is -1/1
bitter pilot
yea
then we can use the fact that both functions have a root
So for the left we can take the slope 3 and shift our function 3x by 7 units to the right so you get:
y = 3(x-(-7)) = 3(x+7) = 3x + 21
Same approach for the right function
leaden crane
wdym by root
bitter pilot
x-intercept
leaden crane
huh
how did you get that as the x intercept?
i've only ever known the x intercept to be a point
or something
bitter pilot
x-intecepts have all the y value 0
y = 3(x+7) ever gets 0 if x = -7 here
so you need (x+7)
linear factor
leaden crane
i see
bitter pilot
Now you integrate
bitter pilot
yes
twin meteorBOT
Someone
bitter pilot
No need for C because you are plugging in 8
leaden crane
true
leaden crane
y = 3(x+7) = 3x + 21
twin meteorBOT
Someone
twin meteorBOT
๐ธdฯn๐ฒยฒs
leaden crane
yeah
bitter pilot
wait lemme check
I did the triangle thing but the area doesnt match something is wrong
I can't figure out the mistake yet but I would also consider the triangle method hehe
leaden crane
bitter pilot
it's not so difficult lol
leaden crane
how do i do it? lmao
sterile quiver
you have a rectangle
2 triangles
and semi circle
you should subtract the area of the circle since it's below the x axis
bitter pilot
just saying
you were right!
severe estuary
would the triangle thing also work here?
Separate each shape, remember circle is gonna be divided by 2 cuz itโs half of it
leaden crane
uhhhh okay i'll give it a try
these types of questions will be on the test, would it not just be easier to not memorize all the areas and just do it the other way?
bitter pilot
these types of questions will be on the test, would it not just be easier to not...
which way?
btw i realized the mistake
my approach was wrong in the beginning, we are to define several functions integrals then we have to do it fully
leaden crane
like the math way, not the geometric way
cus knowing me, there's no way i memorize these areas that quickly lol
sterile quiver
brother
severe estuary
these types of questions will be on the test, would it not just be easier to not...
im gonna geometry on my calc bc exam so if it doesnt work we go down together
sterile quiver
these are basic shapes
sterile quiver
you're gonna have to do trig sub to find the area of the semi circle
twin meteorBOT
๐ธdฯn๐ฒยฒs
sterile quiver
and you'll have to memorize cos(x)^2 = (1+sin(2x))/2
leaden crane
oh lord
severe estuary
and you'll have to memorize cos(x)^2 = (1+sin(2x))/2
wait why
bitter pilot
yea the geometric way is way better
bitter pilot
i dont know why i did this silly mistake lol
leaden crane
calc 1
severe estuary
i dont know why i did this silly mistake lol
banned from the server
severe estuary
because you end up with the integral of cos^2x
he can just do pi * 3^2
/2 cuz semicircle
sterile quiver
yes
bitter pilot
cus knowing me, there's no way i memorize these areas that quickly lol
believe it or not but that's way better
they are intuitive
sterile quiver
im just showing how absurd trying to solve it that way is
severe estuary
ohhhh
leaden crane
so for this problem, how would i apply the area triangle stuff?
bitter pilot
You would have 4 triangle areas
leaden crane
yeah
bitter pilot
severe estuary
what he said
leaden crane
okay so
severe estuary
leaden crane
the first one triangle (yellow) is
(1/2)(1)(3)?
severe estuary
yea
bitter pilot
the first one triangle (yellow) is
(1/2)(1)(3)?
minus 3
it's a negative area basically
leaden crane
area can be negative?
bitter pilot
since it's under
severe estuary
dont think about it too hard like geometry
separate it from the graph just find the areas as if you were given the length and height of it
ykwis
bitter pilot
I mean technically, we are not looking for the area really, but for the integral value, otherwise we would have to use absolute values
leaden crane
so on the x-axis do i look left to right making -8 to -7 +1?
bitter pilot
Like sometimes the numerical value of an integral can also be negative, now why is that
leaden crane
so on the x-axis do i look left to right making -8 to -7 +1?
or all x axis (base) should be +?
bitter pilot
i think the best approach is to treat them first all as "normal" triangles
and then afterwards denote the areas with - or + depending if the triangle is below or over x-axis
bitter pilot
yea
leaden crane
3/2 and since it's below x axis, it's -3/2
SO BOOM first triangle done
next one
leaden crane
(1/2)(5)(6) = 15
bitter pilot
yea
leaden crane
(1/2)(4)(4) = 8, but below x axis, so -8
bitter pilot
yo
leaden crane
andddd
(1/2)(6)(4) = 12
i'm actually crazy with it
okay now what do i do ๐
just add em?
bitter pilot
yea
leaden crane
(-3/2) + (15) + (-8) + (12) = 35/2
leaden crane
im actually insane
bitter pilot
Here the non-geometric way
leaden crane
2nd graders got nothing on me
leaden crane
Here the non-geometric way
yeah i'll do it the second grader way
bitter pilot
XD
leaden crane
for square, can i just do rectangle?
i cba to memorize square
even though it's x^2
it should work
leaden crane
rectangle equation is like 11 * 11 which is just 11^2, so it should work
bitter pilot
where width x height also works but since all lengths are equal you can do either width x width or height x height
bitter pilot
rectangle equation is like 11 * 11 which is just 11^2, so it should work
no
only if it's a square
a rectangle can also have a different height compared to it's width
leaden crane
nono i'm talking about using the rectangle equation for square
leaden crane
let's goooo
i'm not gonna memorize eclipse or trapezoid, there's just no shot they put those on the test
bitter pilot
because technically a square is also a rectangle but not all rectangles are squares
trapezoid is basically a combination of triangle rectangle most of the times
for eclipse, if that comes in your 2nd grade test then you'd be unlucky to not now its area
leaden crane
xD
bitter pilot
but eclipse is just A = ฯ x r1 x r2
severe estuary
the integral and derivative cancel out
leaden crane
wot
severe estuary
ok
leaden crane
so just sinxcosx
severe estuary
yea
bitter pilot
basically this but be careful
leaden crane
lmao
severe estuary
imagine derivative brings you foward one prime and integrating brings you backwards one prime
so they cancel out they r inverse
twin meteorBOT
๐ธdฯn๐ฒยฒs
bitter pilot
But you still need to be careful because what if the bounds were from 2 to sqrt(x)
You would need chain rule
severe estuary
chain rulleeee
leaden crane
but this one can't be sinxcosx
leaden crane
wotttt
leaden crane
what is FTC ๐ญ
severe estuary
fundamental theorem of calculus it says that the derivative and integral are inverse basiically
twin meteorBOT
๐ธdฯn๐ฒยฒs
severe estuary
i dont like the textbook FTC definition its confusing
leaden crane
thanks for that martial
bitter pilot
wth is happening lol
leaden crane
troll
probably
i'm still a bit confused, maybe i need to do it without the shortcut of cancelling out the integral and derivative
unless that's not a shortcut and just how you do it
bitter pilot
unless you also want points taken off
leaden crane
this is test practice, so no grading 
bitter pilot
leaden crane
@supple dawn go troll someone else pls n ty
bitter pilot
but this one can't be sinxcosx
Try to practice this using FTC
<@&268886789983436800>
leaden crane
surprisingly the first time i've seen a troll here
bitter pilot
mods do a great job here that's why
leaden crane
that's true
bitter pilot
leaden crane
i've done like 3 modmail, always get an instant response
okayyyyyy so
can you walk me through this one using FTC?
just so i can get a grasp, then try on the rest of them?
bitter pilot
can you walk me through this one using FTC?
I would suggest the other one with chain rule
because (a) is trivial
severe estuary
bitter pilot
**๐ธdฯn๐ฒยฒs**
.
bitter pilot
to the bounds
bitter pilot
here take a look at bounds a and b
leaden crane
right
bitter pilot
and differentiation give you the integrand basically but in terms of x now
F(2) is just a constant that results to 0
bitter pilot
but this one can't be sinxcosx
This is example is much more interesting
leaden crane
okay wait wait
leaden crane
i'm still so lost on just how to do any of it
severe estuary
i dont like the textbook FTC definition its confusing how i think of it is they are inverses so they cancel each other out
leaden crane
my tiny brain can't grasp what F(2) is even equal to
severe estuary
i dont like the textbook FTC definition its confusing how i think of it is they ...
and you dont have to worry about the constant on the bottom of the integral bc of +c magic
bitter pilot
it's not really inverses if the bounds have you use chain rule
severe estuary
right @bitter pilot
bitter pilot
only works for such example
i think the best way is to just start
and then we analyze what you did right or wrong
because too much theory and no application
yields almost nothing
also
FTC may seem confusing
but you been using this all the time when you calculated integrals/areas
bitter pilot
you are basically plugging in the bounds into the antiderivative which we want to give a different notation from the integrand function
usually it's big cap letters
leaden crane
i'm actually completely lost