💀 my handwriting is unintelligible i guess
#help-17
hushed pewter
And what about this question?
hushed pewter
So is that 1²*x²?
hushed pewter
What does this say?
vocal sleetBOT
@golden imp Has your question been resolved?
hushed pewter
i hope its intelligible now
Oh so you're saying the limit is equal to y, and are trying to find what y is?
@golden imp so what was the purpose of this approach?
What's concerning you exactly?
Oh
Is that they're equal regardless of the value of y?
golden imp
it should be 1
not -1
but turns out that i can't just send the x^2 to the other side as a multiplier
hushed pewter
it should be 1
You haven't solved for y
You've simply found that both sides are equal to -1
whole wing
How does the equation implies that
vocal sleetBOT
How does the equation implies that
worthy citrus
you have no x^2 term on the right hand side, so the coefficient of the x^2 term on the left hand side must be zero
whole wing
you have no x^2 term on the right hand side, so the coefficient of the x^2 term ...
Oh, I see now, I got (a+d)(cx^2 + (d-a)x - b) = 0, but the equality above always hold true, for no matter what x, hence the (a+d) must be equal to 0 as it does not depend on x, is that it?
ty
.close
vocal sleetBOT
vocal sleetBOT
Available help channel!
scarlet oak
does this truth table look right ?
vocal sleetBOT
does this truth table look right ?
forest elk
3x9
twin meteorBOT
Cain
civic pendant
also you need to have 2^3 rows to cover all possible values of p q and r
scarlet oak
also you need to have 2^3 rows to cover all possible values of p q and r
i need 8 rows?
civic pendant
yes
scarlet oak
can you show me ur truth table version of this if u dont mind ?
civic pendant
I do mind
scarlet oak
thanks
appreciate it
do u have any youtubers that u recommend to learn discrete math from ?
tulip canopy
vocal sleetBOT
tulip canopy
can't use l'hopitals for this question so idk how to solve it otherwise
flat whale
Multiply top and bottom by the denominators conjugate
tulip canopy
ye and then i get something like this
then the bottom will simplify to sqrt(35x^2+3x-2) - 1
and idk where to go from there
lol
gaunt sparrow
Denominator's conjugate, not numerator.
vocal sleetBOT
vocal sleetBOT
Available help channel!
sage ocean
Does anyone understand mohr's circle and how to draw it?
vocal sleetBOT
Does anyone understand mohr's circle and how to draw it?
@sage ocean Has your question been resolved?
tawny nacelle
vocal sleetBOT
tawny nacelle
!status
vocal sleetBOT
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
for part a), 2
havent done the other parts yet
so for part a, i let v be an eigenvector of S o T corresponding to a non-zero eigenvalue λ. then S(T(v)) = λv by definition. now, we apply T to both sides to get T(S(T(v)) = T(λv) = λT(v)
so we know that T(v) is an eigenvector of T o S corresponding to λ as long as it is not equal to 0
issue is that i've been unable to prove that T(v) is not zero
i also dont know how to show that the dimensions of the two eigenspaces are equal
sly sierra
if Tv were zero, then STv would also be zero, so v would not be a valid eigenvector of ST
sly sierra
offhand i'm not sure about the eigenspace dimensionality part, i'll think about it a bit and let you know if i have a hint
@ing ah i see why it's true
@tawny nacelle let't try to ping correctly
suppose v1 through vn are linearly independent eigenvectors of ST associated with lambda
suppose Tv1 through Tvn are not linearly independent
then there is a nontrivial linear combination
sum (ai Tvi) = 0
now consider
S(sum (ai Tvi))
tawny nacelle
suppose v1 through vn are linearly independent eigenvectors of ST associated wit...
do they span E(λ, ST)?
or just lin ind?
sly sierra
well you can just suppose they're any LI set of eigenvectors of ST (for lambda)
that'll be true of a basis of E(lambda, ST) of course
the argument above will show that Tv1 through Tvn are also LI
so that would give you
dim E(lambda, ST) <= dim E(lambda, TS)
so you just need the opposite inequality
which i assume you can obtain through a similar argument, lemme think about it briefly
sly sierra
ah yea, the reverse is proved the same way
tawny nacelle
so that would give you
dim E(lambda, ST) <= dim E(lambda, TS)
why can we conclude this?
v1, ..., vn are elements of E(λ, ST) and T(v1), ..., T(vn) are elements of E(λ, TS)
sly sierra
why can we conclude this?
the above argument says that if there are n LI eigenvectors of ST (for lambda) then there are at least n LI eigenvectors of TS (for lambda)
tawny nacelle
but we have that both sets are LI
sly sierra
so in particular, let n = the dimension of E(lambda, ST)
tawny nacelle
the above argument says that if there are n LI eigenvectors of ST (for lambda) t...
hmm, lemme process this
ah i see
its kinda obvious now that i think about it
sly sierra
you can go in the reverse direction too, basically the same proof
show that if w is an eigenvector of TS then Sw is an eigenvector of ST (for the same lambda)
basically interchanging the roles of S and T
tawny nacelle
i see
sly sierra
and the reverse direction leads to the opposite inequality
thus the two spaces have the same dimension
vocal sleetBOT
@tawny nacelle Has your question been resolved?
tawny nacelle
thus the two spaces have the same dimension
i got it!
writing out the details took me a while
sly sierra
nice!
tawny nacelle
i'm quite verbose with my proofs lol
sly sierra
nothing wrong with that
tawny nacelle
okay, let me try b
tawny nacelle
Let 0 be an eigenvalue of S o T. Let x ∈ E(0, S o T). Then x =/= 0, since it's an eigenvector, and (S o T)(x) = 0x = 0. Then we see that S o T is not injective (because ker(T o S) =/= {0}), thus not bijective (since the domain/codomain are the same hence have the same dimension), thus not invertible.
I think this means that at least one of S or T is not invertible?
is that true?
tawny nacelle
I *think* this means that at least one of S or T is not invertible?
also, if we have two functions, and one is not invertible, then is any composition of those two functions not invertible as well?
@sly sierra sorry for pinging you
vocal sleetBOT
@tawny nacelle Has your question been resolved?
tawny nacelle
<@&286206848099549185>
tawny nacelle
Let 0 be an eigenvalue of S o T. Let x ∈ E(0, S o T). Then x =/= 0, since it's a...
for context, i'm doing part b (in this image), and this is what i have so far
i need my two questions asked above answered
cinder oyster
vocal sleetBOT
teal vapor
8^-3x^3
but idk why u wrote as 8^-3x^3
well the final result would be (8x)^3/(7y)^3
@mild flower why
mild flower
the -3 gets applied to the 8 and 7 as well
vocal sleetBOT
@cinder oyster Has your question been resolved?
vocal sleetBOT
vocal sleetBOT
Available help channel!
silk wind
The line L : (6, 4, 0) + t(1, 2, −1), t ∈ R, intersects the plane x + y + 2z = 5 at the point
P. Enter the line L1 that passes through P, lies in the plane and is perpendicular to L.
vocal sleetBOT
The line L : (6, 4, 0) + t(1, 2, −1), t ∈ R, intersects the plane x + y + 2z = 5...
silk wind
I have found p = (1,-6,5) how do I find the line L1 now?
vocal sleetBOT
@silk wind Has your question been resolved?
silk wind
<@&286206848099549185>
vocal sleetBOT
@silk wind Has your question been resolved?
cyan crest
what does "lies in the plane" mean? Does it mean that L1 intersects the plane at infinitely many points?
silk wind
I am not entirely sure it just says lies in the plane
But we don’t have to focus on that for now
I can figure out what that means later
silk wind
Okay!
cyan crest
let's start with $$(1,-6,5) + vt$$ where $v$ is a direction vector and $t \in \mathbb{R}$
twin meteorBOT
_Kookie
silk wind
Yes okay
cyan crest
this line represents L1
and basically we guarantee that this line will pass through P
which is the point you will find
since when t = 0 we get P
Generally speaking
line L travels with direction vector (1,2,-1)
silk wind
Yes!
cyan crest
so if we want to make L1 perpendicular to L
then we need to make sure that $$v \cdot (1,2,-1) = 0$$
twin meteorBOT
_Kookie
silk wind
Yes!
cyan crest
wonderful
now you can write v as (a,b,c) where each entry is in the reals
and now you have to solve this equation
not exactly the easiest thing given you have three variables (a,b,c) to solve for
cyan crest
however, you can get another equation from the question
it says that L1 must lie in the plane
Have you seen the normal equation for a plane?
silk wind
Yes n = (1,1,2)
cyan crest
beautiful
now given that this line L1 lies in the plane
then I believe it will intersect with this plane infinitely many times
that means the direction vector v will be perpendicular to the normal n
can you see why?
silk wind
I think so
cyan crest
and then use one finger in the other hand to indicate the line running directly through the plane
silk wind
Yes!
cyan crest
notice how the direction your palm is facing is perpendicular to the finger in your other hand
silk wind
Yes!
cyan crest
that's how we can be sure that the dot product $v \cdot n = 0$
twin meteorBOT
_Kookie
silk wind
Okay great!
cyan crest
so now we have two equations
this is good
you can now solve for a,b and c
keep in mind though that you will have at least one free variable, but this will be no problem
cyan crest
yep very good
cyan crest
okay good
now you can see that c is completely free
you can set it to whatever you want
I don't recommend 0 though
silk wind
Yes okay so 1?
cyan crest
yep sure
silk wind
Okay so c = 1, b = 3 and a = -5?
silk wind
Okay so v =(-5,3,1)
cyan crest
yep
silk wind
Okay so now we take (1,-6,5) +t(-5,3,1)?
cyan crest
sure
silk wind
Okay so is that the line?
cyan crest
wdym
silk wind
I mean L1
cyan crest
yep
silk wind
Okay and now we can check that it is perpendicular to L?
silk wind
Yes I think so wait I will try
I know how to do it with two vectors but not when t is involved
silk wind
Oh okay!
cyan crest
(-5, 3, 1) • (1,1,2) = 0 as well
unfortunately no since I gotta go to bed
sorry about that
vocal sleetBOT
vocal sleetBOT
Available help channel!
vast shale
If A is invertible, then Ax = b has a unique solution for every b?
vocal sleetBOT
If A is invertible, then Ax = b has a unique solution for every b?
@vast shale Has your question been resolved?
surreal basin
If A is invertible, then Ax = b has a unique solution for every b?
yes, because you can then invert A to get that unique solution from b which is x = A^-1 b
cobalt sparrow
vocal sleetBOT
cobalt sparrow
how do i know which i slarger
vocal sleetBOT
how do i know which i slarger
cobalt sparrow
larger
inner osprey
how do i know which i slarger
just by looking at the two integrals, how are they different?
cobalt sparrow
just by looking at the two integrals, how are they different?
they odnt look different to me
cobalt ocean
try looking at the graphs or something
agile meteor
i’d either use an identity to turn sine into cosine or vice versa
or turn both sine n cosine to exp form
inner osprey
😅
inner osprey
no…
silent arrow
The answer is on the left solved in rad
inner osprey
not good
agile meteor
looking at the graph isn’t very rigorous tho isn’t it?
inner osprey
if you can avoid, it is not good practice to suggest calculator solution…
inner osprey
looking at the graph isn’t very rigorous tho isn’t it?
it says “Without evaluating the integrals” and “Justify your answer”
i see no indication that this problem requires a high level of rigor
this seems like you want to make use of the functions that the integrals are composed of in your justification
agile meteor
tbh my whole experience during calc 2 was me being decent at kinda complex problems and struggling w easy/obvious stuff so 💀
im kinda the king of over complicating shit 😭
inner osprey
😔 so real sometimes
vocal sleetBOT
@cobalt sparrow Has your question been resolved?
vast shale
vocal sleetBOT
vocal sleetBOT
vocal sleetBOT
Available help channel!
vast shale
How do you work this question out
vocal sleetBOT
How do you work this question out
vast shale
hushed pewter
Does that say $5\left(\frac{n}{4}-2\right)=15$?
twin meteorBOT
SWR
vast shale
Yes
hushed pewter
I presume you are trying to solve for n?
hushed pewter
The task is to start at the equation given to you, and algebraically manipulate into an equivalent expression where n is isolated.
It's worth discussing what I mean by equivalent expression. That's the root of what makes algebra useful.
vast shale
Equivalent expression means the same expression just rearranged ?
hushed pewter
Two equations are equivalent if one equation is true if and only if the other equation is true
hushed pewter
Equivalent expression means the same expression just rearranged ?
Exactly correct
There are a number of operations that you can do to an equation to keep the result equivalent:
- Add/subtract any number to both sides
- Multiply/Divide by any non-zero number to both sides
- Add 0 to either side
- Multiply by 1 to either side
vast shale
Two equations are **equivalent** if one equation is true if and only if the othe...
Ok so simplified it’s basically like saying you have y-x/4 = 23+x-9 (made up) and they’re only equivalent if they’re both true or in other words make sense and are correct
hushed pewter
Not quite. y-x/4=23+x-9 is one equation
I'm talking about when two equations are equivalent
For example, if x+2=3, then that equation is equivalent to x=1
vast shale
hushed pewter
There are a number of operations that you can do to an equation to keep the resu...
Hey very good so far
Image is cut off, but you seem to have it
vast shale
So then you just do 5n/4 = 25 then 5n = 25 x 4 , 5n = 100, n = 100 /5 n= 20
@hushed pewter Do you mind helping with some other algebra questions?
hushed pewter
So then you just do 5n/4 = 25 then 5n = 25 x 4 , 5n = 100, n = 100 /5 n= 20
perfect
hushed pewter
<@374795910032523266> Do you mind helping with some other algebra questions?
If I have the time, sure
vast shale
I don’t understand how to get z when it is in the denominator
hushed pewter
$\frac{1}{4z}+\frac{1}{5z}=-9$?
vast shale
1/4z + 1/5z = -9
twin meteorBOT
SWR
vast shale
Yes
hushed pewter
Yea this one is a bit trickier.
vast shale
Do you make them have the same denominator
hushed pewter
Here's a hint though, recall that $\frac{1}{a\times b}=\frac{1}{a}\times\frac{1}{b}$
twin meteorBOT
SWR
rare ginkgo
resend the ques pls
rare ginkgo
resend the ques pls
.
rare ginkgo
what do we have to find
vast shale
are you stuck here?
Not really i’m just not sure of the answer/working out
vast shale
what do we have to find
n
spiral turtle
is that $5\qty(\frac{n}{4} - 2) = 15$
twin meteorBOT
全能の存在
vast shale
Yes
rare ginkgo
n=20?
spiral turtle
Not really i’m just not sure of the answer/working out
you haven't made any errors up to this point
rare ginkgo
ok
vast shale
Ok so the answer i got is -1/20 and then i flip it because it has a minus in front
rare ginkgo
n/4-2= 15/3
vast shale
Oh sorry
rare ginkgo
n/4-2= 3
vast shale
I sent the wrong question
rare ginkgo
n/4= 5
vast shale
I meant this one
rare ginkgo
n= 5*4
spiral turtle
Ok so the answer i got is -1/20 and then i flip it because it has a minus in fro...
So -1/20 is correct for the first question.
spiral turtle
and n = 20 is indeed correct for the second one
vast shale
Oh ok they’re both 20 then
vast shale
-1/20 isn’t 20? i thought you flip it
spiral turtle
you're thinking of (1/20)^-1
vocal sleetBOT
n=20
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
spiral turtle
-1/20 is negative 1/20
vast shale
So then you multiply the fraction by -1?
spiral turtle
you have z = -1/20
vast shale
Oh my bad
spiral turtle
if you multiply the fraction by -1, you would also need to multiple the left hand side by -1
then you'd get -z = 1/20
which is no longer solved.
the question 🙂
vast shale
Okay
spiral turtle
so, if you get rid of all of the fractions then you just have a bunch of numbers and xs, which we know how to solve
what can you multiply both sides of this equation by to remove all of the fractions?
vast shale
8 because it’s the LCM
noble spindle
Can you multiply with something better / and thats not lcm
spiral turtle
8 will work, but the LCM you care about is between 4 and 2, not 4, 2, and 8
vast shale
Ok so 4
spiral turtle
👍
vast shale
4 x 1=4 so it’s a multiple, 4 is the LCM
spiral turtle
LCM(a, b) = ab/GCD(a, b). GCD(4,2) = 2, so LCM(a, b) = 4 * 2 / 2 = 4
vast shale
I got x= -8/5
vast shale
LCM(a, b) = ab/GCD(a, b). GCD(4,2) = 2, so LCM(a, b) = 4 * 2 / 2 = 4
I don’t understand that since I haven’t learnt that
vast shale
-6*
spiral turtle
not -6 either, it's just 6
spiral turtle
I agree with that
might want to rewrite the final line just for clarity
if I were grading your work, I would personally appreciate it.
vast shale
x = 6/5
Oh it’s fine bc i just read it out to my tutor
but thank you for the suggestion
spiral turtle
oh, then that's fine 🙂
vast shale
Could anyone help with this question please
tall rose
You can check your answer by plugging it into the original equation
vast shale
Okay thanks, I was more so asking if the working out was correct
spiral turtle
You messed up on the third line going from the left to the right
can you find your mistake?
vast shale
can you find your mistake?
Not simplifying the fraction?
spiral turtle
it is simplifying the fraction, that's the only thing you did going from the left to the right on the third line 🙂
what mistake did you make?
you did two things
you added 3 and 17 to get 20
and you added -5y and y to get 4y
which one of these is incorrect?
vast shale
-4y it should be
spiral turtle
👍
vast shale
But about the simplifying fraction is it necessary?
spiral turtle
you can simplify the fraction
vast shale
I think it just makes it easier/less steps but still get the same answer
spiral turtle
and it makes it easier
but yes, you'll get the same answer in the end.
So now once you finish solving this, you will have y=3
how do you know this is the right answer?
spiral turtle
ok! so do that and double check it
vast shale
ok! so do that and double check it
Do you do that by just putting 3 into one side of it
spiral turtle
you replace all of the ys with 3
vast shale
I did that but i didn’t get 3 when I did it with both sides of the equation
spiral turtle
you don't need to get 3 = 3
you just need to get "x = x" for some value of x
you just need some sort of true statement
vast shale
Could you give an example?
spiral turtle
ok so let's do an easy one really quick and then check it
5x + 4 = 24
we can solve this quickly and find that x = 4
vast shale
Yes
spiral turtle
to check, we just replace x with 4
5x + 4 becomes 5(4) + 4
so we have 5(4) + 4 = 24
20 + 4 = 24
vast shale
✅
spiral turtle
24 = 24
spiral turtle
it doesn't have to be 3 = 3, it can be any number is equal to itself.
in this case you should find that -3 = -3
(3 - 5y)/4 = 2 - (y + 17)/4
vast shale
okay i will plug 3 in again
spiral turtle
(3 - 5(3))/4 = 2 - (3 + 17)/4
-12/4 = 2 - 20/4
-3 = 2 - 5
-3 = -3
spiral turtle
the same reason I got 24 = 24 when the answer was not 24
you don't need any particular number to equal itself, you just need any true statement
rare ginkgo
what is the ques?
vast shale
you don't need any particular number to equal itself, you just need any true sta...
Okay, thank you for your help
vast shale
Yes sorry
vast shale
Could anyone help with this question
vast shale
Not an answer but explanation please
spiral turtle
At this point xvn, you should know how to approach problems like this as we just walked you through 2 or 3 of them at least, so can you explain how you would approach this?
rare ginkgo
exactly
vast shale
It’s because this one looks different so i sent it
rare ginkgo
these are all very simple ques
spiral turtle
I wouldn't put it like that
vast shale
Well i’m not an undergraduate lol am I now
vast shale
ayo me too
Your role
I guess we are all undergraduates bc we haven’t graduated yet if we think about it lol
rare ginkgo
i am an indian...the math they teach us here is way too advanced
spiral turtle
So you think that this one looks different, can you explain how?
vast shale
Because there’s 3 fractions
spiral turtle
ok, so you need to find the least common multiple of three values
rare ginkgo
Because there’s 3 fractions
you can always combine the LHS and then solve
take lcm of the LHS
and then cross multiply
vast shale
LHS means what?
spiral turtle
between 4, 8, and 3, what is the smallest number that is a multiple of all of these?
LHS means left hand side
RHS means right hand side
vast shale
24
rare ginkgo
sorry for using abbrevations
spiral turtle
ok, so you need to make all three fractions have a denominator of 24.
rare ginkgo
ok, so you need to make all three fractions have a denominator of 24.
you are a great teacher dude
spiral turtle
(then you can just multiply both sides by 24 to clear all of the fractions entirely)
rare ginkgo
i mean i can solve the ques but explaining it is very difficult for me
vast shale
So when multiplying by 24 the numerators will change
spiral turtle
Thank you for the compliment. I've had a bit of practice, and I think given some practice you can learn how to teach effectively too.
vast shale
Do you just do a big bracket around all the fractions on the LHS, and the RHS a bracket there too?
spiral turtle
So when multiplying by 24 the numerators will change
don't multiply by 24 at first.
vast shale
I changed the denominators first
spiral turtle
what you do is multiply each fraction by a number over itself.
so for instance, for the term with something/4
you multiply by 6/6
because 6 * 4 = 24
or I guess you can just multiply everything by 24
yeah that works too
rare ginkgo
this is correct
spiral turtle
yup
vast shale
or I guess you can just multiply everything by 24
Oh but wouldn’t that not work without changing them first bc it won’t get a common denominator
spiral turtle
it would work, you'd just have to be careful about how the denominators and the numerators cancel
vast shale
Multiplying by 24 without changing denominators can work? just trying to double check
spiral turtle
yeah
remember, as long as you do legal things with the equation (always multiply individual terms by 1 or always add 0 to either side, and if you do something that isn't multiplying by 1 or adding 0 to one side, then you have to do it to the other side too), then you'll always get a true statement from a true statement.
vast shale
That makes sense because what is on the LHS is equal to RHS anyway it’s just about solving the value of the letter
do you simplify the fraction on RHS or must they have the same denominator
spiral turtle
you eventually want to clear the fractions
so if you have everything over 24
you can just multiply both LHS and RHS by 24
to get rid of them
vast shale
Okay
vast shale
That makes sense because what is on the LHS is equal to RHS anyway it’s just abo...
is this correct in response to what you said about true statements
drifting pollen
no matter how many times i do this, i get -11/30 pi, and its marked wrong. anyone know where i might be going wrong?
vocal sleetBOT
no matter how many times i do this, i get -11/30 pi, and its marked wrong. anyon...
tulip dirge
no matter how many times i do this, i get -11/30 pi, and its marked wrong. anyon...
can you show what you have done?
drifting pollen
It’s never negative
integration can give negative areas and volumes even if they dont have real-world sensibility no? think of the definite integral on [0, 1] of y = -x, that is surely a negative area, and so volume can be negative too.
also, someone already told me the answer, and while i dont actually remember what it is, it was negative. and this grading is done automatically, so i am sure it is negative
solid sage
The formula for this is based of the formula for a circle which can’t be negative.
It’s the sum of multiples pi r^2 where r changes
Although maybe I’m wrong, could you show your working?
drifting pollen
yes
excuse my handwriting lol
sent one twice woops
two different workings of problem btw
loud walrus
The integral is wrong
tulip dirge
The integral is wrong
yeah shouldnt it be the other way?
tulip dirge
oh
drifting pollen
oh
tulip dirge
i was going to say x2-1 - 1-sqrtx
loud walrus
The inside the parenthesis
Make the other way
And you will get the positive same result now
loud walrus
You saying 3^2 - 2^2 is equal to 2^2 - 3^2?
loud walrus
Ok let me check the rest
drifting pollen
is it backwards because the axis is y = 1, so you need to take the bottom curve first instead of the top one (because the axis is above the two curves)
tulip dirge
And it’s x^4
no its right, there is a negative sign outside the parenthesis
loud walrus
I don’t see the parenthesis
loud walrus
x^2 first
tulip dirge
yeah
loud walrus
Sqrt later
tulip dirge
yup that would give 11pi/30
tulip dirge
i was going to say x2-1 - 1-sqrtx
nah mb, i thought i was wrong for saying this
and then i started looking for mistakes in the integral
loud walrus
No that was the whole thing
tulip dirge
yea i get it now
loud walrus
It’s R^2-r^2
drifting pollen
yeah thank you
this was helpful and 11/30 * pi is the correct answer apparently, thx for the help
.close
vocal sleetBOT
vocal sleetBOT
Available help channel!
crystal glacier
got a quick question:
tan120 => tan(180-60) => tan60 or (-tan60) ?
noble spindle
Its tan(-60) = -tan(60)
crystal glacier
Its tan(-60) = -tan(60)
gotcha thanks
can anyone explain where i went wrong here?
-cos(360 + 30) - tan (360 - 60) + tan (180 - 60) => -cos(30) + tan(60) + tan(60) = tan(45)
- sqrt(3) / 2 + sqrt(3) + sqrt (3) + 1
vocal sleetBOT
@crystal glacier Has your question been resolved?
vocal sleetBOT
@crystal glacier Has your question been resolved?
crystal glacier
why do you write "=>"
just showing my thought process
i dont have the final answer but
thats why im asking before i continue on and develop some bad habit
tall rose
you have the bad habit of writing "=>" instead of "="
you can only use "=>" ("implies") for statements
numbers are not statements
but a=b is a statement
fair prawn
vocal sleetBOT
fair prawn
Is this right?
vocal sleetBOT
Show your work, and if possible, explain where you are stuck.
lone linden
Is it 8 then
yeah
vast shale
vocal sleetBOT
vast shale
i thought using FTC1 would just give tan(u^3)du
so then why and how is this the answer
hushed pewter
- This is a definite integral
- Chain rule
vast shale
how does FTC 1 apply to definite and indefinite integrals differently
hushed pewter
The basic setup of the theorem is that $\frac{d}{dx}\left[\int_a^x f(u)du\right]=f(x)$ where $a$ is any arbitrary constant.
twin meteorBOT
SWR
vast shale
whats an arbitrary constant
vocal sleetBOT
vocal sleetBOT
Available help channel!
autumn coral
vocal sleetBOT
formal pond
What method did you use in class?
autumn coral
welll
worth a shot
at least
id assume there arent THAT many methods
i assume its
characteristic equation method
autumn coral
my friend was doing something like
r^2-2=0
then facotized it
n stuff
factorized*
thin vale
Find the roots of the characterstic equation
They tell you the complementary solution
use the initial value to solve for the undetermined constants
since these are all homogeneous equations there will be no particular solutions
formal pond
r^2-2=0
Yep we get something like this
But we r using lambda
Instead of r
Do you know how we get this
Just do what I said above
autumn coral
Nah
formal pond
Do what I said
autumn coral
so y=e^lambdaX
formal pond
yes
autumn coral
then differntiate twice
autumn coral
e^2x
autumn coral
e^2x * 2
autumn coral
2 right
thin vale
yes
thin vale
lambda is a constant
formal pond
Would it?
thin vale
?
thin vale
it isn't + lambda
autumn coral
oh
thin vale
it's multiplied by lambda
autumn coral
right
thin vale
By the chain rule
thin vale
$e^{\lambda x} \cdot \lambda$
twin meteorBOT
Austin
formal pond
Take your time
I haven’t solved IVPs for ages so I don’t know any of the names for these steps
I just remember how to do them
thin vale
normally you don't actually do this whole e^lambda shennanigans
formal pond
Yea
thin vale
but since he seems to not know the method
formal pond
You just go straight into it
thin vale
Okay we're explaining to you how to do it, and almost more importantly why it works
you'll realize once we get through a few steps this is the same method you are somewhat familiar with
thin vale
$e^{\lambda x} \cdot \lambda^{2}$
autumn coral
Lamda squared e^lamda*x
twin meteorBOT
Austin
thin vale
yes
autumn coral
ye
thin vale
Now recall what we were doing
we had a trial solution
y=e^(lambda x)
now we know what y' and y'' would be
Let's plug all of them in
to the equation
thin vale
we haven't yet shown that our trial solution works
autumn coral
oh
thin vale
but we will see that it does
thin vale
and then we will also find lambda
autumn coral
ok we know y and y''
so we plug them in the equation i have?
ok so
(e^lamdax * lamda^2) - 2 ( e^2x)
=0
formal pond
y = e^lambda *x?
thin vale
why would 2*y square it?
formal pond
i just gave e^2x as an example of how to differentiare it
formal pond
you
thin vale
you tell us
autumn coral
ok whats y
thin vale
No way
autumn coral
Bro u guys confuse the hell out of me
thin vale
I feel like you're not taking this seriously
thin vale
What was our trial solution
autumn coral
the first derivative i got?
thin vale
What
autumn coral
what r u talking abt man
formal pond
Look at the first equation i gave
autumn coral
e^lamdax
formal pond
Try a trial solution
.
thin vale
,w trial
twin meteorBOT
autumn coral
ok so when i got the derivative of it
and then 2nd derivative
thats our y''
then why is the very first one not y
thin vale
it is..
thin vale
no you said e^2x
autumn coral
typo
thin vale
also your refusal to use parenthesis is annoying
autumn coral
..
autumn coral
(e^lamdax * lamda^2) - 2 ( e^lamdax)
formal pond
equals what? 5 elephants?
thin vale
it equals 100 limes and 2 oranges
thin vale
no need to apply log
they already share a common factor
also log (0) is undefined
so you can't apply log
which you would see if you wrote it out completely
formal pond
in your 2 brackets, what occurs in both
thin vale
:|
formal pond
?
thin vale
lambda does not appear as a factor in the second one
formal pond
2 is not even in the bracket
thin vale
You should probably write it out fully
autumn coral
u can expand it...
thin vale
like we've said
it still doesn't have a factor of lambda
maybe if you used parenthesis you'd realize it is e^(lambda * x)
autumn coral
ok we factor the e out
thin vale
Not the e
autumn coral
then what bruh
thin vale
idk how would you factor just the e out
autumn coral
legit said everything in the equation
thin vale
without also factoring out its exponents
autumn coral
ok thats what i meant
formal pond
maybe a better visualiser might help you
autumn coral
maybe if u stop acting like a smartass and just be proper
thin vale
Do you want me to write it out for you?
formal pond
$\lambda^2 e^{\lambda x} - 2e^{\lambda x} = 0$
twin meteorBOT
Mortta
autumn coral
ok so we factor the e out
thin vale
not the e
autumn coral
e^lamdax ( lamda^2 - 2) = 0
thin vale
$$e^{\lambda x}(\lambda^2-2)=0$$
twin meteorBOT
Austin
autumn coral
...
formal pond
e^lamdax ( lamda^2 - 2) = 0
Good
thin vale
Now it'd be nice if we could get rid of the exponential
lucky for us e^(x) is never equal to 0
so we can do what next
autumn coral
Do we set x equal something
formal pond
There is no x
thin vale
There is no x
formal pond
Yes
autumn coral
do we divide by e^lamdax
thin vale
Yes
formal pond
Ye!
thin vale
Which we can do, because e^(x) is never 0, for all x
formal pond
And we’re left with?
autumn coral
lamda = sqrt 2?
autumn coral
whats 0/e^lamdax
thin vale
there will be 2 solutions
thin vale
whats 0/e^lamdax
0
autumn coral
ok so
formal pond
0/ anything that is no 0 is 0
thin vale
you're left with $$\lambda^{2}-2=0$$
twin meteorBOT
Austin
autumn coral
ok ye
formal pond
Solution
thin vale
yeah
autumn coral
ye
formal pond
Solve for lambda
thin vale
your friend called lambda r
autumn coral
ok where do we go from there
thin vale
Solve for lambda
thin vale
amanda
autumn coral
LOL
thin vale
XD
autumn coral
anyways
formal pond
??? What do you expand
autumn coral
ok sqrt 2
autumn coral
o
thin vale
there will be two solutions
formal pond
Nearly
thin vale
every time you guess you get closer
autumn coral
+- sqrt 2
formal pond
Yea
autumn coral
man math while hungry sucks
autumn coral
+- sqrt 2
autumn coral
e^lamdax