^
#help-17
crude scaffold
yeah ill try that
silk osprey
cool, have a nice night
vocal sleetBOT
vocal sleetBOT
Available help channel!
balmy sierra
how would I go about b), c), d)
vocal sleetBOT
@balmy sierra Has your question been resolved?
balmy sierra
.close
vocal sleetBOT
vocal sleetBOT
Available help channel!
honest orbit
how do I find the full area of r = 5cos(3θ)
vocal sleetBOT
how do I find the full area of r = 5cos(3θ)
vocal sleetBOT
vocal sleetBOT
Available help channel!
upper zodiac
vocal sleetBOT
upper zodiac
so would the integral setup be from pi/6 to 5pi/6 since that's when the outer circle completes?
honest orbit
I'm not a helper, but this looks exactly like my hw too
mild flower
i think it would be everything else? pi/6 to 5pi/6 would be the inner circle
upper zodiac
they take forever
really?
so would have to find the next point?
wouldn't that be 13 pi/6?
mild flower
you want thisi area right?
upper zodiac
just the outer loop
mild flower
what area?
upper zodiac
area of outter loop
hmm
im confused i thought it would be when = 0 the point has stopped
.close
vocal sleetBOT
vocal sleetBOT
Available help channel!
plain leaf
Does anyone know how to use desmos/ any graphic calculator? I have a project to draw using 10 linear equations and 15 quadratic equations. and I still doesn’t understand how it works
vocal sleetBOT
Does anyone know how to use desmos/ any graphic calculator? I have a project to ...
upper zodiac
on desmos
f(x) = (your equation)
x^2 or y^2 for quadratics
mx+b for linears
@plain leaf
vocal sleetBOT
@plain leaf Has your question been resolved?
dark nebula
Does anyone know how to use desmos/ any graphic calculator? I have a project to ...
what don’t u understand about desmos?
weak terrace
Dhey deposits 5000 pesos at the end of each month into a retirement account that earns 4.5% interest conpounded monthly. How much will she have in the account after 35 years
vocal sleetBOT
Dhey deposits 5000 pesos at the end of each month into a retirement account that...
lavish stag
hello! whenever we have something which grows at some rate r (in this case your r is 4.5%) we can compute its size with n(1+r)^t
n is your base number, and t the number of time periods 🙂 hope that helps
oop but thats not gonna do it is it
weak terrace
😭
P = 5000
n = 4.5% or 0.045
n= 12
t = 35 years
FV = P x (1+r/n)^nt - 1/r/n
is this thee right formula
keen siren
You need to account for the deposit
lavish stag
be careful though, you wont need r/n because we dont have annual onterest we have monthly
so you might want to make some changes to that formula before computing 🙂
weak terrace
so how
lavish stag
hahaha fair enough
wherever it says r/n you just want to replace with r
and for t take 12x35
make sense?
vocal sleetBOT
lavish stag
churr
weak terrace
.reopen
vocal sleetBOT
✅
weak terrace
one last
convert -150 degrees to radians leave answer in terms of pi
<@&286206848099549185>
keen siren
Do you know the general equation for converting degrees to radians?
I'll give you a hint 360=2pi
weak terrace
pi/180?
keen siren
So what would be your answer in terms of pi
weak terrace
-150/180
weak terrace
-150/180 x pi
keen siren
In terms of pi doesn't mean replacing with pi, it means expressing it with pi in the expression
Ok how can you write that more simply
keen siren
Looks good!
weak terrace
aight thanks
keen siren
The key takeaway: When you want to go from degrees to radians, you multiply your angle (let's say P) by pi/180, so that P*pi/180=P in radians
vocal sleetBOT
plain leaf
<@860480047534047232>
Thx
zinc stratus
Is there any other information? Do a b c d have to be integers or something?
Unless I'm missing some clever trick here it's not obvious to me that there's a closed solution here, when there are 4 variables for 1 relation
I'm not at home though so trying to work things out in my head
What have you tried?
The first thing that is coming to my mind is like, I can express that top function as X + Y / X - Y where X = bd + AC and Y are the other two terms but I don't see how that's immediately useful because the bottom equation doesn't use those same sets of terms
Also can try replacing the bottom of the second fraction with 4 * (b-c)(d-a) but also not obvious to me how that's helpful
tribal dawn
i may be getting close to something
hexed kestrel
||3/4||
hexed kestrel
(a+c)(b+d)=(a+b)(c+d)-(d-a)(b-c)
hexed kestrel
that's what i'm getting too
Yaaayyy!!
tribal dawn
i did it in a different way but got the same answer
hexed kestrel
Yaaayy
hexed kestrel
(a+c)(b+d)=(a+b)(c+d)-(d-a)(b-c)
Use this
hexed kestrel
Also can try replacing the bottom of the second fraction with 4 * (b-c)(d-a) but...
And this
hexed kestrel
i did it in a different way but got the same answer
Could u show it, I wanna c
tribal dawn
could I dm to you? don't want to give someone all my work (academic honesty)
hexed kestrel
Use this
Subtitute this, u will c
Show what u get
Yes, and what can you see from what’s left
No
||1-(d-a)(b-c)/(a+b)(c+d)||
And look at what’s given
hexed kestrel
Oh, ignore the arrow
hexed kestrel
||1-(d-a)(b-c)/(a+b)(c+d)||
Continue from here, and look at what’s given
Oki, and left one is?
You mean right one?
Don’t forget the negative sign, and now what u get?
And left one?
So…
Yes
Yoooo, we get it!
Which one
This is factorizing
Idk how to teach it
I guess u need to ask other for factoring, idk how to teach
vocal sleetBOT
@vast shale Has your question been resolved?
vocal sleetBOT
vocal sleetBOT
Available help channel!
stray pumice
need help with a proof question:
f is a function from [0,1] to R such that:
for every x in [0,1] there exists d>0 such that f is bounded in {[0,1] intersect (x-d,x+d)}
show that f is bounded in [0,1]
vocal sleetBOT
need help with a proof question:
f is a function from [0,1] to R such that:
for ...
stray pumice
someone partially answered me but i didn't really understand
full hatch
someone partially answered me but i didn't really understand
Would you like to explain the proof and we'll see if/what you understood?
vocal sleetBOT
@stray pumice Has your question been resolved?
vocal sleetBOT
vocal sleetBOT
Available help channel!
vocal sleetBOT
@untold arrow Has your question been resolved?
halcyon valley
I have a doubt regarding using the "method of undetermined coefficients" for solving second order differential equations.
The question is as follows: y'' - 2y' + y = 35x³e^x + x²
vocal sleetBOT
I have a doubt regarding using the "method of undetermined coefficients" for sol...
halcyon valley
Please ping me before replying, thankyou
vocal sleetBOT
@halcyon valley Has your question been resolved?
halcyon valley
<@&286206848099549185>
wraith mist
Please ping me before replying, thankyou
In order to solve such ODEs, first consider the homogeneous version:
y'' -2y' + y = 0
It can be easily solved, and let's denote z(x) the solution of that equation.
Note that if w(x) is some solution of the original equation, then z(x) + w(x) is also a solution.
Now that task is reduced to finding w(x)
Since RHS is an exponential polynomial, w(x) is also an exponential polynomial. The coefficients of that polynomial are unknown, so we keep them as variables
We let w(x) = P(x)e^x + Q(x)
halcyon valley
Mhm mhm
wraith mist
Hold on I messed up with the powers
wraith mist
Anyways w(x) = P(x)e^x + Q(x)
Where P(x) and Q(x) are polynomials
w'(x) = P'(x)e^x + P(x)e^x + Q'(x)
w''(x) = P''(x)e^x + 2P'(x)e^x + P(x)e^x + Q''(x)
w'' - 2w' + w = P''(x)e^x + Q''(x) - 2Q'(x) + Q(x)
wraith mist
Where P(x) and Q(x) are polynomials
.
wraith mist
The goal is to find P and Q
halcyon valley
Mhm ok ok
I'm kinda confused cuz they use y_p and y_h here
Just a sec
I'm processing stuff
wraith mist
w'' - 2w' + w = P''(x)e^x + Q''(x) - 2Q'(x) + Q(x)
Ok so we want this to be equal to 35x³e^x + x²
halcyon valley
We let w(x) = P(x)e^x + Q(x)
Uh can u make this a bit more clear?
Mhm
Ok so
My doubt here is
My sir told me to consider like
If u see x^n then u assume d+ax+bx²+cx³+....+zx^n
And for x^n*e^x
U multiply that thing by e^x
Is that correct?
wraith mist
Not quite, because if you want w'' - 2w' + w = x^n e^x + ... then in w(x) we must have x^(n+2) e^x + ...
wraith mist
w'' - 2w' + w = P''(x)e^x + Q''(x) - 2Q'(x) + Q(x)
It's clearly visible from this:
halcyon valley
Okok so
Like
For every x^n u go till x^(n+2) or
Is there a particular case where u go till there
wraith mist
For every x^n u go till x^(n+2) or
For every x^n e^x you go till x^(n+2) e^x
halcyon valley
Ah I see
wraith mist
For every x^n (without the exp) you go till x^n
But that's only for second order linear ODE
wraith mist
For every x^n e^x you go till x^(n+2) e^x
For third order you'd have n+3 here
halcyon valley
Ohh
wraith mist
Anyways, this is the main observation:
So P''(x) = 35x^3,
Q''(x) - 2Q'(x) + Q(x) = x^2
And now we use undetermined coefficients
wraith mist
For example Q(x) = a x^2 + b x + c
Q' = 2a x + b
Q'' = 2a
Q'' - 2Q' + Q = 2a - 4a x - 2b + a x^2 + b x + c = x^2
wraith mist
yes
halcyon valley
And put it back in w=p+q thing
wraith mist
Well, that's only Q(x) but yes
wraith mist
To find P(x) we use the fact that P'' = 35x^3
halcyon valley
Yep
wraith mist
P looks pretty simple
ye
wraith mist
yep
halcyon valley
Right?
Mhm
So we get w from there
And we already have z
So we get the answer
That's it?
wraith mist
That's it?
yes
halcyon valley
Ok so one last thing
Example 3
What is that multiplicity thing
@wraith mist
halcyon valley
<@&286206848099549185> someone?????
wraith mist
What is that multiplicity thing
Multiplicity of $\lambda$ in a polynomial is how many times $(x-\lambda)$ appears in the polynomial factorization
twin meteorBOT
EQUENOS
halcyon valley
Mhm
wraith mist
So for example in $(x-1)(x-2)$ root $1$ has multiplicity 1, but in $(x-1)^2(x-42)$ it has multiplicity 2
twin meteorBOT
EQUENOS
vocal sleetBOT
vocal sleetBOT
Available help channel!
sweet coyote
can someone explain to me how to approach such expressions?
vocal sleetBOT
can someone explain to me how to approach such expressions?
jagged cargo
can someone explain to me how to approach such expressions?
log that shit
halcyon valley
log that shit
Yo I'm already explaining...
jagged cargo
$x = e^{\ln x}$
twin meteorBOT
jagged cargo
the cool thing about this technique is that you can yoink whatever exponent on x down since its in log
so 1^inf or 0^0 or whatever turns into something lhopitalable
jagged cargo
do what i said first
vocal sleetBOT
@sweet coyote Has your question been resolved?
vocal sleetBOT
@sweet coyote Has your question been resolved?
vocal sleetBOT
@sweet coyote Has your question been resolved?
flat whale
twin meteorBOT
riemann
full summit
how can i prove $\sqrt{2}+\sqrt{3}$ is irrational
vocal sleetBOT
how can i prove $\sqrt{2}+\sqrt{3}$ is irrational
twin meteorBOT
pppoopoo
full summit
by contradiction
i know i've seen this proof somewhere but i just can't remember how it was done
queen root
u can try proving the sum of a rational number and an irrational number is irrational
full summit
$(\sqrt{2}+\sqrt{3})^2$?
twin meteorBOT
pppoopoo
potent beacon
Yes
full summit
what should i do after squaring it
potent beacon
Prove square is irrational
vocal sleetBOT
vocal sleetBOT
Available help channel!
obsidian tiger
Hello I have a small issue
vocal sleetBOT
Hello I have a small issue
oak magnet
What is it ?
obsidian tiger
What is it ?
I need to translate it to English, else I get bothered about it
oak magnet
Which language are you speaking ?
obsidian tiger
Examine the following functions with regard to continuity, zeros and reversibility, specify the definition and value range and sketch the curve:
German
Df = R
oak magnet
Ok
obsidian tiger
Wf = [0,2]
oak magnet
Wf is range ?
oak magnet
Ok
obsidian tiger
Continuity is true for all points
oak magnet
Indeed
obsidian tiger
There are no zeros, because f(x) is not =0
And now to my issue
From what I see there is no real reversibility for the function
oak magnet
y = 1 + x/sqrt(x²+9)
obsidian tiger
You won't find a reverse function for it, as it doesn't exist
My question is, can I straight out say, that there doesn't exist a reversible function for it
Or do I have to proof, that there is none?
vocal sleetBOT
@obsidian tiger Has your question been resolved?
oak magnet
I think you have to prove theres none
But i can't remember if there is actually a theorem that says there is no reversible if...
plain finch
it is tricky actually 🥲
obsidian tiger
it is tricky actually 🥲
I can understand that
Try to proof, that something doesn't exist, without it existing
Still thanks
.close
alpine jay
Hi. Im Brian. Im needing help to start an excercise. Idk from where start to calculate. The problem says:
Resolve with Integral Triple: The solid of the image has the form of an hiperboloid of one sheet. The total mass is M and the density is uniform. The lateral surface, in cilindrical coordinates is the one in the image.
With that I gotta get the volume. How do i start? idk how to set the limits
vocal sleetBOT
Hi. Im Brian. Im needing help to start an excercise. Idk from where start to cal...
alpine jay
I dont understand how to write the triple integral. What are the limits?
tried doing this but i dont see much sense on it
<@&286206848099549185>
thick ermine
tried doing this but i dont see much sense on it
Your z bounds should just be +- h/2 I think. Because the bottom of the shape is at -h/2 and the top is at h/2.
thick ermine
yeah I'm wrong sorry
I would integrate over z and theta, and then inside that integrate over r
You can immediately convert the theta integral into a factor of 2pi, so it's just 2pi* (integral dr dz)
better to integrate dr and then dz I think. You're integrating up to the radius of the surface at fixed z, which you have an expression for
and bounds for z are +- h/2
You could probably integrate dz then dr, but it would be a lot worse and you'd have to integrate the upper section's volume then double it rather than just having one integral
Alternatively you could just skip a step and integrate the circle area pi* r^2, with respect to z
alpine jay
You could probably integrate dz then dr, but it would be a lot worse and you'd h...
there shouldnt be problem with the order cause the work says i can use software to calculate it. I gotta know how to put the integral only and then calculate with a software
so by what u say it would be: 0<=theta<=2pi, b<=r<=a and -h/2<=z<=h/2
the limits i mean
but what do i do with the lateral surface? cause i think is an important data to take
thick ermine
so by what u say it would be: 0<=theta<=2pi, b<=r<=a and -h/2<=z<=h/2
no, because that would be just the cylinder, not the surface given
alpine jay
oh right
thick ermine
there shouldnt be problem with the order cause the work says i can use software ...
The order definitely does matter
You can express z bounds with respect to r, or r bounds with respect to z. One is a lot easier
alpine jay
You can express z bounds with respect to r, or r bounds with respect to z. One i...
ok, good to know
thanks
alpine jay
no, because that would be just the cylinder, not the surface given
but how do i set the limits for the surface?
thick ermine
but how do i set the limits for the surface?
I mean like
What I would do is integrate dr inside and dz outside, because the surfaces is rotated about the z axis - easier to express r bounds in terms of z
theta bounds are 0 and 2pi
z bounds are -h/2 to h/2
and then r bounds are +- 1/h sqrt(b^2h^2 + 4(a^2-b^2)z^2)
thick ermine
It's easier to just notice that you're integrating the circles at each z slice
if that makes sense
alpine jay
hm ok...
but the thing is that the problem asks me to represent a triple integral. If i integrate before and then i use those limits as u said i would have a double integral
right?
thick ermine
and then r bounds are +- 1/h sqrt(b^2h^2 + 4(a^2-b^2)z^2)
If the problem says that then yeah, just do bounds like this
That's the method I would use in practice though. It's essentially skipping a step of working out the r integral and conceptually easier to understand I think
alpine jay
and then r bounds are +- 1/h sqrt(b^2h^2 + 4(a^2-b^2)z^2)
+- 1/h*sqrt(b^2h^2 + 4(a^2-b^2)z^2)? or how
sorry if i ask too much. I wanna understand the thing well. Im strugling for a while :p
thick ermine
yeah that's the correct r bound I think
thick ermine
sorry if i ask too much. I wanna understand the thing well. Im strugling for a w...
no worries it's a tricky topic
alpine jay
yeah that's the correct r bound I think
Ok. Do u mind if i write it correctly and i send it to u to confirm i did it right?
alpine jay
no worries it's a tricky topic
thxx
thick ermine
sure
alpine jay
like this u mean?
thick ermine
dr inside dz outside
alpine jay
what exactly u mean with outside and inside?
thick ermine
Because what you've done there is put z in the bounds for the z integral
thick ermine
what exactly u mean with outside and inside?
Like you have a triple integral right
alpine jay
yes
thick ermine
What you're doing really is integrating an integral
and then integrating that
you do an r integral
then integrate that over z
then integrate that over theta
By "inside" I mean what you integrate first and by "outside" I mean what's integrated next. what the inner thing is expressed with respect to. sorry
give me a sec
alpine jay
ah okok got it
thick ermine
like this u mean?
what you've done here is put a z expression in the bounds for a z integral. which isn't okay in a regular, 1 dimensional integral and isn't okay here. the order matters
alpine jay
what you've done here is put a z expression in the bounds for a z integral. whic...
OHH yes
yes i get it now
ok my bad my bad
right so the order should be drdzdtetha can be?
or neither that
alpine jay
yes, just did it
it gives 0 as result tho
like, doesnt make sense that it has V=0 I think
thick ermine
Yep, wolfram is being dumb
Idk why but it doesn't like this integral - maybe because of the constants?
You can just see what result it's going to give you by looking at it, though.
alpine jay
no but it does gives 0
cause
it will be r^2/2
then using Barrow u have
F(b)-F(a)
and cause the square
thick ermine
yeah
alpine jay
is gonna be substracting itself
thick ermine
wait
It's zero
You're integrating r from zero up to the maximum radius in terms of z
And that gives you 2pi * r^2/2 = pi*r^2
circle area
as expected
sorry about the confusion
thick ermine
radius is positive, I was being dumb
alpine jay
yes u right. Usually radius starts from 0 cause the origin haha
well good then
lemme see how the other calculus of the next problems goes then to see if they make more sense now
vocal sleetBOT
@alpine jay Has your question been resolved?
alpine jay
Ok so i calculated the moment of intertia Iz and it gives me this result:
which is great, i think is correct
but when i try to calculate the moment of inertia Iy it gives me error
like the software cant calculate it
its what happened with the one i showed u at the beggining
i dont know if its a software problem, the integral problem
thick ermine
i'm not sure how to calculate those
alpine jay
i have the formulas. If u dont mind helping me i can show u, they kinda simple
if u busy no problem
this is what im trying to calculate
basically the limits of integration are the same
think the S as a constant
thick ermine
is that dirac delta?
alpine jay
so it would be this:
alpine jay
is that dirac delta?
dirac?
alpine jay
delta is a constant
i have it but its not important for the calculus
it just goes out and u multiply at the end of the integrals
delta is the density
like
M is masa
V is volume
Its a constant basically so ignore. It doesnt have any r,z or tetha
thick ermine
I'm not sure why it wouldn't be able to work out that integral
alpine jay
i dont know to be honest
it calculates the first integral
but the second just cant
the first gives this
alpine jay
yes h is a constant
cause the image that the problem gives is this
h is the height so
should be a constant
cant be 0 cause otherwise there wouldnt be a surface lol
maybe the software is undestanding that wrong?
thick ermine
yeah i'm wrong was being dumb
I would just work out the integral by hand if you can
alpine jay
it made sense at first sad
alpine jay
I would just work out the integral by hand if you can
hm ok
maybe if i do them separately
wait
vocal sleetBOT
alpine jay
.reopen
vocal sleetBOT
✅
alpine jay
i did it part by part
with wolfram
gave me this
can be? or looks nonsense
just found out the error it gives me is this
it takes too much time and since i have the free version it just wont calculate it
plucky trout
maybe ,w not sure if texit bot has paid version
alpine jay
texit can calculate?
plucky trout
do ,w [prompt]
alpine jay
what kind of sintaxis does it uses
,w Integrate[((rCos[θ])^2+z^2)r,{r,0,Sqrt[b^2h^2+4(a^2-b^2)*z^2]/h},{z,-h/2,h/2},{θ,0,2π}]
,w Integrate[(40)Power[(40)r*Cos[θ](41),2]+Power[z,2](41)*r,{r,0,Divide[Sqrt[Power[b,2]Power[h,2]+4(40)Power[a,2]-Power[b,2](41)*Power[z,2]],h]},{z,-Divide[h,2],Divide[h,2]},{θ,0,2π}]
twin meteorBOT
alpine jay
hm
alpine jay
well im just gonna put this result
its supposedly correct
thanks for all the help
i really appreciate it
u guys have a good weekend
🫶
.close
vocal sleetBOT
vocal sleetBOT
Available help channel!
upbeat locust
Find the largest set of values of x such that the function $f(x)=\sqrt{ \frac{{9x-3}}{x-2} }$ takes real values.
Can anyone explain the process behind doing a question like this? To my understanding, the "largest" set of values is the entire domain of f(x), so you must find the restrictions to the domain (e.g x cannot be 2 here)
however beyond that is a big question mark
vocal sleetBOT
Find the largest set of values of x such that the function $f(x)=\sqrt{ \frac{{9...
upbeat locust
twin meteorBOT
outletproblems
lone linden
Hint: What is the domain of $\sqrt x$?
twin meteorBOT
Civil Service Pigeon
upbeat locust
x >= 0
but now what can i do
like how should the answer be displayed as
So i know, the domain of f(x) is x >= 0 for any real X and x != 2
but what is the 'set'
lone linden
but what is the 'set'
Are you asking how to notate it?
upbeat locust
Are you asking how to notate it?
yeah kind of like the final answer
what should i actually write now?
lone linden
set builder notation ig?
upbeat locust
what is that
upbeat locust
ok noted
ok i have another question
Since its the "set"
Would be it be, (2, inf) or [1/3,2) and (2,inf) (i think there is some unity sign or something for the latter)
lone linden
,w \frac{9x-3}{x-2} \geq 0
twin meteorBOT
upbeat locust
yeah
because its asking for the largest set of values
is the largest set of values the applicable domain or is it the kind of largest independent range
lone linden
is the largest set of values the applicable domain or is it the kind of largest ...
largest possible domain
upbeat locust
so [1/3,2) and (2,inf)
lone linden
yeah
upbeat locust
then i can use that unity sign right
since they are subsets
or something idk if im over complicating it
lone linden
$\left[\frac{1}{3}, 2 \right) \cup (2, \infty)$ is fine
upbeat locust
yes exactly
twin meteorBOT
Civil Service Pigeon
lone linden
yeah
upbeat locust
.close
vocal sleetBOT
vocal sleetBOT
Available help channel!
lost lodge
I did 9x8x7 and got 504 three letter words
vocal sleetBOT
I did 9x8x7 and got 504 three letter words
onyx sage
but you should do 9 * 9 * 9
because it doesn't state that the letters can't be the same
marble sluice
9p3
lost lodge
.close
vocal sleetBOT
vocal sleetBOT
Available help channel!
hexed vortex
Hey there, I need help:
I have this thing: y = x / 2 + 3
I need to derivate it, so I did: y = x + Dx +3 : Dy = x
hexed vortex
Oops, I send it before I finished, let me edit it...
Hey there, I need help:
I have this thing: y = x / 2 + 3
I need to derivate it, so I did: y = x + Dx +3 : Dy = x + Dx / 2 + 3 - x / 2 - 3
Dy = Dx / 0 Am I correct? :(
Also, I haven't finished all the procedure, but I'm stuck here since it's indefinite.
jagged cargo
i honestly dont know what the hell youre doing
i assume you mean differentiation?
can you send the original question?
hexed vortex
can you send the original question?
I don't know... It's a new topic for me, my professor told us this is how you obtain a derivate of a function f (x).
The original question is just a simple exercise, this one.
Y = x + 3
2
jagged cargo
idk why you are adding dx dy everywhere
and somehow obtaining dy = dx/0
which is uh interesting
hexed vortex
idk why you are adding dx dy everywhere
The professor also told us that thing is "Delta", Delta x and Delta y.
Is this how you actually get a derivate? It was kinda sus when our teacher explained us this thing.
jagged cargo
there are two ways to differentiate
either you use first principle
$f'(x) = \lim_{h\to0}\frac{f(x + h) - f(x)}h$
twin meteorBOT
jagged cargo
or you apply already known differentiation of simple fucntions
in your case im suspecting the former
jagged cargo
it would be unproductive to actually explain what a derivative is etc
i suggest you watch this
hexed vortex
it would be unproductive to actually explain what a derivative is etc
The exercise is this:
Y = x + 3
2
Any ideas how to derivate it? I'm doing what my professor told us, but I'm confused.
jagged cargo
https://www.youtube.com/watch?v=5yfh5cf4-0w
aka you need to watch this
or, you can send your professor notes or whatever
hexed vortex
or, you can send your professor notes or whatever
Gonna send the notes that my professor gave us:
1.- Increase x
2.- Express the increase of y
3.-Algebraically subtract the initial function
4.- Dy / Dx
This is literally what they told us.
hexed vortex
Let me type an example that my professor did with us:
f (X) = 3x^2 + 4
y = 3x^2 + 4
y = 3(x + Dx)^2 +4
y + Dy = 3(x + Dx)^2 +4
y + Dy = 3(x^2 + 2xDx + Dx^2) + 4
y + Dx = 3x^2 + 6xDx + 3Dx^2 + 4
-y = -3x^2 - 4
Dy = 6xDx + 3Dx^2
Dy 6xDx + 3Dx^2
-- = ----- -------
Dx Dx Dx
Dy
-- = 6x + 3Dx
Dx
Then they obtain the limit of this thing:
lim Dy / Dx = 6x + 3(0) = 6x
hexed vortex
I don't know... It's a new topic for me, my professor told us this is how you ob...
And I'm trying to do this exact thing with this thing.
vocal sleetBOT
@hexed vortex Has your question been resolved?
stone gazelle
<:noporfa:1013503552477483079>
What’s the og question and what have you done so far?
ruby ingot
Hi
hexed vortex
What’s the og question and what have you done so far?
Original: #help-17 message
What my professor told us: #help-17 message
Example by my professor: #help-17 message
hexed vortex
So what have you done so far
stone gazelle
Little bit hard to follow what you did there, but it looks like you are missing the y on the LHS
What you want to do is essentially nudge the y and x values by some dy and dx
So you replace all y’s with (y + dy) and all x’s with (x + dx)
So you have
[ y = \frac x2 + 3 ]
[ (y+ \dd y) = \frac { (x+ \dd x) }{2} + 3 ]
twin meteorBOT
shsgd
hexed vortex
Yeah, I have that.
stone gazelle
From here you want to expand the parentheses if needed (not in this case) and separate the x terms from the dx terms and the y terms from the dy terms
On the LHS they are already separated. How about the RHS?
stone gazelle
Left hand side and right hand side
hexed vortex
**shsgd**
After I get this, I subtract the original one (y = x/2 + 3) then it gives me Dy = Dx / 0
untold sun
After I get this, I subtract the original one (y = x/2 + 3) then it gives me Dy ...
are u differentiating both variables with respect to time/third variable??
stone gazelle
After I get this, I subtract the original one (y = x/2 + 3) then it gives me Dy ...
You cannot just subtract x/2 just yet
hexed vortex
are u differentiating both variables with respect to time/third variable??
To be honest, I do not understand what I am doing.
stone gazelle
The /2 is affecting both x and dx. So you have to split them. How do you do that?
untold sun
To be honest, I do not understand what I am doing.
what is the question in the textbook?
hexed vortex
what is the question in the textbook?
It's a homework, I need to derivate this: y = x/2 + 3
untold sun
It's a homework, I need to derivate this: y = x/2 + 3
you mean differentiate?
have you been taught chain rule yet :/
stone gazelle
Crufias what are you on about
stone gazelle
This is differentiation by first principles
untold sun
This is differentiation by first principles
using limit definition?
stone gazelle
No, they’re prof wants them to do it slightly different, same principle though
thick ermine
Let me type an example that my professor did with us:
f (X) = 3x^2 + 4
y = 3x^...
Do they teach first principles differentiation, first?
Before standard derivatives?
stone gazelle
Usually yes
thick ermine
Dumb
hexed vortex
you mean differentiate?
Well... My professor told us this is derivate, the first user told me this is first principle, but I'm confused.
Is this how you derivate? Or what the hell am I doing?
thick ermine
wait, what's principles 😭
It means like, differentiation from the definition
hexed vortex
And why so much people here. 😭
untold sun
It means like, differentiation from the definition
right so limit def. then no?
stone gazelle
The /2 is affecting both x and dx. So you have to split them. How do you do that...
@hexed vortex focus on this
hexed vortex
hexed vortex
The /2 is affecting both x and dx. So you have to split them. How do you do that...
Common term?
stone gazelle
right so limit def. then no?
Differentiation by first principles is the limit definition, yes. This guy’s prof has taught them a slightly different, more informal, way of this
stone gazelle
Common term?
Yeah, so what do you get on the RHS
thick ermine
-
Derivative represents slope of a function at a point
-
Limit as you get closer of the difference in y, divided by the difference in x.
-
dy/dx.
-
What would you expect the derivative of a straight line to be, with the intuition that it's the slope?
-
If you have y as a function f(x), then the y difference between x and x+dx is f(x+dx) - f(x). And the x difference is (x+dx) - (x) = dx
-
Find expressions for those and see if you can take a limit
thick ermine
After I get this, I subtract the original one (y = x/2 + 3) then it gives me Dy ...
No, because you still have the dx/2 term
[(x+dx)/2 + 3] - [x/2 + 3] = dx/2
stone gazelle
Dumb
First principles are taught first because before calculus you are taught how compute the slope of a line, which is what first principles converges to
untold sun
First principles are taught first because before calculus you are taught how com...
yk I never learned this principle stuff. u have a neat source to read up on it?
thick ermine
it's so much better to teach standard examples and get intuition before the "first principles" definition
untold sun
it's so much better to teach standard examples and get intuition before the "fir...
dude icl I think "first principles" is what I learned first too it's just my teacher didn't call it that
thick ermine
Usually just binomial expansions when you're differentiating polynomials like this instead of the fast way
thick ermine
dude icl I think "first principles" is what I learned first too it's just my tea...
We did power rule & differentiating polynomials like a year before first principles
untold sun
We did power rule & differentiating polynomials like a year before first princip...
that's mental
I learned "first principles" first
stone gazelle
We did power rule & differentiating polynomials like a year before first princip...
You went to a shit school then cant lie
thick ermine
Nah, first principles isn't on gcse syllabus. No UK school does it that way
untold sun
was sorta lost so I read in the textbook about the epsilon-delta definition to rly understand it only to get the only question on it wrong in the test 😭
untold sun
You went to a shit school then cant lie
shit might be a stretch. But yeah, not learning principles first blows my mind
thick ermine
If you do first principles first, you have no idea what a derivative is and get really confused. As has happened to Tortilla.
hexed vortex
Yeah, so what do you get on the RHS
This is the original: y = x/2 + 3
I need to add the D so: y + Dy = x + Dx / 2 + 3
Right? Then I need to subtract the original one: y + Dy = x + Dx / 2 + 3 - x/2 - 3
thick ermine
This is the original: y = x/2 + 3
I need to add the D so: y + Dy = x + Dx / 2 +...
y + Dy - y = x + Dx / 2 + 3 - x/2 - 3
And better to use brackets around sums you're dividing
untold sun
If you do first principles first, you have no idea what a derivative is and get ...
if you do it this weird, simplified way. The way in the website you send me works just fine
stone gazelle
Before subtracting the original, separate the dx and dy terms
hexed vortex
Before subtracting the original, separate the dx and dy terms
What do you mean?
stone gazelle
From here you want to expand the parentheses if needed (not in this case) and se...
.
thick ermine
Like this? y + (Dy) = x + (Dx) / 2 + 3 - x/2 - 3
You're dividing the x by 2, not just the Dx
stone gazelle
[ \frac {x+\dd x}{2} = \frac x2 + \frac{\dd x}{2} ]
twin meteorBOT
shsgd
hexed vortex
thick ermine
Like this? y + (Dy) = x + (Dx) / 2 + 3 - x/2 - 3
y + (Dy) = (x/2 + 3) + Dx/2
left bough
this looks painful what kind of calculus is this
thick ermine
this looks painful what kind of calculus is this
Derivatives by the formal definition
untold sun
idk why you are adding dx dy everywhere
this is so real 😭
untold sun
this looks painful what kind of calculus is this
intro to differentiation
untold sun
it's not bad at all Tortilla is just being taught a weird, weird way
stone gazelle
So...
y + (Dy) = x / 2 **+**Dx / 2 + 3 **-** x / 2 **-** 3
It's Dx / 2?
Now just divide both sides by dx
hexed vortex
it's not bad at all Tortilla is just being taught a weird, weird way
I have exams every month, I need to learn 5 topics of each subject in 1 month.
untold sun
I have exams every month, I need to learn 5 topics of each subject in 1 month.
gotcha. I take it you're in calc 1 or an equivalent, then?
untold sun
Hey there, I need help:
I have this thing: **y = x / 2 + 3**
I need to derivat...
be sure to say "differentiate" in order to find the "derivative"
I made the same mistake numerous times when I first started so dw
unless y'all use different terms idk
but it'll help when you ask future questions
hexed vortex
Now just divide both sides by dx
Dy Dx
-----
2
-- = ------
Dx Dx
hexed vortex
be sure to say "differentiate" in order to find the "derivative"
I made the sam...
My professor told us this thing is derivate, I still confused...
thick ermine
Dy Dx
-----
2
-- = ------
Dx Dx
Dy = Dx/2
Dy/Dx = 1/2
thick ermine
My professor told us this thing is derivate, I still confused...
It's called the "derivative". That's the object. The process to get it is called "differentiating"
hexed vortex
untold sun
idk why you are adding dx dy everywhere
it's a different way to differentiate. Took me a minute to realize but he's essentially rewriting the function in terms of whatever point would be on the graph, utilizing "dy" and "dx" as change in y/x.
hexed vortex
Wait, can I do this?
Dy Dx
-----
2
-- = ------
Dx Dx
----
1
stone gazelle
@hexed vortex Here’s a little guide following your prof’s way:
Step 1: Replace x’s and y’s with (x + dx) and (y + dx)
Step 2: Distribute/Expand the parentheses with coefficients if there are any
Step 3: Separate x’s from dx’s and y’s from dy’s so that each one is only multiplied by some constant, not added
Step 4: Subtract original equation
Step 5: Move dy and dx terms onto the left hand side
thick ermine
Honestly I think reading examples is probably best
untold sun
Wait, can I do this?
Dy Dx
-----
2
-- = ------
Dx ...
yes, you will get dy/dx = 0.5 which is the answer
thick ermine
https://www.le.ac.uk/users/dsgp1/EXERCISE/MATHSEX/FSERIES/f2ans.pdf
some examples of how to do this. Please ignore "MATHSEX". It's meant to say "maths ex" I assume.
untold sun
tortilla, is the reasoning behind all this making sense?
b/c that's really the point of principles
98% after you learn other forms you will never use this method
hexed vortex
yes, you will get dy/dx = 0.5 which is the answer
Well... My professor also told us the "Sandwich" method to do this:
untold sun
bro
stone gazelle
I suggest you learn using online resources from here on out
untold sun
what is this professor 💀
hexed vortex
This gives me:
Dx / 2 (You need to multiply)
untold sun
and here I thought my calc teacher was rough...
hexed vortex
what is this professor 💀
Education system where I live is not the best... I'm looking for help by myself, I don't even speak english...
thick ermine
Education system where I live is not the best... I'm looking for help by myself,...
The main idea is to find an expression for Dy/Dx using algebra
untold sun
Education system where I live is not the best... I'm looking for help by myself,...
damn mate no worries then. Don't hesitate to ask questions and develop a firm base in this stuff
stone gazelle
Education system where I live is not the best... I'm looking for help by myself,...
Where are you?
hexed vortex
Where are you?
Mexico.
stone gazelle
Pobrecito
hexed vortex
😭
untold sun
si quieres puedo ensenarte en espanol
necesito la practica tambien
pero es como esto:
estas cambiando la ecuacion
hexed vortex
The rules of the server are "Speak english", I also need to practice my english, no worries about the language.
untold sun
The rules of the server are "Speak english", I also need to practice my english,...
shoot didn't know that
alright then I'll end my explanation there then
hexed vortex
Well... My professor also told us the "Sandwich" method to do this:
Going back to this thing, I got this, now what?
😭
thick ermine
I mean, you're basically done
You have an expression for Dy/Dx right? It's (Dx/2) / Dx which is 1/2
And then you want to see what that goes to as Dx gets really small. But 1/2 is just a constant, so it has a limit of 1/2
The whole thing about straight lines is that they have a constant derivative. Because they're straight. The slope is always the same
If you do more problems and get a bit more comfortable with it, it's become quite clear that it's just a bit of algebra disguised as something nasty
The next problem probably won't take as long
hexed vortex
This is really painful, I think I'm understanding but still kinda confused, right now I need to solve this:
The "sandwich" method is correct?
That thing gives me Dx /2
untold sun
yes you're dividing a fraction by another fraction
Dx/2 divided by Dx/1 is just Dx/2 times 1/Dx
therefore you get 1/2 on the right (Dx cancels)
hexed vortex
Dx/2 divided by Dx/1 is just Dx/2 times 1/Dx
So... Am I correct? Now I need to get the limit: Dx / 2 = (0) / 2 = 0
untold sun
this bocadillo method is just dividing both sides by Dx (Dx/1 = Dx, hence you're doing the same thing to both sides)
hexed vortex
Gonna take a short break, brb. 😭
untold sun
Gonna take a short break, brb. 😭
nw, eat some chocolate
you're almost done though. See quater's note below
thick ermine
So... Am I correct? Now I need to get the limit: Dx / 2 = (0) / 2 = 0
No, you didn't divide by Dx :(
If you divided by Dx and did get Dx/2, then that would go to zero. But currently what you have is the limit of Dy, not Dy/Dx
stone gazelle
this bocadillo method is just dividing both sides by Dx (Dx/1 = Dx, hence you're...
Bocadillo ≠ sandwich
untold sun
Bocadillo ≠ sandwich
rly? that's what my teacher taught me
stone gazelle
It’s just sandwich with an accent. Bocadillo is a type of sandwich
untold sun
then again she speaks Spain Spanish
untold sun
It’s just sandwich with an accent. Bocadillo is a type of sandwich
alr cool then
stone gazelle
Spain spanish is awful imo
thick ermine
Bro hates everything that's not how he was taught
untold sun
Spain spanish is awful imo
my first ever teacher was venezuelan and she only somewhat introduced me to it; second teacher was just a long term sub that made us do duolingo, third was pretty much the same
my fourth was Spain spanish which is when I actually started learning
fifth teacher spoke Peru-an spanish(?)
current teacher idrk maybe Mexican or Spain
next year is back to Spain
untold sun
Bro hates everything that's not how he was taught
i can't blame him. I was never taught Torilla's method so I started off hating it
thick ermine
I still hate it honestly
He's dressed up "algebra with variables" in 5 different trenchcoats
untold sun
I still hate it honestly
it's easier than what I'm learning rn which is good imo
then again I don't have a passion for math like y'all
stone gazelle
my first ever teacher was venezuelan and she only somewhat introduced me to it; ...
Venezuelan spanish is good imo. The Venezuelan’s i’ve heard in my time are pretty easy to understand
Peruvian btw
stone gazelle
To be clear, i’m not Peruvian. I was just pointing out it’s “Peruvian” not Peru-an
thick ermine
cool
hexed vortex
I'm back...
hexed vortex
No, you didn't divide by Dx \:(
Thinking about it, Dx / 2 / Dx / 1 = 0.5, but in the exercises I've done with my professor, I never get the 0.5, I always get something like Dx / 2.
I have another similar exercise, with frac, give me a second...
Sorry for the quality.
But this is the most similar exercise as the original that I'm doing, in fact, they used the "Sandwich" method.
@stone gazelle @thick ermine @untold sun (Sorry for the ping, just to let you guys know I'm back)
thick ermine
Sorry for the quality.
That's correct
hexed vortex
That's correct
This is another exercise, really similar as the original.
untold sun
This is another exercise, really similar as the original.
yeah it looks right
hexed vortex
yeah it looks right
So, following the same theory, my Dx / 2 is correct, my limit is Dx / 2 = (0) / 2 = 0
thick ermine
So, following the same theory, my Dx / 2 is correct, my limit is Dx / 2 = (0) / ...
No
sorry
Dy = Dx/2 right
can we agree on that
hexed vortex
Isn't Dx = 0?
hexed vortex
can we agree on that
I think we do.
thick ermine
Dx tends to zero
The difference in y, Dy, will tend to zero for any continuous function
that's not very interesting
you care about the slope at the point which is the limit of the ratio Dy / Dx
hexed vortex
The end of my exercise is Dx / 2, right?
This is before the limit, just making sure I'm correct.
untold sun
not before "sandwiching"
thick ermine
The end of my exercise is Dx / 2, right?
This is before the limit, just making...
What are you trying to take the limit of here?
hexed vortex
untold sun
you want Dy/Dx because that is change in y values over the change in x values
untold sun
After doing this, my result is Dx / 2
it is 1/2
thick ermine
After doing this, my result is Dx / 2
No! You didn't divide the right hand side!
untold sun
No! You didn't divide the right hand side!
😭
thick ermine
The left hand side you have divided by Dx
The right hand side you haven't divided it by Dx
hexed vortex
thick ermine
You're not doing the same thing to both sides
thick ermine
After doing this, my result is Dx / 2
What if I told you
$a = b/2$
twin meteorBOT
Quaternionic [3375600]
thick ermine
How would you solve for $a/b$
twin meteorBOT
Quaternionic [3375600]
thick ermine
just regular algebra
hexed vortex
By the sandwich method you do this:
You multiply them.
untold sun
uhhh
hexed vortex
Dx X 1
2 X Dx
thick ermine
Yes, so Dx / 2Dx which is 1/2
Sandwich method is so dumb
sorry if i'm coming across as aggressive
hexed vortex
Sandwich method is so dumb
Bro, that thing has carried 2 years of my life in math. 💀
So, going back, the result is:
Dx / 2Dx
untold sun
sorry if i'm coming across as aggressive
maybe it's a latin america thing for how they deal with improper fractions
untold sun
wdym?
untold sun
I though Dx X 2 was 2.
eh you're fine
hexed vortex
So... Lim = Dx / 2Dx = (0) / 2 (0)?
untold sun
So... Lim = Dx / 2Dx = (0) / 2 (0)?
nope
hexed vortex
untold sun
dx/dy is a constant
dx/dy = 0.5
there is nothing to plug 0 into
you plug zero into dx because you are trying to find the limit as it approaches x
thick ermine
You work with Dx as a nonzero number, that you think of as being small
thick ermine
So... Lim = Dx / 2Dx = (0) / 2 (0)?
This is a good question
You can cancel out the variable on the top and bottom of the fraction when Dx is nonzero
Then you get a meaningful answer
untold sun
You can cancel out the variable on the top and bottom of the fraction when Dx is...
this a very good point @hexed vortex
hexed vortex
This is a good question
Dx / 2Dx = 1 / 2
untold sun
Dx approaches zero but it is not 0
untold sun
and even if it was 0/2(0), it'd be undefined
hexed vortex
and even if it was 0/2(0), it'd be undefined
Then I need to factorize... Right? (I don't know the translation for the term)
untold sun
Then I need to factorize... Right? (I don't know the translation for the term)
no
dx/2dx
just cancel the dx's
untold sun
Then I need to factorize... Right? (I don't know the translation for the term)
perhaps it is getting lost in translation b/c by factorize i think you may mean
(dx/dx) times (1/2) ?
untold sun
I have a question here, is:
= 1 / 2
or
= 0 / 2
1/2
hexed vortex
(dx/dx) times (1/2) ?
dx/dx is 1.
thick ermine
I have a question here, is:
= 1 / 2
or
= 0 / 2
The limit of a constant is the constant
So lim (1/2) = 1/2
untold sun
the limit as that function approaches zero is 1/2
it may be undefined at that point, but it is approaching that point
when you're "plugging in 0" to find the limit, you are not literally plugging in 0
you are just plugging in an incredibly small number
because that number is getting smaller and smaller, closer and closer to 0 (hence, limit as dx approaches/arrow towards 0), then you can treat it like 0 like your other example
thick ermine
because that number is getting smaller and smaller, closer and closer to 0 (henc...
This
untold sun
the limit, or the output is trying to reach that value as the x tries to reach 0
hexed vortex
Okay so, just checking this out:
y = x/2 + 3
y + (Dy) = x / 2 + Dx / 2 + 3 - x / 2 - 3
Dy Dx
-----
2
-- = ------
Dx Dx
= Dx / 2DX = 1 / 2
hexed vortex
May I ask what's that app? It's to check limits, right?
thick ermine
but because it is never actually 0, you can cancel into one like this example
A lot of the time i.e. continuous functions you can just put in a value of x = 0 and see what you get for the limit
Doesn't always work and you have to know it's continuous, and you have to simplify stuff first e.g. Dx/2Dx --> 1/2 but at this level most of the time you can just put in x = 0 and see what you get
untold sun
May I ask what's that app? It's to check limits, right?
it's a graphing calculator website
it does not "check limits" , but if you know how to use it, you can use it to help
thick ermine
It's mainly focused on graphing functions and producing visuals
Very easy to use if you know how to use it, and lets you visualise things very well
I had this problem the other day where it's like "You can approximate this function by adding together a bunch of cosine waves, show that the error doesn't go below this value
untold sun
recall the approaches to finding a limit if you're still confused. It may help with wrapping your head around derivatives
thick ermine
I was able to make a visual of the function in the middle of a meeting, really hand
untold sun
untold sun
I was able to make a visual of the function in the middle of a meeting, really h...
that does sound handy. You said you're majoring in maths tho right?
thick ermine
Yeah
this is the question, it's about these things called "Fourier Series" where you add together sine and cosine functions to get other functions
thick ermine
the desmos link I posted explains what it tries to do really well
the point is it's really handy for visualisation
untold sun
the desmos link I posted explains what it tries to do really well
mate, you think I'll be doing that in my series unit?
thick ermine
sorry if I was aggressive or confusing
thick ermine
mate, you think I'll be doing that in my series unit?
no way
thick ermine
This is cambridge university homework, additional problems that you don't have to do
In second year, of majoring in maths
untold sun
This is cambridge university homework, additional problems that you don't have t...
alright cool. Also, you go to cambridge?! Mental
thick ermine
I don't want to say "it's easy" so you don't feel dumb
The gist of it is "Give me a periodic function" like s_N on the left
untold sun
I don't want to say "it's easy" so you don't feel dumb
my 7th grade math teacher said all math is easy once you learn how
so dw
thick ermine
I can choose these coefficients An and Bn that give you that periodic function as a sum of sines and cosines
untold sun
having a firm grasp of anything makes it easy since you know it
hexed vortex
Sorry for disturbing, how can I close this? 😅
thick ermine
Oh just do ".close", sorry about our conversation
untold sun
it's like if I were to ask you Fisherian or Neo-Quantity theory
untold sun
Calculate those for each n and put them in the sum
haven't learned integrals yet.. that's next unit
hexed vortex
Oh just do ".close", sorry about our conversation
No worries, it's good to see my future pain.
untold sun
Sorry for disturbing, how can I close this? 😅
and dw it's our bad for talking too much
it's . then close
smushed together tho
vocal sleetBOT
thick ermine
Oh lol, they're so important that it's just hard to assume people don't know about them 😅 at a certain level in applied maths half of what you do is integrals/derivatives.
untold sun
Alright, gonna close it, thanks thanks thanks a lot.
anytime. don't be scared to ask more questions if you're still confused or something else trips you up
untold sun
What are the help channel names for
it's whoever claims it
elder aspen
Oh ok
elder aspen
Im new mb
elder aspen
Oh i just wanted to ask about the name thing; its the weekend but i definitely will try to ask for help on how sin cosin and tan works later
They confuse me
And complex numbers hurt my brain
untold sun
And complex numbers hurt my brain
just you wait. But, have a nice weekend!
elder aspen
Thanks you too
vocal sleetBOT
Available help channel!
tawny nacelle
I'm going insane: why does such a chart (W, theta) exist?
vocal sleetBOT
I'm going insane: why does such a chart (W, theta) exist?
tawny nacelle
isn't p already covered by both U and V?
why do we know there's going to be such a W?
nvm I'm a moron
.solved
vocal sleetBOT
tawny nacelle
nvm I'm a moron
uniqueness wasn't specified
we could have (U, phi) = (V, psi) = (W, theta) for all we care
vocal sleetBOT
Available help channel!
young aspen
Give me some hints
vocal sleetBOT
Give me some hints
inner osprey
Give me some hints
L = e^ln(L)
young aspen
L = e^ln(L)
Idk but now I see the real answer is 2e^pi/2-2
torn timber
Idk but now I see the real answer is 2e^pi/2-2
what he meant was to let
$$L=\lim_{n \to \infty}\left[\Pi_{k=1}^{n} \left(1+\frac{k^2}{n^2}\right)\right]^{\frac{1}{n}$$
twin meteorBOT
convergence
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
vocal sleetBOT
@young aspen Has your question been resolved?
marble steeple
i cna solve this to get pi/3 and 2pi/3 but there are still two more solutions of pi/2 and 3pi/2 and i don't konw how to get that
vocal sleetBOT
i cna solve this to get pi/3 and 2pi/3 but there are still two more solutions of...
wind geyser
how did you solve it?
wind geyser
When you cancelled $\cos\theta$ you assumed $\cos\theta\neq0$
twin meteorBOT
@wind geyser
