which is a total mess that I'm not sure how to deal with
#help-27
devout snowBOT
@gleaming grove Has your question been resolved?
crimson drift
gleaming grove
that's just the partial of (2x + y + z - 2) w.r.t. x multiplied by its sign to account for the abs, right?
to avoid the messiness of that I thought to split it into two parts
devout snowBOT
@gleaming grove Has your question been resolved?
devout snowBOT
@gleaming grove Has your question been resolved?
gleaming grove
I'll give it a shot
devout snowBOT
@gleaming grove Has your question been resolved?
devout snowBOT
@gleaming grove Has your question been resolved?
devout snowBOT
@gleaming grove Has your question been resolved?
tame parrot
f(x,y,z) is not differentiable at 0 so maybe you should consider finding extrema of $$(2x+y+z-2)^2$$ instead? Also note that the two constraints are symmetric wrt y and z, $$y^2 - xz - (z^2-xy) = 0$$ gives you some nice cases to try
woven radishBOT
5kbps
devout snowBOT
@gleaming grove Has your question been resolved?
supple knot
.close
devout snowBOT
devout snowBOT
Available help channel!
grizzled lagoon
devout snowBOT
@grizzled lagoon Has your question been resolved?
sharp holly
Is someone able to help me with this?
devout snowBOT
Is someone able to help me with this?
tough dust
With what?
devout snowBOT
@sharp holly Has your question been resolved?
echo owl
$$ \tan \frac \pi 6 = \frac 1 \sqrt 3 $$
woven radishBOT
jaydamani
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
frank summit
devout snowBOT
frank summit
can someone help with that
devout snowBOT
@frank summit Has your question been resolved?
devout snowBOT
@frank summit Has your question been resolved?
bleak grail
How can I find question a and b
devout snowBOT
How can I find question a and b
silk zephyr
Do I have use pythagoras to find OT
Yeah, that's what the question asks
bleak grail
I got square root 16^2 - 8^2 = 13.85
silk zephyr
Yup
bleak grail
But the answer is 6cm?
For both a and b
main gull
For part a, you didn't even apply Pythagorean theorem
bleak grail
For part a, you didn't even apply Pythagorean theorem
I did what do you mean?
main gull
For both a and b
Should be the same for both, since it asks for the radius using two different methods
main gull
I did what do you mean?
No you didn't
What's Pythagorean theorem?
What is the formula?
bleak grail
What is the formula?
a + b = c
silk zephyr
It should be PR.PQ=PT²
4.(4+2x)=8²
silk zephyr
The secant is the whole PQ and not just PO as you did over here
main gull
With the little 2 on top
Plug in the values
You know one of the sides, the other in terms of x, and the hypotenuse
bleak grail
Plug in the values
Is x correct in the image
bleak grail
It should be PR.PQ=PT²
4.(4+2x)=8²
That's what I did to find x
main gull
x is the unknown
bleak grail
x is the unknown
Yeah I labelled it as a pronumeral
to find the other side of the triangle
main gull
Use that and Pythagorean theorem to find x
silk zephyr
x is the unknown
You did (4+x), I'm saying it's (4+2x)
bleak grail
You did (4+x), I'm saying it's (4+2x)
Wait 2x from where?
bleak grail
Use that and Pythagorean theorem to find x
I got 12
Bro why is everyone offline
.close
devout snowBOT
silk zephyr
It should be PR.PQ=PT²
4.(4+2x)=8²
@bleak grail
bleak grail
.reopen
devout snowBOT
✅
bleak grail
<@675552040050753539>
I got x=6
bleak grail
<@675552040050753539>
Is 2x double the sides of x RO = OQ
silk zephyr
Is 2x double the sides of x RO = OQ
Yeah, RO = OQ, both are the radii of the circle
bleak grail
Yeah, RO = OQ, both are the radii of the circle
But I have to correspond this to a and b
silk zephyr
But I have to correspond this to a and b
So for part b, you are using the secant property right?
minor crypt
can you help me with a math problem? like help me determine the formula not the answer
bleak grail
So for part b, you are using the secant property right?
Yeah
silk zephyr
can you help me with a math problem? like help me determine the formula not the ...
This channel is in use, use a different channel from math help (available) channel
minor crypt
ohh okay sorry abt that
bleak grail
Yeah, so the secant is PQ
Yeah because a secant is a straight line cutting a curve at two or more points of a circle
silk zephyr
Yeah because a secant is a straight line cutting a curve at two or more points o...
Yup
So what's your doubt exactly
The formula is PT²=PR.PQ for part b
Which you used and you got the correct answer
bleak grail
that's good but how am I suppose to use Pythagorus in order to get the solution
silk zephyr
And for part a, take OR as x
Therefore OT is also x (Since both are radii)
And by pythogoras PO²=PT²+OT²
bleak grail
And for part a, take OR as x
Therefore OT is also x (Since both are radii)
And ...
I got 10
bleak grail
And for part a, take OR as x
Therefore OT is also x (Since both are radii)
And ...
squared root 8^2 + 6^2
silk zephyr
Gimme 1 min
bleak grail
You need to do something? in IRL
Okay I'm waiting but I don't have long until I go sleep
silk zephyr
And then I minus it from 4 Giving me 6
Yup, that's right
silk zephyr
You need to do something? in IRL
Yeah lol
silk zephyr
So my proof of x was wrong then?
Which one?
silk zephyr
Now you're done with both the parts right?
bleak grail
Yeah but I just got some questions that's all
silk zephyr
Now you're done with both the parts right?
Yeah that's cool, my internet is a bit wonky that's it
bleak grail
Yeah that's cool, my internet is a bit wonky that's it
Oh ok
silk zephyr
Yup this one is wrong
It should be
8²=(4+2x).4
2x since you have to take PQ and not PO
bleak grail
Yup this one is wrong
Oh that makes sense because I thought of it as a whole line but x is just the radius
silk zephyr
Oh that makes sense because I thought of it as a whole line but x is just the ra...
Yup yup
remote hornet
how do I find the minima of $x^{2}-\sin5x$
devout snowBOT
how do I find the minima of $x^{2}-\sin5x$
woven radishBOT
ホタル
remote hornet
y' is 2x-5cos5x
small field
,w plot $x^{2}-\sin(5x)$
woven radishBOT
pallid vector
,w 5*cos(5x) = 2x
woven radishBOT
pallid vector
what can I do from here?
You We can't
remote hornet
could I use the taylor approximation for cos5x?
tough dust
You need to find the minima
What do you need to do to find the extrema of any function?
remote hornet
y' = 0
tough dust
yeah
small field
tough dust
Oh then whats the problem?
small field
π/10 lol
remote hornet
need to solve cos5x=2x/5
remote hornet
huh?
tough dust
need to solve cos5x=2x/5
without a calculator?
small field
bro
tough dust
if you wanna approx it then taylor is a solution yeah
tough dust
Yeah thats a correct taylor polynomal
remote hornet
12.5x^2+0.4x-1=0
tough dust
but idk if youre supposed to solve it with trig or with taylor
remote hornet
but idk if youre supposed to solve it with trig or with taylor
I don't think it's possible to solve with trig
tough dust
it is but then you need to go with the complex notation etc its pretty complicated
tough dust
12.5x^2+0.4x-1=0
that is correct
tough dust
perfect
remote hornet
that gives me x = -0.3, 2.7
I only got 2 solutions from this, but y' = 0 has more than 2 how could I guarantee that the 2 solutions I get will have the global minimum?
tough dust
If you think you dont have enough solutions, it means that you need to expand your taylor polynomal
tough dust
Wait give me a sec, ill try to get the answer myself first
tough dust
that gives me x = -0.3, 2.7
Are you sure about these solutions
remote hornet
,w solve 12.5x^2+0.4x-1=0
woven radishBOT
tough dust
You see the thing is you have 5 extrema in your function
So a Taylor polynomal of degree 2 is not enough
remote hornet
You see the thing is you have 5 extrema in your function
how am I supposed to know that without wolfram tho
💀
tough dust
how am I supposed to know that without wolfram tho
good question
You can use nothing but pen and paper?
wooden veldt
where is this question from? who's making you find ugly extrema
tough dust
This is ridiculous to solve without a calculator
unless youre focussing on writing trig in exponential form and solving it that way
remote hornet
where is this question from? who's making you find ugly extrema
a friend 💀
tough dust
a friend 💀
these people they arent friends
wooden veldt
well then stop wasting your time lmao
tough dust
just tell him its almost impossible without a calculator
short nacelle
I’m trying to find the equation of the line passing through 5,-3 and -4,8
short nacelle
I know I get y=-11/9x+b
I am forgetting how to find the y intercept when 0 is not given
restive river
substitute
short nacelle
So 8=-11/9(-4)+b
I end up with 8=44/9+b
I had that and wanted to make sure but I’m lost now
Do I multiple both sides by 9 to get rid of the denominator?
restive river
u just find for b
short nacelle
Is it 28?
I brought the b to one side so I got 8-44/9
Multiply 8 by 9 because of the denominator
So I got 72-44/9
restive river
yes so b = (72-44)/9 = 28/9
fossil moth
devout snowBOT
fossil moth
i just have a question when it comes to deriving this
the answer I came to is 1/((10)(.5x^.5))
like this
how is my answer differing than the actual answer
fossil moth
i changed the square root of x to x to the power of 1/2 than brought it down, subtracted one from the exponent and than derived the other x by the same thing leaving me with just 1
I kept the bottom as 10 and did not change it
wooden zodiac
$\frac{1}{10} + \frac{1}{20\sqrt{x}}$
woven radishBOT
$\frac{\tau \alpha lo}{gist}$
wooden zodiac
this is what u should become splitting into two fractions
1/(20sqrt(x)) can be rewritten as
$\frac{\sqrt{x}}{20x}$
woven radishBOT
$\frac{\tau \alpha lo}{gist}$
wooden zodiac
my answer should be correct u can verify it
I have no idea how u came to your answer
fossil moth
this is whats on the textbook
but i can understand a little better why the did that
just for context, this is what the question is (7B)
so i was trying to derive U
wooden zodiac
did u find where u did wrong
orchid pike
Can someone graph this logarithmic equation
devout snowBOT
Can someone graph this logarithmic equation
orchid pike
vagrant mural
,rotate
woven radishBOT
vagrant mural
what's the problem?
woven radishBOT
orchid pike
I’m not sure if everyone uses this method
but I put the parent form (for exponentials) -1, 0, 1 for x then the 1/b, 1, b, for y. Then I reversed x and y
Then I added h to x and multiplied y by a and added k
(also I slashed the first one cuz there was decimal)
is mine right? or does it depend on what dots you put
Teacher’s
<@&286206848099549185>
devout snowBOT
vagrant mural
okay ig
devout snowBOT
Available help channel!
graceful dock
Hi, I have two concurrent (same times) series of rotation matrices of an object rotating in two different coordinate systems. Is there a way to align them? I tried finding the rotation between the first element of each series, but it doesn't seem to work
devout snowBOT
Hi, I have two concurrent (same times) series of rotation matrices of an object ...
remote hornet
Hi, I have two concurrent (same times) series of rotation matrices of an object ...
wdym by align them
graceful dock
so I have a recording of an object in two different coordinate systems
I want to align them, as in find a matrix to rotate an orientation from one system to the other
the rotations recorded should match up closely (some error up to 10 deg expected)
during the recording there is also a long section where the object is not moving, so the rotations estimated are quite accurate, if that helps
devout snowBOT
@graceful dock Has your question been resolved?
devout snowBOT
@graceful dock Has your question been resolved?
devout snowBOT
@graceful dock Has your question been resolved?
devout snowBOT
small field
🤨
devout snowBOT
warped crescent
. reopen
. reopen
wtf i did, <@&286206848099549185> its ok or should i move to other channel?
pseudo basin
move to another channel so the bot can pin your question
echo owl
It's already closed
devout snowBOT
Available help channel!
vale wadi
how would i show this subtraction method in symbolic algebra o.o
vale wadi
would it be correct to write something like
x+a - b-a
small field
$$69 x = 420 \Rightarrow x=\frac{420}{69}$$
woven radishBOT
Chunkin
vale wadi
the task was to show this method working with symbolic algebra
so i guess it isto write a formula to show that it works
devout snowBOT
@vale wadi Has your question been resolved?
vale wadi
<@&286206848099549185> is it possible to make it into symbolic algebra
devout snowBOT
@vale wadi Has your question been resolved?
devout snowBOT
@vale wadi Has your question been resolved?
devout snowBOT
@vale wadi Has your question been resolved?
restive river
<@&286206848099549185> is it possible to make it into symbolic algebra
What is symbolic algebra
devout snowBOT
Available help channel!
celest patrol
devout snowBOT
celest patrol
I need help in maths
fading grail
from #❓how-to-get-help :
• Don’t ask permission to ask a question, or ask if anyone knows how to solve a type of problem. Just ask your question.
jagged ore
so there is a relationship with length and width
if 2/3 of length is equal to 3/4 of its width
and u defined length as l and width as w
what do u think u can do?
jagged ore
and btw can u send solutions just to make sure im teaching this right?
devout snowBOT
@celest patrol Has your question been resolved?
devout snowBOT
@celest patrol Has your question been resolved?
trim nebula
Without calculations how can i prove that BC is equal to CD
devout snowBOT
Without calculations how can i prove that BC is equal to CD
timber patio
Try using geometry
trim nebula
I tried but I cant figure out how to prove that
timber patio
You know that BA=AD=AC, right?
trim nebula
indeed
timber patio
Then you should try proving that the 2 triangles are equal, because if you do that you also prove that BC=CD
random lantern
What function gives a curve like this?
devout snowBOT
What function gives a curve like this?
random lantern
It's used when calculating changes in population
And this is why I should buy myself some of the past math textbooks I've used, because I can't remember anything
I thought it was like y=x(200-x) but that can't be it because that makes no sense. I also tried with y=x(x-200)
weak tundra
And this is why I should buy myself some of the past math textbooks I've used, b...
maybe a logistic curve fits your purposes. it is of the form y = A/(B+Ce^(-Dx))
weak tundra
It was like x times something
kk
random lantern
It was completely symetrical
Or like
The further the middle you go, the higher inclination it gets
weak tundra
I'm thinking of a sigmoid function
i suggested it but they said it was much simpler 🕵️
maiden briar
Oh right that was it ok
random lantern
Yeah it LOOKS like it, but it wasn't 1 divided by a cluster of e's and x's.
weak tundra
Yeah it LOOKS like it, but it wasn't 1 divided by a cluster of e's and x's.
lol that's no cluster of e and x. it's just one e, and one x. y = 1/(1 + e^-(x-5)) 🕵️
random lantern
Yeah but it was not that
I think the derivative of the function can be written on the form of y=x * (k - x) or something.
weak tundra
I think the derivative of the function can be written on the form of y=x * (k - ...
that's not possible. a polynomial goes to infinity or -infinity when x goes to infinity, which isn't possible for your function because it tends to a finite value
random lantern
That's true.
random lantern
I am not deliberately lying but I have severely misremembered something.
weak tundra
I am not deliberately lying but I have severely misremembered something.
oh okay, but anyway, it can't be a polynomial. maybe you're thinking of dy/dx = ky(1-y) or something like that. this has an exponential function as solution
random lantern
One example of real life use was the number of pheasants on an island. They're brought there, and the number slowly increases becomes higher, and then the increase comes to it's peak, only to increase slower and slower until they reach a point where they don't increase at all and there's just a horisontal line again.
weak tundra
One example of real life use was the number of pheasants on an island. They're b...
sounds like dy/dx = ky(1-y), where y is the population (this one has an exponential solution)
you've misremembered it as dy/dx = kx(1-x), which isn't the same (this one has a cubic solution)
a population would have a carrying capacity, so it can't be unbounded. so a cubic can't be a good model
random lantern
That's would make sense!
Yeah I thought the cubic model did not make sense because it went into the infinite positive
I think I need to figure out the rest on my own from now, so thank you very much for helping me out
!close
devout snowBOT
@random lantern Has your question been resolved?
supple raven
slender badger
I am struggling with this proof, particularly visualising it. If i can visualise it maybe it can help with the proof? maybe I am wrong but i have been trying for so long 😦
devout snowBOT
I am struggling with this proof, particularly visualising it. If i can visualise...
@slender badger Has your question been resolved?
leaden coral
Hello how to properly solve this, my friend said it's wrong
devout snowBOT
Hello how to properly solve this, my friend said it's wrong
@leaden coral Has your question been resolved?
leaden coral
<@&286206848099549185>
crystal rune
@leaden coral You can write it as $g(x) = - \int_{2}^{\frac{1}{x}} f \left ( \frac{1}{t} \right ) \dd{t} + \int_{0}^{\sqrt{x}} 2f(t^{2}) \dd{t}$
woven radishBOT
Sup?
crystal rune
Now apply FTOC
leaden coral
How should I properly apply FTOC or FTC?
crystal rune
You will have g'(x) = -f(1/(1/x)) + 2f(sqrt(x)^2)
leaden coral
oh
crystal rune
Well basically, you just remove the integral sign and plug in the variable bound in your integrand
leaden coral
You will have g'(x) = -f(1/(1/x)) + 2f(sqrt(x)^2)
so just solve it then as it should be? like substituting without changing the function
crystal rune
Just simplify a bit
so that's part (a)
for part (b), you will need to use g'(1) = 4
leaden coral
so I just typed the equation, so simply we need to just substitute the upeer bound and just let it be
leaden coral
You will have g'(x) = -f(x) + 2f(x)
oh
crystal rune
$\frac{1}{\frac{1}{x}} = x$
woven radishBOT
Sup?
crystal rune
$(\sqrt{x})^{2} = x$
woven radishBOT
Sup?
leaden coral
I get it now
crystal rune
Okay nice
leaden coral
so for b, just g'(1)=4 and based on the equation from item (a) we should have f(1)=4
crystal rune
Yes
leaden coral
hey @crystal rune, I have another question that I am stuck with too, do you have the time?
crystal rune
Sure
crystal rune
Np
crystal rune
First sketch
leaden coral
I will do Region 1 (R_1)
leaden coral
Can you do R2?
crystal rune
Yes
The left area
Now just use integration
Now I'm not sure if they want the net signed area or just the net area
cause the first area will be negative, second one positive
leaden coral
crystal rune
Why are you subtracting x in the integral?
leaden coral
because of the upper function minus the lower function, I think
crystal rune
Just integrate f(x) from -1 to 0, take absolute value, and then add the integral of f(x) from 0 to 4
There is no other function
leaden coral
Now I'm not sure if they want the net signed area or just the net area
to clarify maybe just the net area
crystal rune
Just integrate f(x) from -1 to 0, take absolute value, and then add the integral...
.
leaden coral
crystal rune
Or that
You don't need to take the absolute value now
First integrate
Then take the absolute value of w/e number you get
leaden coral
Oh okay I would do that first
celest patrol
.reopen
leaden coral
crystal rune
integral is -(x^3 / 3) + 3x^2
crystal rune
I have to go, I hope you can do this yourself
crystal rune
If it's negative
Then you don't need the absolute value
Then just integrate directly from -1 to 4
$\int_{-1}^{4} (-x^2 + 6x) \dd{x}$
woven radishBOT
Sup?
leaden coral
okay @crystal rune thanks for the help
devout snowBOT
devout snowBOT
Available help channel!
night ingot
Can someone help me on this
devout snowBOT
Can someone help me on this
orchid sierra
Hey @night ingot y, is this a test or a quiz?
night ingot
Nah its HW
orchid sierra
K
night ingot
Its my last one on the hw but i really dont understand or get it
orchid sierra
What shape is the base?
orchid sierra
Ok
So spoiler, I dont remember if the area formula holds for nonsymmetric trapezoids (I am almost sure it does though)
So let's think about how to make that shape using simpler shapes
What easier shapes would make that shape?
night ingot
for the 4 faces we can use a rectangle
rectangles*
for the bases we can use a trapezoid?
im not even sure if its a trapezoid
orchid sierra
focus only on the trapezoid
night ingot
Ok
orchid sierra
How can we make that trapezoid using easier shapes?
orchid sierra
Yes
night ingot
Or square
orchid sierra
One tri and a rec
orchid sierra
Tell us the dimensions of the rectangle, and the dimensions of the triangle
night ingot
Ok I think im wrong about the bases being trapezoids
Im so confused the figure is throwing me off
because on one hand it looks like a square but the top is slanted
so it cant be
Are you sure its a trapezoid because im not sure at all
orchid sierra
Just read the dimensions you already have
night ingot
15 is length?
night ingot
Im not sure
orchid sierra
When you cut the side of length 12, how much was left over for the triangle?
night ingot
Yeah im sorry im not understanding because i dont know to cut from
orchid sierra
This is one of the many problems with online math homework
Let's draw the figure
Like, draw it. On a napkin or whatever. Ill wait
night ingot
Im doing it on paper
my hw
Its just my teacher posted it online
Ill snap a pic
orchid sierra
Ok, let's draw a "cut" ( line segment) parallel to the side of length 15, so that our top piece is split into a rectangle and a right triangle
night ingot
When I think of a trapezoid I think of this
orchid sierra
Like that, kinda
night ingot
Ok
Think I did it
So length 15 should be the length of the right linesegment of the trapezoid?
orchid sierra
Yes, it would be
But how much width belongs to the rectangle and how much for the triangle?
orchid sierra
Uhh 8 is width
You got the width of the rectangle earlier
So how much does the triangle have
Width
orchid sierra
I drew it like the figure we were given
We also could have just drawn the top trapezoid or whatever
You were doing that, it looks
Anyway, let's get the areas of these new shapes.
Tell me when you got it.
I forgot to draw the height:
night ingot
Is the area of the trapezoid 10
orchid sierra
No
Also I was asking you to calculate the areas of the rectangle and the right triangle in the figure above
orchid sierra
Check the triangle again
night ingot
30
orchid sierra
Ok, so what is the area of the trapezoid?
night ingot
150?
orchid sierra
Great
night ingot
Tysm I just didnt know how to split the shapes properly I guess
orchid sierra
Yup
Keep good notes
Copy the general formulas before you use them
And you will memorize them.
night ingot
Ty man
orchid sierra
Yup
knotty jolt
Is this what this question is asking for
devout snowBOT
Is this what this question is asking for
hybrid snow
Yeah
wild creek
i think you need to actually compute the integral tho
knotty jolt
Yeah I didn’t solve it. Just wondering if I set that up right.
devout snowBOT
devout snowBOT
Available help channel!
lean socket
How do you get to this result?
devout snowBOT
How do you get to this result?
devout snowBOT
@lean socket Has your question been resolved?
devout snowBOT
@lean socket Has your question been resolved?
devout snowBOT
@lean socket Has your question been resolved?
devout snowBOT
devout snowBOT
Available help channel!
candid sequoia
devout snowBOT
https://math.stackexchange.com/questions/1677094/prove-the-following-boolean-ide...
candid sequoia
the expression that the person gets at the end is the same expression i get
but how does it simplify to the last four terms is my question
wooden veldt
What's wrong with the answers from the link you sent?
candid sequoia
it doesn't simplify to the last expression that needs to be proven
one step is missing, which I'm stuck at
i couldn't find it anywhere else on the internet
devout snowBOT
@candid sequoia Has your question been resolved?
devout snowBOT
@candid sequoia Has your question been resolved?
merry stream
I'm stuck one sec
devout snowBOT
I'm stuck one sec
merry stream
I tried using integral test but
I couldn't figure out how to integrate it
would a basic comparison test work better?
eager fox
are you trying to find what p values make the series converge?
merry stream
no I'm trying to find if it converges
and show what test I used
like I know it converges bc its a p series
eager fox
you have a whole family of series there
patent tartan
what did you try to integrate?
merry stream
what did you try to integrate?
using the integral test
I'm trying something else
so I'm going to guess its covnergent bc it looks like a convergent p series
and if I use another series that asymptotitc and less than our given series
patent tartan
yeah whats the problem with the resulting integral?
merry stream
yeah whats the problem with the resulting integral?
oh I'm just not great at integration so the exponent of p throws me off
patent tartan
$\int \frac{1}{x (\ln x)^p} \dd{x}$
merry stream
...
woven radishBOT
nvx
merry stream
oh
okay but
I think I found success with basic comparison test just now
is that possible?
like my work is
patent tartan
comparison should work as well yea
merry stream
i) guess convergent bc an looks like convergent p series
ii) find bnthats asymptotic to an and bn>an (in this case bn=1/(lnn)^p with p>1
iii) bn converges definitely because p>1
iv) therefore an must converge as well
yes
can a non alternating series have absolute/conditional convergence?
merry stream
shit how do I check for absolute convergence
devout snowBOT
@merry stream Has your question been resolved?
keen sundial
all of the above
keen sundial
try substituting u = ln(x)
do this
keen sundial
what would you do if it was 2?
what would you do if it was 1.0001?
whatever you would do to those powers, do the same to p
keen sundial
that is incorrect
merry stream
yeah wait
keen sundial
how do you integrate $$\int \frac{1}{x^2}, dx$$
merry stream
u^-p+1
woven radishBOT
JamesH
merry stream
yes
keen sundial
otherwise you might have to use natural log, for instance, in the case where p = 1
ok so now you can evaluate the improper integral
but make sure you either sub back in terms of x or change your limits to match your u-sub
merry stream
I hate integrals smh
um
lim t-->infinity [((lnx)^-p+1)/(-p+1)]
wait
no
[((lnx)^-p+1)/(-p+1)] with bounds 2 and t
so
keen sundial
since p > 1...
keen sundial
here is what we are looking at:
keen sundial
$$\lim_{t\to\infty} \frac{\ln(x)^{-p+1}}{-p+1} \bigg\vert_2^t$$
woven radishBOT
JamesH
merry stream
yes
keen sundial
which we can rewrite $$\frac{1}{1-p}\left(\lim_{t\to\infty} \ln(x)^{-p+1} \bigg\vert_2^t\right)$$
woven radishBOT
JamesH
keen sundial
since p > 1, what can we say about the exponent -p + 1
merry stream
merry stream
since p > 1, what can we say about the exponent -p + 1
its positive?
keen sundial
-2 + 1?
keen sundial
which means we can rewrite
$$\frac{1}{1-p}\left(\lim_{t\to\infty} \frac{1}{\ln(t)^{p-1}} - \frac{1}{\ln(2)^{p-1}}\right)$$
woven radishBOT
JamesH
merry stream
but the first term in the parentheses goes to 0
keen sundial
great
merry stream
so you're left with some random mystery number
keen sundial
do we care what that mystery number is?
keen sundial
the only thing we care about is whether it is finite
merry stream
which it definitly is
keen sundial
great
keen sundial
what did you plan to compare it to?
merry stream
isn't that what you want to show something is convergent?
keen sundial
no. you need to show that it is LESS than something convergent
merry stream
and less than to show comparison diverges
keen sundial
nope
keen sundial
LMAO
merry stream
I mean comparison test > integral test any day
merry stream
it can definitely work here
keen sundial
idk about that
keen sundial
no it doesnt 🙂
merry stream
keen sundial
the series we are looking at is an
if we can show that an > bn, where bn is convergent, then we can say an is convergent
merry stream
no it doesnt 🙂
okay I'm saying given our an I need to find bn such that bn>an and if bn converges so does an
you're saying I'm wrong
keen sundial
you may be saying the same thing actually, but backwards
merry stream
oh I mean to prove that an is convergent choose bn such that an<bn and they are asymptotic to one another
keen sundial
you need to show $$ \text{our series} \le \text{a convergent series}$$
woven radishBOT
JamesH
merry stream
yes thats what I'm saying
our series is an
and if our series is an=1/(n(lnn))^p then another convergent series would be bn=1/(lnn)^p and an<bn
keen sundial
so you want to compare to the series with $$b_n = \frac{1}{(\ln(n))^p}$$
woven radishBOT
JamesH
merry stream
**JamesH**
yes
keen sundial
does that series converge?
merry stream
for p>1 yes sir
keen sundial
how do you know?
merry stream
because p series
keen sundial
the p-test only applies to series of the form $\frac{1}{n^p}$
woven radishBOT
JamesH
keen sundial
and unfortunately, $\ln(n) \le n$
woven radishBOT
JamesH
keen sundial
for n > e
merry stream
you sir are playing with my emotions and I don't like it 😭
keen sundial
which means that $$\frac{1}{n} \le \frac{1}{ln(n)}$$
woven radishBOT
JamesH
keen sundial
which in turn, implies that $$\frac{1}{n^p} \le \frac{1}{ln(n)^p}$$
woven radishBOT
JamesH
merry stream
so I chose a bad second series
keen sundial
right
merry stream
okay so
keen sundial
this is why we cant use comparison
merry stream
I can't just tack on an n^n can I?
keen sundial
where do you want to tack it on?
merry stream
this is why we cant use comparison
okay on one hand I am dumb but on the other hand my professor almost eveyr class is like "don't do limit test its long and a pain in the butt" so my stubborness is coming from a real place
merry stream
where do you want to tack it on?
like just
1/[(n^n)(lnn)^p]
keen sundial
you would have to show that converges in order to use comparison
plus that thing is smaller than the one we are looking at, so it doesnt help
because it has a bigger denominator
merry stream
okay so in short, I need to give up
keen sundial
you need to just use integral test
merry stream
okay can I ask another question before you go
keen sundial
this is a classic example of a case where integral test is the right way to go
sure
merry stream
so like I do a root test
or I tried to
I got lim n-->infinity for (1-1/n)^n
and I know its the answer is going to be something involving e
like maybe e^-1
but did I start off correctly by doing the root test
keen sundial
it looks like yo udid
merry stream
my issue is the limit becomes weird
becuase you don't totally get rid of the n power
keen sundial
suppose that the limit exists, and denote it L
merry stream
is it possible to have like the root have an index of n^2?
to fully remove the power
or do I have to use lhopital
keen sundial
then $$ L = \lim_{n\to\infty} \left(1-\frac{1}{n}\right)^n$$
woven radishBOT
JamesH
keen sundial
take ln of both sides
merry stream
lnL=n*ln(1-1/n)
keen sundial
good
keen sundial
yes it is
merry stream
so you get e=n(1-1/n)
keen sundial
lnL=n*ln(1-1/n)
you forgot about the limit here
keen sundial
you are prob right
merry stream
how do you people know
keen sundial
by practicing and doing stuff a bunch of times
merry stream
like I was only given the tests, whetehr they're useful for terms or factors
distant harbor
Write 1 - 1/n as (n-1)/n then recall that exponentiation distributes over multiplication and division
merry stream
but what do you see that made you go "ratio test"
distant harbor
but what do you see that made you go "ratio test"
A ratio. Lmao
keen sundial
Write 1 - 1/n as (n-1)/n then recall that exponentiation distributes over multip...
that is clever
when you see a sum or difference of fractions things get ugly
combining into a single fraction makes your life much easier
you do still need to deal with that power of n^2, but its not that bad
distant harbor
you do still need to deal with that power of n^2, but its not that bad
Everything should cancel out nicely
merry stream
Everything should cancel out nicely
the beauty of ratio test
keen sundial
im writing down the ratio test and actually it also gets kinda ugly?
in particular, $$\left\vert \frac{a_{n+1}}{a_n}\right\vert = \left\vert \left(\frac{n}{n+1}\right)^{(n+1)^2} \cdot \left(\frac{n}{n-1}\right)^{n^2}\right\vert$$
woven radishBOT
JamesH
keen sundial
i know
distant harbor
Yeah then expand (n+1)^2
merry stream
can you add the exponents instead of distributing them?
like (n+1)^2 + n^2
oh you did that nvm
I'm dumb you're a genius
wait
I'm unsure about your final exponent for (n+1 in the denominator)
keen sundial
notice that i combined all of the n-1s with most of the n+1s to get n^2 -1 in the denom
whether that was useful or not is TBD
merry stream
I'm lost on your arithmetic
keen sundial
i think we would need a chain of inequalities to make this manageable, which may or may not be simpler than just doing root test
keen sundial
**JamesH**
this is pretty doable by rewriting the limit and using lhop
sonic trellis
I play tennis against a friend. The probability I win a point is a, and the probability she wins a point is b. These probabilities stay constant throughout the game. Also, a does not equal to b.
A game is only won when a player wins two consecutive points.
devout snowBOT
I play tennis against a friend. The probability I win a point is a, and the prob...
sonic trellis
.close
vital pike
devout snowBOT
vital pike
Not sure what exactly is wrong
I think RS being congruent to SP may not be true, but not sure why
graceful cosmos
RQP is similar to STP
(Assuming ST is parallel to QR)
That's a problem though.
torpid cairn
(Assuming ST is parallel to QR)
Not really
Just RP/SP=QP/TP and angle SPT=RPQ shows that both triangles are similar
SAS similarity specifically
graceful cosmos
Fair! Don't even need to assume then.
vital pike
Just RP/SP=QP/TP and angle SPT=RPQ shows that both triangles are similar
Oh okay that makes sense. I was wondering if we needed the lines to be parallel
So it is wrong because if PT = QT = x
Then 11/20 = x/2x
11/20=10/20
torpid cairn
Oh okay that makes sense. I was wondering if we needed the lines to be parallel
The lines are parallel
vital pike
Which is false
vital pike
The lines are parallel
How do we know that?
torpid cairn
Just RP/SP=QP/TP and angle SPT=RPQ shows that both triangles are similar
Using SAS we showed that both triangles are indeed similar. So, now we can say angle PST=PRQ which does show ST is parallel to RQ
torpid cairn
So the contradiction here is that 11/20 = 1/2 right?
Yep
buoyant ferry
Hi, me again. This is for my calculus course. I am to use direct substitution to show that direct substitution leads to the indeterminate form 0/0. i think understand this part; just plug in 2 where x is.
but i was wondering…all the videos i’ve watched solves the problem but don’t explain why the final figure is rarely ever equal to 0/0 after factoring. if i’m right, all these steps aren’t simple substitution. can someone explain that to me though please? is it because x≠a in this instance?
devout snowBOT
Hi, me again. This is for my calculus course. I am to use direct substitution to...
buoyant ferry
My course notes say the factoring are under a limit evaluation technique for indeterminate forms 0/0. but why all these steps that don’t result in 0/0? just trying to figure it out incase i am to use this technique.
torpid cairn
Hi, me again. This is for my calculus course. I am to use direct substitution to...
Yes. The factoring and cancellation is done on the basis of x≠2
All of that is done to obtain an expression which has the same limit as the original but does not result in 0/0 form at the limit point and additionally being continuous there
buoyant ferry
thank you!
devout snowBOT
@buoyant ferry Has your question been resolved?
kindred vigil
devout snowBOT
devout snowBOT
@kindred vigil Has your question been resolved?
devout snowBOT
kindred vigil
.reopen
devout snowBOT
✅
kindred vigil
<@&286206848099549185>
torn vessel
what triangle?
devout snowBOT
@kindred vigil Has your question been resolved?
kindred vigil
what triangle?
devout snowBOT
@kindred vigil Has your question been resolved?
kindred vigil
<@&286206848099549185>
devout snowBOT
@kindred vigil Has your question been resolved?
charred pebble
https://cdn.discordapp.com/attachments/968467663837863947/975591514723995668/IMG...
height and base of shaded triangle and the total triangle are given
calculate their areas
divide them and equate it to the ratio given
and solve for x
vague obsidian
Hey. I'm having a problem with cotangent. So when I put the cotangent in the calculator (I put (1)/(tan(1\sqrt{3}))), I get a diff. answer (99.smth).
devout snowBOT
Hey. I'm having a problem with cotangent. So when I put the cotangent in the cal...
vague obsidian
I see
wooden veldt
Because cot(x) = 1/tan(x)
agile narwhal
devout snowBOT
agile narwhal
how to prove this ?
wicked turtle
maybe compare the sum with the corresponding integral?
agile narwhal
its problem from exam
i doubt that we should use integrals
but anyway how it would work with integral ?
pseudo basin
the sum divided by sqrt(n) would be equal to 1/n sum[k=1,n] 1/sqrt(k/n), which converges to the integral of sqrt(x) dx from 0 to 1
agile narwhal
As sqrt(1/k)
?
agile narwhal
the sum divided by sqrt(n) would be equal to 1/n sum[k=1,n] 1/sqrt(k/n), which c...
how did you come up with this ?
pseudo basin
experience with riemann sums i guess 🤷♀️
balmy quest
$f(x) = \frac{1}{\sqrt{x}}$ is decreasing, so $\frac{1}{\sqrt{k}}=\int_{k}^{k+1} \frac{1}{\sqrt{k}} ,dx > \int_{k}^{k+1} \frac{1}{\sqrt{x}} , dx$
woven radishBOT
秋水
agile narwhal
and is there any way without integrals ?
balmy quest
you can use $\frac{1}{\sqrt{k}}=\frac{2}{2\sqrt{k}}<\frac{2}{\sqrt{k-1}+\sqrt{k}}=2 \left(\sqrt{k}-\sqrt{k-1} \right)$
woven radishBOT
秋水
balmy quest
$\frac{1}{\sqrt{k}}=\frac{2}{2\sqrt{k}}>\frac{2}{\sqrt{k}+\sqrt{k+1}}=2 \left(\sqrt{k+1}-\sqrt{k} \right)$
woven radishBOT
秋水
balmy quest
$\sum\left(\sqrt{k+1}-\sqrt{k} \right)$ ，the middle term will disappear
woven radishBOT
秋水
wicked turtle
telescoping sums ftw!
agile narwhal
telescoping sums ftw!
?
wicked turtle
?
a bunch of terms cancel, this is called telescoping. example when n=4: $\sum_{k=1}^{4}(\sqrt{k+1} - \sqrt{k}) = (\sqrt{2} - \sqrt{1}) + (\sqrt{3} - \sqrt{2}) + (\sqrt{4} - \sqrt{3}) + (\sqrt{5} - \sqrt{4}) = \sqrt{5} - \sqrt{1}$
woven radishBOT
OurBelovedBungo
wicked turtle
and in general, $\sum_{k=1}^{n}(\sqrt{k+1} - \sqrt{k}) = \sqrt{n+1} - \sqrt{1}$
woven radishBOT
OurBelovedBungo
wicked turtle
(in words, the length of a telescope equals the sum of the lengths of its individual segments)
wet coral
Integ of x/(x^2-9) from 0 to 5 diverges right?
devout snowBOT
Integ of x/(x^2-9) from 0 to 5 diverges right?
hidden bobcat
yes
still pendant
Help