you can write it as (9^x)((x^x)²)
#help-0
alpine sable
whole otter
What
alpine sable
you can write it as (9^x)((x^x)²)
ig theres where product rule comes from
alpine sable
a^b)^c = a^bc
whole otter
Oh
alpine sable
so u can switch em up
whole otter
x^2x
alpine sable
yup
whole otter
Absol now
alpine sable
id do (x^x)², get derivative of x^x then either use product rule or chain rule
then derivative of 9^x and another product rule
alpine sable
id do (x^x)², get derivative of x^x then either use product rule or chain rule
actually maybe x^2x is easy to differentiate like this too
hm
try it out
alpine sable
implicit differentiation
alpine sable
y= x^2x
lny = 2xlnx
oak perch
f(x)^g(x)=e^(g(x)In(f(x)). Chain rule first , inside product rule on the product g(x)ln(f(x)) I think that’s what the author meant by saying product rule
lone heartBOT
@whole otter Has your question been resolved?
tribal monolith
lone heartBOT
tribal monolith
how did i get this wrong
left isle
doesn't it want you to divide 5i+12j by the magnitude?
oak perch
That’s what I thought but it’s saying
Using the unit vectors, not write it as a unit vector
left isle
"express it using standard unit vectors"
idk i mean, it's a 5, 12, 13 triangle, they probably didn't do that coincidentally
tribal monolith
that still doesn't work
gleaming ridge
5i+12j
5 i + 12 j
5 * i + 12 * j
5î+12ĵ
5î + 12ĵ
it's correct, i think it's just the format
Try
Especially the two last ones, because the unit vectors have a hat
lone heartBOT
@tribal monolith Has your question been resolved?
lone heartBOT
lone heartBOT
Available help channel!
simple saddle
lone heartBOT
simple saddle
why subtracting 2^8 -1
vale wigeon
to disallow an empty icecream cone
i.e. a cone without any toppings
because then you have no ice cream at all and there is no point
simple saddle
so is it the one case where all are NO so removing this: 1x1x1x1x1x1x1x1
vale wigeon
sure if you want
simple saddle
sure if you want
can you alos apply permutation in such cases?
if they ask in question obviously
vale wigeon
would have to see the question in order to tell you.
simple saddle
would have to see the question in order to tell you.
so like how many arrangements of icecream toppings possible kind of this
vale wigeon
arrangements?
simple saddle
arrangements?
is it wrong?
vale wigeon
so like you don't just put the ice cream scoops in a cup but instead stack them?
that would be 8P1 + 8P2 + ... + 8P8 maybe.
simple saddle
so like you don't just put the ice cream scoops in a cup but instead stack them?
no like which topping comes first which comes second likewise
simple saddle
that would be 8P1 + 8P2 + ... + 8P8 maybe.
oh
hidden pier
how do i integrate f(g(x))
lone heartBOT
how do i integrate f(g(x))
mortal trellis
in general you don't
muted relic
is there a question like that
vale wigeon
muted relic
you can just expand the function im pretty sure
muted relic
you can use a useful trig identity
hidden pier
to expand sin^2?
muted relic
sin^2x = 1/2 - 1/2cos2x
muted relic
yeah and that is easy to integrate
similar for cos^2x you need to memorise basically
hidden pier
alr ty
last ether
1-cos2x-sin^2x
I mean you'll eventually get the same thing he was talking about
You're just having an extra step
vale wigeon
why is it that in general you dont
sin(x^2) has no elementary antiderivative.
last ether
But e^(2x) has an elementary antiderivative
mortal trellis
(where elementary means that you can write it as a combination of all the usual functions)
last ether
f(g(x)) is not guaranteed to be elementary when integrated
urban hazel
I need help with this
lone heartBOT
I need help with this
urban hazel
lone heartBOT
@urban hazel Has your question been resolved?
dreamy pendant
i ned help
lone heartBOT
i ned help
dreamy pendant
what 7x6x4
echo socket
,w 7 * 6 * 4
ocean sealBOT
echo socket
.close
elfin vector
Hi! Can someone pls help me check this? Not sure if it's correct, thx.
lone heartBOT
Hi! Can someone pls help me check this? Not sure if it's correct, thx.
@elfin vector Has your question been resolved?
elfin vector
<@&286206848099549185>
elfin vector
.close
lone heartBOT
lone heartBOT
Available help channel!
vagrant agate
how do i graph this?
lone heartBOT
how do i graph this?
tepid drum
find asymptotes extremas intersections with y and x axes along with concavity and inflection points if there are any
then plot them and connect
these will be more than enough
vagrant agate
what does the root 2 do to the graph?
tepid drum
plot tan x
to plot tanx you should find things like what i said above
cerulean talon
lone heartBOT
cerulean talon
this is the answer
i’m unsure of the process it goes through to get to this point
cerulean talon
definition of the logarithm
what you mean by this?
upbeat hornet
what you mean by this?
Do you know how the logarithm is defined?
cerulean talon
no
vale wigeon
$\log_a(x)$ is the answer to the question ``to what power must $a$ be raised to get $x$?''
ocean sealBOT
Ann
vale wigeon
or to put this another way
$\log_a(x)=y$ is \textbf{by definition} equivalent to $a^y = x$
ocean sealBOT
Ann
vale wigeon
and in your case you are dealing with the common logarithm, which is log_10
cerulean talon
wait but to move the log to the left side
vale wigeon
no
cerulean talon
wouldn’t you divide it?
vale wigeon
there is no such thing as "moving" anything anywhere,
cerulean talon
wait
vale wigeon
and also log is a function and not a number.
cerulean talon
move the base of the log to the other side and push whatever to the power
vale wigeon
log(R/S) is not the product of R/S with some mystical number called "log"
vale wigeon
move the base of the log to the other side and push whatever to the power
very mechanistic, and thus rotten, description.
vague coral
but it works
but thats not how it works in reality lol
cerulean talon
i think i understand it slightly.. not really tho
vague coral
it might be an easier way to explain but thats not how it works
cerulean talon
how would u define it then?
vale wigeon
see my above msgs
vague coral
^
vale wigeon
or ask me "I choose not to read things. Can you repeat this again 17 more times, please?"
cerulean talon
oh i see
cerulean talon
or ask me "I choose not to read things. Can you repeat this again 17 more times,...
i’m sorry
i read what you said but i didn’t understand it
looking at it now i understand it
okay i think i fully understand it now, it can be converted both ways
i see
thank you for the help @vale wigeon
hope you have a good rest of your day :)
.close
elfin vector
Hi! Can someone pls check if this is correct? Thx.
lone heartBOT
Hi! Can someone pls check if this is correct? Thx.
@elfin vector Has your question been resolved?
sweet igloo
Could I please have some help with Q3 no idea how to start
lone heartBOT
Could I please have some help with Q3 no idea how to start
upbeat hornet
Could I please have some help with Q3 no idea how to start
Do you know what double integer root means?
sweet igloo
one of the factors is (x+k)^2?
sweet igloo
ok so its (x+a)^2(x+k) as it is to a degree of 3?
upbeat hornet
ok so its (x+a)^2(x+k) as it is to a degree of 3?
k is already used, you could use b
torn knoll
Which one has to be solved
sweet igloo
oh ok im being stupid
torn knoll
3rd or 2nd?
upbeat hornet
3rd or 2nd?
Q3
torn knoll
OK have u solved or should I?
upbeat hornet
ok so its (x+a)^2(x+k) as it is to a degree of 3?
Then, you can multiply out and equate coefficients
You should be able to write a,b, and k in terms of each other
and then solve for each of them
sweet igloo
ok i'll give it a go
torn knoll
First u will have to do hit and trial method
upbeat hornet
First u will have to do hit and trial method
Not necessary
oak perch
He doesn’t have to
upbeat hornet
They already have the cubic factored
and just need to find a and b by expanding and considering coefficients
torn knoll
So use alpha beta gamma
oak perch
He doesn’t have to, what rysro… said is already on the right track
upbeat hornet
So use alpha beta gamma
Notice that the question says that two of the roots are equal, so without loss of generality, alpha is equal to gamma
and then you’re doing what I said
torn knoll
It is lengthy
oak perch
I finished calculating, we are just supervising the person asked to do it
sweet igloo
yeah I got it horribly wrong
upbeat hornet
yeah I got it horribly wrong
can you show your work?
sweet igloo
I just erased it because the number was so far off
but how do I do it with alpha, beta, gamma because that's what we have been doing in class, if you would'nt mind explaining
oak perch
You just compare coefficients as he said
(Substitute b with an expression of a, you will get a factorization of -7 in the end)
sweet igloo
so k = -3?
oak perch
Yes
tacit scroll
lone heartBOT
tacit scroll
within the final term, should it not be n=1
oak perch
within the final term, should it not be n=1
Yes
tacit scroll
thx
tacit scroll
.close
alpine sable
Hi! I'm creating a curve(or more, getting points on a curve) using this formula:
curveSegments = curveAngle / segmentDistance
step = curveAngle / curveSegments / 180 * PI
x = radius * sin(i * step)
y = radius * cos(i * step)
Where i is a number from 0 to curveSegments to create a curve of the points.
The above works perfectly fine, however I would like to introduce another parameter bend.
bend would be how far we've bend the curve, between 0 and 1.
If bend is 0, I want a straight line of points ony and x would stay on -radius .
If bend is 1, I want the curve to be unchanged.
I have no idea how to change the formula to accomplish this, I've played around with the values for the whole morning, but I don't think I understand what needs to happen to do this. Could somebody point me in the right direction? thanks!
lone heartBOT
Hi! I'm creating a curve(or more, getting points on a curve) using this formula:...
@alpine sable Has your question been resolved?
crisp pine
I don't quite follow what you are trying to do, could you some more details/maybe a picture
alpine sable
I don't quite follow what you are trying to do, could you some more details/mayb...
Drawing it in paint
Trying to get a point on a curve provided the angle and radius and segment distance (curveSegments is how many points the curve consists of, I get that using the distance that should be between the segments segmentDistance)
alpine sable
I want some confirmation
This channel is occupied 😄
crisp pine
!help
lone heartBOT
Please read #❓how-to-get-help
crisp pine
Have a look at that if u haven't got help before ^
crisp pine
Drawing it in paint
Trying to get a point on a curve provided the angle and radi...
What is the angle here, is the curve a segment of a circle
alpine sable
What is the angle here, is the curve a segment of a circle
yup
it's a part of a circle
alpine sable
the light blue points I get now, and that is what you would get if bend is 1
and the red ones if bend is 0
so the red line of points is the same lenght as the light blue line
crisp pine
The image is taking a while to load for me, gimme a sec
Okay yeah
So what would you want if bend = 0.5
alpine sable
yup
crisp pine
So after traveling bendx100 percent along the line
We can calculate the tangent to the circle at that point based on the angle
And then continue on with a straight line
Does that make sense?
alpine sable
mhm
crisp pine
May I ask what you are trying to code? I'm assuming this is programming of some description
alpine sable
indeed, working on visualizing the process of bending metal bars
I'm a bit new to these kinds of math
crisp pine
Ohh cool okay so you are showing it wrapping around the circle
Does my explanation make sense?
alpine sable
it does- though I have no idea how to make that into a formula 😄
curveSegments = curveAngle / segmentDistance
step = curveAngle / curveSegments / 180 * PI
x = radius * sin(i * step)
y = radius * cos(i * step)
This would do half of the work already- get a point on the line around the circle at that step
but you can't put if statements in a formula, and I need to stop curving when we've hit the percentage, and continue in the direction of the tangent
crisp pine
Why does it have to be a formula exactly?
Making it a formula is going to require some shenanigans but I can have a think
alpine sable
Why does it have to be a formula exactly?
Eh hard to explain, but I can't just 'loop until I hit bend percentage, get tangent, loop the rest'
I have to be able to get the point at any step at any time
crisp pine
Ahh I see
You can do it a hacky way like this
This function is 0 when x is negative and 1 when it's positive
It can be simplified
You can shift the threshold point
alpine sable
ah so I gotta shift by bend first of all
crisp pine
So if you make your threshold point the bend distance, then you can have the sum of the straight line and the curve, multiplied by this and it's opposite, so only one part is actually used at any given time
There is probably a nicer way to do this but that's my initial thought
crisp pine
I have to be able to get the point at any step at any time
This could still be achieved with an if statement inside a function I think btw but I don't know your exact situation
crisp pine
looks scary xD
It can definitely be simplified im just being a bit lazy
alpine sable
This could still be achieved with an if statement inside a function I think btw ...
how do you imagine that?
alpine sable
oh wait I have an idea-
crisp pine
Well you don't need to loop, just pass in the step
alpine sable
I already have a function technically for the whole curve
it's a bit more complex because it's 3d
Vector3 GetPointOnTorus(float curveRadius, float ringRadius, float curveStep, float ringStep, float bended)
{
float ringPos = ringRadius * Mathf.Cos(ringStep);
return new()
{
x = ringRadius * Mathf.Sin(ringStep),
y = (curveRadius + ringPos) * Mathf.Sin(curveStep),
z = -(curveRadius + ringPos) * Mathf.Cos(curveStep) + curveRadius,
};
}
So the z here is the x we were talking about
to make it more confusing, my bad xD
But yeah if we simplify that to our 2d
crisp pine
Okay cool yeah that sounds promising
alpine sable
this creates 'torus' around the part of a curve
so we are only talking about the z and y axis for the curve
Vector2 GetPointOnTorus(float curveRadius, float curveStep, float bended)
{
return new Vector2()
{
y = curveRadius * Mathf.Sin(curveStep),
x = -curveRadius * Mathf.Cos(curveStep) + curveRadius,
};
}
Excluded the rings that go around the curve
the x negation and adding curveRadius are just adjustments for my situation
so let's say bended is 0.5
hmm mayb first I should figure out how to get the percentage we are on the curve at this point
I think this is correct: percentage/100 = curveStep / Mathf.PI / curveRadius * 4
nope
alpine sable
percentage/100 = curveStep / ((PI / 360) * curveSegments * curveAngle) * 10
meh I give up xD
I was thinking maybe I could use a second circle that was offset by the percentage * radius
lone heartBOT
@alpine sable Has your question been resolved?
alpine sable
.close
lone heartBOT
lone heartBOT
Available help channel!
deep ember
lone heartBOT
glad basalt
gets added
zealous lichen
$-y^2-\frac{y^2}{2}=\frac{-3}{2}y^2$
ocean sealBOT
WhereWolf
glad basalt
yea that
deep ember
i thought -3/2 came from integrating the second term (3y) and factoring out the -1 from y^2
wait i got it
thank you!
.close
subtle gate
lone heartBOT
subtle gate
how do i do b)
vale wigeon
divide 2x^3 - x^2 - 13x - 6 by 2x^2 + ax - 3 symbolically
tardy stag
yep
subtle gate
but idk what a is though
gray isle
you should've noticed something based on the work you did in a)
gray isle
depends on what exactly you're comparing
gilded crypt
i have a doubt for a fourier transform of a function
lone heartBOT
i have a doubt for a fourier transform of a function
ocean sealBOT
sNOw
naive valley
what does the "s" subscript mean?
gilded crypt
that it is the fourier sine transform
so it basically comes down to solving this integral
which is where i'm struggling
,tex yes i tried $t = \sqrt{x}$
gilded crypt
it seems like a step in the right direction but i must be going wrong somewhere
should i share my handwritten attempt?
lone heartBOT
@gilded crypt Has your question been resolved?
gilded crypt
help😭
lone heartBOT
@gilded crypt Has your question been resolved?
naive valley
*sin wt
this doesn't appear to be an integral that you can solve using elementary calculus methods. See for example the answers here: https://math.stackexchange.com/questions/404119/proof-that-1-sqrtx-is-itself-its-sine-and-cosine-transform
lone heartBOT
@gilded crypt Has your question been resolved?
lyric spire
Determine the Boolean value of the following statements. Justify (prove) your
answer!
1: ∀xϵ R ∃yϵ Rx+ y<xy
Hello! Im stuck on this boolean statement. I have to prove that this is either true or false. I'm a first year IT student and I've been struggling with figuring this out for an hour now. Could anyone give an example or show where to start?
lyric spire
R means real number, like 1, 2, -1, 45, -123, whatever.
Atleast that's what I think.
subtle gate
how can k(k-24)=h(h-24), h not equal k, be factorised to give h+k = 24 ?
wild umbra
alright
firsly you should open the brackets
and multiply
what would be the equation now?
subtle gate
firsly you should open the brackets
of ?
wild umbra
um like
its written k(k-24)
so multiply k with both terms
do the same with rhs
yes?
wild umbra
alr, now whats the equation?
subtle gate
k^2-24k = h^2-24h
wild umbra
good now can you see anything which can be factored out somehow
wild umbra
$k^{2}-h^{2}=24k-24h$
ocean sealBOT
yajatk07
subtle gate
ah yeah
wild umbra
can you see now?
subtle gate
(k-h)(k+h) = 24(k-h)
wild umbra
factor out 24 in rhs, and apply the identiey which i told you above in lhs
where you lost?
$(k+h)(k-h)=24(k-h)$
ocean sealBOT
yajatk07
torpid bay
can someone help me understand the logic behind this?
lone heartBOT
can someone help me understand the logic behind this?
vale wigeon
behind what
you have here the definitions of "odd function" and "even function" and the solution checks your function against both, what exactly is escaping you
torpid bay
I don't understand how to use the top table alone to know that f(x) = f(-x)
wait nvm I got
can someone explain why the maximum is reached when pi/2 + 2pi
mortal trellis
knowledge about sin(x)
lone heartBOT
@torpid bay Has your question been resolved?
lone heartBOT
@torpid bay Has your question been resolved?
granite radish
can someone explain why the maximum is reached when pi/2 + 2pi
So, with sin(x), the max value that it can get is 1. To achieve that max value, it will need to be at pi/2. And, with every 2pi added on, it rotates around the unit circle once more, landing you on pi/2 once again.
Therefore, we can say that it will achieve its max at pi/2 + 2pi(n), where n is an integer.
proud lynx
Anybody wanna do this for me? <@&286206848099549185>
lone heartBOT
Anybody wanna do this for me? <@&286206848099549185>
vapid shuttle
Anybody wanna do this for me? <@&286206848099549185>
!15mins
lone heartBOT
Anybody wanna do this for me? <@&286206848099549185>
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
lone heartBOT
Please read #❓how-to-get-help
alpine sable
width 47, length 84
vapid shuttle
width 47, length 84
!nosols
lone heartBOT
width 47, length 84
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.
vapid shuttle
Seriously?
proud lynx
width 47, length 84
barren bolt
It’s actually W 1 and H 2
alpine sable
if you want I cold explain how to do it
proud lynx
Yeah sure
proud lynx
Check dms
alpine sable
ok
lone heartBOT
@proud lynx Has your question been resolved?
alpine sable
lone heartBOT
alpine sable
status:1
lone heartBOT
@alpine sable Has your question been resolved?
alpine sable
<@&286206848099549185>
alpine sable
<@&286206848099549185>
alpine sable
<@&286206848099549185>
boreal inlet
yes?
lone heartBOT
@alpine sable Has your question been resolved?
alpine sable
@boreal inlet
lone heartBOT
@alpine sable Has your question been resolved?
lone heartBOT
@alpine sable Has your question been resolved?
lone heartBOT
lone heartBOT
Available help channel!
cerulean talon
Quadrant 3 notes say to do 180+ where as the solution says 180-
cerulean talon
is there an error?
i’m talking about Example 1: b) ii)
Find the exact value of sin(225)
<@&286206848099549185>
cerulean talon
ASTC
alpine sable
Good good
alpine sable
Quadrant 3 notes say to do 180+ where as the solution says 180-
It's the exact same thing
cerulean talon
and the degrees added to the angle
alpine sable
add or subtract 180 you get to the same place
cerulean talon
It's the exact same thing
but why would it be different on the thing?
alpine sable
So what will the sign be in the quadrant when the angle is 225°?
cerulean talon
is there a preferred way of doing it?
cerulean talon
So what will the sign be in the quadrant when the angle is 225°?
-cos
since its in the tan quadrant
alpine sable
is there a preferred way of doing it?
I don't think so
Now what do you know about sin(pi/3)?
cerulean talon
I don't think so
would it be better to use + since it stands out?
cerulean talon
Now what do you know about sin(pi/3)?
it equals sin(30) and also sin(1/sqrt(3))
depends how you look at it
alpine sable
it equals sin(30) and also sin(1/sqrt(3))
Right
alpine sable
would it be better to use + since it stands out?
maybe
But do you know the exact value of sin(pi/3)?
also it would be good to remeber this
alpine sable
also it would be good to remeber this
^^^
the angle between the line and x axis is 45 degrees
cerulean talon
But do you know the exact value of sin(pi/3)?
isn’t it sin(1/sqrt(3)) ?
alpine sable
isn’t it sin(1/sqrt(3)) ?
The exact value of sine at pi/3 is sqrt(3)/2
also
do you know all of the common trig values?
cerulean talon
what you mean?
alpine sable
do you know all of the common trig values?
^
alpine sable
would it be better to use + since it stands out?
30° 45° 60° 90°
I’ll let you take this one from here
cerulean talon
30° 45° 60° 90°
oh yeah i know that
alpine sable
so you know the values of $\sin(30°)$,$\sin(45°)$ , $\sin(60°)$ and $\sin(90°)$?
ocean sealBOT
Free Geoffrey
cerulean talon
those are from the right angle?
alpine sable
those are from the right angle?
idk what you mean by that
cerulean talon
okay ik 45, 30 and 60
not the 90
idk if that’s part of the course atm tho
i’ll check
trig identities, radians, trig functions, trig equations, application of trig functions
that’s what’s left
so i don’t think i’ll be doing 90’s yet
alpine sable
you did do 90
cerulean talon
i did?
cerulean talon
oh
i think i’m fine for the moment with what i want to get done today tho
but i’ll probably look at 90’s after i’m done
maybe a yt video on it would be good
finite knot
how can i prove this inequality
lone heartBOT
how can i prove this inequality
oak perch
The common way is like this:
To prove for example the RHS, x-1-ln(x)>=0
You calculate its derivative to find the minimal of x-1-ln(x), show that it >=0
Similar for the other direction
finite knot
To prove for example the RHS, x-1-ln(x)>=0
how do i know that x-1-ln(x) >=0
oak perch
I mean to prove this
finite knot
i think that i tried that yesterday and it didn't work, i think im gonna try it again
oak perch
For the other direction there is a simpler method, by using Cauchy's mean-value theorem
finite knot
im supposed to use the mean value theorem cuz this unit is about mean value theorem
oak perch
(ln(x)-ln(1))/(x-1)=1/t for some 1<=t<=x, ln(x)=(x-1)/t, and notice that t<=x
I see
This goes for RHS too
cause t>=1
sonic mural
can someone check my quiz redo answers?
finite knot
what is rhs btw lol
oak perch
Right hand side
ln(x)=(1/t)(x-1) is what we obtained. Now use the facts, that t>=1 and t<=x
finite knot
(ln(x)-ln(1))/(x-1)=1/t for some 1<=t<=x, ln(x)=(x-1)/t, and notice that t<=x
but i dont know the interval how am i supposed to use the mvt
oak perch
The beginning of this message you are replying to, is how we use Cauchy mean value theorem
(f(b)-f(a))/(b-a)=f’(t) for some a<=t<=b. Now a=1, b=x, f(t)=ln(t)
finite knot
(f(b)-f(a))/(b-a)=f’(t) for some a<=t<=b. Now a=1, b=x, f(t)=ln(t)
how do you come to the conclusion that a=1 and b=x and f(t)=lnt, the exercise does not define any function without any said interval
oak perch
…
oak perch
Cauchy mean value theorem is on Wikipedia and your textbook, read again. And read my answer again
finite knot
isnt mvt supposed to be lagranes theorem???
oak perch
I have no idea what you are talking about. I answered your question. And better , used Cauchy mean value theorem as you asked. All steps are given, made everything clear
finite knot
alr, i'll try to figure it out, ill tell u what i might get
oak perch
Sure
finite knot
i think there seems to be a misunderstanding
my book just has the MVT no cauchy mvt
i dont know that
i was so confused
it just has this
alpine sable
be careful using greek letters to define intervals
finite knot
be careful using greek letters to define intervals
my book is greek why would it not have greek letters
alpine sable
my book is greek why would it not have greek letters
can confuse
finite knot
ok]
finite knot
i like ξ cuz it looks aesthetic
oak perch
Now t>=1 what it gives you
1<=t<=x , 1/x<=1/t<=1
And you have ln(x)=(1/t)(x-1)
So?
finite knot
but i wrote
where do i find the 1, this is an open itnerval of t
i apologize if this is infuriating you, but i just woke up i appreciate what you're doiing
oak perch
x positive right
finite knot
yes
oak perch
So (ln(x)-ln(1))/(x-1)=(1/t) for some 1<=t<=x when x>1
finite knot
but x>0, it doesnt specifically specify anywhere that x>1, would u have to add both 1 in x>0, and get x+1>1
oak perch
I will get to when x<1
Now ln(x)=(1/t)(x-1)
Since 1<=t<=x, we have 1/x<=1/t<=1
x-1 is positive so we multiply by x-1
(1/x)(x-1)<=(1/t)(x-1)<=x-1
This is 1-1/x<=ln(x)<=x-1
When 0<x<1
There exists x<=t<=1 such that ln(x)=(1/t)(x-1) still
1<=1/t<=1/x this time
But x-1 is negative
So 1-x is positive
Multiplied by 1-x we have
1-x<=(1/t)(1-x)<=1/x-1
So we still have
1-1/x<=ln(x)<=x-1
(Since a<=b<=c is equivalent to (-c)<=(-b)<=(-a))
x<1 case is exactly like x>1 case I merged them together thought I didn’t need to point out
You can also view ln(x)=-ln(x^-1), x<1 then x^-1=1/x>1 >1 also give you <1 case
finite knot
So (ln(x)-ln(1))/(x-1)=(1/t) for some 1<=t<=x when x>1
i still dont get how you somehow set b=1
oak perch
i still dont get how you somehow set b=1
BECAUSE LN(1)=0
ln(x)=ln(x)-0=ln(x)-ln(1)
Why not set 1
You can even directly derive <1 case using >1 case:
x<1, then y=1/x>1
1-1/y<=ln(y)<=y-1
ln(y)=ln(x^-1)=-ln(x)
So 1-x<=-ln(x)<=1/x-1
So still, 1-1/x<=ln(x)<=x-1
finite knot
oak perch
Why you are still at this step
finite knot
Why you are still at this step
i rewrote this step
oak perch
And 1, not 0
finite knot
WHY
finite knot
then cant i set a limit that gets very close to 0?
finite knot
me neiher, i just dont know anymore, its an open interval
oak perch
…
I used =1 and solved your question
If you find any part of my steps wrong point it out
You not understanding the motivation of any step isn’t a mistake in a step
Okay
Correct is correct
finite knot
so i set b=1 without ever questioning why and move on with solving the exercise?
oak perch
I answered you ln(1)=0, you can’t get it so move on from this and continue
Since not understanding motivation isn’t a mistake in the step. With or without it you can move on
finite knot
im gonna ask one more thing, if i set b=0.5, wouldn't x then be x>0.5 or set b=1/e, wouldnt x now be x>1/e??
oak perch
Why not setting one
(ln(x)-ln(a))/(x-a)
If your a is not 1
ln(a) is not zero
You want …<=ln(x)<=…
Only a=1 can make ln(a) disappear
ln(1)=0
So you are left with ln(x)/(x-1)
Only ln(x) on the numerator
No other things get in our way
This is the motivation, not any explanation used in the steps. You can’t get it then move on
finite knot
oak perch
Okay. So read my answer again. I answered twice how it is proved when x>1, then I directly proved the case when x<1, and also gave you a proof of when x<1 using it’s true when x>1.
oak perch
So (ln(x)-ln(1))/(x-1)=(1/t) for some 1<=t<=x when x>1
From this
oak perch
Now ln(x)=(1/t)(x-1)
Proving the case when x>1
oak perch
When 0<x<1
Proving the case when x<1 directly
oak perch
You can even directly derive <1 case using >1 case:
Proving x<1 true using x>1 true
oak perch
When 0<x<1
Then directly prove it when x<1 if you like
Or
Prove x<1 using x>1 case like I mentioned
You wrote very well, keep going
finite knot
Since 1<=t<=x, we have 1/x<=1/t<=1
but ξ ∈ (1,x) how is 1<= ξ <= x, when its supposed to be 1<ξ<x
oak perch
< implies <=
So it’s not wrong to write <=
I don’t distinguish them just for convenience
finite knot
< implies <=
how?
finite knot
then why do open intervals and closed intervals exist, why not have one type interval and nothing else
oak perch
Again I used it because of convenience. That is not a mistake
If you want to you can definitely use <
finite knot
but i have to prove it with<=
oak perch
how can i prove this inequality
Your question only required <= so I used <=
finite knot
😕
oak perch
It is true that when x doesn’t equal 1, 1-1/x<ln(x)<x-1
But your question requires you to prove 1-1/x<=ln(x)<=x-1
So I used <=
It’s not wrong
Writing as you wrote is better, stronger result. It’s just what I wrote is also correct
Did you finish x<1 part
finite knot
im still tryna mentally grasp this before writting
oak perch
Okay…
finite knot
When 0<x<1
how? but this apllies when x>1 , how is it still true when 0<x<1
oak perch
I gave you both direct proof and a proof using x>1. Let us go through the latter again:
We proved it’s true when x>1 right
Now if 0<x<1, we have x^-1=1/x>1
Let y=1/x
So it’s true for y
Meaning
1-1/y<ln(y)<y-1
Now ln(y)=ln(x^-1)=-ln(x)
So 1-1/y<-ln(x)<y-1
Replace y with 1/x
1-x<-ln(x)<1/x-1
So we have
1-1/x<ln(x)<x-1
See
finite knot
Now if 0<x<1, we have x^-1=1/x>1
what
oak perch
You don’t know that when x<1, 1/x>1?
finite knot
why do you define a y
finite knot
is there a symbol for "let <variable> "
finite knot
You prefer I to use 1/x instead along the whole steps?
no
oak perch
I don’t think so
It’s just we now have it’s true for numbers >1, and 1/x>1, we also have ln(1/x)=-ln(x). So we can directly prove the <1 case using >1 case
fallow sphinx
Can someone help me in so confused
finite knot
Can someone help me in so confused
bro get in your own lane
finite knot
1-1/y<ln(y)<y-1
how do you get this
fallow sphinx
Oh my fault
finite knot
i think im going insane here
finite knot
im gonna write everything again , and i will see everything line by line
oak perch
Great and I will be here, answer any question until you finish the proof
finite knot
how are you able to understand this
oak perch
It’s simple…
finite knot
no
oak perch
You will get better when you read further
We all stuck at somewhere . Me too. You will pass, and you will find this very simple too one day soon
finite knot
i literally have no teacher im just reading this school book pdf 1 month before school opens
oak perch
Me too, I read everything myself
lone heartBOT
@finite knot Has your question been resolved?
finite knot
Now if 0<x<1, we have x^-1=1/x>1
then wouldn't 0>1/x>1
how is 0>1?
wouldn't this be false?
finite knot
I said given 0<x<1, we have x>1
didnt u say we have 1/x >1?
finite knot
but x is inbetween 0 and 1
finite knot
then 1/0 > 1/x > 1
finite knot
because 1/0 is mathmatecian's nightmare we ignore it?
finite knot
i wasn't really taught inequalities
finite knot
ok fair enough i understand now
oak perch
Okay
finite knot
but to replace x with y , x has to be equal with y , but isn't because y=1/x
oak perch
Replace y with 1/x
.
oak perch
Yeah
finite knot
so does this mean 1/x > 1?
oak perch
Now if 0<x<1, we have x^-1=1/x>1
You said you are going to read this proof , line by line, more than 40 mins ago
oak perch
But we keep circling at same places
finite knot
because i've gone insane
oak perch
Great
oak perch
Yeah since it becomes 0=0=0 when x=1
finite knot
Yeah since it becomes 0=0=0 when x=1
so i use the union symbol to say that x>1 ⋃ x<1 => x>0?
oak perch
No
finite knot
intersect?
oak perch
They don’t imply x=1 is true
It’s just that we can verify it’s true when x=1 directly
1-1/x ln(x) x-1
Plug in x=1
1-1/1, ln(1), 1-1
They are 0,0,0
So the inequality holds when x=1, we just verified it directly
finite knot
ah
all makes sense now
i am grateful for all your time
but
we have proved that this applies for x>1 and x<1 and x=1, so i just rewrite the whole inequality in the end with <= and be done with it? @oak perch
inland oriole
How do you derive the formula for norm of vector?
lone heartBOT
How do you derive the formula for norm of vector?
median oar
Well you just make it up
It needs to satisfy some conditions
But otherwise any such function can be considered a norm
mortal trellis
well in 2d and 3d you can take inspiration from the real world and use pythagoras
inland oriole
Im guessing it's just like pythagorean formula in 3 dimensions but applied to vectors
mortal trellis
in any higher dimension its a definition
inland oriole
okay
mortal trellis
and there are other definitions that are also viable
median oar
Hence, “make it up”
inland oriole
Hence, “make it up”
lol
median oar
No I’m serious
inland oriole
and there are other definitions that are also viable
What are other definitions?
inland oriole
idk what that is
median oar
I want to go from a to b
But I need to go along the block
So I can’t go diagonally
I need to go in perpendicular directions always
inland oriole
ofc
median oar
(And a lot of cities look very blocky so that’s why it’s called the taxicab or Manhattan distance)
It has to satisfy these 3 properties
inland oriole
inland oriole
NICE
median oar
d(x, x) = 0
inland oriole
d(x, x) = 0
what does that mean?
inland oriole
and x and x are two terminal and initial points?
median oar
Yeah
Ah wait
N(x + y) <= N(x) + N(y)
N(x) <= 0
N(x) = 0 iff x = 0
N(𝛂x) = |𝛂|N(x)
N is the function, norm
If N satisfies the above conditions we call it a norm
inland oriole
Nice
median oar
So you can pretty much make it up
(Up to some restrictions)
But the one you will see most often is the $||\cdot||_2$ norm
ocean sealBOT
Frosst
median oar
That will be
Let $\mathbf x=(x_1,x_2,…,x_n)$, then $||\mathbf x||2 = \sqrt{\sum{i=1}^n x_i^2}$
ocean sealBOT
Frosst
inland oriole
But the one you will see most often is the $||\cdot||_2$ norm
What is that called?
inland oriole
Haha that’s cool
median oar
Since it’s pretty good for Euclidean geometry
But think about it right
If you were on a sphere
It’s not quite Euclidean
inland oriole
Okay
median oar
(You live on earth, it’s kinda a sphere)
inland oriole
Yeah
median oar
The shortest path (distance) from 1 point to another isnt a straight line
Rather it is curved
inland oriole
Mhm
median oar
Hence it is why plane flight paths are not straight in the normal sense
inland oriole
Hence it is why plane flight paths are not *straight* in the normal sense
Is a curve the geometric interpretation for a 2 norm
median oar
Uhh
Norms just have to satisfy the above mentioned qualities
There’s no condition that it be “the best”
There’s no notion of “the best” either
It just so happens that we live in Euclidean space in a day to day setting
inland oriole
It just so happens that we live in Euclidean space in a day to day setting
Yeah thankfully lol
median oar
Well if you want to go from 1 corner to the opposite corner of a block
You’d realise both ways is the same
That’s an example of the taxicab norm
median oar
1 norm 2 norm is there also a 3 norm?
Of course
When mathematicians got numbers they can’t stop counting
Wait I’m right
inland oriole
Hahaha
median oar
It should be 1 norm
Because!
The p norm is like this
Let $\mathbf x=(x_1,x_2,…,x_n)$, then $||\mathbf x||p = \sqrt[p]{\sum{i=1}^n |x_i|^p}$
Oh I forgot to add abs vals
ocean sealBOT
Frosst
median oar
We even have the infinity norm!
inland oriole
Lmaoo
inland oriole
**Frosst**
What is tho
median oar
The infinity norm just returns the value of the biggest component
Because any smaller component will grow far slower than the biggest component
So in comparison they won’t matter
inland oriole
**Frosst**
I don’t understand what that is
fathom orchid
lone heartBOT
fathom orchid
a lil bit
fathom orchid
ok
zealous lichen
and another property of e is that $e^{i\theta}=cos\theta+isin\theta$
ocean sealBOT
WhereWolf
fathom orchid
ooh
zealous lichen
the proof is the difficult part
fathom orchid
oh
zealous lichen
**WhereWolf**
this is a important formula so you might want to memorize it
zealous lichen
btw another important thing is that $|e^{i\theta}|=1$ (it lies on unit circle)
ocean sealBOT
WhereWolf
zealous lichen
using that every complex number can be written as $re^{i\theta}$ where r is its radius and theta is its angle
ocean sealBOT
WhereWolf
zealous lichen
d/dx e^x = e^x
that's what motivates e
e can be also defined as $e=\lim_{x\to 0}(1+x)^\frac{1}{x}$
ocean sealBOT
WhereWolf
zealous lichen
hmm idk how to explain e intuitively
fathom orchid
oh ok
no problem
1/x with x tending to 0 is a very big number right ? how do they calculate it ?
zealous lichen
we can substitute a small x to approximate e