fathom flicker
This calculus course covers differentiation and integration of functions of one variable, and concludes with a brief discussion of infinite series. Calculus is fundamental to many scientific disciplines including physics, engineering, and economics.
Course Format
This course h...
bold bane
If you are interested in learning trigonometry, you can find textbooks online by googling --> trigonometry filetype:pdf
fathom flicker
free of course
sage dagger
I don’t have a solid idea of where I was going to learn from. But I know there are some reputable YouTubers out there and khan academy and all that stuff but I’m very open to trying other things as well
bold bane
The benefit of a textbook is that it will have math problems that you can work on.
fathom flicker
I'd recommend the OCW over khan or youtubers 100x more, although they are options aswell
sage dagger
If you are interested in learning trigonometry, you can find textbooks online by...
Oh that works?
sleek mural
So just to clarify, the top and bottom angle pairs will always be congruent?
Not the right and left ones?
bold bane
Oh that works?
Yup, try to out for any subject or even book title.
sage dagger
Oh sweet
sage dagger
https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/
Ooh
This looks interesting
I’ll look into it
Thanks!
fathom flicker
np
obtuse pebbleBOT
@sage dagger Has your question been resolved?
cold star
Not the right and left ones?
yeah
cold star
Is that always the casev
not always, right and left arcs could be congruent to one another is two lines were parallel
sleek mural
not always, right and left arcs could be congruent to one another is two lines w...
But I mean in this case
cold star
if line ab and cd were parallel, the arcs that are opposite from one another would be congruent
cold star
But I mean in this case
in this case its just those two arcs congruent
sleek mural
in this case its just those two arcs congruent
Ty
sleek mural
if line ab and cd were parallel, the arcs that are opposite from one another wou...
So all of them?
Right, left, top, and bottom?
cold star
yeah
sleek mural
Thank u
sleek mural
And so would top and bottom?
cold star
always opposite
cold star
youre welcomee
sage dagger
.close
sudden rivet
Hi! I have a question about factoring with the distributive property. The problem:
Apply the distributive property to factor out the greatest common factor
12 + 80
obtuse pebbleBOT
Hi! I have a question about factoring with the distributive property. The proble...
sudden rivet
I am trying to arrive at it by prime factorization of the above two numbers. It seems the answer is 4(3 + 20), but I don't know how exactly to arrive at it via prime factorization.
12 = 2 * 2 * 3
80 = 2 * 2 * 2 * 2 * 5
But then how do I combine the above prime factorizations to equal the 92?
undone rover
I am trying to arrive at it by prime factorization of the above two numbers. It ...
80 nor 12 are prime numbers
oh nvm
frosty spoke
are you asking how to find the greatest common factor of 80 and 12?
sudden rivet
Yes, but via prime factorizations. I read that it's a valid method.
frosty spoke
do you know what the greatest common factor means?
sudden rivet
It's the largest factor that the two (or more) numbers share, I also saw it read as the greatest common divisor.
undone rover
try putting them both in an equation
sudden rivet
I know you can just list them out like:
12: 1, 2, 3, 4, 6, 12
80: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80
And find the largest number that the two share
frosty spoke
okay, so yeah if you want to find them via prime factorization
suppose you write out 12 = 2 x 2 x 3
undone rover
12 = 2 * 2 * 3
80 = 2 * 2 * 2 * 2 * 5
But then how do I combine the above prime...
I'd say some back to this
frosty spoke
and 80 = 2 x 2 x 2 x 2 x 5
undone rover
^
frosty spoke
then you just want to take what's in common between them
so that would be 2 x 2
but honestly this is a terrible method for finding the greatest common factor lol
sudden rivet
Oh my god, I see what I did wrong
I don't know why but I was trying to manipulate multiplying the factors of 12 & 80 in a way to equal 92 lol
frosty spoke
yeah that doesn't make much sense
sudden rivet
but honestly this is a terrible method for finding the greatest common factor lo...
What's your preferred method?
undone rover
GCF
frosty spoke
division
you can use the Euclidean algorithm to do it
and this means you don't actually have to compute the prime factorizations of the numbers
which is sorta important if the numbers get large
but also important with things like polynomials, which you might not necessarily be able to easily factor
sudden rivet
(What level of math is that? I am so undeveloped mathematically that I'm still on 6th grade math, catching up via Khanacademy)
frosty spoke
there's not really a prerequisite, but then again, I didn't take high school math, so I wouldn't know when it's taught
sudden rivet
I see! Well, thank you guys for answering my question and for introducing me to some alternate methods, I greatly appreciate it.
obtuse pebbleBOT
@sudden rivet Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
worldly trench
how do I model this curve?
obtuse pebbleBOT
how do I model this curve?
worldly trench
It looks like an increasing sine function, but is this the best way to model it?
deft crescent
nvm
clear condor
is the question just modeling the curve?
worldly trench
is the question just modeling the curve?
yeah, I'm trying to perform some regression so that I can predict future data values from an equation
I basically want an equation that most accurately represent this
obtuse pebbleBOT
@worldly trench Has your question been resolved?
obtuse pebbleBOT
@worldly trench Has your question been resolved?
quiet drum
obtuse pebbleBOT
quiet drum
idk where to start
timid silo
what do all the angles in any quadrilateral add up to?
timid silo
interior should be 360
quiet drum
theyre both 360
timid silo
we have all the angles in terms of x and y, so we can add them together to make one equation
quiet drum
but theres two variables
ive made the equation
its just 360 = all of that summed
but how do we get rid of the brackets
burnt vector
Since ABCD is a trapezium, what do we know about the relationships between angles DAB with ADC, and angle DCB with ABC?
burnt vector
im not sure
Interior angles property. Have you learnt that?
quiet drum
they aall add up to 360?
burnt vector
Nope
That's not the one I'm referring to
We know that AB is parallel to DC, don't we?
quiet drum
yes
burnt vector
From there, what can you deduce from the sum of angles ADC and DAC?
If you aren't aware of this property.... u should look up the angle theorems/properties before attempting these questions
brave pivot
left 2 angle add up to 180, same as right 2 angle, because parallel
quiet drum
From there, what can you deduce from the sum of angles ADC and DAC?
supplementary angles
so they should add to 180?
burnt vector
supplementary angles
Yes. Do you know why?
quiet drum
because they are on a horizontal line
and all angles on horizontal lkine will equal to 180
burnt vector
Not quite
brave pivot
no because parallel line
quiet drum
not neccesrily horizontal
burnt vector
Okay think of it this way
If I extended line AB to the left
And named the extention to be point "Y"
Angle ADC is equal to angle YAD, right?
quiet drum
because
burnt vector
As such, it's trivially true that since YAD + DAB = 180, so does CDA + DAB
quiet drum
c and y are corresponding
brave pivot
so they should add to 180?
look at this if you can understan man
burnt vector
So from there
We can create 2 simultaneous equations
And from there solve for x and y
quiet drum
i understan why the left side is 180
but whats wih the righ side
how did we deduce that
quiet drum
im asking how we got to the point where we knew that the right side is equal to 180
obsidian heron
then x+4 +x-4=180
quiet drum
ill try doing the simulatenous equation
obsidian heron
you can get 2x=180 which x=90
and on the right you can solve for y with your x= 90
try it
i think the question only ask for x and y, dont waste time on solve the angle
obsidian heron
tahts right
quiet drum
alright
obsidian heron
anything else?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
teal turret
obtuse pebbleBOT
eager pulsar
help
fathom flicker
, rotate
warm shaleBOT
fathom flicker
!status
obtuse pebbleBOT
What step are you on?
1. I don't know where to begin
2. I have begun but got stuck midway
3. I got an answer but I'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
1
eager pulsar
yes
fathom flicker
What have you tried?
eager pulsar
i tried to - 3.2 under 3.2 and 1.7p
fathom flicker
The goal should be to get all the "p"s on one side
and the numbers on the other side
eager pulsar
how
fathom flicker
So right now you have -0.9p and 3.2 on the same side
How could you get the -0.9p to the other side?
eager pulsar
i have no idea
fathom flicker
Could you add 0.9p to both sides?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
random spade
What is the solution of 7x congruent to 10 mod 12
obtuse pebbleBOT
What is the solution of 7x congruent to 10 mod 12
timber island
wdym by 'congruent'
congruency only applies to geometry, no?
it should be proportional (i think), no it should equals
random spade
The sign equal but with another line so 3 lines
random spade
congruency only applies to geometry, no?
No
timber island
The sign equal but with another line so 3 lines
ah
wooden cipher
Congruent is the right word lol
random spade
How to solve this basically
wooden cipher
Since 7 is coprime to 12, we can usedivision
random spade
Since 7 is coprime to 12, we can usedivision
Yea i found they were coprime in the question just before
wooden cipher
My way of doing this and iirc the only is the keep adding 12 until you get a multiple of 7 and then divide
Just keep going in the seqquence 10, 22, 34, ...
What is the first multiple of 7
random spade
Hold on 1 sec i don’t have a calculator
wooden cipher
Take your time
random spade
Well it’s 70
warm shaleBOT
GarlicBredFries
wooden cipher
Then divide by 7
random spade
So x congruent to 7 mod 12
random spade
So that’s the answer?
random spade
Why in my textbook the answer is 3 and -5 did the professor got it wrong??
jolly ginkgo
you can also do it by substituting x=0,1,2,...,11
wooden cipher
Oh true
jolly ginkgo
3?
wooden cipher
Why in my textbook the answer is 3 and -5 did the professor got it wrong??
Weird, are you sure you got the question right?
jolly ginkgo
how is 3 even right
jolly ginkgo
ye
wooden cipher
You can check pretty easily
jolly ginkgo
,w 7x = 10 mod 12
wooden cipher
Lmao
warm shaleBOT
random spade
I so im confused maybe this isn’t the professor correction and just me who did a mistake
wooden cipher
Yeah thought so
wooden cipher
I was laughing at the bots weird interpretations
warm shaleBOT
jolly ginkgo
nic
wooden cipher
Can you check to make surte you wrote the right question down?
Because our solution is wolframalpha verified
random spade
Can you check to make surte you wrote the right question down?
Resolve the equation 7x congruent to 10 mod 12
jolly ginkgo
yeah no 10 is the answer then
random spade
That’s the exact question
And the question before was show that 7 is inversible mod 12
So basically 7 and 12 are coprime
jolly ginkgo
yep
timber island
context?
royal basin
,rccw
warm shaleBOT
obtuse pebbleBOT
@maiden compass Has your question been resolved?
uneven night
simplify lg(1000^3)
obtuse pebbleBOT
simplify lg(1000^3)
next reef
!status
obtuse pebbleBOT
What step are you on?
1. I don't know where to begin
2. I have begun but got stuck midway
3. I got an answer but I'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
i don't know how to begin
vestal marsh
Do you know log laws?
vestal marsh
Well to the left
vestal marsh
And what's 1000 in exponent form
next reef
so wet get 3lg1000
Careful
uneven night
10^3
next reef
Are you dealing with log or ln rn?
uneven night
lg
next reef
Right
vestal marsh
Doesn't matter though that rule holds true still...
next reef
Simplify 3*lg(10^3)
uneven night
oh i see
move the 3 to the left again
we get 9
and lg and 10 takes out each other
thanks
next reef
We did nothing
uneven night
.close
mild zodiac
obtuse pebbleBOT
fossil crag
!status
obtuse pebbleBOT
What step are you on?
1. I don't know where to begin
2. I have begun but got stuck midway
3. I got an answer but I'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
1
fossil crag
First, start by writing the result of the matrix/vector product on the left hand side
royal basin
First, start by writing the result of the matrix/vector product on the left hand...
might not be worthwhile.
@mild zodiac how familiar are you with systems of linear equations?
mild zodiac
I'm bad
royal basin
that doesn't really answer my question one way or the other.
unless you mean that you are bad at doing things with linear systems.
ok, so are you familiar with things such as the determinant?
mild zodiac
.close
obtuse pebbleBOT
fossil crag
He got scared of the bloodthirsty determinant
obtuse pebbleBOT
Available help channel!
uneven night
obtuse pebbleBOT
uneven night
i dont understand how the denominator changed
from sqrt(x^2+1) +x ---> x+x??
did he factor out sqrt(x^2)?
fossil crag
Yes, exactly what he did
uneven night
but then we get sqrt(x^2)(1+1/sqrt(x^2)?
fossil crag
Leaving sqrt(1 + 1/x²), which goes to 1 as x -> +infinity
uneven night
but what did he factor out x^2
or sqrt(x^2)?
because if he factored out sqrt(x^2) it will leave (1+1/x^2)
fossil crag
for x positive, sqrt(x²) = ?
uneven night
x
fossil crag
Yes
uneven night
because if he factored out sqrt(x^2) it will leave (1+1/x^2) am i right
fossil crag
So we're now left with xsqrt(1 + 1/x²))
The whole denominator is now x(1 + sqrt(1 + 1/x²))
wild swallow
confusion due to skipped steps 
uneven night
i'am confused
fossil crag
Yes, what you sent is lacking a bit of rigorousness
uneven night
So we're now left with xsqrt(1 + 1/x²))
i understand this
uneven night
The whole denominator is now x(1 + sqrt(1 + 1/x²))
but not this
wild swallow
[ \lim_{x \to \infty} \f {\s{x^2 + 1} + x} {x + x} = 1 ]
warm shaleBOT
wild swallow
this lets you swap out the denominator
wild swallow
mmm yes that's evident
uneven night
oh okay
so i think we should first multiply with conjugate
then we get (1/sqrt(x^2+1 +x))
fossil crag
Yes
uneven night
and now i will factor out sqrt(x^2) so i will get 1/x(1+(1/sqrt(x^2)) +x)
fossil crag
Be careful
uneven night
Be careful
is this true
uneven night
and now i will factor out sqrt(x^2) so i will get 1/x(1+(1/sqrt(x^2)) +x)
is this right
fossil crag
then we get (1/sqrt(x^2+1 +x))
First the brackets here are shaky
You mean 1/(sqrt(x²+1) + x)
uneven night
You mean 1/(sqrt(x²+1) + x)
yes ofc
fossil crag
If you're factoring out sqrt(x²) = x here :
uneven night
Should it be sqrt of the parentheses as well?
fossil crag
What you wrote in parentheses is not good
sqrt(1 + 1/x²) is very different from 1 + 1/sqrt(x²)
uneven night
I think this is correct maybe or what do you think @fossil crag
fossil crag
Yes the last thing you wrote is correct
uneven night
thank you so much
and now i will get x+x
i understand
I have a similar problem
This one I don’t know how to move on
Should I factor out the sqrt(x^2)
So I get this maybe?
yes thanks again
.close
broken wedge
Hello, bombed my midterms today, what's the first step here? Is it long division?
obtuse pebbleBOT
Hello, bombed my midterms today, what's the first step here? Is it long division...
broken wedge
Oh it is but i just dont know how to long divide 💀
impulsively posted here woops
.close
timid silo
obtuse pebbleBOT
timid silo
Is this integration correct?
fossil crag
I think the coefficient in front of tan^-1(...) is 1/3, not 2/3
Differentiate the result you got to make sure you get back on the same function you started with
If you're off by a coefficient, you know what to change in your result
obtuse pebbleBOT
@timid silo Has your question been resolved?
timid silo
I think the coefficient in front of tan^-1(...) is 1/3, not 2/3
This is the solution
Where did half come from?
fossil crag
Where did half come from?
Do u-sub u = (2x+3)²
$du = (4x+6)dx$, then you're left with $\int \frac{1}{9+u²}du$
warm shaleBOT
rafilou2003
fossil crag
Where did half come from?
There shouldn't be half
<@&286206848099549185> feel free to correct me if I'm wrong here
fossil crag
OOOOHH
timid silo
Yes but don’t you have to factor out 2 from the numerator to make it derivative
fossil crag
Let me correct what i said up above
$du = 2(4x+6)dx$, then you're left with $\int \frac{1}{2(9+u²)}du$
warm shaleBOT
rafilou2003
fossil crag
Then you get on the answer we get here
timid silo
Can you do it without u sub please
fossil crag
Yeah
timid silo
Thank you for your time and help🙏
fossil crag
Np
timid silo
.close
next lintel
vector proof
obtuse pebbleBOT
vector proof
next lintel
of a kite
i need vector proof of a kite
how can i prove a quadrilateral is a kite using vectors
obtuse pebbleBOT
@next lintel Has your question been resolved?
obtuse pebbleBOT
@next lintel Has your question been resolved?
obtuse pebbleBOT
@next lintel Has your question been resolved?
drifting wraith
@next lintelis conqueur very bad?
obtuse pebbleBOT
@next lintel Has your question been resolved?
willow sierra
i need help
obtuse pebbleBOT
i need help
plain stag
and yet you haven't posted your question
obtuse pebbleBOT
@willow sierra Has your question been resolved?
willow sierra
I need help
obtuse pebbleBOT
@willow sierra Has your question been resolved?
obtuse pebbleBOT
@willow sierra Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
timid silo
Ok I need help
obtuse pebbleBOT
Ok I need help
timid silo
I'm very confused
in this example
they use the
formula
to find the area bounded by the x axis
so you start of with 0 and that subtracted by the other equation of the line
ok fine whatever
but then in this example
even though it is again boudned by the x acis
we don't use the formula and subtract the 0 by f(x)
as they did in the first example
why
timid silo
Total area, not net area
wdym
fierce lagoon
From [-3,0), the cubic is on top, but from (0, 1], the cubic is on the bottom
timid silo
From [-3,0), the cubic is on top, but from (0, 1], the cubic is on the bottom
okay gotcha
fierce lagoon
Net area is not the same as total area
timid silo
can you explain the difference in this context
fierce lagoon
Total area is the integration of |f(x)|, and the integral will always be psotive
Within integrals, if the curve is beneath the x axis, the area is negative
timid silo
yep gotcha
timid silo
From [-3,0), the cubic is on top, but from (0, 1], the cubic is on the bottom
isn't the cubic 0 to 1 identical to the top example
I still don't understand why we don't use 0 - f(x)
timid silo
Because 0-f(x) will make the integral negative
from 0 to 1 will it though
because we are subtracting by that integral
so if it is negative
negative * negative = postive
timid silo
It's the same because 0 - f(x) = - f(x) = -1 • f(x)
ah I see
so they just re-arranged
fierce lagoon
Basically yeah
timid silo
and for the initial -3 to 0 do we not have the subtract by 0 because the function is above the x axis hence it would make f(x) - 0 which is just f(x)
obtuse pebbleBOT
@timid silo Has your question been resolved?
Available help channel!
Available help channel!
bright root
What's the difference between a metric and a euclidean R^n space?
obtuse pebbleBOT
What's the difference between a metric and a euclidean R^n space?
haughty coyote
All dot products generate norms
Not all norms are generated by dot products
Therefore a metric space has more generality
For instance the norms 1 and $\infty$, defined by
$|(x, y)|1 = |x| + |y|$
and
$|(x, y)|{\infty} = max(|x|, |y|)$
(here for $\mathbb{R}^2$) don't correspond to a dot product for any dimension
warm shaleBOT
mateo713
bright root
hmm, I'm not super comfortable with these concepts, is a norm of a vector in essence it's length from the zero vector?
haughty coyote
||v|| = d(0, v) yes
It's weird you'd see metrics without norms though
But then
Do also note that all norms generate distances, but not all distances generate norms
So technically a metric space is bit more general than a normed space
bright root
ok so the base-level answer for my question is that euclidean spaces are a specific subset of metric spaces, right?
haughty coyote
Basically yes. You limit yourself to the metrics generated by norms generated by dot products, but in return you have a lot more theory
bright root
How do these two relate to a "coordinate system"?
haughty coyote
That's just about giving your vector space a basis
Which doesn't require a metric, a metric doesn't require a basis
So pretty independent
bright root
What's a metric precisely?
bright root
Yeah ok. Is then a norm a kind of metric?
haughty coyote
Do also note that all norms generate distances, but not all distances generate n...
Basically
Because you can define d(x,y) = ||x-y||
bright root
What do you mean precisely when you say x generates y?
Does that mean that if you have a precise x, then that forces a precise y?
haughty coyote
Because you can define d(x,y) = \|\|x-y\|\|
This is the distance canonically generated by a given norm
So we'll talk about the distance associated to that norm
Of course you could also define d(x,y) = 2 ||x-y|| but it's not the usual one anymore, and all properties linking them are most likely gone
Same with the norm defined as ||v|| = sqrt(v.v), change that and all the nice properties go out the window
bright root
Right so you can either deviate from the standard definition of the distance, or the norm and lose interesting properties.
bright root
Do also note that all norms generate distances, but not all distances generate n...
But from this I'm inferring that the latter case should differ in some distinct way.
haughty coyote
Example: the parallelogram identity is verified by a norm if and only if it comes from a dot product
So there's differences that emerge yes
north anchor
someone can demostrate antiderivate rules
obtuse pebbleBOT
someone can demostrate antiderivate rules
devout rune
Anti-derivative rules are usually just the inverse of derivative rules. Is there a particular one you're having trouble with?
north anchor
are integrals?
devout rune
I'm not sure what you mean, can you give an example of a problem you're trying to work through where this is relevant?
devout rune
Well, there's the fundamental theorem of calculus which basically states the following:
$F(x) = \int_0^x f(x) dx$ and $\frac{d}{dx}F(x) = f(x)$
warm shaleBOT
actinium226
north anchor
Are there cases that do not exist F(X) = f(x)?
devout rune
Usually the only difference between antiderivatives and integrals is that the antiderivative will be a function, whereas if you have an integral then its either a definite integral, i.e. a constant, or an indefinite integral, meaning you'll always have that +C as in your picture
You mean d/dx F(x) = f(x)? Yes of course. The theorem requires that f is continuous on [0, x].
So the unit step function, for example, does not have an antiderivative in the sense described above. I'm not quite sure if this helps answer your question but please let me know if you need more.
north anchor
cWell, another doubt that I have about other theorems is that there are theorems that say that they must be continuous in (a,b) and differentiable in [a,b] and that there is at least one point c.
why it must be continuous in that interval and not in another combination like (a,b] or (a,b)
devout rune
I'm not sure what question you're asking
It sounds like maybe something related to the mean value theorem, but I just don't quite understand what it is that you're asking, and what it has to do with antiderivative rules, which is what you started asking about.
north anchor
I know is that I have an examination of the mean value theorem and Rolle's theorem, it is that I do not understand why the definition was made like this, I do not understand the part of the ends of intervals because it is differentiable in [a,b] and not it's ab)
devout rune
I don't think I have a good answers for you on that one. I know I had similar questions when I studied this material and didn't quite get a satisfactory answer.
obtuse pebbleBOT
devout rune
Good luck @north anchor !
obtuse pebbleBOT
Available help channel!
proven dragon
why do they start with that? if f is differentiable, i thought we should start with the definition? sorry if its a dumb question
obtuse pebbleBOT
why do they start with that? if f is differentiable, i thought we should start w...
brisk matrix
you are trying to show continuity, not differentiability
devout rune
I don't think there's any real meaning to it. I think it's maybe a little more intuitive to start, as you suggest, with what you know and build up to what you want to show, but here they did it the other way around
brisk matrix
i think the opposite is more intuitive
proven dragon
I don't think there's any real meaning to it. I think it's maybe a little more i...
yeah i think thats why im confused
it does make sense, but its a bit weird idk
brisk matrix
i think the opposite is more intuitive
well in this case
if they just gave you the last equation, you’d be confused as to where it came from
they’re just saying “this is what we need to prove”, and then rearrange it to make use of differentiability
starting from f’(z) * 0 = 0 is not very helpful imo
devout rune
yea now that i see it a little more closely, it might be a little more difficult to understand that way, but in general the way trying to prove a lot of things is that you start by playing with what you want to show and seeing if you can break it down into what you know, and then when you write the proof you usually write those steps backwards, which is why a lot of proofs can look so strange.
well you'd start from lim (f(z+h) - f(z))/h = f'(z), then since lim h = 0 (i.e. converges), you can multiply h by both sides and take the limit
and then you'd get lim f(z+h) - f(z) = 0, hence lim f(z+h) = f(z) hence the function is continuous
But it really does work either way
proven dragon
in my head this i would be thinking 'we know its differentiable' then start with the definition of diffeentiable and then from there proving its continuous? is that a wrong way of thinking about proofs in general? i realise in this case, it would be more difficult to do it this way?
ahhh okay okay
devout rune
You might like Michael Penn's youtube channel. He does various proofs like this, and he explains it in a way that's pretty easy to follow.
brisk matrix
in my head this i would be thinking 'we know its differentiable' then start with...
if the proof is correct it doesn’t matter which way you approach it. though starting from what you know is often good
but consider something like showing injectivity
regardless of what you know, you’d usually start with f(x) = f(y)
and in some intermediate step you’d apply your assumption to arrive at x = y
proven dragon
You might like Michael Penn's youtube channel. He does various proofs like this,...
cool ill watch, ive always struggled with proofs aha
proven dragon
and in some intermediate step you’d apply your assumption to arrive at x = y
yeah makes sense, its similar to what they did here ig
proven dragon
and then you'd get lim f(z+h) - f(z) = 0, hence lim f(z+h) = f(z) hence the func...
cool i got it this way now as well, ty
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proven dragon
so they are saying the top line has derivative of that matrix? can someone explain why we can represent derivates as a matrix? not sure when they even covered that idea in class
obtuse pebbleBOT
so they are saying the top line has derivative of that matrix? can someone expla...
proven dragon
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wispy lynx
Would it be correct to simplify A÷(BX+BY) as A÷B(X+Y) or do you have to write it as A÷(B(Y+X)) ?
obtuse pebbleBOT
Would it be correct to simplify A÷(BX+BY) as A÷B(X+Y) or do you have to write it...
wispy lynx
They aren't though, since you would do division first in A÷B(X+Y) but in A÷(B(X+Y)) you would end up multiplying before dividing.
proven zephyr
A÷(B(Y+X)) would be more appropriate tbh
A÷B(X+Y) could either mean
A÷(B(X+Y))
A÷B * (X+Y)
better to just use
A÷(B(X+Y))
dark stirrup
Division and multiplication are resolved left-to-right
wispy lynx
A÷B(X+Y) could either mean
> A÷(B(X+Y))
> A÷B * (X+Y)
Yes exactly what I thought, it would be better/necessary to add those extra parentheses to avoid amibiguity.
dark stirrup
So A÷B(X+Y) will do A÷B first, then multiply that result by (X+Y). That is, it'll do (A÷B)(X+Y)
proven zephyr
that's why fractions > division symbol 🙂
wispy lynx
Agreed, but unfortunately I can't write in fractions on mobile
clear condor
oh i didn't see the ()
exotic palm
I’m not sure on what to do after this to make it into a Cartesian equation
dark stirrup
$r=5\sec{\theta}$?
warm shaleBOT
SWR
exotic palm
yeah
restive wharf
write out the definition of $$\sec{\theta}$$
warm shaleBOT
Jeeves
restive wharf
Then think about how you generally convert polar equations into cartesian ones
exotic palm
restive wharf
One is the fact that x^2+y^2=r^2, which you don't need here
There are another 1/2 identities that would be worth considering
obtuse pebbleBOT
restive wharf
np
clear condor
can you show the picture
more clearly
what is the problem
the 3 angle can be moved up to be next to the 1 angle
yep
if you move m and n together
obtuse pebbleBOT
@timid silo Has your question been resolved?
short holly
i need help confirming a physcis question
obtuse pebbleBOT
i need help confirming a physcis question
short holly
An airport baggage handler pulls a baggage trolley 19.1 m along the flat tarmac. He pulls in a direction 22.5o above the horizontal. The trolley wheels exert a resisting frictional force averaging 34.3 N. Find the magnitude of the force the baggage handler must supply to pull the trolley at a constant speed.
obtuse pebbleBOT
@short holly Has your question been resolved?
obtuse pebbleBOT
@short holly Has your question been resolved?
untold gate
Can someone please explain this? I don't get it
obtuse pebbleBOT
Can someone please explain this? I don't get it
untold gate
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obtuse pebbleBOT
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safe plank
obtuse pebbleBOT
safe plank
I know I need to change the n on top, but I can't figure out a solution that works
I see that from f(x) to f'(x) you multiply by 1, f'(x) to f''(x) *2, f''(x) to f'''(x) *3
frigid prism
I see that from f(x) to f'(x) you multiply by 1, f'(x) to f''(x) *2, f''(x) to f...
yes
so 1, 1*2, 1*2*3...
obtuse pebbleBOT
@safe plank Has your question been resolved?
safe plank
so 1, 1\*2, 1\*2\*3...
So just change n to n!?
frigid prism
So just change n to n!?
yes
manic sorrel
Determine whether the series is convergent or divergent. If it is convergent, find its sum.
3 + 2 + 4/3 + 8/9 + ...
obtuse pebbleBOT
Determine whether the series is convergent or divergent. If it is convergent, fi...
manic sorrel
I got the equation to be (2^(n-1))/(3^(n-2))
now, I got confused about finding the sum
I assumed r is 2/3
same with a
putting that into the sum thing, I get S = (2/3)/(1-(2/3))
which gives me 2
but that's not the sum
just wondering what I'm doing wrong here
I can kind of tell what's wrong here, because the equation isn't just (2/3)^n
but i'm not sure what r would be here instead of 2/3 (or a for that matter)
manic sorrel
it gave me the list of numbers above and I had to find the equation
jolly ginkgo
You did a/1-r right
Yes
manic sorrel
one is n-1 and the other is n-2
manic sorrel
You did a/1-r right
yes to this
frank horizon
If a is 3 and r is 2/3 wouldn’t u get 9
trim portal
one is n-1 and the other is n-2
That's true, but keep in mind that e.g. 3^(n-1)=3*3^(n-2)
jolly ginkgo
Yes
manic sorrel
why is a 3? i thought a would be 2/3
frank horizon
A is first term right
trim portal
3*2/3=2, 2\*2/3=4/3, 4/3\*2/3=8/9...
Watch this, I started with 3, and got 2, 4/3, 8/9...
frank horizon
Lol
manic sorrel
thanks lmao
frank horizon
Np
manic sorrel
Watch this, I started with 3, and got 2, 4/3, 8/9...
i wish i knew how to use that notation bot thingy
writing it out like that is so hard to read
but i don't think I follow what you're doing here
trim portal
$3 \cdot \frac{2}{3} = 2, 2 \cdot \frac{2}{3} = \frac{4}{3}$
warm shaleBOT
MathIsAlwaysRight
trim portal
it would be written like that, what I meant is that it infact is a geometric series
just the starting term changed
But since you understand poopy's explanation, it's not important anymore
obtuse pebbleBOT
daring sedge
Can someone help me with part C line 1
the parametric equations are
x = 1 + t
y = 2
z = 1 + 8t
normally you're meant to set the equations all equal but its tripping me up because the y part doesn't have any value of t
jolly ginkgo
Hi logan
jolly ginkgo
normally you're meant to set the equations all equal but its tripping me up beca...
That's why it is a line?
daring sedge
how do you find the general equations
jolly ginkgo
Eliminate t
daring sedge
daring sedge
this is the form it wants, i did it for the other line
jolly ginkgo
It will be like x-1 = (z-1)/8 , y=2
This will be it
daring sedge
@jolly ginkgo like that
obtuse pebbleBOT
@daring sedge Has your question been resolved?
jolly ginkgo
No idea
daring sedge
should be
8x - z = 7
and do calculate the angle between the two lines i would just use the directional vector part of the line equation
jolly ginkgo
and do calculate the angle between the two lines i would just use the directiona...
Yes
tranquil willow
Could someone help me to find dy²/dx²?
obtuse pebbleBOT
Could someone help me to find dy²/dx²?
jolly ginkgo
Could someone help me to find dy²/dx²?
You mean d²?
cold gale
Hi! How would you find the first derivative?
jolly ginkgo
Hi! How would you find the first derivative?
Change your channel buddy
tranquil willow
You mean d²?
Nope , its dy²/dx²
cold gale
I literally sent my message when you did. Keep a friendly manner please
jolly ginkgo
Nope , its dy²/dx²
I doubt
jolly ginkgo
I literally sent my message when you did. Keep a friendly manner please
Sorry it was my bad
It was misunderstanding
tranquil willow
I doubt <:catThink:526983414226747393>
Second derivative
jolly ginkgo
I thought you were about to ask a question not help
jolly ginkgo
Second derivative
Then it's written as d²y/dx²
tranquil willow
Then it's written as d²y/dx²
Yep
jolly ginkgo
First as axion asked you have to find the first derivative
jolly ginkgo
I literally sent my message when you did. Keep a friendly manner please
I thought you were someone else trying to ask a question in their channel
jolly ginkgo
I just find 4.3x².(-sin(x³))
Yeah correct
tranquil willow
And how to find the second derivative
jolly ginkgo
Find the derivative again
cold gale
I thought you were someone else trying to ask a question in their channel
It's alright man
tranquil willow
I'm still confused
jolly ginkgo
I'm still confused
You have to find the derivative of $-12 x² \sin (x³)$ this time
warm shaleBOT
fäf
tranquil willow
You have to find the derivative of $-12 x² \sin (x³)$ this time
How it can be -12x²?
jolly ginkgo
4×3 and the minus
tranquil willow
Not 12?
Okay ik
I still haven't found the second derivative 😭
How to calculate it @jolly ginkgo
jolly ginkgo
You have to find the derivative of $-12 x² \sin (x³)$ this time
.
tranquil willow
But how to find the derivative
jolly ginkgo
But how to find the derivative
How did you find it the first time
Use product and chain rule
jolly ginkgo
Don't think so
tranquil willow
So
tranquil willow
-24x . sin(x³) - 36x⁴ . cos(x³)
jolly ginkgo
,w differentiate -12x² sin x³
warm shaleBOT
obtuse pebbleBOT
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Available help channel!
timid silo
Hello just curious where the ke comes in from
obtuse pebbleBOT
Hello just curious where the ke comes in from
timid silo
timid silo
c is just some constant, so e^c is just some other constant
so k is another name for a constant
kind hawk
they are using it here to denote a constant, yes
timid silo
k
thick gyro
k
drifting badger
As a side-note, e^c > 0. The reason they can change this to an arbritray constant K, is because of the absolute value that has been removed after simplying e^ln|x + 1| to |x + 1|.
timid silo
As a side-note, e^c > 0. The reason they can change this to an arbritray constan...
bet thanks also they did e to both sides because you cant just do it to one right
kind hawk
yes
whatever you do to one side you have to do to the other
otherwise they wont stay equal
timid silo
of course thank you guys
thick gyro
on a side note, you can also do whatever you want if you have two equations
like
if you have
a = b
and c = d
then a^c = b^d
a/c = b/d
a + c = b + d
thick gyro
oh yh true
drifting badger
(make sure you arent dividing by 0 when you do stuff like this)
You suck all the fun out of division 
obtuse pebbleBOT
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Available help channel!
ashen elk
I have to solve a bunch of puzzles for my informatics degree, but a lot of them have to do with highschool maths. I haven't practiced algebra or calculus for a fat 2 years so i've forgotten most of the techniques and term, anybody have any hints to help me further? no answers please!
obtuse pebbleBOT
I have to solve a bunch of puzzles for my informatics degree, but a lot of them ...
ashen elk
This is as far as i came. I just saw that the Laptas examples are set in increments of 700, but that's it
harsh lantern
Yes
main rose
So you can substract equations from each other, and add them
You can also multply any one equation by a constant
One concern here is that you have 4 variables but only 3 equations, so there wont be an unique solution
ashen elk
well if i multiplied 700L id have to multiply 2N, 2K and 6M, but that wouldnt match the second line with 1400L, so i got confused
main rose
Oh but you have to express it in L so that works
So then you get 1400L + 4N = 4K + 12M
and 1400L + 43N = 50K + 60M that doesnt necessarily contradict right?
You need to multiply every term if you multiply by a constant
ashen elk
how does that not contradict?
oh wait, because I added 4N in one and 43N in the other..?
main rose
Yeah
Wait
not
thats wrong sorry
Okay so systematically what you wanna do:
Seperate the M, and multiply, add and subtract such that just L remains on the right hand
Do you think you can figure that out?
ashen elk
yeahh i can try! how long does this channel stay open? I wanna come back if i don't work it out
main rose
Just keep checkign on it every once in a while
it autocloses after like an hour or something
Just ping me, I'm at my pc
ashen elk
okay, thanks!
obtuse pebbleBOT
@ashen elk Has your question been resolved?
ashen elk
@main rose I just looked up gauss's pivot, i think i understand how it works, however, in all examples the fractor is just a number and doesn't have a variable attached to it. but in my case, i do. Does gaussian elimanation still work then?
2100L + 60K - 3N = 90M
1400L - 50K + 43N = 60M
700L - 2K + 2N = 6M
See how the right hand still has the variable M? not sure what to do with that
main rose
So I am not exactly sure what they my by gauss' pivot since I havent studied this material is english
but you can reduce the problem to an expanded matrix as such;
(2100 60 -3 -90 | 0 )
(1400 -50 43 -60| 0 )
(700 -2 2 -6 | 0 )
and then you can apply gaussian elimination
but you shouldnt worry to much about that
I can give you a simple example
ashen elk
ahh like that
could u give me an example, cuz i have an oppointment in 5 minutes so i have to. I'll look at it after my appointment! thanks for helping me!
main rose
Say you have x + 2y + z = 3
2y - x + z = 2
With the second equation z = 2 + x - 2y, plugging this into the first we find
x + 2y + 2 + x - 2y = 3
Then 2x + 2 = 3, 2x = 1, x = 1/2
This time it has a solution because the y dissappears but this is not true in general
I hope that illustrates the process
obtuse pebbleBOT
@ashen elk Has your question been resolved?
night echo
obtuse pebbleBOT
night echo
I dont know how to obtain the correct response
(practice question)
oh
i think i see it now
my fault
.close
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lean spear
obtuse pebbleBOT
lean spear
how am i supposed to know which graph comes first?
i just watched the video where he teaches it but its not very intuitive to finding the first function
dim mulch
pick one of them, assume its g and see if the other two are consistent with g' and g''
there are only 6 possible combinations so it shouldnt take long to find it
lean spear
could you help me with the first graph? i dont know how to draw the slope/derivative from just looking at it
dim mulch
ah okay
lean spear
dim mulch
when the graph seems to be tending upward, the slope is positive
lean spear
like that?
lean spear
the next part is where im confused, how do i draw the slope line?
dim mulch
you sorta guess it, use your intuition
you arent given any values so you have to extimate
lean spear
im not sure what it would look like
its positive until it gets to 0
then it becomes negative until it gets to zero again
then it becomes positive
lean spear
i think maybe graph 3
dim mulch
have a look at where the marks are along the x axis and see if they correspond to where oud expect g' to be 0
dim mulch
so if we assume graph 1 to be g, we would expect g' to cross 0 at just beyond the 6th mark on the x axis, right?
pulsar breach
obtuse pebbleBOT
obtuse pebbleBOT
@pulsar breach Has your question been resolved?
obtuse pebbleBOT
@pulsar breach Has your question been resolved?
obtuse pebbleBOT
@pulsar breach Has your question been resolved?
obtuse pebbleBOT
@pulsar breach Has your question been resolved?
haughty cliff
Does anyone know how to solve this? I've been trying to write it down for several minutes and equate to zero but i have no idea how to do it.
obtuse pebbleBOT
Does anyone know how to solve this? I've been trying to write it down for severa...
haughty cliff
in the range of [0,3pi/2]
uneven flicker
ok so do you know the formula for sin(a+b)
haughty cliff
yea
uneven flicker
what is it?
haughty cliff
sin(a+b)=sinasinb+cosasinb
uneven flicker
good, so we can say that sin(2a) = 2sin(a)cos(a)
hence sin(4x) = 2sin(2x)cos(2x)
so lets expand the original equation
,, cos(2x) + 2sin(2x)cos(2x) = 1 + 2sin(2x)
warm shaleBOT
Davit
uneven flicker
did you get everything until this part?
haughty cliff
yep
uneven flicker
good
now look at the left hand side
we can factor cos(2x) yes?
lets do that
,, cos(2x)(1+2sin(2x)) = 1 + 2sin(2x)
warm shaleBOT
Davit
uneven flicker
nah theres no need to
haughty cliff
ok
uneven flicker
we can substitute 2x for u, then find the solutions for u, then plug in for x
for simplicity sake
,, cos(u)(1+2sin(u)) = 1 + 2sin(u)
warm shaleBOT
Davit
uneven flicker
here solutions would be the following group
uneven flicker
so maybe now we can throw 1+2sin(u) on a left side?
i did not do that, since you would naturally think that you can divide 1+2sin(u) and get cos(u) = 1, but we can also solve for cos(u) = 0 and 1+2sin(u) = 0
you are familiar with the unit circle right?
haughty cliff
yea
uneven flicker
i meant circle not table
fossil crag
<@&268886789983436800>
uneven flicker
<@&268886789983436800>
thank you
plucky swan
whack ass montage
haughty cliff
i meant circle not table
we only had a little about it so I I don't understand it well
uneven flicker
we only had a little about it so I I don't understand it well
its all right, but i would recommend you go over it a little, because youll need it
its basically a circle with a radius of 1
on an xy plane
y value is the same as sin(x) and x is the value of cos(x)
haughty cliff
you mean this yea?
uneven flicker
you mean this yea?
not exactly but this will do too
ok so given the prob says in the range of [0;3pi/2]
we take into consideration that the solutions are within the first and second quardrant
referring back to my previous message, for solutions
first step would be to solve for cos(u) = 0, which yields u = pi/2 under these circumstances
oh also dont forget that this range is only for x values and not u
haughty cliff
so just u= pi/2 without +npi?
uneven flicker
lets solve with 2x not u ok
cos(2x) =0 hence 2x = pi/2 + kpi
k is an integer
x = pi/4 + kpi/2
haughty cliff
why do we use k if we have given a number range?
uneven flicker
1+2sin(2x) = 0 sin(2x) = -1/2
sin(30 deg) = 1/2 thus sin(210 deg) = sin(330 deg) = -1/2
uneven flicker
why do we use k if we have given a number range?
well... there are infinite amount of solutions
sometimes youll be fine with writing just one
but if you dont, you give the impression that you maybe dont know it
or thats from my experience
haughty cliff
oh ok
