or ax + b
#help-13
kind magnet
faint perch
Actually if you know the equation of the tangent line, it will be done
Yes, both are the same actually. You know m (or a), which is the slope of the tangent, known as the value of the derivative at the point we study.
kind magnet
right
so far I have y = 39x + b
if I plug in any point I can find the equation of the secant line
faint perch
And to find b, we just take one point that we know it is on the tangent line and replace x and y with its coordinates to find b.
faint perch
if I plug in any point I can find the equation of the secant line
Secant?
vernal shell
What I mean is that you should read carefully. They give you the slope at (2, f(2))
faint perch
which is (2,78)
Yes!
vernal shell
that's the secant line, not tanget π§
faint perch
Actually yes, the slope at X=2 is given in the exercise, and is not the value you computed before
kind magnet
well I found the velocity of the secant line which is really its slope
but I am still missing an equation to find the tangent
that's what I am not understanding correctly
faint perch
You found the average velocity, not the actual velocity when he was at 2 hours
kind magnet
oh
faint perch
This one is given at the beginning of the exercise
kind magnet
wouldn't it just be 45 km / h?
vernal shell
wouldn't it just be 45 km / h?
yes ππ½
kind magnet
so that is the actual velocity
which would be the slope
so y = 45x + b
78 = 45(2) + b
78 - 90 = -12
y = 45x -12
so is the slope of a tangent line always the actual velocity? I'm struggling to picture the graph in my head
faint perch
You can actually prove that for any function f (derivable), the equation of its tangent at a is:
y=f'(a)(x-a)+f(a)
And that's what you have.
faint perch
so is the slope of a tangent line always the actual velocity? I'm struggling to ...
The slope of the tangent line at a point is the value of the derivative at this point, so it must be the real velocity yes. The average velocity doesn't represent at all the value at this time.
kind magnet
ok
so if I'm trying to find the slope of another tangent line, i should look for the derivative at that point
faint perch
so is the slope of a tangent line always the actual velocity? I'm struggling to ...
Just to draw the graph of the distance, you may see it better
kind magnet
I don't think they provided the graph right?
faint perch
No but you can imagine an example
kind magnet
I'm not really good with graphs, I know the basic ones like quadratic, rational, log, exponential
but when they ask me to graph it in my head or draw it out, I always struggle
faint perch
I understand, I'll see if I can draw something clear
kind magnet
But thanks for explaining the slope of the problem
I was really trying to understand how to find the slope
.closer
.close
near rose
Hello
onyx dove
yeah
crimson sedge
how would i use this formula in a economics? (micro)
cedar kilnBOT
how would i use this formula in a economics? (micro)
crimson sedge
so like it can be used for value of a product
could it calculate stuff like
quantity demanded?
or just like any kind of value
is there anything it cant get?
i see
so id assume it would work for stuff like supply too then
yeah
i see
last question then if you dont mind
Why for this is the top -200
why do both of them fall?
i mean like i thought i would do .90-1.10
instead of the -200
i see
so the 200 units in this case would be the entire 200 cents
ok
ah i see
i didnt realize those numbers were negative
since they didnt have a(-) sign
π
thanks
also for this yeah
but a-b is -0.2 in this case
because 0.9-1.1
ok
π
.close
cedar kilnBOT
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Available help channel!
wicked badge
cedar kilnBOT
wicked badge
zero's are 2, -2, 1, -1 aren't they
none of them work when trying to set the equality
idk where I fucked up
0 = x^4 - 5x^2 + 4 right????
onyx dove
this is the entire equation?
wicked badge
yes
wraith daggerBOT
chlamydia
wicked badge
then mult left by 2x-3
wraith daggerBOT
chlamydia
wicked badge
um yeah in my solution tests it didn't equal 1 or -1
obsidian coral
0 = x^4 - 5x^2 + 4 right????
Where did this come from?
wicked badge
um
I mult. left by 2x-3 and right by (x-1)(x+1)
solve then subtract 2x-3 from one side
wait
I multiplied left by x^2-1 my bad
but you get x^2-1 = x^4 -4x^2 +3
then subtract and ummmmmm
wicked badge
Where did this come from?
yeah um I don't see where you're confused
obsidian coral
but you get x^2-1 = x^4 -4x^2 +3
No that right side is wrong
wicked badge
really?
obsidian coral
If you multiplied by (x^2-1) on both sides
obsidian coral
Then the x^2-1 on the right cancels out
wicked badge
yeah I know
obsidian coral
(x-1)(x+1) = x^2-1
obsidian coral
If it cancels out, why do you have x^4 -4x^2 +3?
wicked badge
??????
obsidian coral
what about the x^2-3?????
It stays this
wicked badge
because (x^2 - 3) (x-1) (x+1) = x^4 - 4x^2 +3
obsidian coral
$(x^2 - 1)(\frac1{x+1}+\frac1{x-1}=\frac{x^2 - 3}{x^2 -1})$
wraith daggerBOT
dldh06
obsidian coral
Then you distribute that
wicked badge
god I need a drink
obsidian coral
And the x^2 - 1 cancels out on the right side
wicked badge
I KNOW
I REMOVED IT
what are you on about??!!!
it becomes x^2-3
on the right side
are you okay
I don't understand the issue
x^2-3/1
obsidian coral
because (x^2 - 3) (x-1) (x+1) = x^4 - 4x^2 +3
Then why did you do this?
obsidian coral
If you knew it canceled, why did you expand that?
wicked badge
it should be um x^2-1 equals x^4......
what
holy shit dude
I EXPANDED BECAUSE YOU SUBTRACT x^2-1
I'm so lost
am i doing this wrong????
to get zero's one side needs to be zero
to do that you subtract one side
to do that you have to multiply by the denominator's
are you trying to make me sound crazy???
I really don't need your help for that I need your help for math
oh my god
okay whatever I'll keep throwing my head at the wall till I get it I guess
is there no solution maybe??? they're all extranious???
obsidian coral
$(\frac1{x+1}+\frac1{x-1}=\frac{x^2 - 3}{x^2 -1})$
That's the original problem, step one is finding the common denominator, right?
wraith daggerBOT
dldh06
wicked badge
step 1 is making a common denominator
by multiplying the left side by the right side's denom and vice versa
obsidian coral
Are you saying (x + 1)(x - 1)(x^2 - 1)?
wicked badge
no
omg
idk if i can do this
I've been doing homework for like 3 hours and I'm on problem 2
obsidian coral
Then what do you mean by multiplying the left side by the right side
wicked badge
i don't fucking know man
1+1(x^2-1)
is that right???
I am desperately trying here
I should just sleep on it
obsidian coral
$(x^2 - 1) \cdot \left(\frac1{x+1}+\frac1{x-1}\right)=\left(\frac{x^2 - 3}{x^2 -1}\right) \cdot (x + 1)(x - 1)$
wraith daggerBOT
dldh06
obsidian coral
You mean like that?
cedar kilnBOT
wicked badge
have a goodnight
cedar kilnBOT
Available help channel!
stone forum
how do i factoirse basic equations
cedar kilnBOT
how do i factoirse basic equations
obsidian coral
Do you have an example
stone forum
yes
ill bring it up now
3yz - 18xy + 6xy^2z
thats the question thaqt i need help with
hello?
onyx dove
look for common factors
dusty snow
cedar kilnBOT
lunar elbow
There's a formula to find the distance between two points
You can also graph the points and use the Pythagorean theorem
surreal cave
You can also graph the points and use the Pythagorean theorem
^ do this and also recall that the distance formula is just a generalized Pythagorean Theorem for two points
lunar elbow
yes
dusty snow
15 units?
dusty snow
?
crimson sedge
the answer is correct yes but working is more important
crimson sedge
I just searched it up tbh
oh
dusty snow
I agree with you
crimson sedge
do you know the distance formula
crimson sedge
$$d = \sqrt{(x_2-x_1)^2 + (y_2 - y_1)^2}$$
wraith daggerBOT
sen
crimson sedge
right
now substitute and tell me what you get
lmk if you get stuck
also @dusty snow what would be x1 y1 and x2 y2?
crimson sedge
nop
crimson sedge
also <@685628085441396755> what would be x1 y1 and x2 y2?
.
lets start with this first
dusty snow
In the equation x1 is 5 and y1 is 4
crimson sedge
So you have these two points. (5,4) (-4, 16)
dusty snow
Yea
crimson sedge
In the equation x1 is 5 and y1 is 4
ok and x2 and y2
dusty snow
-4 and 16
crimson sedge
ok
crimson sedge
also here
what is (-9)^2=?
dusty snow
225 square root 15
crimson sedge
dusty snow
I do have a couple more questions if your up for it
I posted a new one thank you tho so much !
.close
late wadi
for d, is just proving it isn't a function from R->R enough to prove it cannot be a bijection?
long swan
yes
in order for it to be a bijection from R to R, it must be a function from R to R (lol?)
late wadi
yeah sorry its kind of a dumb question i was just confused because it seemed quite obvious
violet jay
You might have some trouble proving that it's not a function R -> R
long swan
show that D is never negative
late wadi
You might have some trouble proving that it's not a function R -> R
why would that be
violet jay
Because it is π
late wadi
oh im so dumb
sorry
the denomiator isnt subtracting...
im really tired π
would i just say something like for all integers, x^2 + 1 and x^2 + 2 are positive integers
reals*
so therefore x^2 + 1 / x^2 + 2 is positive for all reals, so it is not onto, therefore it is not a bijection?
violet jay
Sounds good
crimson sedge
Iβm trying to confirm my answer for this question.
βFind and simplify the difference quotient for the function f(t) = 3/t-6β
Would the answer be -3/(t+h-6)(t-6) or would it be -3/t-6^2
cedar kilnBOT
Iβm trying to confirm my answer for this question.
βFind and simplify the diff...
@crimson sedge Has your question been resolved?
south tundra
The former
versed fulcrum
Hello
versed fulcrum
How do I simplify cosh(arcsinh(x))?
rare vault
probably write them in their other form
I didn't try but that seems like the logical step
versed fulcrum
probably write them in their other form
What other form?
versed fulcrum
arcsinh(x) = log(sqrt(x^2+1)+x)
Ohhhh
Where can I find the other forms?
For like arccosh or arctanh, arccoth, etc
rare vault
look them up
versed fulcrum
What's the name for it?
rare vault
I don't know the derivations, I either memorize or look them up
I have no idea, just "alternate form" probably
use wolfean
,w alternate form sinh(x)
wraith daggerBOT
rare vault
there we go
jaunty mural
if u werent given their other form
how are these functions defined for you
you should use whatever defn uve been given
rare vault
yeah if you have more context from class or whatever use that - I gave the solution I would use, doesn't make it the only one or even the one they want
jaunty mural
We weren't given any but
howww
jaunty mural
is this some course
jaunty mural
We weren't given any but
like theres no way you dont have a definition sinh otherwise how can anyone be expected to use it.
heres some symbols, but we wont tell u what they mean...
.close
cedar kilnBOT
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Available help channel!
lucid orchid
Hey, so I'm a first year doing discrete and I have alot of trouble around proofs and generally just cant do them. We're doing some set theory stuff and its quiet proof heavy. I'm really stuck with it and I just don't know how to actually write a proof or generally how to tackle these kinds of questions.
lucid orchid
Here's what I've tried doing, but its probably just a bunch of garbo I really dont know what im talking about with this stuff
idk why its so low quality on discord, open it on chrome or something to see it ig
crystal raptor
If a was onto, that would mean you could write any integer into the form 4n-3, is this true?
lucid orchid
Oh is it not? How would I show that algebraically for all values instead of some specific case
Do I just need to try specific values to find that out
crystal raptor
If you are trying to disprove something, you only need to provide 1 counter example
lucid orchid
so because we are multiplying by 4 we end up skipping lots of integers in Z right
crystal raptor
Sure
crystal raptor
Yep 2 is a fine example
lucid orchid
so this would be considered a proof by counter example right
crystal raptor
Yes
lucid orchid
what about question B, I'm really confused by that one
since reals are uncountably infinite
you should be able to make any value in the reals from a product of any other two real values
crystal raptor
Your written argument is not enough
You have said that it's true because multiplying two reals gives you a real
lucid orchid
yes, I know, that is why I need some help haha
crystal raptor
But you need to justify that you can get any real by multiplying two reals
lucid orchid
I cant think of a value where that would'nt be the case
crystal raptor
Well yes bc spoiler alert it is onto
lucid orchid
ok but how would you show that algeberically, thats kind of what ive been asking about
since thats what I cant do
crystal raptor
How could you get 5 using g?
lucid orchid
itself and one
crystal raptor
Okay and Ο?
lucid orchid
same thing, you can get any value with itself and one ohh
but how would I express that
crystal raptor
Right so how would you get a generic real number b
crystal raptor
Yeah
So given b in R you've found x,y in RxR such that g(x,y) = b, this is exsctly being onto
For disproving one to one you need to give a specific example of what you're saying
lucid orchid
so one to one is another proof by counter example
while the onto is a direct proof or would it be considered something else
crystal raptor
Direct
Counter example proofs are only when you are trying to disprove something
Hence the word counter
lucid orchid
Okay, so I would need to write two different combinations of x * y both resulting in the same real value
crystal raptor
Yes
lucid orchid
alr thank you
in question C, why is the second Z nonneg
and not Z+
how are those two things different sets
crystal raptor
Because they are different things
lucid orchid
weird they never showed us Z nonneg in the leactures
crystal raptor
0 is nonnegative, it is not positive
crystal raptor
Yes
lucid orchid
gosh and I thought Z+ contained 0 this whole time
alright thank you, I'll get all this down
I'll try the third question on my own I have a pretty good idea for it
I think
it would have to do with that second set containing 0
I'll close this thread then?
crystal raptor
If you don't have anymore questions currently yeah do .close
night anchor
.close
cedar kilnBOT
Available help channel!
foggy tendon
Didn't really understand the last line there
cedar kilnBOT
Didn't really understand the last line there
@foggy tendon Has your question been resolved?
crystal raptor
They replaced d/dt with what it's equal to in 8.39
crystal raptor
Differentiate 1/u using quotient rule
foggy tendon
I mean how they got the u^2 into the derivative
crystal raptor
Differentiate 1/u using quotient rule
Do this
keen nova
i found the perpendicular equation to l, intersected it with both l and m and thus got two points, then did pythagoras for those two points and got this answer, but it is still wrong, in what way should i think about this question
muted timber
show your work?
jaunty mural
i think u need to draw and show us picture.
muted timber
,rotate
keen nova
,rotate
wraith daggerBOT
keen nova
it's a bit all over the place sorry
muted timber
okay first off that drawing is really poor, you're given -1/6 slope
you're drawing the line as slope -1
keen nova
oh fuck that first drawing is irrelevant i forgot to remove
the bottom right is relevant although still poor
jaunty mural
u dont need a perpendicular equation really
you can form a right angled triangle with the y axis i think
yh, draw the 2 lines. do some geometry near the y axis
muted timber
okay, so from what i've seen, you're cobtaining the line perpendicular to l that intercepts it on the y-intercept
keen nova
yep
muted timber
and you call that l'
keen nova
yep
muted timber
then on the bottom right you're getting the coords of the intercept of m and l'
keen nova
yeah, i intersect l' and l (which is obviously 0,-9 i figured), and l' and m
muted timber
and with that you obtain Q as intercept of m and l'?
keen nova
yeah
muted timber
however Q is not correct
keen nova
hmm
muted timber
the x coord of Q is in fact x=2, but the y coord is not corectly computed
6*2 is not 18
muted timber
most mistakes usually seem like so
that's why you try to draw stuff as close as possible
to see if the analytical answer is close to the drawing
i'd recommend you graph all 3 lines in geogebra or something and compare with your drawing
keen nova
do you recommend doing so often? i thought i'd try to visualize it all with pen and paper as often as possible to get intuitive understanding
sometimes i do wolfram but not always
cedar kilnBOT
Available help channel!
slender mango
Hey, just need some help understanding this
cedar kilnBOT
Hey, just need some help understanding this
slender mango
I can make the numerator into 10-8x^7
and those two will take out eachother
and you are left with -3/10
am I correct so far?
nevermind I think I got it
you take out x^7 from both, the 3 and 10 goes to infinity and you are left with -8/8 and since its squred evenly you are left iwth 1
.close
cedar kilnBOT
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Available help channel!
lethal jackal
What is the actual, precise definition of something like $Z = \mathbf E_X[g(X, Y)]$? I've always been told that the X is "the variable over which you're taking the expectation," and then given some examples involving discrete or continuous random variables, i.e. something like $f_Z(y) = \int g(x, y) f_X(x) , dx$, but this doesn't make a lot of sense with the formal definition of expectation as: $\mathbf E[g(X, Y)] = \int g(X, Y) , d\mathbf P$, where P is a probability measure over some product space. In other words, you don't get to choose the random variable to integrate over; expectation is integration over the entire probability space.
I would think something like $\mathbf E_X[g(X, Y)]$ is some alternative notation for $\mathbf E[g(X, Y)|Y]$, but this does not appear to be true even for the continuous case, because the conditional density $f_{X|Y}$ shows up in the integral, instead of $f_X$?
wraith daggerBOT
Saccharine
cedar kilnBOT
@lethal jackal Has your question been resolved?
lethal jackal
<@&286206848099549185> this question is probably going to close since I have to go, but I would appreciate a reply that I can go over asynchronously. Thanks!
jaunty mural
<@&286206848099549185> this question is probably going to close since I have to ...
ur better off in the topic specific channels
cedar kilnBOT
@lethal jackal Has your question been resolved?
lime wagon
can someone please explain to me the notation of the bounds of this integral?
cedar kilnBOT
@lime wagon Has your question been resolved?
crimson sedge
How to solve this?
cedar kilnBOT
How to solve this?
crimson sedge
I need explanation please not answer
broken mist
You can solve for x by setting the sum of those expressions to 180 (interior angles of triangle sum to 180)
Then plug x back into them to figure out the actual value of each angle
crimson sedge
I still don't get it im slow here
primal socket
Yeah I know that
broken mist
You can figure out the side lengths relative to each other from the angles though
primal socket
Yeah
broken mist
I still don't get it im slow here
(2x + 3) + (x - 1) + (x + 6) = 180
crimson sedge
Okay yeah I know that part then what tho
broken mist
You can solve for x from there, then put x into each angle to figure out how big they are
crimson sedge
I can do 2x-x+X if I don't know what X equals to I am overthinking here ain't i
crimson sedge
Yeah then what?
primal socket
I don't know what you are trying to do with that
Do what M. Frost said
When you solve for x, you can substitute x for its value into every angle and figure out the size of each angle
Then just arrange them from smallest to largest
crimson sedge
Well can you show me an example?
primal socket
This one?
primal socket
Angle at P = (x-1)Β°, at Q (x+6)Β°, at M (2x+3)Β°
We know that the sum of the 3 interior angles in a triangle equals to 180Β°
crimson sedge
Yeah
primal socket
So we write (x-1) + (x+6) + (2x + 3) = 180
And solve for x
$(x-1) + (x+6) + (2x + 3) = 180$
wraith daggerBOT
USS-Enterprise
primal socket
$x - 1 + x + 6 + 2x + 3 = 180$
wraith daggerBOT
USS-Enterprise
crimson sedge
I get that part but how do I solve for x thats what I don't get
primal socket
$4x - 8 = 180$
wraith daggerBOT
USS-Enterprise
primal socket
$4x = 172$
wraith daggerBOT
USS-Enterprise
primal socket
$x = 43$
wraith daggerBOT
USS-Enterprise
primal socket
You've just solved for x.
crimson sedge
$4x - 8 = 180$
How did you get that part?
primal socket
Collected like terms
First, we use commutativity to get
$x + x + 2x - 1 + 6 + 3 = 180$
wraith daggerBOT
USS-Enterprise
crimson sedge
Oh so you did X+X+2x=4x
primal socket
Then collect like terms
crimson sedge
But how about 8 how did you get that?
primal socket
-1 + 6 + 3 = -1 + 9 = +8
crimson sedge
Oh so we get the X and y
crimson sedge
What is 8 equal to?
wraith daggerBOT
USS-Enterprise
primal socket
**USS\-Enterprise**
I made a mistake here, it's supposed to be +8 not -8***
primal socket
We know the angle at P is (x-1)Β°, at Q (x+6)Β° and at M (2x + 3)Β°
So we just plug x in
The angle at P is (43-1)Β°, at Q (43+6)Β°, at M (2*43 + 3)Β°
Or the angle at P is 42Β°, at Q it's 49Β° and at M it's 89Β°
crimson sedge
Okay now I order them from smallest to largest
primal socket
You can check your work at this point and check if the 3 angles really do add up to 180
And we get 42Β° + 49Β° + 89Β° = 180Β°
So we are correct
crimson sedge
Exactly
primal socket
And yes, now you just arrange them from smallest to largest
crimson sedge
But there was like a rule from smallest to largest
For example
If pq was the smallest
Then it would be the one right Infront of it
Or something like that
primal socket
The longest side in a triangle is opposite to the largest angle
crimson sedge
How do I know the longest side
primal socket
(and the shortest side is opposite to the smallest angle)
You check which side is accross the largest angle
crimson sedge
And shortest side
primal socket
We see the angle at M is the largest
And the side opposite the angle M is PQ
Which is the longest side
crimson sedge
Pq
primal socket
Yes
crimson sedge
Okay I think it's B
primal socket
You are correct
primal socket
You don't have to worry about the lengths of sides in your case though
crimson sedge
Okay π
primal socket
If you are sure you've arranged the angles correctly you can see there's only one option with your arrangement of angles
So you don't need to check the lengths of the sides
crimson sedge
Well I mean checking is good right
primal socket
Of course
crimson sedge
Why not just check
primal socket
No problem, you too π
errant tide
I'm sorry in advance for asking about something thats probably easy and I should know. Im dealing with exponential equations. Photomath says that I should factorize a part of this equation and shows me the result, but I don't understand how the factorization works here.
errant tide
sudden schooner
,rotate
wraith daggerBOT
sudden schooner
is that 3*4^2 or something else in the last line you wrote
errant tide
its (1+4+3Γ4Β²)
errant tide
I dont understand how photomath got that
sudden schooner
and we solve for x
errant tide
that equates to 212?
Yes, its supposed to be a exponential equation
errant tide
and we solve for x
Photomath says that I should factorize that first line, and thats how I got the second line. But I dont understand how the factorization works here
sudden schooner
Photomath says that I should factorize that first line, and thats how I got the ...
no idea either π . Seems wrong if i'm being completely honest, gonna have to expand it to find out
errant tide
π π
surreal cave
$4^{x-1}=\frac{4^x}{4}$
wraith daggerBOT
MrFancy
surreal cave
Do you see how we can factor $4^{x-1}+4^{x}+3\cdot4^{x-1}$? :)
wraith daggerBOT
MrFancy
errant tide
Im going to be honest, Im still not pretty sure about how that ends up being photomath says
surreal cave
Im going to be honest, Im still not pretty sure about how that ends up being pho...
substitute 4^x/4 for the all 4^(x-1)'s in the equation
cedar kilnBOT
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Available help channel!
dreamy oyster
Help please
cedar kilnBOT
Help please
dreamy oyster
.close
cedar kilnBOT
obsidian coral
Help please
Okay
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Available help channel!
worldly wagon
cedar kilnBOT
latent bloom
Hey so what is the issue with your question?
worldly wagon
i dont know which diagram it is
i think itis D
but theres more than one
so im not sure which one they are
pls
?
worldly wagon
8 sides?
worldly wagon
9 faces
worldly wagon
so what diagrams would it be
worldly wagon
ok
undone sage
After seeing it's figure
undone sage
And D too
worldly wagon
ok
undone sage
C can't since both faces are above
So if u fold it
The bottom face will not be there
worldly wagon
cedar kilnBOT
lunar lynx
ik one is C
How did you work that out?
tropic oxide
try working out which ones CAN'T be cube nets
wicked badge
cedar kilnBOT
wicked badge
somebody please tell me you know how to solve this
I don't know where to start, I think i have to find the function given the points but I don't know how to do that since it's not a linear function
obsidian kindle
ok so that notation is saying you need to solve the following function operation: g(f(3))
knowing that, and looking at the plot points you were given should help you find the value of f(3), which you can then use to find the value of g(f(3))
cedar kilnBOT
@wicked badge Has your question been resolved?
errant seal
Is there an equation for the inverse of a 4x4 matrix?
cedar kilnBOT
Is there an equation for the inverse of a 4x4 matrix?
foggy merlin
you can try to find the formula yourself but goodluck
errant seal
π₯²
tropic oxide
you will go through approximately 25 metric fucktons of pain and suffering
errant seal
how do games calculate it then
errant seal
damn
sage forge
Is there an equation for the inverse of a 4x4 matrix?
As mentioned the general formula would be pain to find. Is there a specific matrix that you want to find the inverse of?
errant seal
I think in video games, it's usually a 4x4 matrix that has the 3x3 top left area used
and only the last row and last column as the w component
so like this
a b c 0
d e f 0
g h i 0
0 0 0 1
$\left[ \begin{array}{cccc|cccc} a & b & c & 0 & 1 & 0 & 0 & 0 \ d & e & f & 0 & 0 & 1 & 0 & 0 \ g & h & i & 0 & 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \end{array} \right]$
wraith daggerBOT
Herels
foggy merlin
and reduce it to get the identity matrix on the left
white latch
Hi guys I am currently learning about application of derivatives and had a question on concavity, so I understand concavity, but to picturize it better I made up a scenario. Here is my idea. Lets say we have a wonky function with numerous turns where the Y axis represents Revenue relative to time. The goal is to sketch the function efficiently. If we had a humongous polynomial function it would be difficult as we would have to input numerous points to get an accurate representation of the graph which is not efficient. However if we take the first derivative and we will get information on the behavior of the function between certain intervals for when the derivative is equal to 0 or DNE. Now we come across a problem although we find the behavior of the function between certain intervals we could argue for each individual sector (interval) that the function can be drawn concave upwards or concave downwards and the behavior would remain the same. To understand whether or not its concave downward or concave upwards we need to take the second derivative which will give us the instantaneous rate of change of Revenue. Now throughout the function there exists a point where we have possible inflection points at f''0 or f'' DNE. Now after the possible inflection points we test points nearby the interval to observe the behavior of the slopes. If f'' > 0 I picturize as instanteous change in revenue being positive, so I think of that as our loss in revenue minimizing and becoming increasingly positive. If f'' < 0 the instataneous change in revenue is negative which means are losses will be increasing. After we conclude the concavity for all the intervals for f'', we can then determine the inflection points.
white latch
The importance of inflection points is that if we think about it in terms of revenue we can determine the moment we start making positive revenue and the moment when revenue becomes negative.
I'd like some feedback. Perhaps I could have simplified what I wrote as it does seem a fair bit long, but in essence when we find the behavior of the function for intervals there are 2 arguments we can make, concave up or concave down and we cannot assume as it could be incorrect and we would be changing the data of the function which would lead to incorrect conclusions, this is where the concavity test is vital. I associate f'' > 0 in terms of revenue I picturize that as revenue becoming increasingly positive and f'' < 0 as our revenue dropping, instantaneous change in revenue negative, hence losing.
cedar kilnBOT
@white latch Has your question been resolved?
cedar kilnBOT
@white latch Has your question been resolved?
rare vapor
Why does problem 8 use the shell method
cedar kilnBOT
Why does problem 8 use the shell method
cedar kilnBOT
@rare vapor Has your question been resolved?
gentle egret
am i doing this right? if you need the work let me know
cedar kilnBOT
am i doing this right? if you need the work let me know
sturdy shoal
am i doing this right? if you need the work let me know
looks right!
gentle egret
looks right!
im stuck between b and d. they both look right to me
sturdy shoal
im stuck between b and d. they both look right to me
remember you're law of exponents: $a^b\cdot a^c=a^{b+c}$
wraith daggerBOT
Stumpman
gentle egret
.close
warm glacier
cedar kilnBOT
warm glacier
Could someone tell me why the chance is not 0.18?
Is it because I did not account for the 12% error
In that case would I be more correct if I did 0.18 * 0.78 * (1-0.12) / 0.78 = 0.18 * 0.88?
cedar kilnBOT
@warm glacier Has your question been resolved?
warm glacier
<@&286206848099549185>
cedar kilnBOT
@warm glacier Has your question been resolved?
tropic tree
the line about 18% is indicating the entire population. a random person takes the test. there's an 18% chance they have it
P(D) is indeed 18%
P(P) is not 78%
P(D β© P) is not 18%*78%
let's start by list all that we know
what is P(!D) = ?
P(P|D) = ?
P(P|!D) = ?
cedar kilnBOT
@warm glacier Has your question been resolved?
warm glacier
P(P|D) = ?
I know
P(!D) = 1-0.18
but not too sure about P(P|D), I think it is 0.78 and then P(P|!D) = 0.12?
So I had difficulty recognizing that this problem is a bayes theorem problem
so I didn't think about P(P|D) and P(P|!D)
I've solved it thanks
.close
opaque shadow
can i have help with this squeeze thereom question?
cedar kilnBOT
can i have help with this squeeze thereom question?
opaque shadow
i do not know how to do it
they havent taught me how to properly do it in my classes
cedar kilnBOT
@opaque shadow Has your question been resolved?
ancient lodge
Do yk what the squeeze theorem states
opaque shadow
ok
ancient lodge
So for this limit, what is the main part that's causing issues rn
opaque shadow
it is in 3 parts
the exponential the root and cosine
we havent been taught how to deal with it in 3 parts
ancient lodge
yk direct substitution tho right
what's the part that's preventing us from using direct substitution
opaque shadow
what's the part that's preventing us from using direct substitution
the cosine?
ancient lodge
mhm
can you think of how you would write an inequality to bound the cosine?
Hint: Consider ||the range of the cosine function||
opaque shadow
can you think of how you would write an inequality to bound the cosine?
no not really im still fresh to limits because i didnt do add maths
ancient lodge
no not really im still fresh to limits because i didnt do add maths
What I'm asking rn isn't limits
it's basic function knowledge
opaque shadow
my teacher sped through functions and limits in 1 week i do not know anything by heart or by mind
im still fresh to it all ;-;
ancient lodge
ok you should know that sin and cos have ranges of [-1,1]
opaque shadow
yes i know that one
ancient lodge
so writing this as an inequality, what do you get
ancient lodge
so writing this as an inequality, what do you get
anyway back to this
opaque shadow
so writing this as an inequality, what do you get
uhhhh
ancient lodge
well let's say we had y was in the interval [-5,3]
how would you write that with inequality notation
don't overthink it, you probably learned this in basic algebra
opaque shadow
im so confused its annoying me im sorry bro ;-; π
like nothing is coming to mind
ancient lodge
$-5 \leq y \leq 3$?
wraith daggerBOT
Civil Service Pigeon
opaque shadow
i give up on life
ancient lodge
does that mean yes it looks familiar or no it doesn't
opaque shadow
yes it looks familiar
OHHHHHH shit
so since its cosine
it would be -1 <= cos(pi/x-2) <= 1
?
ancient lodge
btw I'm answering you rn so idk if you want to delete this to be considering of other people's time
ancient lodge
it would be -1 <= cos(pi/x-2) <= 1
But yes, we have $$-1 \leq \cos \left(\frac{\pi}{x-2} \right) \leq 1$$
wraith daggerBOT
Civil Service Pigeon
ancient lodge
now what we want to eventually do is bound the entire thing we're taking the limit of
opaque shadow
btw I'm answering you rn so idk if you want to delete this to be considering of ...
thats the wrong question mb i wanted to show them a limit question inbetween -inf and +inf
ancient lodge
oop
ancient lodge
**Civil Service Pigeon**
anyway can you think of how we would introduce the factor of $e^x \cdot \sqrt[3]{x-2}$
wraith daggerBOT
Civil Service Pigeon
opaque shadow
But yes, we have $$-1 \leq \cos \left(\frac{\pi}{x-2} \right) \leq 1$$
ok
opaque shadow
anyway can you think of how we would introduce the factor of $e^x \cdot \sqrt[3]...
hmmmm
since its an exponential
the fixed number is 2.71828
right?
or that isnt needed
opaque shadow
Ok
ancient lodge
instead, consider multiplying all three sides of this inequality by something to bound the function you're taking the limit of
opaque shadow
so multiply all 3 sides by cos(pi/x-2)?
ancient lodge
yes you're getting to the right idea now
$\sqrt[3]{x-2} \leq \sqrt[3]{x-2} \cos \left(\frac{\pi}{x-2} \right) \leq \sqrt[3]{x-2}$
wraith daggerBOT
Civil Service Pigeon
ancient lodge
π
$e^x \sqrt[3]{x-2} \leq e^x \sqrt[3]{x-2} \cos \left(\frac{\pi}{x-2} \right) \leq e^x \sqrt[3]{x-2}$
wraith daggerBOT
Civil Service Pigeon
ancient lodge
and now we can apply the idea from here
opaque shadow
ok....
ancient lodge
in this case, take the limit as $x \to 2^+$
wraith daggerBOT
Civil Service Pigeon
ancient lodge
Which yields $$-\lim_{x \to 2^{+}} e^x \sqrt[3]{x-2} \leq \lim_{x \to 2^{+}} e^x \sqrt[3]{x-2} \cos \left(\frac{\pi}{x-2} \right) \leq \lim_{x \to 2^{+}} e^x \sqrt[3]{x-2}$$
wraith daggerBOT
Civil Service Pigeon
ancient lodge
shoot I forgot that negative sign on the left the entire time lol
ancient lodge
so then we input 2 into the function?
but yes
opaque shadow
shoot I forgot that negative sign on the left the entire time lol
no dont worry i put it in already
ancient lodge
do you see what happens now
opaque shadow
do you see what happens now
yes and no
we are basically trying to make the function equat to 2?
is that how you spell equat? ππ
opaque shadow
hmmmm
opaque shadow
OHHHH
so just put 2 into the left side
and the right side
and if both answers are the same
then the whole equation is equal to that?
ancient lodge
not 'equation', but limit
But after direct substitution, yes, you do get $$0 \leq \lim_{x \to 2^{+}} e^x \sqrt[3]{x-2} \cos \left(\frac{\pi}{x-2} \right) \leq 0$$
wraith daggerBOT
Civil Service Pigeon
ancient lodge
which means the original limit must be zero
you can think of it as "squeezing" the limit between the same value, in this case 0
which is why it's called the squeeze theorem
ancient lodge
π
ancient lodge
opaque shadow
.close
cedar kilnBOT
opaque shadow
.reopen
cedar kilnBOT
β
opaque shadow
there is one last one
i forgot about
this one
what is throwing me off is that the numirator of the limit
.close
cedar kilnBOT
opaque shadow
.close
cedar kilnBOT
Available help channel!
void sand
b)
void sand
!status
cedar kilnBOT
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
cedar kilnBOT
@void sand Has your question been resolved?
void sand
<@&286206848099549185>
cedar kilnBOT
@void sand Has your question been resolved?
cedar kilnBOT
@void sand Has your question been resolved?
cedar kilnBOT
@void sand Has your question been resolved?
cedar kilnBOT
cedar kilnBOT
Available help channel!
vale barn
pls help
vale barn
someone pls help
foggy merlin
do you know binomial ?
warm bone
Binomial theorem
warm bone
Explain
vale barn
wait ill send a pic
foggy merlin
find the coefficient of x^6 and x^5 in (3x+1)^9 π€
vale barn
foggy merlin
$\binom{9}{r} (3x)^{r}$
wraith daggerBOT
Herels
vale barn
shouldnt it be (3x)^9-r
foggy merlin
not always
I mean, it can be (3x)^{9-r} or (3x)^{r}
its just that addition is commutative, so we can use the later
foggy merlin
yes
foggy merlin
multiply by -x
foggy merlin
$\binom{9}{5} (3x)^5 (-x) = -\binom{9}{5} 3^5 x^6$
wraith daggerBOT
Herels
vale barn
what happens to the 2 in (2-x)
foggy merlin
$2\binom{9}{6}(3x)^6$
wraith daggerBOT
Herels
vale barn
no im talking abotu the 2
vale barn
cant i multiply it with -x and 2
wild grove
cedar kilnBOT
wild grove
Can any one help me out with it?
cedar kilnBOT
@wild grove Has your question been resolved?
crimson delta
did you have a result of the form 1=(something)?
crimson delta
can you factor that
wild grove
Ih
crimson delta
Ih?
wild grove
(m-n)(m+n)=1
crimson delta
yes
cedar kilnBOT
Available help channel!
dreamy lava
Answer to this problem was apparentley x-8/3x-12... what makes my method wrong?
