y1 = 2(x+1)*x1 +c
#help-23
dull sequoia
That’s not the equation of a line
A line needs both an x and y
Thus far I see an x and what is this c doing
How about I will do this example, then you repeat the steps back to me
dull sequoia
Okay first step we get a generic straight line, y = mx + c
dull sequoia
are you referring to y-y1= m(x-x1) though?
Kinda, but that’s a formula and I don’t particularly like memorising formulas
This straight line has gradient = 2(x+1) where x is the x coordinate of the point at which it’s tangent
We know it’s tangent at (x₁, y₁)
Thus the gradient will be 2(x₁ + 1)
So our line is now y = 2(x₁ + 1)x + c
rugged surge
yes
dull sequoia
We also know that this line has to pass through the point at which it’s tangent
It is the point that the tangent line and the curve always shares
(Otherwise they wouldn’t be tangent or touching)
Hence y₁ = 2(x₁ + 1)x₁ + c must be true
rugged surge
yes
dull sequoia
It is easily observed that c must therefore be y₁ - 2(x₁ + 1)x₁
Hence the equation of our tangent line is y = 2(x₁ + 1)x + y₁ - 2(x₁ + 1)x₁
Even better we know what y₁ is in terms of x₁ from
rugged surge
yess
dull sequoia
Hence the equation of our tangent line is y = 2(x₁ + 1)x + (x₁ + 1)² - 2(x₁ + 1)x₁
Now we apply the constraint that this line needs to pass through the origin
This means the constant is 0, as demonstrated here:
0 = 2(x₁ + 1)0 + (x₁ + 1)² - 2(x₁ + 1)x₁
Implies
0 = (x₁ + 1)² - 2(x₁ + 1)x₁
Before this we considered any arbitrary x in the domain of our curve y = (x+1)²
Now we consider just the x₁’s that meet the condition:
The tangent lines at these points (x₁, y(x₁)) pass through the origin
0 = (x₁ + 1)² - 2(x₁ + 1)x₁
We now simply have to solve for all x₁ ∈ domain of y that satisfies this above equation
Does that make sense
rugged surge
(x1+1)^2 = 2(x1+1)x1
x1^2 + 2 +2x1 = 2x1^2 +2x1
-x1^2 =-2
x1 = +-sqrt2
rugged surge
that our line is tangent to the curve at point sqrt2,(sqr2+1)^2 and -sqrt2,(-sqrt2-1)^2
dull sequoia
What is our line
rugged surge
So our line is now y = 2(x₁ + 1)x + c
this only right
dull sequoia
That’s not a line
That’s a line with 2 free parameters
We don’t even know what x₁ and c are
rugged surge
but we just figured x1 is +-sqrt2 and c is 0 right
dull sequoia
But I’m sure you can find some x₁ and c such that this line would be tangent to some point on the curve
Which is really not saying much at all
dull sequoia
but we just figured x1 is +-sqrt2 and c is 0 right
But that wasn’t in the equation of the line you gave me
dull sequoia
this only right
This equation has x₁ and c’s in them and I don’t know what those are
rugged surge
y = 2(sqrt2 + 1)x
dull sequoia
What about this line
rugged surge
i dont get it, we found x1 and c so cant we just input in y =2(x1+1) + c and get equation for the line?
dull sequoia
What about this line
I’m asking what’s special about this line
Why should I care about it (in the context of this question)
rugged surge
Why should I care about it (in the context of this question)
oh
dull sequoia
(Don’t forget to read the question! What are you trying to do?)
rugged surge
(Don’t forget to read the question! What are you trying to do?)
Wait can I please get back to you in 10 mins. Got to eat dinner
dull sequoia
Mhmm
rugged surge
Why should I care about it (in the context of this question)
Because the line y=2(sqrt2+1)sqrt2 is tangent to the curve y=(x+1)^2 at points +-sqrt2 and the above equation of line I stated is tangent to the curve at x=sqrt2
dull sequoia
Woah
At points?
This 1 line is tangent to multiply points at once?
This is possible, but are you sure this is what you want to say? (Possible for some functions but not this one)
rugged surge
Yeaaa sorry I didn't actually mean that
I meant that line is tangent to sqrt2,(sqrt2+1)^2
rugged surge
Why does this matter
Matters because we are tasked with finding equation of the line which is tangent to the curve such that it passes through origion and applying all this conditions we got that the line should be tangent at sqrt2 and -sqrt2
So we can use this to get equation of our tangent which is the final answer
safe radishBOT
@rugged surge Has your question been resolved?
upbeat ridge
Hello
upbeat ridge
What defines two vectors being orthogonal?
Does the cross product have to equal 0?
Or is it the dot product have to equal 0?
Or is it both?
drifting mountain
look at defs of both
upbeat ridge
look at defs of both
Would you like to provide a link or somewhere??
frigid spruce
The dot product of two orthogonal vectors is zero.
upbeat ridge
Because I am seeing both would equal to 0
frigid spruce
this is enough
frigid spruce
orthogonal means perpendicular to each other
frigid spruce
Good to see you
hello
quasi bison
orthogonality is dot product = 0.
upbeat ridge
orthogonal means perpendicular to each other
What about cross product?
Parallel?
Equals being 0?
frigid spruce
its orthogonal to both the vectors
quasi bison
cross product is 0 iff either one of the input vectors is 0 or they're parallel to ach other.
each*
upbeat ridge
cross product is 0 iff either one of the input vectors is 0 or they're parallel ...
Ohhh okay
I understand now
Thanks so much both
.close
lean otter
The point (2, 3) is on the graph of Determine the
corresponding coordinates of this point on the graph of
y =-2(f (2(x + 5)) -4.
safe radishBOT
The point (2, 3) is on the graph of Determine the
corresponding coordinates of t...
@lean otter Has your question been resolved?
jovial linden
What do I have to do here exactly?
safe radishBOT
What do I have to do here exactly?
quasi bison
find the area of triangle PTS
marble ravine
Can I have some help with these questions
safe radishBOT
Can I have some help with these questions
marble ravine
safe radishBOT
@marble ravine Has your question been resolved?
static yacht
Spectral radius once again.
safe radishBOT
Spectral radius once again.
static yacht
How do I put a lower bound on the spectral radius?
The upper bound / the actual spectral radius must be 5.
What does the lower bound mean?
And how do I find those from the 1-norm, before I draw the disks?
My work so far, updated:
safe radishBOT
@static yacht Has your question been resolved?
static yacht
.close
dusky harbor
Warning: Physics Question incoming.
safe radishBOT
Warning: Physics Question incoming.
gusty saddle
hey i need help
dusky harbor
haha clAIMed
gusty saddle
ok so scalar product of two vectors
broken yew
ok so scalar product of two vectors
#linear-algebra might have better luck
gusty saddle
is the value of a scalar product included in a predifined constant interval for watever vectors
gusty saddle
<#540211747613704221> might have better luck
ok thx
dusky harbor
So... I'm quite confused on this friction question in physics. Perhaps somebody her could help me out?
gusty saddle
So... I'm quite confused on this friction question in physics. Perhaps somebody ...
wat iz it
dusky harbor
I managed to answer question a), but not b).
Here's what we know:
m = 12500kg,
μ_s = 0.42
θ = 25 or 24.8346 (for precision) (degrees)
Maximum static friction = 51502.5N
Weight = 122625N
For question b, we're given:
F = 6.0 * 10^4 N (I get confused here whether if this is the resultant force or if this is the pulling force).
Find acceleration of the object/train.
I'm quite confused how I should handle this F value... Anyone have a hint?
gusty saddle
sorry cant help u with that cuz i dont study that in english so idk what some words mean
dusky harbor
no problem hehe, if you don't mind me asking, what language do you study physics?
gusty saddle
no problem hehe, if you don't mind me asking, what language do you study physics...
french
dusky harbor
Maybe google translate can help us? xD
gusty saddle
lemme try translating some woreds
dusky harbor
J'ai réussi à répondre à la question a), mais pas à b).
Voici ce que nous savons :
m = 12 500 kg,
µ_s = 0,42
θ = 25 ou 24,8346 (pour précision) (degrés)
Frottement statique maximal = 51502,5N
Poids = 122625N
Pour la question b, on nous donne :
F = 6,0 * 10^4 N (je ne sais pas ici si c'est la force résultante ou si c'est la force de traction).
Trouvez l'accélération de l'objet/du train.
Je ne sais pas trop comment gérer cette valeur F... Quelqu'un a-t-il une astuce ?
gusty saddle
J'ai réussi à répondre à la question a), mais pas à b).
Voici ce que nous savons...
ok so uh
sec
gusty saddle
J'ai réussi à répondre à la question a), mais pas à b).
Voici ce que nous savons...
can we dm
i have an idea but im not so sure about it
dusky harbor
oki :3
safe radishBOT
@dusky harbor Has your question been resolved?
dusky harbor
<@&286206848099549185> Perhaps any physics experts online?
dusky harbor
ye it's about Friction, i mentioned it above-
dusky harbor
dis
tidal fern
Max friction = R μ
dusky harbor
yeah, or max static in this case?
dusky harbor
yee, btw, I found the answer for question a), but need some help with question b)
fyi
dusky harbor
i got θ = 25 or 24.8346 (for precision) (degrees), and the answer sheet said the same
tidal fern
Nice
So you know that now we can negate the force of friction with the acceleration no?
dusky harbor
negate the force of friction?
tidal fern
So
You can ignore the friction effectively
As if the train is travelling on a surface where it won't roll back but also won't be stopped from going uphill
dusky harbor
I see, because the maximum static frictional force is still keeping the train from rolling back at that angle.
dusky harbor
no I'm not getting the value equal to the answer sheet--- mmmm.
So I know a = F/m, but the issue is that I get confused / don't know what "F" should go up there.
Ill paste what we know here fyi.
m = 12500kg,
μ_s = 0.42
θ = 25 or 24.8346 (for precision) (degrees)
Weight = 122625N
For question b, we're given:
F = 6.0 * 10^4 N (I get confused here whether if this is the resultant force or if this is the pulling force).
Find acceleration of the object/train.
@tidal fern doot :3 Apologies if I'm bothering you here.
The answer just states 0.68...
I tried to go backwards and multiply 0.68 with 12500, getting F = 8500... It did not help... what am I forgetting--
I have to get to sleep
safe radishBOT
@dusky harbor Has your question been resolved?
near wren
Given a circle in a grid of squares, how can I calculate the percentage (or cm, whatever) of the surface that is overlapped with the circle in each cell?
safe radishBOT
Given a circle in a grid of squares, how can I calculate the percentage (or cm, ...
near wren
Context, I'm making a game and each tile has an special bonus and I want to know the sum of different bonuses the character has when walking in those regions haha
tulip saddle
let's assume center of the circle of radius r has coords (x,y), where that's the distance of the center from the intersection
then we have 4 right triangles + 4 arc sections - arc triangles
for light blue for instance
hmmm
nah you need to brute force it I think? Find the intersections of the circle with the edges, get the 4 angles, and use that to solve for difference in x and y between each pair of adjacent angles is the x and y of the right triangles
then use the difference in actual angles * area of circle to get the curved arc beyond the straight edge of the triangle
yeah that works
so given ø1, ø2, ø3, ø4 from top to left to bottom to right
we would do (|(cosø1-cosø2)(sinø1-sinø2)/2|r^2+(ø1-ø2)r^2/2)/πr^2
ASSUMING RADIAN VALUES FOR ANGLE
which gives us simply %betweenø1Andø2 = (2|(cosø1-cosø2)(sinø1-sinø2)|+ø1-ø2)/2π
I think? someone check my work
atan2 is a general programming function for finding angle across a rang of [0,2π] given ± either input
@near wren
make sense?
I hope it does bc I have to eat gl
near wren
I'm reading it and trying to make sense of it haha
tulip saddle
wait nvm that doesn't work
argh
lemme put it this way
what I calcualted was light green + light yellow / area
what you need to subtract before dividing by area is orange, which is the weirdly shaped triangle going between center of circle and hte two points (does NOT include the curved slice beyond blue)
ok that should be enough ttyl
tbh I would take a shortcut and simply use the angles directly
ie (ø1-ø2)/2π = %ø1toø2
since the slice between the angles is similar in proportion to the right triangle-ish area / total area
vastly simpler and nearly the same effect
safe radishBOT
@near wren Has your question been resolved?
safe radishBOT
@near wren Has your question been resolved?
leaden scaffold
Could someone help me with the final one? I just don't understand why I'm getting the final one wrong
safe radishBOT
Could someone help me with the final one? I just don't understand why I'm gettin...
safe radishBOT
@leaden scaffold Has your question been resolved?
leaden scaffold
<@&286206848099549185>
copper apex
Am I wrong or is your f_y is wrong?
leaden scaffold
Am I wrong or is your f_y is wrong?
where
leaden scaffold
Question c
This is my most recent updated work:
First part of c is still incorrect
copper apex
Yeah, actually could you explain how you got this result for f_y?
leaden scaffold
Yeah, actually could you explain how you got this result for f_y?
I'm multiplying fx*fy.
copper apex
I'm multiplying fx*fy.
Multiplying? I assumed that there was an error in the exercise and that the question c was just asking you to compute the partial derivative f_y
copper apex
I'm multiplying fx*fy.
But if you're doing this multiplication, that means you've computed f_y, so it may be done already
leaden scaffold
But if you're doing this multiplication, that means you've computed f_y, so it m...
Am I even supposed to be doing fx*fy?
copper apex
I guess no
Looking at the following questions, you just need fy to be able to do them
But if you've written fx*fy it's still incorrect, that wouldn't be the correct value (even if it's not what's asked)
leaden scaffold
But if you've written fx*fy it's still incorrect, that wouldn't be the correct v...
Where do we go from here?
copper apex
Just compute fy I guess, like you did for fx
And assume the question is just to compute fy and not "fy*fx" I think
safe radishBOT
@leaden scaffold Has your question been resolved?
safe radishBOT
safe radishBOT
Available help channel!
plain bobcat
yo alright so for #6
safe radishBOT
yo alright so for #6
light hatch
worthy hemlock
What's the link part? The words "exercise 2" is blue meaning it's hyperlinked
safe radishBOT
@plain bobcat Has your question been resolved?
silent crest
i have no clue how they found the velocity
safe radishBOT
i have no clue how they found the velocity
silent crest
where did they get the 92 and 3
quiet juniper
2nd line is wrong
drowsy moss
the 92 is a typo
silent crest
fucking hell just found a chegg answer where they dont skip 90% of the work and i see now
drowsy moss
otherwise it's just the derivative and power rule
static token
how to solve 5^(x-12) = 7^(x+8)
safe radishBOT
how to solve 5^(x-12) = 7^(x+8)
twilit spindle
how to solve 5^(x-12) = 7^(x+8)
may I suggest taking the natural logarithm of both sides :)
safe radishBOT
@static token Has your question been resolved?
safe radishBOT
safe radishBOT
Available help channel!
static token
may I suggest taking the natural logarithm of both sides :)
sorry im stuck in a part
safe radishBOT
sorry im stuck in a part
stray socket
Show work
static token
so uhh
xlog(5) - 12log(5) = xlog(7)+8log(7) right
then
xlog(5) + xlog(5) = 12log(7)+8log(7) ??
stray socket
Check the left side
broken yew
how to solve 5^(x-12) = 7^(x+8)
stray socket
Yeah I saw the original question
broken yew
thats for me.
static token
which left side u mean
stray socket
xlog(5) + xlog(5) = 12log(7)+8log(7) ??
This one
broken yew
bottom left
static token
i dont see whats wrong?
static token
uhh ohh the exam going to start
broken yew
so uhh
xlog(5) - 12log(5) = xlog(7)+8log(7) right
this is right
stray socket
where did the extra xlog(5) come from
broken yew
xlog(5) + xlog(5) = 12log(7)+8log(7) ??
no idea how u get this
static token
i hope i can do it
safe radishBOT
safe radishBOT
Available help channel!
wind abyss
safe radishBOT
wind abyss
Hi i am doing this problem now
And i am stuck at this spot i will take a pic
I am now stuck here
hazy elbow
factorize the numerator
broken yew
simplify
then do as suggested
ngl idk why u didnt try this 
you try simplifying as far as possible before being stuck
hazy elbow
should i re arrange?
like the numerator is a quadratic. It is pretty easy to factor it
wind abyss
well the only thing i am thinking about doing idk
Is doing x-2x
then left with -2x^2-x
is good?
and -x(2x+1)
broken yew
well the only thing i am thinking about doing idk
well yes? (like if theres something to do, do it, until u have nothing to do)
hazy elbow
no
hazy elbow
are you aware of wavy curve method
orchid ginkgo
Go on youtube
Make a search for the wavy curve method it will come in handy
Trust
orchid ginkgo
would something like this be similar to it
What is this exactly
safe radishBOT
To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.
flat frigateBOT
safe radishBOT
@wind abyss Has your question been resolved?
safe radishBOT
@wind abyss Has your question been resolved?
safe radishBOT
safe radishBOT
Available help channel!
wraith prism
Any direct option discarded method
safe radishBOT
Any direct option discarded method
sacred canyon
(3/2)^4 + (3/2)^3 + (3/2)^2 + (3/2)^1 + (3/2)^0 ......... (3/2)^k-1
in this series we have used a(r^n -1 ) / r-1
but
why did he use this fomula
little mesa
is it a limiting sum?
sacred canyon
can you elaborate
wraith prism
Geometric series sum
sacred canyon
yes
sacred canyon
no but why we used that here
(3/2)^4 + (3/2)^3 + (3/2)^2 + (3/2)^1 + (3/2)^0 ......... (3/2)^k-1
in this series we have used a(r^n -1 ) / r-1
wooden garnet
bit of a strange one, but I'm trying to set up lean on my computer for by installing lean4 on vscode, it works with their examples. but when i copy the code from the lecture slides it gives errors
safe radishBOT
bit of a strange one, but I'm trying to set up lean on my computer for by instal...
wooden garnet
safe radishBOT
@wooden garnet Has your question been resolved?
safe radishBOT
safe radishBOT
Available help channel!
upbeat ridge
Hello
upbeat ridge
What trig identity can I use for
harsh parcel
is that the original q?
reef estuary
But how do I derive?
i would equate that to some variable x, and then sin() both sides. use sin(A+B) = sinAcosB + cosAsinB to get a value for sinx, and then find x
but that seems long, let me think of something simpler
broken yew
were u reading what i wrote smfh 
upper rivet
you can use this directly tbh
reef estuary
you can use this directly tbh
bro that is the exact same thing that @broken yew and I said
reef estuary
were u reading what i wrote smfh <:pandaScreams:513070256550445083>
my bad bro 😅
broken yew
i was @ OP.
gentle thorn
Let x = arcsin 1/root 10, y=arcsin 1/root 5
sin(x+y) = sin x cos y + sin y cos x
= sin x * sqrt(1-sin^2 y) + sin y sqrt(1-sin^2 x)
= 1/root 10 * 2/root 5+ 1/root 5 * 3/root 10
= 5/root 50 = 5 root 50/50= 25 root 2/50= root 2/2
So x+y = arcsin(root 2/2), not sure if its in the right range, but you can easily infer from here what the value is
reef estuary
Let x = arcsin 1/root 10, y=arcsin 1/root 5
sin(x+y) = sin x cos y + sin y cos ...
bro aint no one gonna have that formula readily available
just do it by the derivation of the formula
gentle thorn
what formula
gentle thorn
only thing I use is pythagorean identity and angle addition
safe radishBOT
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.
broken yew
how hard is it???
broken yew
Let x = arcsin 1/root 10, y=arcsin 1/root 5
sin(x+y) = sin x cos y + sin y cos ...
do NOT give solutions, the OP has clearly been given enough hints already
if you readup
upper rivet
bro aint no one gonna have that formula readily available
its well known here lol, it'd be expected to have it memorized
bur fair enough if you want to do it the way you two mentioned
broken yew
it comes from somewhere and if you understand where it comes from, you will have no problem doing the original question
upper rivet
of course, but it's good to know that a general form exists so you dont have to reinvent the wheel every time
broken yew
it is better to rederive it than hard memorize it
upper rivet
anyways let's stop derailing, let OP do whichever suits them
broken yew
otherwise you can throw another 100 trig formulas for people to pointlessly remember
reef estuary
of course with enough practice and usage of that formula you would grow into using it, but it's always good to understand how the formula came into existence and to use it in it's original form
reef estuary
otherwise you can throw another 100 trig formulas for people to pointlessly reme...
exactly
safe radishBOT
@upbeat ridge Has your question been resolved?
idle hamlet
find overlapping area between 3y^{2}=3x\ +5 and y^{2}=-2x
safe radishBOT
find overlapping area between 3y^{2}=3x\ +5 and y^{2}=-2x
idle hamlet
i don't get this
drifting mountain
draw a diagram
normal moss
$3y^2 = 3x + 5$ and $y^2 = -2x$
flat frigateBOT
USS-Enterprise
safe radishBOT
@idle hamlet Has your question been resolved?
lean otter
yall math is too hard
safe radishBOT
yall math is too hard
obtuse plover
why
safe radishBOT
@lean otter Has your question been resolved?
odd spoke
safe radishBOT
mortal sandal
!show
safe radishBOT
Show your work, and if possible, explain where you are stuck.
odd spoke
solid shell
Try a u-substitution
solid shell
Nope they’re doing integration by parts there
odd spoke
Nope they’re doing integration by parts there
is it not possible to do it by parts?
mortal sandal
How familiar are you with u-substitution
You can think here, what should your du be
probably it should be related to the x^4
solid shell
You technically can but it’s a lot more advanced than any class I’ve taken as far as I can tell base don wolfram re: doing int by parts
mortal sandal
so your u ends up being x^5-11
odd spoke
How familiar are you with u-substitution
ive done it a few times just a bit rusty
mortal sandal
if you see a sqrt it's usually the thing in the sqrt
mortal sandal
how can u tell when do by parts or sub
Well here they tell you
But in general, you might try by parts and realize the sqrt(x^5-11) is pretty much impossible to integrate
or you might spot the u=x^5-11 sub quickly
amber shore
Hi, I am a bit confused.
What is the difference between a polynomial and an algebraic sum?
safe radishBOT
Hi, I am a bit confused.
What is the difference between a polynomial and an alge...
quasi bison
an algebraic sum can have all sorts of shit being added (or subtracted) together.
in a polynomial you can only include certain kinds of shit
amber shore
What kinds are there to include?
quasi bison
monomials
i.e. a number multiplied by a product of some (potentially none) letters, each raised to a nonnegative integer power
amber shore
Well, what you just said is the definition of an algebraic sum from my textbook.
amber shore
Ok, give me a sec
quasi bison
polynomials disallow shit like 1/x or sqrt(y) or e^z
amber shore
quasi bison
jednomian = monomial?
amber shore
"An algebraic expression, formed by summing monomials is called an algebraic sum, e. g. ..."
amber shore
jednomian = monomial?
exactly
quasi bison
right hm
in that case there appears to be no difference,
unless your book only wants to look at polynomials in one variable
in which case algebraic sums are what allows multiple variables
amber shore
unless your book only wants to look at polynomials in one variable
From a quick look, it doesn't want to look at them at all.
amber shore
in which case algebraic sums are what allows multiple variables
I guess that's the difference too
quasi bison
amber shore
<:thonk:406575732563902485>
Which one were you recting to though?
quasi bison
i wasn't reacting to any message in particular
amber shore
unless your book only wants to look at polynomials in one variable
In the entire textbook, there's nothing about one-variable expressions, so I can't know for sure.
I guess I'll have to work it out myself, but thanks anyways.
.close
solar needle
if something is given in a proof it is wrong to assume it right
safe radishBOT
if something is given in a proof it is wrong to assume it right
static hedge
huh
cosmic grove
if something is given in a proof it is wrong to assume it right
if its proven, then its right
logic moment
solar needle
im trying to do a question thats like for all sets X and Y we have that Y \ ( Y \ X) = X iff X subset of Y
which means i need to prove both directions
if i wanted to prove <= i know that X is a subset of Y and i want to assume a in X right
cosmic grove
you are trying to prove an equivalence
you suppose X is a subset of Y then prove that Y \ (Y \ X) = X
after that, you do the opposite
you suppose that Y \ (Y \ X) = X and prove that X is a subset of Y from this
solar needle
so you can assume x is a subset of y even tho its given
cosmic grove
so you can assume x is a subset of y even tho its given
its a statement
X is a subset of Y
idk what you need to prove here
solar needle
i thought assume and suppose were interchangeable
cosmic grove
they mean the same thing ? Idk my main language isnt english
cosmic grove
You want to prove an equivalence between two statements.
You dont need to prove that each statement is true (and I dont know how you would do that honestly)
solar needle
you can prove an equivalence even if they arent necessarily true statements?
cosmic grove
you can prove an equivalence even if they arent necessarily true statements?
yes
But generally, if both statements are false, the equivalence should be false. Been a while since ive done logic
Oh wait false is equivalent to false so the equivalence is true
🤔🤔
solar needle
if x is a subset of Y then an arbitrary element of x belongs in Y right
solar needle
Y / (Y / X) is basically saying a is in Y and a is not in Y?
because by definition its like x in Y and x not in (x in Y and x not in X)
cosmic grove
Y \ X is the set of of elements of y which are not in X
solar needle
yeah but if you simplify it
you get Y \ (the set of elements of y which are not in X)
which means
ohw ait...
cosmic grove
so Y \ (Y \ X) is the set of all elements of Y which are not in (Y \X)
solar needle
i think i see
but then you get Y \ the set of all elements of Y which are not in (Y \X)
cosmic grove
so its just X
solar needle
so x is in Y and x is not in the set of all elements of Y which are not in (Y \X)
so it must be X
cosmic grove
yes
solar needle
this is a weird statement to me ngl
cosmic grove
how ?
solar needle
i think im sort of confused how i tie in that if x is a subset of y and a is in X then a is also an element of Y
cosmic grove
sorry im in class rn
safe radishBOT
Available help channel!
dreamy salmon
Could I get help on this?
safe radishBOT
Could I get help on this?
static yacht
I think you're supposed to use logarithms ^^
mortal sandal
ye isolate the x using logarithms
dreamy salmon
log_12_10?
mortal sandal
yeah
dreamy salmon
how do I calculate that 😭
mortal sandal
usually we don't write the second underscore
mortal sandal
how do I calculate that 😭
with a calculator
static yacht
My physical calculator, TI-30XS can do logarithms. You can use symbolab.com perhaps, although I think it has CAS-capabilities
dreamy salmon
aa yeah ofc i have ti nspire
but these types of questions r supposed to be solved without cas
like in an exam setting
i have an exam tm hence why im asking
😭
mortal sandal
time to learn taylor series approximation
yeah any calculator more complicated then a 4 function one should have logs
dreamy salmon
yeah any calculator more complicated then a 4 function one should have logs
what the format?
log_12 (10)
?
mortal sandal
Yeah
flat frigateBOT
mortal sandal
,calc log_12(10)
flat frigateBOT
The following error occured while calculating:
Error: Undefined function log_12
mortal sandal
,calc log(10)/log(12)
flat frigateBOT
Result:
0.92662840802913
log change of base formula
crude nest
look at the power rule on this site ^
gusty saddle
hi im trying to become good at maths anything i should do
dreamy salmon
https://www.chilimath.com/lessons/advanced-algebra/logarithm-rules/
i have these in my book i just didnt know how to apply
dreamy salmon
log change of base formula
yeah but issue is that how the hell am i supposed to type it out on a non cas calculator
crude nest
ok lets do the first one together - then you should be able to get it down
we want to find x for 12^x = 10
dreamy salmon
log_12(10)?
crude nest
the power rule for logs states this: (idk latex so this will be ugly)
log(M^k) = k * log(M)
so you take the log of both sides, you would get -> log(12^x) = log(10^1)
and using that power rule formula, you rewrite -> x * log(12) = log(10)
dreamy salmon
moving log(12) to the other side?
dreamy salmon
but how do i get rid of log T_T
crude nest
no you'll want to divide log(12)
dreamy salmon
if the * belongs to the log it should be divided
crude nest
yes
crude nest
nah it would still just be log(10) / log(12)
dreamy salmon
a
a
lmao
wait so x = log(10) / log(12)
and then?
can u divide x by log and then u can eliminate all
😭
crude nest
sadly no
dreamy salmon
XDDD
dreamy salmon
righttt
crude nest
so its just 1/log(12) which is perfectly fine as a final answer
you should NOT need to expand any logs if ur taking an exam without a calculator that can compute them
because if they aren't basic, there is no way to approximate them without using a taylor series expansion which is way more complicated
crude nest
what does the answer page say?
dreamy salmon
because if they aren't basic, there is no way to approximate them without using ...
right thats what i was thinking X_X
crude nest
oh lol i think i overlooked something
lets use log with a base of 12
log_12(12^x) = log_12(10)
my bad
dreamy salmon
its okk
crude nest
so it is really simple then. Take log base 12 of both sides bc that way the left side ends up being 1 * x
dreamy salmon
log_12(12^x) = log_12(10)
should I writethis down?
crude nest
yeah
dreamy salmon
jepp
crude nest
2nd question, do the same thing except use a log with base 10
crude nest
so because it's the 12 that has the exponent, we want to take the log_12 of both sides, meaning we will take the log of 12^x and the log of 10
dreamy salmon
so because it's the 12 that has the exponent, we want to take the log_12 of both...
so like literally log of 12?
im confused
crude nest
ill write it
dreamy salmon
sorry
crude nest
girl don't apologize, i love doing math 😂
does that kinda make sense? you choose the log base, based on what you need to get rid of
dreamy salmon
aa right
i think i get it now
wait so thats just the answer?
dont u have to solve it aswell
the right side
crude nest
nope, because trying to solve that would literally just get you back to the start with 12^x = 10
dreamy salmon
aa
crude nest
writing log_12(10) is good enough if you don't have a calculator, because it isn't a simple logarithm
crude nest
yes, the only difference is you take the log with base 10
dreamy salmon
log_10(25^4x)?
crude nest
mmmm try that again, pay attention to the pattern in the first question
remember the power rule as well: log(M^k) = k * log(M)
dreamy salmon
i accidently saw the answer 😭
crude nest
its okay if you saw the answer, you still gotta know how they got there
dreamy salmon
remember the power rule as well: log(M^k) = k * log(M)
that looks
different
what is M?
dreamy salmon
aa so log_k(a) isnt the same
dreamy salmon
its ok
crude nest
the power rule is a formula for logarithms though, its basically saying that if you take the log of a number being raised by x, the result will be x * log(number)
crude nest
yes
crude nest
ok, so what does the left side turn into when you take the log of it for number 2?
crude nest
use log with a base of 10
dreamy salmon
x= log_10(25^4)
crude nest
where are you getting 25^4?
crude nest
start with the equations:
https://cdn.discordapp.com/attachments/903481106819612714/1155918401127063682/image.png
dreamy salmon
x* log(10) = log_10(25)^4
crude nest
so its 10^4x = 25
crude nest
then take the log of both sides
dreamy salmon
log(10)^4x = log(25)
crude nest
almost, look at how we did the first the problem and compare it to this one
its basically the same, the only difference is that its 10^4x instead of 10^x
crude nest
it doesn't disappear, it stays there
the right side is a constant
meaning its just a number. any number, imagine it was just 5 or 10 or 1 billion
dreamy salmon
so no log on the right side
crude nest
no you do log it
crude nest
but thats it, no exponents or anything
you do plain log(25) because standard log base is 10
dreamy salmon
log(10)^4x = log(25)
.
crude nest
wym?
dreamy salmon
u said almost
crude nest
when you take the log of both sides, it becomes 4x = log(25)
oh, maybe that was from before you had edited it
dreamy salmon
log of 10 just eliminates itself?
crude nest
yes! since we chose a log with base 10, taking the log_10(10) = 1
dreamy salmon
aa right
crude nest
which just leaves whatever exponent was up there
crude nest
yes
crude nest
no, you can't simplify logs unfortunately
dreamy salmon
lmao
crude nest
in a perfect world lol
crude nest
exactly!
im not sure why, but its pretty much standard practice and notation that a log without a defined base has base 10 by default
dreamy salmon
aaa
i guessed 14.12c) would be log_2(5)
but idk why
and idk what happens to x+1
crude nest
treat the (x+1) exactly how you would treat the (x) or the (4x)
just pull it out front
dreamy salmon
x*log_2(5)= 1?
crude nest
hmm work out the simplification more. you should end up getting x+1 = log_2(5) at some point
dreamy salmon
what do i move first
aaaa
am i supposed to just move x+1 completely to the front
and the move 1 back?
crude nest
if you're solving for x, you need to isolate it on it's own
so how get rid of the 1 from the left side?
dreamy salmon
so how get rid of the 1 from the left side?
divide it by 1?
dreamy salmon
times?
crude nest
subtract from both sides
crude nest
heck yeah
dreamy salmon
aaaaa oke
ive never done that before
do people go about this a different way or is that the only way
subtracting from both sides
crude nest
it is the only way
if you were to divide by 1 it wouldn't go anywhere, but even if you were to divide a different number (like 5), you would end up getting something like x/5 + 5/5 which still doesn't fix the issue of x needing to be by itself
dreamy salmon
ooh right its supposed to be isolated
do you mind helping me with another type of question..
crude nest
it depends on the question, but ill give it a shot
dreamy salmon
assuming the base is 10?
dreamy salmon
does it complicate it?
crude nest
no, this should be pretty easy. I just need to find the right formula
most of these questions don't require any actual computation, you're really just manipulating the equation to find an answer using theorems and laws
dreamy salmon
oh okay
but if given the possibility to use a calculator, you could just plot in the question?
dreamy salmon
aa
dreamy salmon
one of them is this -> log(M) + log(N) = log(M*N)
is this one of the
fractional indices or w/e
crude nest
its one of the rules for logs
dreamy salmon
i still dont remember all of them lmao
crude nest
eh i dont think anyone does
for this first question, use that law in your reply above and simplify the left side of the equation
crude nest
yeah
the goal here is to get one log term on both sides
so we have a log_5(something) = log_5(something else)
crude nest
multiply the equations togethe
crude nest
log_2((x-2)(x-3))
dreamy salmon
and
dreamy salmon
log(2x+6)?
wait was I wrong
dreamy salmon
log_2((x-2)(x-3))
doesnt this equal to the same as mine
crude nest
you're multiplying them so it will expand into a polynomial: x^2 - 5x + 6
crude nest
do you know what the FOIL method is
dreamy salmon
no never heard
dreamy salmon
ooh
didnt know it had a name but
cant u just do a to b and then a to d
like a spider or w/e
crude nest
no u gotta do outer inner
dreamy salmon
foil
crude nest
yeah
crude nest
dont forget the 6
crude nest
right
dreamy salmon
why is it minus but not plus
crude nest
because when you are foiling, you have to be mindful of the signs. if a number has a minus sign in front of it, consider it negative when multiplying
crude nest
and now the cool part,
since both sides have logs of the same base, you can just write whats inside the log and get rid of them now as they are arbitrary
so you will get -> x^2 -5x + 6 = 6
dreamy salmon
but u didnt get rid of the other values lmao
u mean that the log has now disappeared so
we need to isolate the x now?
dreamy salmon
we're left with x^2 -5x?
crude nest
ok, so that means its x^2 - 5x = 0, the next step is to add 5x to both sides so we can reduce that square
dreamy salmon
ok, so that means its x^2 - 5x = 0, the next step is to add 5x to both sides so ...
so in total we will have 3 5xs'?
crude nest
not necessarily, once you complete the operations you will end up with one 5x term on the right side
crude nest
yeah -> x^2 - 5x + 5x = 0 + 5x
dreamy salmon
eyah
crude nest
yeah, you divide by x
dreamy salmon
no before that
sorry im really fkn slow
but here
x^2 = 5x
what happened to 5x+5x
crude nest
the 5x on the left side was negative
x^2 - 5x = 0
by adding 5x to both sides, you're just swapping it from the left side to the right side of the equation
so after you add it, it becomes ->
x^2 = 5x
crude nest
yes
dreamy salmon
and then x^2 = 0 + 5x
dreamy salmon
aaa okei so then we isolate x?
crude nest
yes, isolate x by dividing both sides by x
dreamy salmon
am i over complicating this
i feel like this is harder than it should be
shouldn't take me half an hour
crude nest
it's alright, you just gotta practice a ton to get this stuff down. i've had my fair share of hair pulling frustration with math...
dreamy salmon
does the exponent leave aswell?
dreamy salmon
it's alright, you just gotta practice a ton to get this stuff down. i've had my ...
ive been here the whole day if u havent noticed already
feel like ive barely progressed
and i have so much ahead of me
crude nest
yes -> one thing with exponents is that they subtract if the terms are being divided. This means that x^3 / x^2 -> = x^(3-2)
so x^2 / x = x^(2-1)
what are u cramming for if u dont mind my asking
dreamy salmon
what are u cramming for if u dont mind my asking
as in?
dreamy salmon
yes -> one thing with exponents is that they subtract if the terms are being div...
oh only when divided
crude nest
why all the studying?
dreamy salmon
I have an exam tomorrow that gives me a final course grade
but you can't pass the course without the homework done
so I have like 8 chapters ahead of me still
to be able to pass
its good study too but really frustrating when ur as slow as me
crude nest
have you looked into using AI for assistance yet? ChatGPT can be pretty good at explaining how to do problems (it does not produce correct answers though so don't trust that)
dreamy salmon
I use it for simple questions I know the concept of
but it also gives wrong answers
crude nest
yeah it cant calculate for shit
crude nest
check out the organic chemistry teacher on YT too, he covers a ton of topics and is an awesome teacher
yes x=5
dreamy salmon
check out the organic chemistry teacher on YT too, he covers a ton of topics and...
yeah he's great too but his videos are really long sometimes
i was like ok i wont watch vids since im limited
but i end up spending 1 hr minimum
with each question so idk and i feel like i cant ask questions
so ill just be even more lost
crude nest
gotcha
dreamy salmon
b) is the same?
crude nest
it's a bit different
dreamy salmon
no brackets for the right side
crude nest
instead of using a multiplication law u use a division one
use rule 2 for the left side
dreamy salmon
so log7(x^2/5x)?
crude nest
(x^2 + 5x) / x
inside the log
start working on that and see if you can solve it similarly to the previous problem. I need to go for a bit and move my car
dreamy salmon
yeah sure
dreamy salmon
start working on that and see if you can solve it similarly to the previous prob...
x = log_7(18)-log_7(5)?
do i just divide by log_7?
to get x = 13?
crude nest
so remember this: (x^2 + 5x) / x
is simplified to (x + 5)
-
log(x + 5) = log(18)
-
x + 5 = 18
-
x = 18-5
-
x = 13
so yes that looks right!
u got it
crude nest
x^2 / x is equal to x^(2-1) which equals x
dreamy salmon
okay ill remember that
so u dont divide by log_7
wait so is my way even correct T_T
crude nest
yeah it is actually
dreamy salmon
or were u just proofing it
crude nest
i was just working out all of the steps
dreamy salmon
so i dont have to divide but just
calculate the brackets
or when does log disappear
crude nest
log becomes redundant when both sides of the equation are logs of the same base
dreamy salmon
aa so it just cancels out
crude nest
for example,
log_4(equation) = log_5(equation) for this one, you would NOT be able to ignore the logs
log_5(equation) = log_5(equation) goes to (equation) = (equation) because the logs have the same base
dreamy salmon
aaaa
can I ask you about oneeeeeeee last problem 😭 its simple i think idk but I just don't understand
crude nest
sure haha but i do have to get off real soon
dreamy salmon
yeah sorryyyy
crude nest
no worries
dreamy salmon
its like the first question of the new chapter so i think it will form my understanding for the rest
crude nest
ok, for these ones u use those laws i sent earlier
so for the first one, remember subtraction becomes division - > log_2(20/5)
you will get log_2(4)
which is a very easy log to solve, as it's just asking what power of 2 is equal to 4, which we know is 2 (2^2 = 4)
crude nest
im not sure what u mean
dreamy salmon
like can b just be turned to the opposite of addition