.close
#help-0
languid bolt
lone heartBOT
languid bolt
my brain is lagging
alpine sable
lol atleast u used ur brain lmfao sry
languid bolt
lone heartBOT
Available help channel!
gilded kiln
lone heartBOT
gilded kiln
how is c wrong?
chances of it being a king and a heart are (1/52)
chances of a card being a king is 3/52
(1/52) / (3/52) = 1/3
lone heartBOT
@gilded kiln Has your question been resolved?
empty plover
A gradient is in the direction in which the function increases the most?
empty plover
So its not the maximum variation?
and if the graph is on a maximum it takes the minimal value of decreasing?
mild copper
A gradient is in the direction in which the function increases the most?
Yes it is
And when u take gradient=0 u got the (x,y) where the function variation is minimal, that's it, it's a critical point
Then can be a minimum, a maximum or none
wide shale
Hi. I'm re-learning projective geometry on my own. I see in a text im following the words "linearly independent" in reference to points in a projective space. I know this means that if two points are linearly independent they cannot be written as a linear combination of eachother. What I'm missing here, with respect to projective geometry is some geometric intuition. Every point of a projective space is a line, which I believe is different than a vector. When vectors are said to be linear combinations of other vectors, their location in space doesnt matter, just their length and direction. But for lines their location does seem to matter. So what is the geometric intuition for two lines being linearly dependent/independent? Can you also show this algebraically? Thanks for the help, hope my question is clear
lone heartBOT
Hi. I'm re-learning projective geometry on my own. I see in a text im following ...
@wide shale Has your question been resolved?
lone heartBOT
@wide shale Has your question been resolved?
novel night
Might help to see the context of the statement, but one possiblity is: points in projective space are lines through the origin in a (one-higher-dimensional) vector space. But lines through the origin are very closely related to non-zero vectors! A non-zero vector determines a line through the origin, and the line determines the vector up to a scalar multiple.
Then, for example, saying projective points A, B and C are "linearly independent" could just mean that for any choice of non-zero vectors a, b, c that point along the lines A, B, C respectively, the vectors a, b, c are linearly independent.
If one choice of a,b,c are linearly independent, then any non-zero scalar multiple of those will also be linearly independent, and so it doesn't matter which a,b,c you pick: you get the same answer as to whether or not A,B,C are linearly independent.
Btw, normal notation here would be to write A = [a] or similar.
lone heartBOT
@wide shale Has your question been resolved?
wide shale
@novel night Ah I did not realize they where lines through the origin, thanks! So intuitively they can be thought of vectors after all, yes?
novel night
Vectors up to a scalar multiple, yep. But again, not 100% certain this is what your source means by linear independence. Just.... probably.
alpine sable
Hello guys
lone heartBOT
Hello guys
alpine sable
I completed 1a b c but am not understanding how to do d and e
Actually I don't even understand 1d properly let alone doing it
And in 1e I tried applying the cosine rule but it didn't work
lone heartBOT
@alpine sable Has your question been resolved?
alpine sable
SOS
lone heartBOT
@alpine sable Has your question been resolved?
alpine sable
<@&286206848099549185>
lone heartBOT
@alpine sable Has your question been resolved?
native venture
any tips for solving hard trig questions? I am really struggling 🥲
chilly lark
What type of trig questions?
native venture
identity questions for proving
chilly lark
Can i see an example?
native venture
yes sure just a min
chilly lark
Ping me when ur back
hazy violet
any tips for solving hard trig questions? I am really struggling 🥲
IF the problem has functions other than sin or cos
Write them in terms of sin and cos
You'll find its easier to with sin and cos
native venture
hazy violet
Ah.
chilly lark
I am not yet at that stage of trigonometry but, maybe start by simplifying the powers.
And also trying to use basic trig identities and formulas to "complify" the questions then simplifying it again in a different way (something i usually do)
Sorry i cant help much, but i hope this helps
marble oxide
there isnt many online rn
marble oxide
honestly it looks like chinese to me
this looks like geometry... you're trying to find the last one?
chilly lark
For me
Best method would be "meet in the middle" where you algebraically finish both sides to meet in the middle ex: sinx=sinx
Dont forget you can rewrite 1 as cos²+sin² and etc
native venture
sometimes there's algebra which seems impossible like no one could have known that you had to do those steps in order to solve the problem
even after seeing the solution i cannot understand how i could have come up on that on my own
marble oxide
good luck
chilly lark
Also dont forget you can split up division
chilly lark
Ciao, gud luck
graceful sparrow
@native venture just try memorizing the formulas and seeing where you can apply them. most of your problems can be solved that way
native venture
i know them by heart
chilly lark
Trig identity hexagon is great memorizing tool too
native venture
Trig identity hexagon is great memorizing tool too
i dont think these identities fit into that
chilly lark
Ye it just cover the basics
chilly lark
I saw your notes here and thought "damn i thought its harder than that"
glacial dagger
Someone please check my work very thoroughly. My teacher is trying to assess our class’s average intelligence and I want to look good.
glacial dagger
lone heartBOT
@glacial dagger Has your question been resolved?
glacial dagger
<@&286206848099549185>
lone heartBOT
@glacial dagger Has your question been resolved?
lone heartBOT
@glacial dagger Has your question been resolved?
tacit arch
Don't make people download a file to help you. Just screenshot and upload
lone heartBOT
@glacial dagger Has your question been resolved?
lone heartBOT
@glacial dagger Has your question been resolved?
languid bolt
Someone please check my work very thoroughly. My teacher is trying to assess our...
as riemann said, maybe just post a screenshot of the file
languid bolt
Someone please check my work very thoroughly. My teacher is trying to assess our...
and also, the teacher is trying to assess your class's average intelligence to find a good teaching pace, so i'd recommend for you to just do it by yourself tbh
desert tusk
and also, the teacher is trying to assess your class's average intelligence to f...
Isn’t helping people for exams against the rules here
trim wagon
Isn’t helping people for exams against the rules here
It isn’t exam
desert tusk
oh
trim wagon
At least it doesn’t look that way
burnt bridge
How to find maximum and minimum of this function
Please help
@lone heart please help
limpid spade
@lone heart help him
limber hill
How to find maximum and minimum of this function
It is a 11th grade quadratic equation chapter question, right?
I solved so many of these questions
Wow nostalgia
alpine sable
Please help
Open your help channel, #❓how-to-get-help
burnt bridge
It is a 11th grade quadratic equation chapter question, right?
Yess
median zinc
let the derivative of the function = 0, and solve for x, then substitute that x back into the original function to find y.
glacial dagger
.close
queen shale
find the value of k for which the line y=kx-2 is tangent to the curve y=2x^2-9x+4
queen shale
help pls
i equated them together and got 2x^2-(k+2)x +6
i dont know how to equate the k
static marsh
move everything to 1 side so you have a quadratic equation
then you need to solve for k such that the equation has 1 and only 1 root
you can use the discriminant to do so
alpine sable
Second one
alpine sable
Yes
alpine sable
your answer is in that form
queen shale
but my answer is in form of ax^2/bx^2+c
brave kayak
hello, how can I find out what the diameter is based on the circumference?
brave kayak
on a circle
grave island
work backwards and find D i mean
brave kayak
work backwards and find D i mean
what
grave island
hello, how can I find out what the diameter is based on the circumference?
do we know how to calculate circumference
brave kayak
yes
grave island
what is it then
brave kayak
D * pi
grave island
yea
d being ur diameter
so
just divide ur circumference by pi
which leaves ur answer
brave kayak
how can I divide it by pi if I don't know what pi is
grave island
use calculator
or estimate pi as
3.141
u should just know first like 3 digits
of pi ig
brave kayak
and how do I get the area?
ocean sealBOT
Mothy
grave island
r being 1/2 D
edgy flare
lone heartBOT
edgy flare
@runic hull
You used the example of {1,2} and let m = 0 and t = -1 would satisfy my example.
Which it does, but when you consider 1/2, that would then show that m = 1/2 and t = 0
Then you would slowly approach 1 until you get m = 1 And anything less than 1 would be become t
That implies THERE EXISTS an m such that that any other t <= a for all a is less than m.
Does it not follow from my work?
What I’m saying is that is what I’m trying to get across but I’m not sure if I did so correctly.
lone heartBOT
@edgy flare Has your question been resolved?
edgy flare
lone heartBOT
@edgy flare Has your question been resolved?
lone heartBOT
@edgy flare Has your question been resolved?
runic hull
Oh, I see what you're trying to say now. I thought you made two different statements, and in the first you just claimed there was a lower bound, and in the second you claimed there exists a particular t.
The way you explained it shows your understanding is correct but it reads as if you're saying there exists a specific t that is lower than m.
A better way to write it would be. A is bounded => there exists a number m such that m<=a for all a in A, and for any lower bound of A, m>=t.
Essentially, replace the "there exists" on for t with "for all t such t<=a for all a in A" and it will be more readable
runic hull
@edgy flare
(won't be around to help rn but I thought I'd quickly clarify rather than add confusion)
mortal trellis
so far you have just shown that max(N, M) is an upper bound for AuB. not that it is the supremum
lone heartBOT
Available help channel!
tawny condor
if f is differentiable on [a, b] and f' is bounded on [a, b], then is f' continuous on [a, b]?
alpine sable
yes
tawny condor
and how would I prove that?
alpine sable
well you can say that the only thing that f can have which would make it discontinuous yet differntiable is a hole at a point
tawny condor
you mean f'? we know that f is continuous
alpine sable
Hold one moment pls
tawny condor
okay
naive valley
if f is differentiable on [a, b] and f' is bounded on [a, b], then is f' continu...
see the first example at this MSE answer: https://math.stackexchange.com/a/423279
tawny condor
okay so it's not necessarily true
naive valley
right
tawny condor
I hoped it would be true, welp
naive valley
the derivative of a differentiable function must obey the intermediate value property, which implies that it can't have jump discontinuities
but it can have "oscillate to death" discontinuities while still remaining bounded
mellow reef
The sum of the 1st and 2nd terms of a geometric sequence is 108, and the sum of the 3rd and 4th terms is 12. Find 2 possible values of the common ratio and the corresponding values of the 1st term.
mellow reef
now i fr dunno where to start
@stiff geyser u mind helping me out bro
its 3am and my brain doesnt work properly
reason why im doing this 3am cus i had training a while ago, i been doing this since the end of training
and tomorrow i have to go out
stiff geyser
Hmm...
It is simple.try it out
U must be aware of the terms of series
That is
a,ar,ar^2,ar^3......
stiff geyser
Yeah
mellow reef
a + ar = 108
stiff geyser
Correct
stiff geyser
Yep
stiff geyser
U got 2 equations right?
mellow reef
yea
stiff geyser
Simplify them
mellow reef
u mean factor
stiff geyser
Yeah
stiff geyser
S
mellow reef
OHH
then i can
for ar^2 + ar^3 = 12
i can do
i can use the thing for the first one
so r^2 (a(1+r)) = 12
then
idk next
stiff geyser
stiff geyser
Observe the 2 simplified equations
neon bronze
108/a
stiff geyser
Or divide the two equations
mellow reef
how
mellow reef
r = +- 1/3
stiff geyser
S
mellow reef
so if its plus minus
i plug the positive first
to
lets say the first equation
then the negative
is that right
stiff geyser
Yeah
mellow reef
ok wait
lemme solve
i got 81 for the first one @stiff geyser
then for the second i got
162
am i correct
stiff geyser
Yep
worthy solstice
Trying to find extrema and where the graph is increasing and decreasing, can somebody walk me through the steps rq:
worthy solstice
My work so far is:
- find first derivative: 3x^2 + 12 = 0
- Factor: 3(x^2 + 4)=0
alpine sable
Hm
worthy solstice
getting hung up on the factoring part. I don't know how to deal with an x^2 in factoring
alpine sable
Is it multiple correct?
solemn juniper
I mean you just solve x^2 + 4 = 0
worthy solstice
Is it multiple correct?
by multiple you mean the 3?
alpine sable
I mean you gotta choose only one option among these or you can choose multiple?
worthy solstice
only one option
worthy solstice
I mean you just solve x^2 + 4 = 0
I end up having to take the square root of both sides right?
so end up with 2i and -2i?
solemn juniper
Right
But since this is (presumably) a function that only cares about real inputs, it has no (real) solutions
So with regards to extrema, the derivative having no reals that make it 0 is equivalent to having... how many extrema?
alpine sable
worthy solstice
oh wait
worthy solstice
So with regards to extrema, the derivative having no reals that make it 0 is equ...
so this means that since its never 0 the curve never changes right?
solemn juniper
There are no sign changes in the slope
alpine sable
This should help
worthy solstice
So it would just be increasing for all x because 1) slope is positive and 2) it has no slope sign changes?
solemn juniper
Quite so
alpine sable
You can eliminate A,B,C option
solemn juniper
No you can't
alpine sable
Now just D and E remains
alpine sable
No you can't
Why not
solemn juniper
Because C is correct, as shrike said
The derivative is always positive, therefore the function is increasing for all x
It's that simple
worthy solstice
But since this is (presumably) a function that only cares about real inputs, it ...
because it all hinges on this statement right?
since the result of factoring was 2i?
solemn juniper
We ignore complex solutions in such problems
alpine sable
The derivative is always positive, therefore the function is increasing for all ...
But increasing for all x?
solemn juniper
The derivative is always positive
alpine sable
Oh
solemn juniper
For every real number x, f'(x) is positive
worthy solstice
My grip on the calc logic is a little fuzzy here. So you're saying that if I run into another one of these problems that has a factoring result of i (say, -5i), I should just treat it as
- -5
- Not real, and I can assume that the derivative curve never changes
alpine sable
Ooh
solemn juniper
These calculus courses always deal with real-valued inputs and outputs, so you can discard all complex solutions to stuff
So 2.
worthy solstice
Okay so the answer would be number 2 then, and I just assume the curve won't change?
yeee
alpine sable
Wait wdym by f'
solemn juniper
Derivative of f
worthy solstice
f' is the derivative
worthy solstice
So to wrap up, my flow to solve a problem like this would be:
- Find first derivative
- Factor that
- If I end up with not real numbers like 2i or something, assume the graph never changes slopes
- If I get real numbers, I plug these factored numbers into the first derivative number line test to solve for when the graph is increasing or decreasing
Right?
solemn juniper
Yes
Well it's not you assuming the graph never changes slope, it's a result of the derivative calculations
But yes
worthy solstice
Lets go, gotcha
and this is a silly question but which coefficient tells me that the slope is in this case positive?
so that in my case's problem, I know that its increasing for all x?
solemn juniper
The one in front of x^3
solemn juniper
Yes
worthy solstice
awesome
solemn juniper
If you factored and got real solutions, the slope would be negative in between them
But positive on the outside
worthy solstice
yeee
That makes sense, I think I'm finally starting to get it
thanks a lot for your and @alpine sable 's help, much appreciated!
.close
alpine vector
Hi, I have been struggling with this question for over an hour and like to think I've got it half figured out. But whenever I get past one point I get stuck at another and at this point I don't even know exactly where I'm stuck. Here is a picture of the question, excuse all the erased pencil marks
alpine vector
I've decided to add 30 degrees in the counterclockwise direction giving me -37,5 degrees compared to the x-axis, I think
Since line k will be -67,5 degrees from the x-axis (again, if I'm not mistaken)
lone heartBOT
@alpine vector Has your question been resolved?
alpine vector
<@&286206848099549185>
serene junco
How'd you get -67.5 degrees?
I get something a little different for line k
@alpine vector
alpine vector
Yeah I've realised now how stupid that is
I thought because a slope of -1 would be 45 degrees
so 2 would be 22.5
I haven't had maths in 3 years, but recently got let into Mechanical engineering. So honestly assume I'm a complete idiot
serene junco
That's actually not quite how it works. Cutting the angle in half doesn't necessarily double the slope
I think we'll have to use a little trig
alpine vector
Which would that be?
serene junco
Well the slope is rise/run, you can imagine a right triangle whose hypotenuse has a slope of -2
The "rise" and "run" are the legs of the triangle
Does that sort of make sense?
alpine vector
I think I'm following yeah
serene junco
So, in general the slope of a line is equal to the tangent of the angle it makes with the x axis
tan(theta) = -2 where theta is the angle between the x-axis and line k
serene junco
You don't actually need to calculate theta exactly
alpine vector
But you do need it for line j right?
alpine vector
Yes I had these, but definitely had forgotten about them haha
So how do we use these for the -2 if I may ask
serene junco
Well, if line k makes an angle of theta with the x-axis
then line l makes an angle of (theta + 30) degrees
Just like how tan(theta) was the slope of k,
tan(theta + 30) is the slope of l
serene junco
Awesome, great
So we can use the tan(a+b) formula
where a is theta and b is 30 deg
alpine vector
Just for my information, we could also do -30 right? since direction wasn't specified?
Iknow it's not relevant, jst curious
serene junco
Yes that's true
The question just asked for one possible slope of l but there are two possibilities
alpine vector
so -2 + 30 / 1 --2 * 30?
serene junco
well
alpine vector
That sounds like a no
serene junco
$\tan(\theta + 30^\circ) = \frac{\tan(\theta) + \tan(30^\circ)}{1-\tan(\theta)\tan(30^\circ)}$
ocean sealBOT
tatpoj
alpine vector
oh theta is 0?
alpine vector
Sorry about that, Im stupid
alpine vector
oh right of course
Sorry for having to explain this like I'm a child, but honestly I am at this point
serene junco
No, you're doing fine
If you studied trig before, you probably memorized a circle full of sines and cosines of common angles
including 30 degrees
So, yeah you may be expected to just have things like this memorized
serene junco
google yes, calculator probably not
you need an exact value, not a decimal approximation
for now I'll just tell you
alpine vector
Thank you
serene junco
$tan(30^\circ) = \frac{1}{\sqrt{3}}$
ocean sealBOT
tatpoj
alpine vector
Oh okay gotcha
alpine vector
**tatpoj**
So we can now substitute tan30 for that in this one, right?
ocean sealBOT
tatpoj
serene junco
Uhhhh, maybe, depending on the type of calculator you have it might be able to simplify that
But according to your directions you're looking for an expression like $a + b \sqrt{c}$, not a decimal
ocean sealBOT
tatpoj
alpine vector
Oh yeah true
serene junco
Just take the numerator -2+1/sqrt(3) first
can you make a common denominator to simplify that?
serene junco
All good. From here it's just a lot of simplifying radicals, I'm not sure how familiar you are with that
If you have something like a ti-89 that you're allowed to use in class, it may be able to do the simplifying for you
alpine vector
Yeah I think I have exactly that calculator actually
simplifying radicals I should be able to do
Thank you so so much
serene junco
No problem! It should come out to something like d+e sqrt(f) , like the problem said, where d, e, and f are all whole numbers
all the ugly stuff simplifies away
alpine vector
Thank you so much, I should be able to do that
serene junco
👍
alpine vector
.close
frigid verge
lone heartBOT
wraith lotus
frigid verge
🤣 it's so weird
lone heartBOT
@frigid verge Has your question been resolved?
frigid verge
<@&286206848099549185>
rustic coral
<@&286206848099549185>
Note $2uv^2=\frac{2uv^3}{v}$
ocean sealBOT
messy circle creation
ocean sealBOT
messy circle creation
rustic coral
$=\frac{2uv^3}{u}=2v^3$
ocean sealBOT
messy circle creation
rustic coral
Dividing by a fraction is the same as multiplying by the reciprocal, \ \ Consider $\frac{a/b}{c/d}$ \ \ Multiply numerator and denominator by $bd$ \ \ Its equal to $ad/bc=(a/b)(d/c)$
ocean sealBOT
messy circle creation
frigid verge
ah ok i see now
keen socket
When dividing by a fraction, u flip and multiply
frigid verge
When dividing by a fraction, u flip and multiply
ye ye
2uv^4/uv which = 2v^3
Sorry I can't write it fancy
Yeah I think I got it now
Thanks so much guys
.close
lone heartBOT
lone heartBOT
Available help channel!
carmine wind
I need help with this problem problem
Every 2 seconds I am am able to advance one out of 10 spaces; after completion of the tenth space I return to the first one. On the 10th space I earn a value of 4 points. Each space always takes 2 seconds to advance. My question is how to calculate the length of time\advances it will take to reach 115 points. I hope to understand how to calculate this
lone heartBOT
I need help with this problem problem
Every 2 seconds I am am able to advance on...
@carmine wind Has your question been resolved?
carmine wind
<@&286206848099549185>
I belive that it will take somthing like 6 mins to reach the goal, I want to understand how do i graph this problem?
abstract fractal
Well, how long does it take to reach the 10th space once?
abstract fractal
How many times do you need to reach the 10th space to get 115 points?
carmine wind
If I divide 115 by four, I will need to cycle through all ten spaces 28.75 times
abstract fractal
But since you can't go 0.75 rounds, since you need a full round to get points, how many times do you need?
carmine wind
29? because of round up?
abstract fractal
Yes
So if it takes 20 seconds to go a full round, and there's 29 rounds, how long is that?
carmine wind
9.666 Mins so 9 and a half mins?
abstract fractal
9 and a half is close
But because the answer converts nicely into minutes and seconds, I don't see any reason to make that approximation
carmine wind
Ight, How should I go about graphing this?
abstract fractal
Wdym graphing it
carmine wind
Is it possable to create an equasion to liniarly see when i will reach other point goals?
abstract fractal
It won't be linear, but sure
carmine wind
Okay, How?
abstract fractal
You can make a nice equation with the floor function
carmine wind
Oooh
abstract fractal
Consider what the graph should look like
Maybe sketch it out on paper or sumn. Get an intuition
Can you see that it kinda looks like a floor function? The scale is a bit off, but it's still very similar
carmine wind
What does a Floor Function look like?
abstract fractal
Gimme a sec
carmine wind
Ight thanks
abstract fractal
It's basically a "round down" function
Well, not basically---that's literally what it is
carmine wind
Ha, With the values we have discovered. How do we Plug them in?
abstract fractal
Well, we need to think how our function is to relate to the floor function
Can you envision that the graph we want is the same as the floor function, but scaled vertically and horizontally to some degree?
carmine wind
I think yes?
abstract fractal
How much should it be scaled horizontally?
carmine wind
I think that the graph should scale horozontaly with the amount Cycles or time?
carmine wind
There will always be 10 spaces to advance that is a fixed number, and It will always take 2 seconds to advance a single space, and after the 10 spaces it will always earn 4 points.
I belive that the 10 spaces will be our fixed number
abstract fractal
Well, it depends on what you want to input
Do you want the function to take the number of spaces as an input, or the amount of time?
carmine wind
Lets go with time
If we go with time, 2 secconds per one space. 20 secconds for a cycle of 10 spaces earning that 4 points right, Where do i put the values and why?
abstract fractal
Notice that in the floor function, it increases every 1 unit
For our function, when do we want it to increase? After how many units?
carmine wind
We want to increase the amount of Points earned, 4, after 10 Cycles.
abstract fractal
But we want it in terms of time, not cycles, right?
We're inputting time, after all
carmine wind
Ah yes, Then We want to increase the amount of Points earned after The 20 seconds of time? I want to discover the next value after each cycle and graph it, First cycle happens within the 20 seconds we gain 4 points, Second Cycle happens and we gain a total of 8 points
abstract fractal
We want it to increase after 20 units, yes
So we should scale horizontally by 20
Now, how much should we scale vertically?
carmine wind
We should scale Points vertically, because we are using the time it takes to complete a cycle horizontally.
abstract fractal
But by how much?
Notice that after the floor function increases, it increases by 1 unit
How much do we want our graph to increase by?
carmine wind
By increments of 4 after one cycle
So
horizontally will increase by increments of 4, and Vertically we will increase our values by 20?
abstract fractal
Other way around
So, if our function was floor(x), how would we scale that horizontally by 20?
carmine wind
Vertically we will increase by increments of 4, Horizontally we will increace our time value by 20 secconds
abstract fractal
We're not scaling both vertically
abstract fractal
So, if our function was floor(x), how would we scale that horizontally by 20?
.
abstract fractal
You have it the wrong way around again
Horizontally is our time axis
We'll scale horizontally by 20
abstract fractal
So, if our function was floor(x), how would we scale that horizontally by 20?
.
carmine wind
So, if our function was floor(x), how would we scale that horizontally by 20?
We should scale with the 4 points right?
carmine wind
With the vertical axis increasing by the value of 4 points, and the horizontal axis increasing by 20 seconds of time, How do we plug these values into the Floor Function?
abstract fractal
Let's say we had a function f(x). How would we scale it horizontally by, say, 2?
carmine wind
Would we plug in our value "2" Into the X variable?
abstract fractal
No
We don't want to evaluate the function at 2
We want another function entirely
carmine wind
We want another function entirely
We need a function different than the floor function or to input the "2" Differently in our function?
abstract fractal
Do you remember (assuming you've already been taught this) function transformations?
How to scale functions, how to shift functions, etc?
alpine sable
Hi
carmine wind
Do you remember (assuming you've already been taught this) function transformati...
I might be familiar with it, Lets assume Minimum knowledge on Function transformations. Although I am learning how to shift linear graphs Y=mx+b In my schooling right now. Flipping, Horizontal translation, Vertical translations and Basic Parent Functions.
abstract fractal
Here's a basic overview
If you have a function f(x), and you want to scale horizontally by some factor a, you get f(x/a)
carmine wind
Okay 👍 Thank :)
abstract fractal
If you want to scale vertically by some factor b, you get bf(x)
So, what happens when we scale floor(x) horizontally by 20? What do we get?
carmine wind
If you have a function f(x), and you want to scale horizontally by some factor a...
f "(x/a)" <----- Is Part fraction?
abstract fractal
Wdym "part a"
Let's say I wanted to scale x² horizontally by 2
I would write (x/2)²
If you graph it, you'll see (x/2)² is the same as x², but scaled horizontally by 2
carmine wind
Okay, I have an online calculator Called Desmos with the floor function. I have my function looking like this
abstract fractal
Why are you squaring it
I mean, why are you flooring it
I was referring to a completely separate example
carmine wind
I mean, why are you flooring it
OOF
carmine wind
I was referring to a completely separate example
Sorry i missed that, What should our next step be?
carmine wind
Plugging in our time into the function like so? (x/2)
carmine wind
Sorry the floor was assumeed
abstract fractal
Still, why over 2?
carmine wind
For our time of twenty secconds
abstract fractal
20 is not the same as 2
carmine wind
Valid point, When using 20 the graph streched and i thought i could use 2 as a substotutions knowing it was really a 20
abstract fractal
Try not to use a number when you really mean a different number
Anyway, what's our function after stretching horizontally by 20?
carmine wind
It should look like floor(x/20) right?
lone heartBOT
@carmine wind Has your question been resolved?
carmine wind
How would we do that?
Hey, I'm back
If we were to scale our points vertically, would plugging our "4" into floor(4/20) look like that?
abstract fractal
No
carmine wind
Now we want to scale _that_ vertically by 4
Apologies for the extended absence
abstract fractal
You aren't evaluating the function at 4
You're finding a new function that's the same as the old one, stretched vertically by 4
carmine wind
https://youtu.be/Tmdrjs9xufc
I will study this one sec
Thank <3
Okay, With some new knowledge We want to stretch it out by 4 vertically. In the video it says to vertically strech it will look like #f(x) If we plug in our values "4" Vertically and our Horizontal Value "20" It should look like this? 4f(x/20)?
abstract fractal
Yes
abstract fractal
Well, 4floor(x/20), but yes
carmine wind
Okay, I wanna make sure I understand this
Our problem was
I wanted to increase a point value by 4 after a cycle of 10 spaces taking 2 seconds to advance in each space. I wanted to find the Amount of time it will take for each value after our initial 4 and graph it.
We took 4 and used a floor function because we needed to vertically stretch it and used 20 seconds because it represents our 10 spaces
And we ended up finding we wanted to input points and time and eventually we got the Super function
Thank dio <3 This was really fun :)
Here is our Super Function in Action :P With Dark Mode
lone heartBOT
@carmine wind Has your question been resolved?
hard bison
how do you convert this recursive formula to an explicit one?
hard bison
its the formula for adding up squares from 1 to n
nocturne dove
It's not clear
ocean sealBOT
alpine sable
is that n^4?
alpine sable
oh
hard bison
and the end is n-1
alpine sable
so over all u are adding up n^2s
- C
where C is any constant
so overall u will get $C+\sum_{i=1}^{n-1} i^2$
ocean sealBOT
does calc even end
alpine sable
yes there is
do you know for the formula for the sum of first n squares
you can also try to find it yourself if u really want a challenge
hard bison
i’d prefer a non-challenge
hard bison
can i also have the video? if possible
alpine sable
can i also have the video? if possible
it has the proof i think
u can easily verify it by seeing the differences
but that wont tell from where the formula comes
placid rock
If I am generating data according to some pdf (or I think I am) what statistical test can I use to test if the data I am generating is distributed according to such pdf?
lone heartBOT
If I am generating data according to some pdf (or I think I am) what statistical...
@placid rock Has your question been resolved?
lone heartBOT
@placid rock Has your question been resolved?
keen plinth
If I am generating data according to some pdf (or I think I am) what statistical...
chi-squared test
placid rock
from my understanding this would involve binning my sampes to form a histogram?
keen plinth
yes
mellow reef
Find the first term and the common ratio of an infinite geometric series whose sum is 5 and each term is 4 times the sum of all the terms that follow it?
mellow reef
idk how to start
lone heartBOT
@mellow reef Has your question been resolved?
mellow reef
<@&286206848099549185> please help
i been tryna do this idk how to get r
<@&286206848099549185> im stuck rn
i started 11 pm and its 4 pm rn
analog basalt
Right so I think you want to calculate $a_1$
ocean sealBOT
Yuese
mellow reef
Right so I think you want to calculate $a_1$
yes a_1 and r
analog basalt
We know that "each term is 4 times the sum of all the terms following it"
mellow reef
yup
analog basalt
Let's call "the sum of all the terms following it" maybe $S_{\geq2}$
ocean sealBOT
Yuese
mellow reef
yup
analog basalt
Note that $S_\infty = a_1 + S_{\geq2}$, does that make sense?
ocean sealBOT
Yuese
analog basalt
Because we just add the first term to every term except for the first term, and then we just get the whole sum
mellow reef
mhm
analog basalt
But we know that the first term is four times as much as the terms following it
mellow reef
following it does that mean the terms before it or after it
analog basalt
In other words, $a_1$ is four times as much as $S_{\geq2}$
ocean sealBOT
Yuese
analog basalt
following it does that mean the terms before it or after it
After
mellow reef
ok
analog basalt
In other words, $a_1$ is four times as much as $S_{\geq2}$
So could you write that relationship in symbols?
mellow reef
So could you write that relationship in symbols?
is this correct
analog basalt
Yes that is true
mellow reef
waht next
analog basalt
So we can easily rewrite that as $\frac{a_1}{4} = S_{\geq2}$
ocean sealBOT
Yuese
analog basalt
And we knew that $a_1 + S_{\geq2} = S_\infty = 5$
ocean sealBOT
Yuese
mellow reef
yep
analog basalt
**Yuese**
Now we take this and we replace $S_{\geq2}$ with $\frac{a_1}{4}$, because we know that they are equal
ocean sealBOT
Yuese
analog basalt
Does that make sense?
analog basalt
So then we get $a_1 + \frac{a_1}{4} = 5$
ocean sealBOT
Yuese
mellow reef
yup
analog basalt
And this is just an equation in one variable which is easily solvable
mellow reef
And this is just an equation in one variable which is easily solvable
a_1 = 4
but how to get r
analog basalt
but how to get r
We can get this really easily from one of the equations you derived above
analog basalt
The first one out of this picture
mellow reef
4/1-r = 5
analog basalt
thank you somuch
No problem!
mellow reef
.close
elder viper
I have a question as for the two following statements,
lone heartBOT
I have a question as for the two following statements,
elder viper
"a symbol 'a' at the i-th position is worth a times B^i" where B stands for the base in some context
"the highest value that can be written in a base B with n symbols is B^n - 1"
I presume that 'a times B^i' or 'B^n - 1' results are in base 10, I'm surprised that it's not specified
anyway, how can I convince myself that those results/outputs are in base 10? It seems like it's taken for granted
spring gust
It's for every base
elder viper
could I ask for an example to get my point across?
Let's just say base 2 is being considered along with three digits,
spring gust
Alright let's work on base 2
The digits are 1 and 0
If we have only 3 digits, we count up:
000
001
010
011
100
101
110
111
The last number we can represent is equal to 7(base 10)
Which is equal to 2^3 - 1, as required
spring gust
The operation of counting and arithmetic in general is divorced from the representation we use
So let's take our three digits in base 2
a b c
The definition of representation is the c means c * 2^0
b = b 2^1
c = c 2*2
Which is indeed equal to 7 (base 10), which is 7*10^0
You can think of the numbers as some abstract entities that we pitiful humans need a representation to see
The number 7 is the same no matter which representation we use to display it
VII in Roman numerals is also a way of representing it, and just as "valid". It's a simple fact that the type of representation we use is just very simple and useful
elder viper
hm
so on one hand you have what you refer as to the representation (7 in base 2), what's the term used to denote the 7
spring gust
Don't know
7 is how you write the number in base 19
10
It gets a bit self referential if you're not careful about how you write it
elder viper
The operation of counting and arithmetic in general is divorced from the represe...
.
.close
lone heartBOT
spring gust
damn, this is very deep
Well it's as deep as you take it
naive sorrel
lol
spring gust
You can check out #foundations when you're more mathematically mature to see how mathematicians define the nitty gritty definitions
lone heartBOT
Available help channel!
rocky ledge
lone heartBOT
vale wigeon
well as a first step you could add 1 and then divide by 2 on both sides to get sin(2θ) = 1/2
lone heartBOT
@rocky ledge Has your question been resolved?
lone heartBOT
lone heartBOT
Available help channel!
mellow reef
Find all x that satisfy x = 1 - x + x^2 - x^3 + x^4 - .....
lone heartBOT
Find all x that satisfy x = 1 - x + x^2 - x^3 + x^4 - .....
runic hull
Find all x that satisfy x = 1 - x + x^2 - x^3 + x^4 - .....
what have you tried?
mellow reef
what have you tried?
idk how to start at all
but our topic is abt geometric sequences and series
mellow reef
so what kind of series is on the right hand side?
the common ratio is
a negative
or
idk..
runic hull
it is!
mellow reef
so where should the commas be
how do i know that the term is 1-x or if its 1
idk where to start tbh
runic hull
you mean the common ratio or the first term?
mellow reef
you mean the common ratio or the first term?
what would the first term be
runic hull
try thinking about it both ways. if it is 1, what would the common ratio be? do the other terms fit that pattern?
mellow reef
is it 1-x then
runic hull
like I said,
try thinking about it
if it was 1-x what would the common ratio be? do the other terms fit the pattern?
mellow reef
this is the example they give us
and look at the items they give us
😭
common ratio is -x?
runic hull
common ratio is -x?
yes
mellow reef
what would i do w the common ratio
limpid tapir
you needed to find the 7th term
runic hull
well, do you know the formula for the sum of a geometric series?
limpid tapir
in order to do that you have to use the formula
mellow reef
you needed to find the 7th term
no its a different question
runic hull
you needed to find the 7th term
the above screenshot is just an example
mellow reef
the picture i sent is easy
limpid tapir
a okay
mellow reef
thats the one they teach us
limpid tapir
sorry
mellow reef
its oke
runic hull
yes
because the geometric series in your question is the sum of the terms 1,-x,(-x)^2, ect
so now you can use that formula to simplify the problem
mellow reef
wait is it (-x)^2 or -x^2
runic hull
(-x)^2 because the common ration is 1
mellow reef
but i thought common ratio was -x
runic hull
yes
mellow reef
what next
runic hull
think about it
what is the equation you're trying to solve once you simplify the right hand side?
runic hull
that's not an equation
you had x=1-x+x^2-x^3+...
so what does it become if you replace the right hand side with the expression you just found?
mellow reef
x=S_infinity
runic hull
what is S_infinity in terms of x though? that would be more useful to you
mellow reef
i dunno
runic hull
okay so
you started with
x=1-x+x^2-x^3+...
then we decided we wanted to simplfy the right hand side
we decided the simplified version was 1/(1+x)
so ...
what should the equation be?
runic hull
idk what r5 or the phi thing is
but do you see how to solve it?
mellow reef
yeah
mellow reef
idk what r5 or the phi thing is
but do you see how to solve it?
is this the right answer
x^2+x-1=0
runic hull
ah sqrt(5).
Seems correct. just check that both of those roots are |x|<1. if one is bigger than 1 then the series doesn't converge in the first place
mellow reef
ah sqrt(5).
Seems correct. just check that both of those roots are |x|<1. if o...
what does converge mean in math
in my class we were taught that if the absolute value of r is greater than 1
then the series doesnt exist
is that what you mean
mellow reef
well one of the roots is greater than 1
but is the other one less than -1
limpid tapir
but is the other one less than -1
its not
write out the two roots separately
$x = \frac{-1-\sqrt{5}}{2}$ $x = \frac{-1+\sqrt{5}}{2}$
ocean sealBOT
kjmkty
limpid tapir
what do you think, which one of these is in the range of [-1,1]
mellow reef
does that mean the series doesnt exist
limpid tapir
what matters is that you should chooshe which one of these roots is less than 1 and greater than -1
mellow reef
what that mean
limpid tapir
can you approximate $\frac{-1-\sqrt{5}}{2}$?
ocean sealBOT
kjmkty
mellow reef
its less than -1
ocean sealBOT
kjmkty
mellow reef
greater than 1
limpid tapir
is it?
mellow reef
my teacher said absolute value so its always positive
so its greater than 1
its 1.6 smething
mellow reef
its 1.6 something
limpid tapir
wait lol
mellow reef
i looked it up it said its the golden ratio
ocean sealBOT
mellow reef
but the golden ratio is 1.6180
my friend said its phi
and phi is 1.6
for the negative its -1.6
limpid tapir
no
limpid tapir
golden ratio is $\frac{1+\sqrt{5}}{2}$
ocean sealBOT
kjmkty
limpid tapir
not $\frac{-1+\sqrt{5}}{2}$