Linear combination of A-C and B-C
#help-10
compact shadow
worthy notch
Thanks, I will do some reading up on how that works
But just to confirm, point X is not the middle/centroid of A, B, and C
compact shadow
Np
worthy notch
.close
fervent nimbus
obtuse pebbleBOT
fervent nimbus
Not sure what I've done wrong here. Obviously I've messed up somewhere on that ratios but I don't understand how.
Or... wait. Is it that AB is 2/3rds of BC?
compact shadow
when AB:BC=r:s, B=(s/(r+s))A+(r/(r+s))C
fervent nimbus
What is r and s in this formula
compact shadow
2,3
fervent nimbus
Right, but where do the variables come from? Just random letters?
Or is there some underlying reason for having chosen them?
compact shadow
r,s are real numbers
compact shadow
You want a proof?
Choose any point P, different from them, origin for example , whatever you like
Let A’B be parallel to PC, for a point A’ on PA
compact shadow
BC’ be parallel to PA for a point C’ on PC
PB (as vectors)=PA’+PC’
Now |PA’|/|PA|=1-|A’A|/|AP|=1-|AB|/|AC|=1-r/(r+s)=s/(r+s)
|PC’|/|PC|=r/r+s is similar
fervent nimbus
Yeah I'm doing basic geometry right now so
compact shadow
B = A(3/5) + C(2/5)
Yeah
fervent nimbus
none of that means anything to me just yet
a further evaluation would be B = (3/5) + (2/7)(2/5)
Correct?
fervent nimbus
I feel that this is an overcomplication of the problem for the sake of proofs and a broadly applicable formula
Which is usually good, but I'm really sleepy lol
compact shadow
Whatever
fervent nimbus
Huh?
compact shadow
If you found calculating -3/5+14/5 and -3/5+4/5 hard then so be it
I don’t really care
compact shadow
Blocked
fervent nimbus
That won't turn people off from trying to improve and progress, I'm sure.
Dope, Gonna see if I can pull a moderator in.
<@&286206848099549185>
<@&268886789983436800>
At your discretion, I believe there's a pretty obvious violation of the following guideline from the server rules:
Respect that other people might be at a different stage in their education than you, what is obvious to you might not be obvious to them.
compact shadow
Bark bark
hot hazel
yeah uh
@compact shadow i am unconvinced you were acting in good faith in this exchange
and instead were just trying to show how smart you were
hot hazel
...
compact shadow
Just my instincts to fight back. They stop I stop
fervent nimbus
Thank you, @hot hazel for giving your attention to this. I'll leave the situation in your capable hands and just head to bed. Math is not on the cards for the rest of tonight haha
Should I close the channel or are you able to close it as Mod?
hot hazel
No attention to argue
it seems you want to frame this as an "argument" as a tactic to avoid it; i dont care who is correct in the "argument", i do care that you took an approach which the user said was currently beyond them, and then immediately implied that they're unintelligent for not immediately getting what you were hinting at
hot hazel
I am pinging you to communicate behaviour that I want you to change.
hot hazel
Should I close the channel or are you able to close it as Mod?
i can close it
fervent nimbus
Excellent. Have a good rest of your day guys.
hot hazel
.close
snow peak
what would be the easiest way to do this and how
obtuse pebbleBOT
what would be the easiest way to do this and how
high lily
elimination
timid silo
yeah elimination
high lily
have you solved these types of questions using elimination before?
obtuse pebbleBOT
@snow peak Has your question been resolved?
obtuse pebbleBOT
@snow peak Has your question been resolved?
solar remnant
you have to use linear combinaisons between equations
(1) : 5x - 2y - 10 = 0
(2) : 3x + 8y - 15 = 0
What about 3 (1) - 5 (2) ?
tall arrow
Alex’s car uses 0.48 Litres per 1 kilometre.
In USA, they measure lengths on the road using miles.
1 mile = 1.609 km. Volume is measured in gallons.
1 US gallon = 4.546 litre.
How many gallons per mile does Alex’s car use?
obtuse pebbleBOT
Alex’s car uses 0.48 Litres per 1 kilometre.
In USA, they measure lengths on the...
tall arrow
This one can be a bit tricky, but bear with me
I've done some calculations and found out that the answer should be equal to around 0.35 Gallons per one entire mile.
(We weren't granted permission to use calculators, so that's why you see me doing a lot of fractions)
Anyway, when checking the answer in the book, it claimed it was 0.17 Gallons... How come?
Have I done something wrong in my calculations?
obtuse pebbleBOT
@tall arrow Has your question been resolved?
scenic matrix
f:[0,1]->R , f is continuous. prove that
obtuse pebbleBOT
f:[0,1]->R , f is continuous. prove that
scenic matrix
preferably without using epsilon-delta definitions( if possible)
obtuse pebbleBOT
@scenic matrix Has your question been resolved?
scenic matrix
<@&286206848099549185>
obtuse pebbleBOT
@scenic matrix Has your question been resolved?
charred plume
Do you know what it means to be uniformly continuous, and if so, are you aware that f is uniformly continuous?
@scenic matrix pinging because I only have a few minutes.
charred plume
It's going to have to be an epsilon delta kind of thing. Basically for any epsilon you'll want a delta that guarantees whenever x and y are within delta of each other, f(x) - f(y) is smaller than blah. Then you pick N so that 1/N is less than delta. Then you say for any n > N, f(k/n) - f((k+1)/n) is smaller than blah, so when you break the sum into pairs, each pair only contributes blah. Now you can figure out what blah had to be, in order to guarantee the sum itself is less than epsilon. There's a few more details to figure out (like what happens when n is odd, and there's one term in the sum that doesn't pair off), but hopefully that's a decent roadmap.
charred plume
You mean 1/N being less than delta? It's so that when n > N, k/n and (k+1)/n are within delta of each other.
charred plume
Oh, like why I have n and N?
scenic matrix
nonono, i get that part, we set an N and then use n>N blah blah. What im wondering is How you ended up getting 1/N
Or is it just something i need to write down and see for myself
charred plume
Probably? The idea is, given any epsilon we must produce an N. So we are going to find a delta, and chose N larger than 1/delta.
I was just saying that in the equivalent form 1/N < delta.
charred plume
No worries, I compressed a lot into that post.
scenic matrix
By the way, do you perhaps have a book/yt vid/whatever you can recommend that Will help me understand such delta epsilon excercises?
charred plume
Nothing on YouTube, no (might exist, I just don't know). I learned a lot of this stuff from baby Rudin. One sec.
Yeah, Google for baby Rudin... :)
scenic matrix
Okay, thank you bro!!
charred plume
No prob!
scenic matrix
Mind answering one more question? Im having trouble with splitting the sum into pairs, How would that work
charred plume
Idea is to do something like (-f(1/n) + f(2/n)) + (-f(3/n) + f(4/n)) + ...
The plus and minus signs come from the (-1)^k in the original formula.
charred plume
One way to write that is with a new summation variable. So sum j = 1 to n/2.
charred plume
And these parts Will cancel ravg other out?
Not fully cancel, but almost! Close enough that when you add them all up, you're still less than epsilon
scenic matrix
Not fully cancel, but almost! Close enough that when you add them all up, you'r...
like the last(n-th) part Will be the only remaining?
charred plume
The terms in the sum will look like (-f((2j-1)/n) + f(2j/n)). You might have to write that out and see... Adding those for j going from 1 to n/2 (assuming n is even) is the same as the original sum over all k.
scenic matrix
And this will sum up to less than epsilon, ending the proof?
charred plume
Here's the intuition for this question... f is continuous, so the value of f at one point is close to it's value at any nearby point (so long as it's near enough). So when you take the sum in the question, you're adding f at one point, but then subtracting at the next point over. So those should almost cancel out, and even when you add up the whole thing, each pair of neighboring points only contributes a little bit. That means the sum itself is very small (and converges to zero as n gets bigger).
scenic matrix
Alright, i get the idea of the whole thing now
charred plume
Yeah, if you choose the "blah" I mentioned correctly. Basically it's the amount each pair is allowed to contribute to the sum in order to guarantee that the sum itself is less than epsilon. blah probably is just epsilon, or epsilon/2, or 2 epsilon. Gotta write it out. But anyways, I've gotta get going!
scenic matrix
So basically, we start off using the continuity definition and progress to the limit
charred plume
Yeah, that's right.
charred plume
Sure, no prob. :)
scenic matrix
Do you have time for another exercise?
charred plume
Haha, not right now, sorry.
harsh sentinel
The value of c for which the pair of equations: cx – y = 2 and 6x – 2y = 3 will have infinitely many solutions is
obtuse pebbleBOT
The value of c for which the pair of equations: cx – y = 2 and 6x – 2y = 3 will ...
harsh sentinel
how do I solve this
I've seen on google that this is the condition for equations to have infinitely many solutions
a1/a2 = b1/b2≠ c1/c2
is this true
upbeat island
infinitely many solutions means no matter what you plug in for x and y the equation will hold true
so basic example
x + 1 = x + c
letting c = 1 will always leave us with a simplified equation 1 = 1
always true
harsh sentinel
yeah but how do I verify that. It's nearly impossible/time consuming to plug in many values to test if that holds true
nocturne minnow
Infinitely many solutions means that one equation is a scalar multiple of the other
harsh sentinel
Infinitely many solutions means that one equation is a scalar multiple of the ot...
so like 2x+3y+2=0 and 4x+6y+4=0?
upbeat island
what dldh06 said and also i'd make a substitution and get a single equation with a single variable
nocturne minnow
so like 2x+3y+2=0 and 4x+6y+4=0?
Yes
harsh sentinel
what dldh06 said and also i'd make a substitution and get a single equation with...
how so
nocturne minnow
So that example you showed, has infinitely many solutions because one equation is a scalar multiple of the other
harsh sentinel
So that example you showed, has infinitely many solutions because one equation i...
how is it a scalar multiple can you expand on that please
cx-y-2=0
6x-2y-3=0
I don't see a connection
nocturne minnow
You have to determine a c, that makes that equation a scalar multiple of the other
harsh sentinel
You have to determine a c, that makes that equation a scalar multiple of the oth...
and what is the solution
harsh sentinel
That's your job to do, not mine
3 is the obvious answer but it hold true for only a and b not c
so 3x-y-2=0
6x-2y-3=0
nocturne minnow
One, stop ping replying
Two, determine if the equations are right
harsh sentinel
3x-y-2=0
6x-2y-3=0
if we multiply the first equation by 2 we get
6x-2y-4=0 the first two terms are the same but the last one is different
nocturne minnow
As mentioned, you could have written the equations wrong
harsh sentinel
I got it from google, it's a multiple choice question and one of the answers is "no value"
upbeat island
i've solved the system for c and i get that in terms of x which is weird but i guess not off limits strictly speaking
harsh sentinel
I guess no value is the correct choice then
upbeat island
oh maybe that's the case then
nocturne minnow
I got it from google, it's a multiple choice question and one of the answers is ...
Then this is the answer
If that's an option
As I mentioned, for it to make infinitely many solutions, one of the equations should be a scalar multiple of the other
And if it's not possible, and one of the choices is "no value", that's the answer
timid silo
can you explain in simple words what’s happening here
obtuse pebbleBOT
can you explain in simple words what’s happening here
timid silo
like i don’t understand how should i memorize this
simple siren
Well well how do I explain this, it's already there... So you want to know why equilateral and isosceles triangle bisects the base line, right?
high lily
consider the basic formula for area of a triangle
simple siren
That'd be 1/2 * b * h, right?
nocturne minnow
In my opinion, that is explained pretty well
simple siren
Thats what seems so, though i wanna know what the area take on this prob
simple siren
Right right, now that i don't know
high lily
consider the basic formula for area of a triangle
A = bh/2
the height of the triangle remains the same
the areas of the split triangles ill be the same iff the bases of the split triangles are also the same,
otherwise they'll be different
timid silo
what i understand from this is that you can make a height in an isosceles and in an equilateral triangle
so it will cut the base
because formula yet we haven’t studied
high lily
that can also be justified from congruence proofs (AAS or HL)
timid silo
yes that’s what the teacher has explained us 2day
alright thanks
4the help
.close
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
hasty plinth
Am doing summer HW for math and forgot everything about polar lol
obtuse pebbleBOT
Am doing summer HW for math and forgot everything about polar lol
hasty plinth
How do I do these types of problems
Sick of trying to reverse engineer a method to do these problems based off the answers lol
obtuse pebbleBOT
@hasty plinth Has your question been resolved?
obtuse pebbleBOT
@hasty plinth Has your question been resolved?
upbeat island
were the green curves there for the original question?
they seem to be a hint
that's where the difference of those two terms comes in
royal solar
what’s the real question
obtuse pebbleBOT
@hasty plinth Has your question been resolved?
hasty plinth
what’s the real question
What’s the equation
hasty plinth
were the green curves there for the original question?
They were
vagrant gale
What is the problem asking
Do you have to type in the equation or is it multiple choice?
royal solar
What’s the equation
rsin(theta) = 10cos^2(theta)
r = 10 cot(theta) cos(theta)
now sub in n * pi/4 or pi/6 to get a general idea of what happens
10 is more of a scale factor. Just try to picture the rest. obviously no value at 0, but both cos and cot are even functions. So you know f(x) = f(-x)
@hasty plinth
obtuse pebbleBOT
@hasty plinth Has your question been resolved?
hasty plinth
Do you have to type in the equation or is it multiple choice?
type in the equation
hasty plinth
rsin(theta) = 10cos^2(theta)
r = 10 cot(theta) cos(theta)
now sub in n * pi/4 or...
I...don't get it how did you even start with rsin(theta) = 10cos^2(theta). I'm supposed to get the equation from the graph not the other way around. I don't know what hints I'm supposed to use from the graph that will indicate certain parts of the equation
obtuse pebbleBOT
@hasty plinth Has your question been resolved?
obtuse pebbleBOT
@hasty plinth Has your question been resolved?
royal solar
I...don't get it how did you even start with rsin(theta) = 10cos^2(theta). I'm s...
times sin(theta)
as theta tends to 0, cos tends to 1 and sin tends to 0, so the whole expression tends to infinite
as it tends to pi/2 it tends to 0,
hasty plinth
I think you misunderstand; this is the answer
We are not given this information
I need to get that based on the graph so I need cues from the graph to arrive at that answer but I don’t know what cues there are
tardy epoch
I need to get that based on the graph so I need cues from the graph to arrive at...
You're supposed to know the graphs of r=csc(theta) and r= sin(theta) and other trig functions probably
fathom wren
I need to get that based on the graph so I need cues from the graph to arrive at...
So a circle directly above the horizontal line is likely to be sin(θ), while directly to the right of the vertical line is likely to be cos(θ) (try to imagine how you'd plot those radial functions as functions of the degree). You can tell the circle in that graph is scaled by 10 by seeing the circle's diameter is 10. Likewise a horizontal line would be sin⁻¹(θ) while a vertical line would be cos⁻¹(θ). You can see it's scaled by 10 under the similar logic as before. Then you're meant to do the subtraction mentally (mainly focusing where the answer becomes 0 and where the answer becomes infinity). I wouldn't consider this an easy problem (although it's simple if you know what you're doing). Thankfully that's why they gave you the circle and horizontal line to guide your thinking.
hasty plinth
I see...
I'll be honest and say I don't really understand it 100% and will definitely need to do more practice but it makes a lot more sense now that you explained. Thank you!
.close
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
toxic mulch
obtuse pebbleBOT
toxic mulch
I legit cant get the answer
I cant get the answer they have
i keep getting smthing else
mathway also gives another answer
😩
rigid agate
Yeah, so the derivative of x*sqrt(64-x²/4) is not equal to any of the expression you gave as the answer
Also that looks suspicious
toxic mulch
the book got it wrong?
toxic mulch
I legit cant get the answer
the course?
rigid agate
That correct, so actually they got that parentesis wrong, and that last transformation
rigid agate
so this is wrong?
yes
toxic mulch
rigid agate
rigid agate
they gave an inversed answer
I checked if that was the problem, but a simple rotation would't perfectly fix it
toxic mulch
what is that +16?
rigid agate
Just to push the function up a bit. So it equals the correct solution at x=0
But you see that that doesn't fit perfectly
So something deeper went wrong with that😅
toxic mulch
my solution was: (-x^2-128)/2*sqr/(64-(x^2/4))
rigid agate
my solution was: (-x^2-128)/2*sqr/(64-(x^2/4))
Can you show some work? Doesn't look right when I plot it
toxic mulch
oke but it is extremely messy
rigid agate
oke but it is extremely messy
Ok
toxic mulch
rigid agate
Have you considered x * sqrt(64-x²/4) = sqrt(64x² - 4)
toxic mulch
when you plot it, it has 1 extreme value
why does it say it has 2
in this excercise
toxic mulch
Have you considered `x * sqrt(64-x²/4) = sqrt(64x² - 4)`
how does that work
rigid agate
how does that work
sqrt(a² * b) = sqrt(a²) * sqrt(b) = a * sqrt(b)
Assuming a is positive
Basically this rule backwards
hexed blaze
o is this finding derivative problem only are you have to find all critical points
rigid agate
Yes which means you don't need the product rule anymore
hexed blaze
This is the type of problem where tou should not use product rule
toxic mulch
Have you considered `x * sqrt(64-x²/4) = sqrt(64x² - 4)`
how did you get - 4 tho
hexed blaze
can't we just use log trick
toxic mulch
can't we just use log trick
wdym?
toxic mulch
Getting tired
np at all
hexed blaze
well log trick is to refrain from using harsh product rule
,tex
$y=x\sqrt{64-\frac{x^4}{4}}\\
\ln y = \ln x + \frac12 \ln(64-\frac{x^2}{4})\\
\frac{y'}{y}=\frac{1}{x}+\frac14 \frac{-x}{64-\frac{x^4}{4}}$
warm shaleBOT
Darkness
toxic mulch
so y'/y equals A'
hexed blaze
y' is A'
toxic mulch
why did you write /y under it
hexed blaze
if you want to get y' only then multiply both side for y
Because derivative of ln(f(x)) is f'(x)/f(x)
toxic mulch
this confused me more ngl
toxic mulch
Yes which means you don't need the product rule anymore
i think am just gonna try this technique real quick
hexed blaze
This is so nice because when you want to find extrema
toxic mulch
if you want to get y' only then multiply both side for y
i dont understand what is meant by this
hexed blaze
you know derivative of ln(x)?
toxic mulch
1/x
toxic mulch
y'x/yx
hexed blaze
yes
hexed blaze
,tex
$y=x\sqrt{64-\frac{x^4}{4}}\\
\ln y = \ln x + \frac12 \ln(64-\frac{x^2}{4})\\
\text{Differentiating both side wrt x}\\
\frac{y'}{y}=\frac{1}{x}+\frac14 \frac{-x}{64-\frac{x^2}{4}}\\
y'=\left(\frac{1}{x}-\frac{x}{4^4-x^2}\right)y$\\
If you want to find extrema, then it is just\\
$y'=0\\
\frac{1}{x}-\frac{x}{4^4-x^2} = 0 \\
\Rightarrow x=\pm 8\sqrt{2}$
warm shaleBOT
Darkness
hexed blaze
it is exactly like this table
toxic mulch
no you have x^4
rigid agate
i think am just gonna try this technique real quick
Remember though that it only works for x >= 0. If you want the cleanest answer, you maybe just have to do the product rule, which isn't so bad here, because one factor is just x.
hexed blaze
oops my mistake a bit at writing down
toxic mulch
darkness you have ^4
hexed blaze
differentiating both side with respect to x
the reason why we do this way is because
you know
(a+b)'=a'+b' (very simple)
(ab)'= a'b+ab' (too much)
toxic mulch
oh yeah
hexed blaze
and ln(a^b)=bln(a)
toxic mulch
yes
toxic mulch
yesyes
hexed blaze
so differentiation would be much more lenient
hexed blaze
well those are just extra step if you want to ask to find max/min value
if your task is just differentiating then you can end at the earlier step
toxic mulch
hexed blaze
yes
i mean you can try, if y=0 then the function is just
a straight horizontal line
toxic mulch
no
hexed blaze
it has no max or min
toxic mulch
bcs y is equal to this:
hexed blaze
yes ik
hexed blaze
sure but remember to write "differentiating both side wrt x" or your teacher might question you
hexed blaze
with respect to
toxic mulch
oke oke oke
hexed blaze
basically it indicates which variable you are trying to differentiate
hexed blaze
$\frac{1}{x}-\frac{x}{4^4-x^2} = 0$ \
(with tons of step...)\
$\Rightarrow x=\pm 8\sqrt{2}$
warm shaleBOT
Darkness
hexed blaze
basically what I mean is that show your work why you end up that
hexed blaze
sure but prefer here because we have bot
plush smelt
If i got EC/DC instaed of CE/CD would that be wrong?
obtuse pebbleBOT
If i got EC/DC instaed of CE/CD would that be wrong?
brisk grove
no, any length between two points A and B can be written AB or BA
the order doesn't matter
pallid flame
the line segment has the same length no matter which direction u go
brisk grove
CE is the same as EC, and DC is the same as CD
plush smelt
in a question like this would i need to find like sinA of both triangles
cause this person did BC/AC and then CE/CD to find the answer
pallid flame
if anything you would look for sin and cos of angle A
since that is what the answer choices consist of
plush smelt
oh
the sin and cos of angle a?
oh is that what sin A means?
and cos B
cos A
i mean
.close
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
next ravine
obtuse pebbleBOT
haughty coyote
radians go from 0 to 2pi. Degrees go from 0 to 360
haughty coyote
therefore, one can go from radians to degrees by saying degrees = radians * 180/pi, as 180° = pi radians
haughty coyote
so 0.85 will just be sine(0.85) in the calc?
radians and degrees play the same role: the input of the trig functions, not outputs
sin(45°) = sin(pi/4) = sqrt(2)/2
next ravine
so basiaccly both question have the same asnwer??
haughty coyote
no, one is in radians, the other in degrees
next ravine
iam confueeed
woeful spruce
radians are an exact messure of angle, compared to degrees
next ravine
So you cant calculate them with help of the calculator??
like the sine(0.85) I mentioned above
woeful spruce
45° is exactly the same as pi/4
I don't mean in this case, I'm speaking of the general radians vs degrees, and exact may be the wrong word to use. It's more "mathematical" to use radians
next ravine
sorry, iam just very cunfused about that
haughty coyote
like the sine(0.85) I mentioned above
sin(0.85 radians) has a value, same as sin(48.7014.... degrees)
calculators handle both anyways because there's no difference at that level. Radians are just more useful when it comes to the formulas produced
next ravine
mhm...
i dont know about the first one, but the second one I think you gave me a clue, sec please
is that correct?
or doest it need to be 85? and why exactly?
haughty coyote
that'd be for degrees -> radians
haughty coyote
or doest it need to be 85? and why exactly?
what are you trying to do ? Cause honestly I don't know
haughty coyote
radians and degrees play the same role: the input of the trig functions, not out...
so don't convert 0.85
they ask for the exact same angle. Once expressed in degrees, once expressed in radians
next ravine
so are we talking for the first, or second question??
oakyyy
so how do we get that?
haughty coyote
you already found 58°, just do the same thing but in radians (normally converting is fine, but here for rounding reasons I don't think 58 is precise enough)
,w sin(58°)
warm shaleBOT
haughty coyote
that was correct
next ravine
why 8pie tho?
warm shaleBOT
haughty coyote
as you can see this is an approximation though
next ravine
okay,so 54degrees is the 8??
haughty coyote
there's no 54 and not really 8, moreso 8/45, and as I said, in actually no because it's an approximation
next ravine
alrighty, but still that doesnt answer my question well.
are those calculations for the first or second question.
next ravine
okayyyy
and for the 2nd one how do I do it exactly?
58 times what?
or do I need to learn to convert somehow?
haughty coyote
therefore, one can go from radians to degrees by saying degrees = radians * 180/...
that's almost the first thing I said
so you also have radians in terms of degrees by inverting the formula
haughty coyote
you already found 58°, just do the same thing but in radians (normally convertin...
but as I said, I don't think this is a good method because it can fail
next ravine
okayy so it will be radians = degrees / 180/pi?
I would rather to try and fail 😄 than not try and still fail, thank you so much for trying to help me thooo ❤️
haughty coyote
okayy so it will be radians = degrees / 180/pi?
yes, but pls avoid unnecessary double fractions
- pi/180 is cleaner
next ravine
so radians = 58/ pi/180?
haughty coyote
,w 58 * pi/180
warm shaleBOT
haughty coyote
,w arcsin(0.85)
warm shaleBOT
haughty coyote
as you can see, this is not a very good method due to the propagation of rounding errors
gets you 1.01 when 1.02 is the actual rounded value
next ravine
when I do sin-1(0.85) I get a different answer?
next ravine
goot it 😄 thank you man
will it still woork if I do the sin(0.85) to get the right answer?
I get 0.75 so I guess not
haughty coyote
will it still woork if I do the sin(0.85) to get the right answer?
again, 0.85 is a length, not an angle
don't confuse them
next ravine
ahaa
haughty coyote
since we're not composing trig functions the distinction between angles and lengths is legitimate
next ravine
@haughty coyote sorry, regarding question 1, how exactly would YOU get 58 degrees with the help of calculator?
obtuse pebbleBOT
<@484702191156985877> sorry, regarding question 1, how exactly would YOU get 58 ...
haughty coyote
arcsin(0.85)
if given in radians, convert to degrees
or directly set the calculator in degrees mode
timid silo
obtuse pebbleBOT
timid silo
Ok guys, in both of these problems I am solving for V(t), but I use 2 different methods, and get 2 completely different answers
I think I found an issue in method#2:
$\int \frac{1}{\sqrt{V}} dV ≠ ln \sqrt{V}$
warm shaleBOT
! R4MP4G3
timid silo
Correct?
daring rock
**! R4MP4G3**
I haven't looked through all your work yet but you're right about this
timid silo
Well then that means method#2 is for sure wrong, and #1 could maybe be right xD
daring rock
Yeah the way you handled the integral in the first one is right
But the fifth line is not correct
$(a+b)^2 \neq a^2+b^2$
warm shaleBOT
tatpoj
warm shaleBOT
! R4MP4G3
daring rock
Right
I don't even know if you need to expand it though unless it asks for the answer in a specific form
timid silo
so V = 1/4 (kt)^2 + C
daring rock
That's still equivalent to your last answer
daring rock
Yeah, that's good
You could expand it if you want, but I don't think you need to. This form is simpler
daring rock
Sure thing
timid silo
Math is awesome, but man there are so many ways to make a tiny mistake that ruins the whole thing xD
timid silo
.close
obtuse pebbleBOT
timid silo
.reopen
obtuse pebbleBOT
✅
timid silo
And a second pair of eyes sometimes
Exactly why I joined this server heh
Ok thanks again heh
.close
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
hallow hatch
A meeting is held with eight people around a circular table. Two of the participants, April and May, want to sit so that there is exactly one other person between them. How many different seatings are possible? (Two seatings are considered the same if one can be rotated to obtain the other.)
obtuse pebbleBOT
A meeting is held with eight people around a circular table. Two of the particip...
hallow hatch
is the answer 2!*5!?
,calc 5!*2!
there one way to place the person that will sit in between april and may
2! ways the place april and may around them
and then 5! ways to order everyone else
warm shaleBOT
Result:
240
Aren't there actually six possibilities for who sits between April and May?
hallow hatch
Aren't there actually six possibilities for who sits between April and May?
oh thats right
so is it
,calc 6*5!*2!
warm shaleBOT
Result:
1440
Yes, or equivalently 6!2!
Because there are 2! ways to seat April and May, and 6! ways to seat everyone else
daring rock
You don't really need to count the seat between them separately because it's a valid seat for any of the other six
No problem!
hallow hatch
You don't really need to count the seat between them separately because it's a v...
ah i see
now i know for next time
.close
zinc cape
obtuse pebbleBOT
zinc cape
how do i do this?
tardy epoch
how do i do this?
• Show your work, and if possible, explain where you are stuck.
obtuse pebbleBOT
@zinc cape Has your question been resolved?
zinc cape
sorry im completely stuck
right now im trying something i found which is to try to express it as a series
tardy epoch
right now im trying something i found which is to try to express it as a series
maybe try reduction of order?
https://tutorial.math.lamar.edu/Classes/DE/ReductionofOrder.aspx
zinc cape
i guess y=0 is a solution
am i correct?
but this has no x
i guess 0*x
im not sure if i can use that since i am not given a solution
@tardy epoch
tardy epoch
y=0 is probably not one of the solutions they want you to find
obtuse pebbleBOT
@zinc cape Has your question been resolved?
zinc cape
.close
devout ivy
You want to place a cable from the top of a 25-meter high tower to a point located 50 meters from the base of the tower. How much should the cable measure?
obtuse pebbleBOT
You want to place a cable from the top of a 25-meter high tower to a point locat...
fierce lagoon
versed cave
Yes it's easier to understand how to use the pythagorean theorem if you make a little sketch of the problem
plush smelt
hey
high lily
converse theorems
high lily
parallel like theorems
do you know about alternate, corresponding, coint angle theorems on parallel lines?
plush smelt
nah
cedar lichen
You've never heard about parallel lines cut by a transversal?
high lily
then you should start by reading up on those
plush smelt
Kk
high lily
google parallel line theorems
cedar lichen
plush smelt
knowing that the vertical angles are congruent and the corresponding angles are the same lets you know that the two lines are parralel?
is that what the video is saying
i get the relationships of the angles
just not what they say about the lines
cedar lichen
Vertical angles are always congruent. It doesn't matter if any lines are parallel, unless you're meaning something else by "vertical angles"
plush smelt
Vertical angles are always congruent. It doesn't matter if any lines are paralle...
Yeah
thats what i meant
cedar lichen
If and only if two parallel lines are cut by a transversal, then you know 1) corresponding angles are congruent and 2) alternate interior/exterior angles are congruent
Which means if you have two lines cut by a third, and either corresponding angles are congruent or alternate interior/exterior angles are congruent, then the lines must be parallel
plush smelt
so in this case
angle 2 and 5 are alternate interior angles
is that how u would get
line K and L are parralel
?
yo
<@&286206848099549185>
daring rock
angle 2 and 5 are alternate interior angles
2 and 5 are alternate interior angles and they're congruent. That's how you know K and L are parallel
plush smelt
5 and 7 are corresponding right
daring rock
Yes
daring rock
Yep
hushed raft
obtuse pebbleBOT
slow sonnet
no, why would you think so?
hushed raft
no, why would you think so?
Because each of them just have x in the denominator
So the vertical asymptote = x = 0 right
slow sonnet
why can't E have a vertical asymptote at x=1 then?
slow sonnet
just because the denominator is 0, doesn't mean necessarily it's an asymptote
hushed raft
but if the denominator ever = 0 then it becomes undefined right
hushed raft
oh
slow sonnet
this is undefined
slow sonnet
0/0 is indeterminate
it's not undefined
it's called indeterminate
undefined means that no value can make sense, we can't define it
indeterminate means we don't have the enough information to determine exactly what is it
could by anything
hushed raft
I remember indeterminate forms from limits
slow sonnet
then x*0 = a
what number x, when multiplied by 0, will give a constant a ?
nothing
you can't
but in the 0/0 case
0/0 = x
so x*0 = 0
what number x, when multiplied by 0, gives us 0?
anything
can't know
slow sonnet
we need more info
hushed raft
How do we do 49 then
slow sonnet
the questions tells you how
hushed raft
ok wait I mean like algebraic ally
if I was asked this on a text w/o a calc how should I do it?
slow sonnet
there are some ways to do it
considering that you are required to sketch them means that you didn't study those methods yet
hushed raft
oh ok thanks 👍🏾 have a great day
slow sonnet
you basically calculate the limit
if the limit exist, then it's a hole
if not, can be asymptote or something else
arctic girder
obtuse pebbleBOT
arctic girder
HELP
obtuse pebbleBOT
@arctic girder Has your question been resolved?
bold bane
The question is using degrees instead of radians so try to solve it with wo full rotations equalling 720 degrees.
arctic girder
i got 60
bold bane
I got the same.
obtuse pebbleBOT
@arctic girder Has your question been resolved?
zinc cape
obtuse pebbleBOT
zinc cape
How do I do this?
I can’t find a tutorial for this type of problem
How do I guess a y=x^m?
how do I check if my guess will work?
obtuse pebbleBOT
@zinc cape Has your question been resolved?
tardy epoch
how do I check if my guess will work?
Plug it in by calculating y' and y''
zinc cape
@tardy epoch so what if i did y=x^2
what does that even mean
or am i supposed to try to get a y that results in 0=0
but im not sure if that is even possible?
i feel like im stuck on guessing part mainly
shut locust
Tried to solve it by linear differential equation but I'm stuck too lol
tardy epoch
<@261933205387477002> so what if i did y=x^2
You're supposed to leave the exponent as a variable and solve for it
tardy epoch
All you've done here is show that m=2 does not give a solution
zinc cape
what do you mean
tardy epoch
what do you mean
Plug y=x^m into your differential equation
zinc cape
but what m do i use
Ok instead of that
How about this other question
This one I did but can you guys please check if it’s correct
I just found out about Bernoulli so I’m a not on solid ground
Ok I see my mistake
Okay back to other problem
tardy epoch
You're supposed to leave the exponent as a variable and solve for it
.
Plug in x^m and find m
zinc cape
ok I see what you are saying
rigid pine
Wolfram says it's a Sturm-Liouville equation so if you've studied that theory you could probably get a solution.
Wolfram gives one solution as x e^(x^3/3) so plugging in x^m will never work anyway. The other solution looks complicated so I assume one would use the reduction of order method to find that other solution.
rigid pine
,w y''-x^2y'-3xy=0
warm shaleBOT
zinc cape
what about the back
rigid pine
If you can reason how x exp(x^3/3) is a solution then the other can be immediately found.
zinc cape
what do you mean by that
rigid pine
Which bit?
zinc cape
why do you say Cxe^(x^3/3) is a solution?
when it has a portion after the + sign
i dont understand what some of the symbols in the back mean too
rigid pine
It can be shown that the general solution of a second order linear differential equation is the linear combination of two linearly independent solution.
zinc cape
rigid pine
Both will be solutions.
zinc cape
why is there plus sign
rigid pine
I'm only talking about the solution labelled with the circled 1 for now.
rigid pine
If you can reason why this is a solution then you can find the general solution.
Wolfram has given us the general solution, in this case, which means it contains every possible solution. If we set c_1 = 1 and c_2 = 0 we get a solution.
zinc cape
ok
rigid pine
Since the RHS side of the equation is zero, we can multiply any solution by any constant as it will still output zero on the RHS so this is why you can attach a c_1 to this.
I sadly can't see a method other than trial and error or insight using the chain rule why you'd want to guess something like y = x e^(ax) is a solution and determine that it works for a = 1/3 but if you can get this single solution you can find everything else with certainty.
This method will always give you a second linearly independent solution if you have one solution.
I assume you'd leave your answer for the second linearly independent solution in the form of an integral which evaluates to gives what WolframAlpha gives.
zinc cape
so i would write the first solution as y=xe^(x^3/3) or y=Cxe^(x^3/3)
rigid pine
I neatened their solution up a bit. They have the third root of x^3 and I included the division of the cube root of 3 in with a new c_2.
Now you can see how it matches the theory above.
Wolfram says this is what it is so replace the a with -1/3 and z with x^3/3.
rigid pine
I'd personally just use y_1 = x e^(x^3/3) and then follow the method for y_2 and find the integral that way.
zinc cape
are you talking about reduction
rigid pine
Yes. Wolfram puts stuff into the bounds which I don't think would come up using the method so I'd have to verify things for myself to be confident.
zinc cape
and instead of t for integral i would use x right\
rigid pine
Just follow the method of the theory I gave.
I'm currently looking at the details now.
zinc cape
well i thought that gives the general solution
so it would just give what wolfram gave
how do i create another solution
zinc cape
lesson 23b?
i did but i did not really understand it so i tried looking up more about it
rigid pine
Here's the complete section for it.
zinc cape
ok it does give second independent solution
rigid pine
Just remember that the linear combination of any two linearly independent solutions will give the general solution so really there's only two solutions you can have as the answers (of course, they can have any constant attached to them too).
$y_2(x) = x \exp(\frac{x^3}{x}) \int u(x) \dd{x}$.
warm shaleBOT
stabulo
rigid pine
We need to differentiate this twice and substitute the results into the original differential equation to find u(x) but since this is a product of three things each depending on x, I'm using a derivative calculator to save time.
The reason is uses the integral of u(x) instead of just u(x) is because it will save time as normally you'd end up with another DE at the end you again need to solve.
rigid pine
Yes. It would take up a lot of space constant putting the (x) part.
I eventually get to this differential equation:
$\dv{u}{x} + \left(\frac{x^3 + 2}{x}\right) y(x) = 0.$
warm shaleBOT
stabulo
warm shaleBOT
stabulo
rigid pine
The book goes through the method to get to the formula but just remembering that y_2(x) = ... completely eliminates the need to remember the formula.
f_2(x) = 1, f_1(x) = -x^2, f_0(x) = -3x.
In the end:
$u(x) = x^{-2} \exp(\frac{-x^3}{3})$
warm shaleBOT
stabulo
rigid pine
y_2(x) can now be determined.
rigid pine
Inside the big integral reduces to the u(x) I gave above.
zinc cape
so this is a second nonlinear solution?
rigid pine
rigid pine
Impossible.
These two solutions form the general solution.
If
doesn't have a elementary solution it is impossible to do so.
The linear combination of these two linearly independent solutions, y_1(x) and y_2(x), form the general solution and contain every solution.
I'd just write this:
$y_1(x) = x e^{x^3/3}, y_2(x) = xe^{x^3/3} \int x^{-2}e^{-x^3/3}\dd{x}$.
warm shaleBOT
stabulo
zinc cape
ok
can i ask you a quick question for another question?
for this
i believe im supposed to use
so x would be 2
a would be 1
i solve for n
but what is f^(n+1)(z)
rigid pine
Funnily, I've actually just studied this a bit.
zinc cape
for f^(n+1) am i supposed to do it sort of like the power series?
where i expand it and find the pattern that gives the derivatives?
and what should z be?
a is 1
x is 2
do i just pick 2
since its the highest
rigid pine
I don't like this remainer since how are you going to determine z.
See here.
f(x) = ln(x) so determine some general formula for f^(n+1)(x).
f'(x) = 1/x
f''(x) = -1/x^2
f'''(x) = 2/x^3
...
f^(n)(x) = something.
zinc cape
and starting index should be 0 right
rigid pine
Yes f^(0) = f(x). But we only need the n+1-th derivative.
Maybe there's some road block ahead by this approach I've literally just proven this theorem myself yesterday and procrastinated since.
rigid pine
this seems to be to find R_0
That stuff below is for proving the theorem.
Focus on this bit.
It can be shown that the integral on the right of (1) is equal to
The formula you are using uses two different applications of the mean-value theorem for integrals to obtain their formula.
timid silo
The formula you are using uses two different applications of the mean-value theo...
Also this
rigid pine
Both are Advanced Calculus by David V. Widder. The white background is my PDF copy of the second edition book and the yellow background version is my PDF of the first edition book. I revert to the original edition since the image to text my other uses messed up formulas.
rigid pine
Big X?
zinc cape
yes
rigid pine
It's basically saying that is g(x) >= 0 (or g(x) <= 0) then you can factor out the f(x) but it's evaluated at some point between the bounds of the integral.
I was just referencing this part as your formula looked like it used this version of the remainder but it seems hard to find X to use this formula so revert back to the original.
The reason we are interested in this form is to prove other theorems. I've not got to these sections but I think it's used for l'hopitals rule
timid silo
We can just take X as the ceiling value of the function at the point you want it at
zinc cape
ok so
rigid pine
We can just take X as the ceiling value of the function at the point you want it...
Is there some new tech I need to know.
timid silo
what would the f^(n+1)X be?
You've to find n
rigid pine
Oh. Seems like a strange path to take. Maybe it's fine, hard to tell for me.
zinc cape
so X
rigid pine
Apparently
rigid pine
$f^{(n)} = (-1)^{n-1}(n - 1)! x^{-n}$
zinc cape
because im trying to solve for ln2
warm shaleBOT
stabulo
timid silo
**stabulo**
Hey what's this
zinc cape
i believe that doesnt work
because at n = 0
it would be (-1)!
so what is X
2
right
but then that would result in 2-2
which would be 0
and delete my whole equation
rigid pine
**stabulo**
for n>= 1.
"(x) = ln(x) so determine some general formula for f^(n+1)(x).
f'(x) = 1/x
f''(x) = -1/x^2
f'''(x) = 2/x^3
...
f^(n)(x) = something."
f''''(x) = -6/x^4
f^(5) = 24/x^4.
rigid pine
$f^{(n+1)}(x) = \frac{(-1)^n n!}{x^{n+1}}$.
warm shaleBOT
stabulo
zinc cape
that is what i had
rigid pine
Oh.
zinc cape
no idea how to get a good answer
zinc cape
oh right i forgot to write n before !
because remainder is this in reality right
it takes 500 terms
for the difference to be
but i was told the answer was 1000
terms
rigid pine
I'll have to explore this a little more closely, this is new territory for me too.
plush smelt
hey so i was confused for a questionlike this why u woudnt do pythagorean theorem cause when i saw this question thats what i woulda tried to do
obtuse pebbleBOT
hey so i was confused for a questionlike this why u woudnt do pythagorean theore...
plush smelt
what in the question tells this person i should use SOHCAHTOA
i thought pythagorean because we have a missing side? idk if thats a good reason or if that even makes sense with a question like this
sage iron
i thought pythagorean because we have a missing side? idk if thats a good reason...
try using it then
plush smelt
u cant use it
i see why pythagorean theorem wouldnt work cause thats for sides and not angles
so what tells them to use sohcahtoa
yo
<@&286206848099549185>
mellow halo
what in the question tells this person i should use SOHCAHTOA
Look