why is your profile almost the exact color of discord dark mode
#help-39
craggy tendon
ok
craggy tendon
how do you define a parabola?
craggy tendon
yep
visual canyon
now for the focus
and directrix i guess
where would those be in this parabola
and why...
craggy tendon
how do you define a parabola?
well answer this
craggy tendon
umm you didn't answer my question
visual canyon
oh sorry thought we moved past that with the actual image of a parabola
craggy tendon
well like
it doesn't help since I think you don't understand what the focus and the directrix are
visual canyon
a parabola is a symmetrical open curve
craggy tendon
so is a hyperbola
visual canyon
ik that every point on the curve is equidistant from a point called a focus and a straight line called a directrix
kind rampart
be more specific
craggy tendon
ik that every point on the curve is equidistant from a point called a focus and ...
okay perfect
so you do know how to derive the locus of a parabola is y^2=4ax
correct?
visual canyon
mhm
craggy tendon
alright so you start from (a,0) as the focus correct?
visual canyon
i didnt before but im getting what ur saying
craggy tendon
and x=-a as the directrix
visual canyon
oook
craggy tendon
now you just derived it for y^2=4ax
now what about x^2= 4ay?
what do you think the focus and directrix are
uhh i need to sleep, I'll let somebody else take over
visual canyon
yeah sorry dude im not too sharp rn
idle cipher
f=sqrt(1/(4a))
visual canyon
pretty much never when im on this server i ask dumb questions when i get tired and cant figure stuff out
idle cipher
this is not the graph
kind rampart
its an example graph
visual canyon
that is just a random parabola
kind rampart
so you know
y^2 = 4ax
and the directrix of that is
y = -a
and the focus is (a, 0)
visual canyon
ok backtracking
to get the directrix i need----
to know the axis of the parabola, and the vertex of the parabola
lets say the vertex is 0,0
the axis would be..
kind rampart
wdym by axis
idle cipher
is there a correct answer for this question?
kind rampart
yes
kind rampart
are you back
kind rampart
ok so
visual canyon
ive been here sorry just not sure what to say
visual canyon
not off the top of my head but i could figure it out
kind rampart
ok
just remember that the focus is inside the parabola while the directrix is out
anyways
8x^2 = y
kind rampart
x^2 = 4ay
this equation is almost this
correct
the only difference is that x^2 has an 8 next to it
do you have any idea how to remove the 8
how would you say remove the 8 in
8x = 64
kind rampart
yes
visual canyon
8x/8 64/8
idle cipher
why not just (0,1/4a)
kind rampart
divide both sides by 8
are you done here
kind rampart
(8x^2)/8 = y/8
visual canyon
i just feel like i've missed a step in getting these values
kind rampart
good job
visual canyon
yeah this makes sense
kind rampart
so now you should have
x^2 = y/8
or x^2 = (1/8)y
it should be quite clear what to do now
x^2 = 4ay
kind rampart
find out what a is
visual canyon
8
visual canyon
i have wasted your time for that i am sorry
even looking at the right problem i am incompetent. think i should cut my losses run away and do some googling
thanks for trying 
.close
pearl pondBOT
pearl pondBOT
Available help channel!
compact pewter
I think I'm just actually incapable of solving this because I am completely lost, I'm able to get partway through but I have no idea what to do after that and how to get what they want
compact pewter
I can understand that the distance along the shore is 8 - x and the distance along the triangle is sqrt(25 + x^2) but I really have no idea what to do from there other than plug into the distance = rate * time formula maybe
compact pewter
yeah I'm supposed to use calculus, wdym by wrt?
charred ocean
With respect to
compact pewter
oh
wouldn't that just give me minimum total distance traveled or am I misunderstanding?
charred ocean
What do you get?
compact pewter
how do you do that nice math formatting on here?
but if I solve for 0 after getting the derivative I get 2sqrt(5)
charred ocean
how do you do that nice math formatting on here?
$f(x)$ and $$g(x)$$
jolly parrotBOT
Pixelius
charred ocean
but if I solve for 0 after getting the derivative I get `2sqrt(5)`
Yeah I did it and I get no real solution
compact pewter
Yeah I did it and I get no real solution
So for this kind of question I can answer wrong and reset to get the "right" answer it expects
if I do that it'll give me a similar problem with different values
would doing that help?
charred ocean
but if I solve for 0 after getting the derivative I get `2sqrt(5)`
Wait no I made a typo this is correct
glass meadow
I can understand that the distance along the shore is `8 - x` and the distance a...
The distance along the shore (from the restaurant) is x, not 8-x
So on the boat it's sqrt(25 + (8-x)^2)
charred ocean
Yeah $\frac{10}{\sqrt{5}}=2\sqrt{5}$
jolly parrotBOT
Pixelius
dawn sand
how do you do that nice math formatting on here?
Its a markdown language called LaTeX. Wrap the math in dollar signs or use ,tex to get the bot to display that way, but i wouldnt recommend it unless you've learnt some LaTeX, since itll just look like a mess if you dont have proper formatting. Plain text suffices.
charred ocean
Yeah so you need to convert it from distance from nearest point to distance from restaurant like what Damian is saying
compact pewter
Its a markdown language called LaTeX. Wrap the math in dollar signs or use ,tex ...
oh okay cool
compact pewter
Yeah so you need to convert it from distance from nearest point to distance from...
when taking the derivative with new values I get x / 2sqrt(x^2 + 9) - 1/3
I think
but solving that for x is rude
I think it's 6sqrt5/5 tho
okay so that was correct
charred ocean
okay so that was correct
15 - 6 sqrt(5)/5 ?
charred ocean
Yeah
compact pewter
which i actually just dont know how to do at all
"if she walks at 3mi/hr find the minimum speed at which she must row so that the quickest way to the restaurant is to row directly (with no walking)?"
charred ocean
So if our equation is $T=\frac{15-x}{3}+\frac{\sqrt{9+x^2}}{row speed}$
jolly parrotBOT
Pixelius
compact pewter
optimizations and related rates I think are the hardest things for me because most of the time visuals aren't provided and I'm really bad at visualizing
charred ocean
Do you see how to solve it?
compact pewter
**Pixelius**
if we're solving for rowspeed though that's already given at 2mi/hr
charred ocean
if we're solving for rowspeed though that's already given at 2mi/hr
No it’s not. It used to be but now we don’t know it
glass meadow
Wouldn't it just be 3mi/hr?
compact pewter
I'm definitely doing stuff wrong because I'm getting negative numbers
Lol
I got -3.9
somehow
compact pewter
Wouldn't it just be 3mi/hr?
it was not 3mi/hr I used one of the attempts to check just in case
charred ocean
It’s definitely not above 3, but it’s probably slightly less
glass meadow
It’s definitely not above 3, but it’s probably slightly less
But anything less than 3 would leave some room for optimization by walking a bit
charred ocean
Yeah it just is 3
compact pewter
which still gives me something negative if I plug in the value I had for x before
compact pewter
Yeah it just is 3
but it just wasn't like I put in 3 as an answer and it told me it was incorrect
glass meadow
"if she walks at 3mi/hr find the minimum speed at which she must row so that the...
You sure that's the whole question?
compact pewter
charred ocean
With the same 3, 15?
charred ocean
But anything less than 3 would leave some room for optimization by walking a bit
My thought was the diagonal would cause the distance to be lower so it could go slightly slower, but that was wrong
compact pewter
no matter what I do I keep getting negative values so um
maybe the x value is different too?
charred ocean
maybe the x value is different too?
What is the x value?
compact pewter
6sqrt(5)/5
charred ocean
6sqrt(5)/5
No that’s what it used to be
glass meadow
I feel like the question is wrong but I'm too tired to double check
charred ocean
Now we’re changing the situation
compact pewter
this is what it tells me when I get that second part wrong
so I guess I need a new T function
charred ocean
Sure, so what’s a good T function here?
compact pewter
don't we do like T = sqrt(234)/x
or something
where x is the speed
but I'm missing part of it
but if no walking is happening isnt the other part just 0?
charred ocean
Yes
That’s how long it would take, but it doesn’t quite have the behavior we’re looking for
compact pewter
yeah especially since
the derivative would never be zero
so what am I missing?
is it
do I have to like add the x/3 for walking even though no walking is happening?
alright I'm lost
charred ocean
Well this is what it says when you put 3?
compact pewter
yes
I guess I can redo it again and see what response it wants from me and maybe that'll help it make more sense
is there anything to try before that?
glass meadow
Do (a) but with a rowing speed of 2.99mi/h
If you find that x is not 0, then the minimum speed for (b) must be more than 2.99mi/h
compact pewter
Do (a) but with a rowing speed of 2.99mi/h
for which variable?
oh nvm
ignore that
glass meadow
@charred ocean that does make sense, right? I'm very sleepy right now 
compact pewter
that gives me like
5.4517...
unless I'm doing things wrong
I started using a calculator because this evaluation is a lot
Idk what I'm doing lol
glass meadow
Idk either
compact pewter
fair enough
idk why I evaluated at 6sqrt5/5
im tired
as the 2 increases
the value where the derivative equals zero goes rapidly to the right
using 2.9 instead as the rowspeed
the value where the derivative is 0 is larger than 10
that value is the x value found earlier, 6sqrt(5)/5
little sundial
are you on part B
compact pewter
are you on part B
Yes, I was advised to evaluate part a with larger values in place of rowspeed
charred ocean
that value is the x value found earlier, 6sqrt(5)/5
Yes, but that’s with 2 for row speed
compact pewter
Yes, but that’s with 2 for row speed
Yes, I agree, when I make the rowspeed larger the value where the derivative is zero rapidly increases
charred ocean
Yes, I agree, when I make the rowspeed larger the value where the derivative is ...
As r->3 the optimal walking distance approaches -inf
compact pewter
uh
yes
I understand
We know that there's no walking happening
I wonder if I need to evaluate the original equation and get the time it takes and compare that using the sqrt(234)/x equation to get a value smaller than the original
charred ocean
But how do you know there’s no walking happening? Because the marginal benefit of walking, even a little bit, $\frac{\partial T}{\partial x}$ tends to negative infinity, when $\lim_{r\to 3}{T}$
jolly parrotBOT
Pixelius
compact pewter
well doesn't it say (with no walking)
charred ocean
Now I’m getting sleeply lol
compact pewter
right in the question
charred ocean
Yeah, but if you just do $T=\sqrt{234}/r$ you get $\frac{dT}{dr}=\frac{\sqrt{234}}{-r^2}$
jolly parrotBOT
Pixelius
compact pewter
yes and that is never zero because of course it isn't
charred ocean
Which never equals zero
compact pewter
mhm 
charred ocean
Btw what level calculus is this?
compact pewter
Btw what level calculus is this?
1...
charred ocean
Oh okay lol
compact pewter
mhm I made the bold choice to take calc 1 and 2 in the summer to catch up on college courses
so everything is very accelerated
timewise
Do you think i should just see what the program wants from me as an answer? since the first part is doable enough
charred ocean
Yeah, all of calc 1 and 2 in like 10 weeks is a lot
compact pewter
mhm
charred ocean
Do you think i should just see what the program wants from me as an answer? sinc...
I mean it has to be 3
compact pewter
I entered that
It didn't like that
I'll just put in 3 and show what happens
and now I try again with new values
charred ocean
what am I looking at here?
The value of r where the derivative of T wrt X is zero
compact pewter
okay but how am I meant to solve this with no calculator or anything- I can do part a with no calculator just fine
how do I arrive at a value like this?
and why did you guys think it had to be 3?
I'm so confused
oh btw all of this calc 1 is meant to be done without calculator (which is weird because some of the homework stuff says to round to the nearest 4 decimals or something similar)
charred ocean
how do I arrive at a value like this?
Okay I have it
You do part a, with a parameter r for the rowing speed
Notice the orange and blue are the same
compact pewter
okay that's something I can do
charred ocean
Then you solve for r where x=0
charred ocean
Okay I think the answer should be 39/sqrt(194)
compact pewter
that... worked...
what on earth am I even doing
ykw one sec
g(x) should be the derivative of f(x)
h(x) should be an equation that gives r based on various x values
charred ocean
Part a:
compact pewter
then you said where the derivative is zero
charred ocean
Part b:
compact pewter
so I did h(g(0))
compact pewter
Part a:
I did part a but not like that
g(x) is the equation I solved for x for part a
charred ocean
In part a set the derivative equal to zero and solve, given a value for s
In part b, take the derivative, then set x=0, then set the simplified derivative to zero, and solve for the rowing speed
charred ocean
<@207713902903951360> that does make sense, right? I'm very sleepy right now <:t...
It was not 3
compact pewter
why did you guys think it was 3?
charred ocean
Because if it was less than 3, there might be some benefit to walking just a little bit since it’s just a little faster, but my argument to begin with was that since going diagonal shortens the distance travelled you can justify a slightly slower speed, and still have a lower total time than any amount of walking would justify, but since the answer was very close to 3 I was convinced that it might actually be 3 by a convincing story and a numerical argument (when we approached 3 things got weird) but the answer is actually pretty simple now that we’ve seen it
charred ocean
.close
pearl pondBOT
jolly parrotBOT
dice
pearl pondBOT
@midnight haven Has your question been resolved?
stoic pecan
tried differntiating yet?
t2 is basically y
find gradient at 0
in the range of 80-100
oh damn
how tf do you know latex without calc
thats crazy
do you know how to find mininum/max point?
what have they tought you
ah ic
im not gonna lie i have no idea how to do this without calculus
no working but like
80 max value 100 min value
if they ask just say " I felt it"
like theres proof here that the bottom grows faster than the top
but thats not the level of math here
do you need working
uh oh
calculus
which im assuming ur not taking
sure
shouldnt it be the same though
oh
dude im too braindead for this rn
i should be able to do this
yeah that formula seems right
i mean you can
just not sure how to proof it
with actual working
midnight haven
hi
@midnight haven
you pinged us ?
im answering someone elses inquery
im afraid you have to contact another helper
next elk
Yes
dusky ocean
Can you explain how you came up with this? we don't need calculus here
dusky ocean
which course are you taking which has this question? 
Have you learned calculus? After reading more in-depth, this might be an optimization question
dusky ocean
So if this equation is true then t2 is a continuous decreasing function so if you plug 80 for t1, you will get smallest t2 and plug 100 will give you largest t2
You can round it
Round it to nearest integer is quite common if you ask me
Because we often model problems into continuous function
So if we need integer, we need to round it
round it to nearest integer
According to rounding rule
Because we need an integer
But the formula for t2 of yours is in real number
So we need to round it
pearl pondBOT
@midnight haven Has your question been resolved?
shell brook
pearl pondBOT
shell brook
what do yall think
vital estuary
what have you tried
vital estuary
yeah both work
im assuming they either want you to count the solutions or circle all possible answers
id circle 0 and 2 to be safe
then ask your teacher about it
shell brook
oh oka
vital estuary
in case youre curious, the solution goes
n×n = n+n
n² = 2n
n²-2n = 0
n(n-2) = 0
n = 0,2
i assume you did something similar
midnight haven
pearl pondBOT
frank goblet
f(n)=f(1)+...+f(1)=n*f(1)=5n
Using this we can rewrite the sum you have to calculate
midnight haven
Yeah
midnight haven
Wait lemme rethink
midnight haven
f(n)=f(1)+...+f(1)=n*f(1)=5n
F(m) =f(1)
?
How can you say that f(2) =f(1)
And each of them is f(1)?
I don't get this
@frank goblet
frank goblet
No
f(5)=f(1+4)=f(1)+f(4)
Then we do it again
And get
f(5)=5*f(1)
I never said f(m)=f(1)
pearl pondBOT
pearl pondBOT
Available help channel!
pearl pondBOT
humble wedge
What
dusky widget
<@&268886789983436800> very weird
unborn abyss
pearl pondBOT
pearl pondBOT
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winged perch
Suppose you have a deck with only three cards. Each card has two sides, and each side is either black or white. One card has two black sides. The second card has one black and one white side. The third card has two white sides. Now suppose all three cards are placed in a bag and shuffled. Someone reaches into the bag and pulls out a card and places it flat on a table. A black side is shown facing up, but you don’t know the color of the side facing down. Show that the probability that the other side is also black is 2/3. Use the counting method (Section 2 of the chapter) to approach this problem. This means counting up the ways that each card could produce the observed data (a black side facing up on the table).
I have:-
waxen condor
So this is conditional probability
waxen condor
See black side showing , so its either other side is black or white
winged perch
I agree.
waxen condor
its saying you to compute using bayes
winged perch
Nope:_
Use the counting method (Section 2 of the chapter) to approach this problem.
One side is black you know ,
waxen condor
its saying you to compute using bayes
If not then shouldnt it be 1/2
winged perch
B1 --> B1, B2
B2 --> B2, B1
B3 --> B3, W1
I guess now from 3 cards, we have 2 possibilities where the second card is black.
That's the answer 2/3.
waxen condor
um !
winged perch
Please correct me, if I am wrong.
waxen condor
no it seems correct
You re considering from a earlier stage
That the side faced can either one of the full black colored card
winged perch
Ohh
But how it will change our outcome:-
Show that the probability that the other side is also black is 2/3
See i have considered we have already picked out a card
And we know it has a black side
Now in our sample set we had three cards
Now we know its either one of the , two
So i considered the probability to be merely 1/2
But the question isnt asking that
waxen condor
we have total 6 posiblities
C1 - B1,B2
c2 - w1,B1
c3 - W1,w2 like this
Now we eliminated c3
and we already have a black face
So the other , can be black in 2/3 ways
winged perch
Yes 🙂
waxen condor
<I havent counted the faces fact in my consideration>
winged perch
No worries.
waxen condor
```
B1 --> B1, B2
B2 --> B2, B1
B3 --> B3, W1
```
So your way is totally correct
winged perch
Thank you for helping me out 🙂
waxen condor
Welcome !
winged perch
.close
pearl pondBOT
random cypress
just checking visually on desmos it seems to be x²=1-2y
you might have gotten that sign wrong
pearl pondBOT
@cinder ledge Has your question been resolved?
vague saddle
The answer should combine both because its center can lie in 1st or 2nd quadrant
cinder ledge
just checking visually on desmos it seems to be x²=1-2y
this doesnt make sense as y is always +ive
so you get x=sqrt(1-2x)
1-2x can be -ve
random cypress
no i said i just checked visually and the minus absolutely makes sense, it looks completely correct on desmos
cinder ledge
wait so wheres the mistake
cinder ledge
but its opening downwards right
random cypress
i misread
cinder ledge
so its going to 3rd and 4th
random cypress
my bad then
oh therefore yours is correct yeah
my bad i thought the circle was inside the unit circle
yours is correct for outside and mine is correct for inside
pearl pondBOT
pearl pondBOT
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obtuse oar
[cos(2001)pi] + [cot(2001)pi/2] + [sec(2001)pi/3] + [tan(2001)pi/4] + [cosec(2001)pi/6]
obtuse oar
i am unable to simplify this question
obtuse oar
i guess part of the argument
old marsh
$\cos (2001\pi) + \cot(2001 \cdot \frac {\pi}{2}) + \sec(2001 \cdot \frac {\pi}{3}) + \tan(2001 \cdot \frac {\pi}{4}) + \csc(2001 \cdot \frac {\pi}{6})$
It’s this right
obtuse oar
yeah
old marsh
Ok
muted shale
wdym?
obtuse oar
like simplify it
muted shale
lets go at it one term at a time
obtuse oar
alright
muted shale
what does cos(2001pi) simplify to?
jolly parrotBOT
obtuse oar
should i divide by 360 or 900?
obtuse oar
like in terms of 2pi?
muted shale
- why those numbers?
old marsh
Is this in degrees or radians
muted shale
remember that cos(x)=cos(x+2pi)
obtuse oar
radians
obtuse oar
remember that cos(x)=cos(x+2pi)
yeah
old marsh
So you wouldn’t use 360 or 900
muted shale
so use that
obtuse oar
So you wouldn’t use 360 or 900
thats something i use in mind
i then proceed with 2pi and 5pi
obtuse oar
if a number is greater than 1000 its better to cut it off by 900
obtuse oar
for simplification
obtuse oar
bcz its multiple of 90??
muted shale
why would oyu use a multiple of 90 and not just 360?
if you take a multiple of 90, then you have to keep track whehter its positive or negative
if you take a multiple of 360 you know that itll always stay the same sign
okay, anyway
use the fact that cos(x)=cos(x+2pi)
obtuse oar
visible confusion
muted shale
to simplify cos(2001 pi)
obtuse oar
use the fact that cos(x)=cos(x+2pi)
yeah next
muted shale
2001 = 2+0+0+1 = 3
what?
obtuse oar
i guess hes trying to do a multiple of 3, the rule of divisibility
muted shale
cos(2001pi)=cos(2001pi-2000pi)=cos(pi)=-1
here you go
this is what i was asking for
muted shale
so how would we do cot(2001 pi/2)?
muted shale
same cot pi/2
can you calculate this further?
obtuse oar
0?
obtuse oar
yup
obtuse oar
make it cos?
muted shale
sure
obtuse oar
667pi?
obtuse oar
then 720 - 53
muted shale
did you forget what we did like 5 minutes ago?
obtuse oar
ig
muted shale
the trigononometric functions are 2pi periodic
so you can always add or subtract 2pi
obtuse oar
yup
obtuse oar
so well subtract something from 4pi?
muted shale
where tf are you getting 4 pi from??
obtuse oar
integral multiple of 2pi????
muted shale
*integer
but eah sure
yeah
but how is 4 pi useful here?
i would think 666pi is much more useful
which is also a multiple of 2 pi
pearl pondBOT
jolly parrotBOT
brisk steeple
,w arctan 2 deg
jolly parrotBOT
brisk steeple
,w in degree arctan 2
jolly parrotBOT
Requested by anshulyolo121
brisk steeple
,w 1.1071487177940905*180/pi
jolly parrotBOT
brisk steeple
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Available help channel!
charred dome
howdo i do this?
pearl pondBOT
howdo i do this?
muted shale
How far do you have to simplify?
muted shale
You could consider it "solved" now
charred dome
i need to subtract them
cunning comet
just like solve i mean
take a calculator.
grizzled dust
just like solve i mean
solve for what? there's no unknown variable, i think you mean simplify?
charred dome
yes simplify sry
cunning comet
i need to do it by hand
what should the result be?
charred dome
2√3 / 9
grizzled dust
do you know how to write roots as fractional exponents?
charred dome
no thats what im trying to get help for lol
jolly parrotBOT
∫oosh (lemonsaurus appreciator)
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reaction for more information.
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grizzled dust
so just use that, and also write 9 in the denominator as 3^2, play around with the exponents according to common rules, get a common denominator and should be straightforward
charred dome
oo yeah
thanks
well i am actually doing this but i have no idea how they got the answer, i highlighted the part we're on
grizzled dust
@cunning comet what makes you so convinced it would be a negative number?
charred dome
<@836252175675818054> what makes you so convinced it would be a negative number?
Does this look right so far?
grizzled dust
ya
charred dome
i have no idea how to get common denominator now
grizzled dust
multiply by 3^(4/3) / 3^(4/3) (because then you add the exponents on the bottom to get 3^(2/3 + 4/3) = 3^2
just algebra
charred dome
wait wait
where did u get 3^(4/3)
i know its just algebra but i never did anything like this "complicated" before
grizzled dust
just generalize what you do usually...how do you get a common denominator in general? if you are given 1 / 2 and 7 / 20 what do you do? multiply the first fraction by 20 / 2 (youre essentially dividing the 2 denominators to decide what to multiply by) so in this case if you divide 3^2 by 3^(2/3) what do you get? 3^(4/3)
charred dome
i never thought of it this way before.. i always think that you multiply it by the LCD
grizzled dust
LCD? the LCD of 2 and 20 would be 2, how would it help to multiply by 2?
charred dome
i mean GCF
sry
isnt that right?
i havent done algebra in a while tbh
oh the LCM
grizzled dust
anyway it's no different than "simpler" fractions
but maybe you are out of practice with fractional exponents
charred dome
well how do i know which denominator is greater in this case?
yes i very much am
oh well i assume 2/1 is greater than 2/3
grizzled dust
you can always just multiply each fraction by the denominator of the other 🤷♂️
charred dome
so then i need to make 2/1 (3)
grizzled dust
then simplify later after the addition\ subtraction or whatever you need to do
charred dome
whicg is 6/3
grizzled dust
if thats easier to think about
charred dome
yes maybe it is
so is that correct then?
i just make the 3^ (2/1)(3)?
do i need to do anything to the numerator ? as well?
grizzled dust
im not entirely sure what you are asking
charred dome
For the denominator on the right side
I am trying to get that denominator to match the left side
grizzled dust
in this?
charred dome
So I need the exponents to have the same denominator correct?
grizzled dust
right it's 2^(6/3) so if you want to bring that to denominator to 2^(2/3) is what youre asking?
charred dome
Bring it? Wat do you mean by that… but no I was asking if I had to change the numerator as well?
grizzled dust
you're losing me
: (
if you want to change the denominator, yeah you would have to multiply top and bottom by something right? so yes you would have to change the numerator if that's what you're asking?
charred dome
Yeah that’s what I meant
grizzled dust
so if the goal is to get a denom of 2^(2/3) on the right-hand term then what would you need to multiply top and bottom of that fraction by?
charred dome
By 3/3?
grizzled dust
...?
grizzled dust
you essentially need to multiply 2^(6/3) by something and get 2(2^3), this is like the equation:
x (2^(6/3)) = 2^(2/3)
x = 2^(2/3) / 2^(6/3)
do you see how this is just dividing the denominators
charred dome
I’m not actually sure tbh
Yeah
Let me try it
Do I need to make it 2^(6/3) or can we just directly change 3^2 to 3^2/3
charred dome
you essentially need to multiply 2^(6/3) by something and get 2(2^3), this is li...
would it just be easier to bring the exponents up to the numerator to simplify that first?
grizzled dust
i don't really have anything to add to this, already dissected it quite a bit, if you are still confused i suggest you watch some khan academy videos on how to get a common denominator or such
i don't think im very good at teaching algebra skills
charred dome
i don't think im very good at teaching algebra skills
i solved much easier that way just to let you know
if you want to know what i did
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gilded talon
context: trying to prove there's a (i dont think necessarily unique yet) function phi(x) that satisfies the differential equation y'=f(x,y) given that f(x,y) and f_y(x,y) are continous around 0
gilded talon
even after assuming these 3, how does phi_n even build itself to phi
like how it's recursively defined doesn't clearly make sense
i think an example of the recursive definition would make it make more sense
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@gilded talon Has your question been resolved?
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viscid pivot
if you have a vector v=αu (u is also a vector) and u=/=0
pearl pondBOT
if you have a vector v=αu (u is also a vector) and u=/=0
viscid pivot
how come u is linearly independent?
tropic saddle
the set {u} is linearly independent, the set {u,v} is linearly dependent
viscid pivot
right, but can't {u} also be linearly dependent?
tropic saddle
its not the zero vector
viscid pivot
how come it can only be linearly independent?
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warm charm
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warm charm
i used this rule
and got (t+9)(6) - (6t+20)*1
all over (t+9)^2
im dumb holdon
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warm charm
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warm charm
im just guessing values, how do i know for sure?
warm charm
yeah
but it could be ANYTHING
quite literally
answer a could've been -1.0000001
not anything
woeful harness
oh i see what u mean
warm charm
but theres infinite answers that are graphically accurate
woeful harness
try using decimals to find the slope? looks like u just used integers
warm charm
so i have to guess?
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warm charm
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west sapphire
what's goes wrong at x=-3 and x=5?
quiet tendon
also, are you sure you meant -3 and 5? those values aren't even displayed on this graph
west sapphire
maybe because they're the endpoints of the displayed graph?
quiet tendon
oh wait is that just where the graph gets cut off 😭
it's assumed that the function continues normally after that, it's not like the function just stops
this is just a zoomed in version of the graph
west sapphire
x=1 is certainly correct since the function is not even continuous there
but there are other points, look for "corners"
warm charm
my anser was 1
and it was wrong
and the clue said it's "not differentiable at corners"
which i assumed meant endpoitns
so i tried to add -3 and 5 on it
and still wrong
quiet tendon
corners are where the function doesn't really smoothly transition
1 is definitely correct
because discontinuous
warm charm
ohhhhh
warm charm
ok i see
west sapphire
"corners" are where there are different slopes to the left and to the right
warm charm
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west sapphire
as it says, it's the greatest integer function
quiet tendon
it's also sometimes called the floor function
look up a graph
and you'll easily see where it's not differentiable
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quiet tendon
it can either be 1 or -1
warm charm
im tweaking out
fluid axle
well shit
quiet tendon
does it not accept either 1, -1, or +/- 1?
warm charm
💀 what the fuck
quiet tendon
bruh
fluid axle
that was my second guess lol
warm charm
the "watch it" video had the exact same problem as the question
i still dont get how thats an acceptable answer
fluid axle
and that's just for one class holy shit
warm charm
one term yea
fluid axle
for 110 I'd expect real assignments and real ppl actually correcting my stuff
not this abomination
rip
warm charm
it's technically the homework + the textbook (ebook)
but i dont use the textbook at all
fluid axle
ah
fluid axle
what do they say in more info
warm charm
im so done with webassign
fluid axle
it's so cursed
fluid axle
ah this works too
warm charm
thanks for your help
warm charm
it's a lot more manageable than the college algebra class i took last summer
it's less homework
thanks again
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north oriole
Sorry but I need help with trig again, anybody 😭
So I tried to do
arcsin (- 1/2 )
I know that sin (π/6) = 1/2.
Then I tried to find the reference angle to π/6. Is my final answer 5π/6 ?
quiet tendon
arcsin has a range of -pi/2 to pi/2
you're right in using the reference angle of pi/6
north oriole
Oh
quiet tendon
sin(5pi/6) is not only not in the right range, sin(5pi/6) is also 1/2
you need an angle below the x axis in the unit circle, and specifically between -pi/2 and pi/2
north oriole
On the unit circle, how do I figure out where the domain -π/2 and π/2 is?
quiet tendon
well the right half of the x-axis corresponds to the angle 0
2pi is all the way around, so pi is halfway around and pi/2 is a quarter of the way around
north oriole
Ok
quiet tendon
so -pi/2 is a quarter of the way around moving down from the angle 0
and pi/2 is a quarter of the way around moving up from the angle 0
hence the range -pi/2 to pi/2 is the right half of the unit circle
north oriole
So only quadrants 1 and 4 are possible answers for arcsin?
quiet tendon
yes
north oriole
So my final answer would be π/6 ?
quiet tendon
sin(pi/6) != 1/2
if you know that your sine value is negative and your options are either an angle in the first quadrant or the fourth quadrant, what can you conclude about the quadrant of your angle?
north oriole
Its probably in the 4th quadrant because ( x , -y ), right?
quiet tendon
yes
now you might be used to representing fourth quadrant angles in the range of 3pi/2 to 2pi
quiet tendon
11pi/6 is the right angle, it satisfies sin(11pi/6) = -1/2
but
range of arcsin is -pi/2 to pi/2
is there a way to equivalently represent the angle 11pi/6 to be in this range?
north oriole
I thought 11π/6 is in the range of that. It's in the 4th quadrant right?
quiet tendon
technically -pi/2 to pi/2 covers angles in quadrants 1 and 4, yes
but 0 degrees is the same as 360 degrees is the same as 720 degrees and so on
north oriole
Oh
quiet tendon
the actual representation of your angle needs to be between -pi/2 and pi/2
angles are equivalent up to an integer multiple of 2pi (rotation by 360 degrees)
north oriole
Ok
quiet tendon
so if you want to find an equivalent angle to 11pi/6 in the range of [-pi/2, pi/2], you can just add or subtract 2pi until you get an angle in that range
north oriole
So if I did
11π/6 + 12π/6 (One period)
Then 23π/6 is equivalent?
north oriole
Oh I need to subtract
If I subtracted it would be
-π/6
I didn't think we can have negatives
quiet tendon
your angle 11pi/6 corresponds to rotating counterclockwise by 11pi/6 radians. to get an output of arcsin, you need to be able to phrase the angle either in terms of a counterclockwise rotation of up to pi/2 radians, or a clockwise rotation of pi/2 radians
and rotating 11pi/6 counterclockwise is the same as rotating by pi/6 clockwise, or -pi/6 counterclockwise
north oriole
Oh my goodness that makes so much more sense
So in that case, arccos (-1/2) would be π/3 ?
north oriole
Oh
quiet tendon
so your options are either a quadrant 1 or quadrant 2 angle satisfying cos = -1/2
you'd expect that to be a quadrant 2 angle
north oriole
Because its a -1/2, right?
quiet tendon
and you can do the same stuff with the reference angles to figure out what the angle is, and if that angle isn't in the range [0, pi], add or subtract 2pi until it is in that range
quiet tendon
yes
north oriole
Ok ok ok
I'm sorry, I have a quiz tomorrow and I've been struggling
Can I try one more with you?
quiet tendon
sure
north oriole
Ok
quiet tendon
no need to apologize btw, we're here to help
north oriole
Thank uu
arcsin (-√2 / 2 )
So because its sin, the range is
[ -π / 2 , π / 2 ]
?
quiet tendon
yes
north oriole
Ok then it must be in quadrant 4
quiet tendon
yes
north oriole
Is it 7π / 4 ?
quiet tendon
right angle, but check the range
quiet tendon
yes
quiet tendon
is that a negative?
quiet tendon
then yeah, arcsin(-sqrt2 / 2) = -pi/4
north oriole
YAY oh my goodness I'm so thankful you helped me
I seriously thought I was goingto fail my quiz tomorrow
quiet tendon
no problem, be sure to memorize your ranges of inverse trig functions
quiet tendon
happy to help, good luck!
north oriole
!done
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north oriole
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