/2
#help-13
bright karma
bright karma
just work it out and you'll be golden 🙂
does this look about right
ima just roll w it
thx
.close
frail swift
would the integral of 3/(50+2t) dt be 1.5ln|50+2t| + c or 1.5ln|25+t| + c?
frail swift
idk how to use latex lmao
work for the first one:
integral ( 3/(50+2t) dt ) = 3 * integral ( 1/(50+2t) dt) = 3 * integral (1/u * du/2) = 3/2 * integral(du/u) = 3/2 * ln|u| = 3/2 * ln|50+2t|
work for the second one:
integral ( 3/(50+2t) dt ) = 3/2 * integral ( 1/(25+t) dt) = 3/2 * integral (du/u) = 3/2 * ln|u| = 3/2 * ln|25+t|
slate lintel
well like the point is that both of those answers are correct
frail swift
but when i plug them into my first order linear differential they give me different answers 😭
slate lintel
,w graph {1.5 ln|50+2t|, 1.5ln|25+t|}
frail swift
Problem: A 100-gallon tank contains 50 gallons of a solution composed of 90% water and 10% alcohol. A second solution containing 50% water and 50% alcohol is added to the tank at the rate of 5 gallons per minute. As the second solution is being added, the tank is being drained at a rate of 3 gallons per minute. What is the concentration of alcohol when the tank is full?
dQ/dt + r2 * Q/(v0 + (r1 - r2)t) = r1q1
dQ/dt + 3Q/(50 + (5-3)t) = 5 * 0.5
P(t) = 3/(50+2t), Q(t) = 2.5
u = e^int(3/(50 + 2t) dt) = e^1.5ln|25+t| = (25+t)^1.5
Q * (25+t)^1.5 = int(2.5 * (25+t)^1.5) + c
Q * (25+t)^1.5 = 2.5 * int((25+t)^1.5) + c
Q * (25+t)^1.5 = (25+t)^2.5 + c
plug in Q = 5, t = 0
5 * 25^1.5 = 25^2.5 + c
(5 * 25^1.5) - (25^2.5) = -2500
tank full when 100 = 50 + 2t, t = 25 mins
Q = ((25+t)^2.5 - 2500)/(25+t)^1.5
Q = ((50)^2.5 - 2500)/(50)^1.5
Q = 42.9289 gallons```
```v0 = 50, r1 = 5, r2 = 3, q1 = 0.5, Q0 = 50 * 0.1 = 5
dQ/dt + r2 * Q/(v0 + (r1 - r2)t) = r1q1
dQ/dt + 3Q/(50 + (5-3)t) = 5 * 0.5
P(t) = 3/(50+2t), Q(t) = 2.5
u = e^int(3/(50 + 2t) dt) = e^1.5ln|50+2t| = (50+2t)^1.5
Q * (50+2t)^1.5 = int(2.5 * (50+2t)^1.5) + c
Q * (50+2t)^1.5 = 2.5 * int((50+2t)^1.5) + c
Q * (50+2t)^1.5 = (50+2t)^2.5 + c
plug in Q = 5, t = 0
5 * 50^1.5 = 50^2.5 + c
(5 * 50^1.5) - (50^2.5) = -15909
tank full when 100 = 50 + 2t, t = 25 mins
Q = ((50+2t)^2.5 - 15909)/(50+2t)^1.5
Q = ((100)^2.5 - 15909)/(100)^1.5
Q = 84.091 gallons```
wraith daggerBOT
slate lintel
they are more similar than you think...
,w graph {1.5 ln|50+2t|, 1.5ln|25+t|} for -25 < t < -20
slate lintel
yes :)
slate lintel
frail swift
and took the 1.5ln|2| out and subtracted it from 1.5ln|50+2t| or something
that's so wacky bruh
slate lintel
they differ by a constant :)
frail swift
WOAH
slate lintel
this shows up a lot in logarithms
also trig sometimes
as for your problem though
there's probably some initial condition
strong remnant
.
strong remnant
WHAT ARE THE CHANCES
frail swift
that's crazy
strong remnant
I click on this server like one every month too
thats so wild
that of all the times its now
frail swift
bro the two logs differ by a constant
frail swift
yeah but i feel like there's like no way of knowing that one is more correct in the context of the question
until you solve it and get like a completely wrong answer
like when i was using ln|50 + 2t| i ended up getting like 84 gallons of alcohol
strong remnant
its pretty sus that one just gives u the wrong answer
i dont think he will use some goofy ah thing like this
this is why you don't make up your own questions lmao
frail swift
like when i was using ln|50 + 2t| i ended up getting like 84 gallons of alcohol
and this shouldn't be possible bc there should only be 50 gallons of alcohol max if the 50% water 50% alcohol fills up
strong remnant
yep
slate lintel
both of them will give you the right answer after you sub in the initial conditions
frail swift
whaaa
strong remnant
its pretty sus that one just gives u the wrong answer
ik it was sus
frail swift
ok yeah hold up
i did get a different c
with the ln|50 + 2t|
but i got a different answer
slate lintel
it was probably different by checks notes
frail swift
ig i just needa try that again
slate lintel
this much
frail swift
i definitely messed up somewhere then
bc my c ended up being like -2500 for the ln|25 + t|
and like -17000 i dont even remember for the ln|50 + 2t|
hm
ig ill try it again
yeah
somethings happening with the arithmetic
still not getting the same answer
with 25+t
dQ/dt + r2 * Q/(v0 + (r1 - r2)t) = r1q1
dQ/dt + 3Q/(50 + (5-3)t) = 5 * 0.5
P(t) = 3/(50+2t), Q(t) = 2.5
u = e^int(3/(50 + 2t) dt) = e^1.5ln|25+t| = (25+t)^1.5
Q * (25+t)^1.5 = int(2.5 * (25+t)^1.5) + c
Q * (25+t)^1.5 = 2.5 * int((25+t)^1.5) + c
Q * (25+t)^1.5 = (25+t)^2.5 + c
plug in Q = 5, t = 0
5 * 25^1.5 = 25^2.5 + c
(5 * 25^1.5) - (25^2.5) = -2500
tank full when 100 = 50 + 2t, t = 25 mins
Q = ((25+t)^2.5 - 2500)/(25+t)^1.5
Q = ((50)^2.5 - 2500)/(50)^1.5
Q = 42.9289 gallons```
with 50+2t
```v0 = 50, r1 = 5, r2 = 3, q1 = 0.5, Q0 = 50 * 0.1 = 5
dQ/dt + r2 * Q/(v0 + (r1 - r2)t) = r1q1
dQ/dt + 3Q/(50 + (5-3)t) = 5 * 0.5
P(t) = 3/(50+2t), Q(t) = 2.5
u = e^int(3/(50 + 2t) dt) = e^1.5ln|50+2t| = (50+2t)^1.5
Q * (50+2t)^1.5 = int(2.5 * (50+2t)^1.5) + c
Q * (50+2t)^1.5 = 2.5 * int((50+2t)^1.5) + c
Q * (50+2t)^1.5 = 2.82843 * (25+t)^2.5 + c
plug in Q = 5, t = 0
5 * 50^1.5 = 50^2.5 + c
(5 * 50^1.5) - 2.82843(25)^2.5 = -7071.07
tank full when 100 = 50 + 2t, t = 25 mins
Q = (2.82843(25+t)^2.5 - 7071.07)/(50+2t)^1.5
Q = (2.82843(50)^2.5 - 7071.07)/(100)^1.5
Q = 42.9290 gallons```ohh i know what i did wrong
i shoulda added a dt in the int((50+2t)^1.5) because then i would've remembered that the integral isn't gonna end up being just (50+2t)^2.5
it's gonna be half that
if i do u sub and stuff it would be like int((u^1.5) * du/2))
lemme try with this newfound knowledge
ok that made it worse
i really shoulda done this on paper
i'm double stupid
wait no i'm not
forget it
i know i did the integral wrong somehow
in this integral of 2.5 * (50+2t)^1.5 dt step
taking the 2 out of the 2t beforehand makes it a lot easier to actually solve the problem
the integral isnt as weird
at least i learned this cool thing abt integrals
thanks
.close
celest otter
split flame
cedar kilnBOT
tidal breach
Where are you at with this?
split flame
I'm looking at the answer and I understand every step except for multiplying each term by 10x + 5
tidal breach
Oh no worries there. Do you mean once each side is under a common factor?
split flame
No I mean how does this work?
exotic furnace
Technically this is illegal lol
tidal breach
If you multiply both sides of an equation by the same amount you get an equivalent equation
exotic furnace
But ig it’s done to get rid of the fractions
tidal breach
Like 5x=10 is the same as 25x=50
split flame
Yeah
exotic furnace
You can do this provided you make sure that when you do solve for x, it won’t make 10x+5 zero
split flame
But ig it’s done to get rid of the fractions
I think I understand this part
But I have a test on this stuff tomorrow and I wont think to do that
cedar kilnBOT
Available help channel!
uneven siren
cedar kilnBOT
uneven siren
what to take as the repeted value as theres no more 1+ in this one
versed kayak
does 2....infinity mean 22222... ?
uneven siren
does 2....infinity mean 22222... ?
it means that the pattern is infinitely repeating
cedar kilnBOT
@uneven siren Has your question been resolved?
surreal star
need help with this
cedar kilnBOT
need help with this
cerulean star
need help with this
take a clearer picture.
cedar kilnBOT
@surreal star Has your question been resolved?
cedar kilnBOT
cedar kilnBOT
Available help channel!
craggy stratus
If I want to find the derivative of sin(x)e^sin(x), what would that be?
craggy stratus
I assume you want to do f'g+fg'
so the derivative of sin(x) is cos(x) and e^sin(x) remains the same, but I'm not entirely sure what happens after that..
cos(x)e^sin(x)+sin(x) ....?
because i know one rule is that e^x remains the same when derived, but then the other day when i derived e^5x it became 5e^5x..
and would then the sin(x) that i possibly take down become cos(x)?
and if so would the final derivative be cos(x)e^sin(x) + sin(x)cos(x)e^sin(x)
dire thorn
and would then the sin(x) that i possibly take down become cos(x)?
Yeah
craggy stratus
so why is the rule that e^x stays as e^x
wouldn't the rule be e^x becomes xe^x?
and yeah sorry for really stupid questions but I've not studied math for so long so i literally have forgotten every explanation i got for this 5 years ago
ah nevermind
dire thorn
Chain rule: if h(x)=f(g(x)) then h’(x) = f’(g(x))*g’(x)
cedar kilnBOT
cedar kilnBOT
Available help channel!
bronze pivot
Why is $P(B) = P(B \cap A_1) + P(B \cap A_2) + P(B \cap A_3)$?
wraith daggerBOT
LeGM
tropic oxide
presumably because the three A_i are disjoint and their union is either the entire sample space or a superset of B at least.
... looks like it's the former.
B is the disjoint union of B&A_1, B&A_2 and B&A_3
cedar kilnBOT
@bronze pivot Has your question been resolved?
rustic steeple
for x where a and b are positive numbers different from 1
cedar kilnBOT
for x where a and b are positive numbers different from 1
foggy merlin
how would you find x
use what i wrote
rustic steeple
erm im still not rly sure tho sorry 😭 only y9 💀
cedar kilnBOT
@rustic steeple Has your question been resolved?
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@rustic steeple Has your question been resolved?
pale furnace
hi
cold briar
hello
frank quiver
hello
pale furnace
how do i find log2(6x-3)=5
cold briar
log base 2?
pale furnace
yrs
finite lion
if log2(6x-3)=5, what is 6x-3 equal to?
pale furnace
i forgot
frank quiver
Remember, if we have something like 2^x = 3, then what do we do?
pale furnace
hmm
sullen spire
$\log_x(a) = b$
\implies
$a=x^b$
frank quiver
the base of the exponent becomes the base of the log
pale furnace
yea
frank quiver
yeaah you got it
wraith daggerBOT
shavet
$\log_x(a) = b$
\implies
$a=x^b$
```Compilation error:```! Missing $ inserted.
<inserted text>
$
l.58 \implies
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.```
pale furnace
what happened to parenthesis
frank quiver
what happened to parenthesis
its just there to make reading the question easier
pale furnace
its just there to make reading the question easier
oh
sullen spire
this is a pretty yucky question ahaha
pale furnace
i have another concern
so
log2(729)
i need to slove it
i wrote 1
i think it i the wrong answer
frank quiver
just put it into a calculator, if you're allowed to
pale furnace
i need to show workings
pale furnace
how many points is this?
easy question
first question
i feel so dumb for asking this
there was a question
pale furnace
oh
pale furnace
20x^2y^5-18xy^2
pale furnace
You sure it's log2(729)?
or log 3
frank quiver
do you mean ${20x^2 . y^5 - 18xy^2$ ?
pale furnace
if it was log 3 wouldnt the answer=1 due to ame base and argument giving the same answer
wraith daggerBOT
SHREK_IS_SENPAI
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frank quiver
**SHREK\_IS\_SENPAI**
Compile Error! Click the <:errors:504266556054962196> reac...
oo okok
pale furnace
yes
frank quiver
so $0 = 20x^2y^5 - 18xy^2$ ?
wraith daggerBOT
SHREK_IS_SENPAI
pale furnace
tes
frank quiver
if it was log 3 wouldnt the answer=1 due to ame base and argument giving the sam...
nono, remember $log_a(a) = 1$
wraith daggerBOT
SHREK_IS_SENPAI
pale furnace
yea
frank quiver
yeah and this question is $log_a(a^6)$
wraith daggerBOT
SHREK_IS_SENPAI
pale furnace
yes
frank quiver
yeah yeah so that would give you what?
frank quiver
**shavet**
```latex
$\log_x(a) = b$
\implies
$a=x^b$
```Compilation error:```! M...
you can use this if you are struggling
frank quiver
20x^2y^5-18xy^2
for this one, what is the question?
frank quiver
okok are there any factors that you can see?
frank quiver
no unfortunately not
pale furnace
it is times right
frank quiver
exponents only decrease when dividing
frank quiver
it is times right
yip its y^2 . y
pale furnace
then what do i do
frank quiver
then what do i do
see if there are any factors you could take out
pale furnace
2?
pale furnace
yes
pale furnace
so 10x- 9y^3
crimson sedge
cedar kilnBOT
crimson sedge
Can someone tell me why each of the options are true/false?
severe nimbus
We don't do that here
You need to explain what you've tried yourself, why you have come to the conclusions you have
We can help you find the answer yourself but won't give it to you
!status
cedar kilnBOT
What step are you on?
1. I don't know where to begin
2. I have begun but got stuck midway
3. I got an answer but I'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
I guess We can't find the anwser to A with the information given
Idk about B, C and D
severe nimbus
Yeah the information given does not implicate A as being true. It could be true if all customers are size ten or are evenly distributed but we don't have enough info to say that for sure. So it doesn't qualify as "must be true"
Do you know the difference between average and median?
crimson sedge
Ohh it's D
Right?
The ones after the median are 10 or larger
Since 10 is in the middle
severe nimbus
Yes
crimson sedge
I completely forgot that the numbers are ordered from small to large
Also why are C or B false?
severe nimbus
The easiest way to disprove that they "must be true" is to find a case where they aren't.
For example, if the smallest shoe is size 9 and the biggest is size 12, the median can still be 10, just depending on the number of each size, but the average would be 9+12/12 = 10.5
The same case disproves C
Generally it is much easier to prove that something isn't true than that it is, since proving it isn't true only requires finding one case, vs proving something is true requires covering all bases
crimson sedge
Thank you so so much
severe nimbus
No problem, happy to help
ruby mauve
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digital cliff
coose
cedar kilnBOT
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deft kiln
Use the formula for the future value of an ordinary annuity to calculate A with the monthly payment R=$250, the annual interest rate r= 7.5%, and the number of monthly payments n = 18. A$
enter your response here
(Round to the nearest cent as needed.)
wraith daggerBOT
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deft kiln
reopen
plucky owl
reopen
You have #help-2 so you don't need this channel, please close this one using .close
cedar kilnBOT
deft kiln
I checked the view example and I still dont know how they got that formula
cedar kilnBOT
Available help channel!
silent kelp
What are the y coordinates of the points where y^3 + x^3 +30y^2 =4x+3 has vertical tangent lines. calc
glossy inlet
You need to find points such that the derivative is undefined
glossy inlet
chain rule
glossy inlet
$\frac{d}{dx}y^n = ny^{n - 1}\frac{dy}{dx}$
wraith daggerBOT
jewels!
silent kelp
so normal chain rule but add dy/dx at the end
silent kelp
ok
glossy inlet
solve for dy/dx
silent kelp
okay
silent kelp
solve for dy/dx
(4-3x^2)/(3y^2+60y) = dy/dx
glossy inlet
Yup looks good
weary current
is there an app like geogebra but for geometry and trigonometry?
fallen moat
GeoGebra Geometry 
crimson sedge
cedar kilnBOT
crimson sedge
It's in french but it's basically saying to find the area total needed to cover the entire ramp with wood.
I don't know if I'm going the right way but so far I've only thought of it as a quarter of a cylinder.
cedar kilnBOT
@crimson sedge Has your question been resolved?
crimson sedge
<@&286206848099549185>
fallen moat
i would really wish to help but i don't understand the question, e.g. I can't see if it's a 2-D or 3-D diagram
crimson sedge
Its a 3d
crimson sedge
i would really wish to help but i don't understand the question, e.g. I can't se...
The Question is just : Find the area total of the ramp to be covered with wood.
The translation at least.
fallen moat
does it mean that the ramp is 1.2m tall, with the ramp is of the shape of a quarter of a circle?
crimson sedge
Nope the ramp is 1.2 meters in width
And its 2.4 meters long
The bottom at least
Is indicated to be 2.4 meters long
While the height is still unknown.
fallen moat
hmmm
crimson sedge
But if I were to think that its a quarter of a cylinder.. then it might be the same height as the length
crimson sedge
Yep basically like that.
The only thing I need to find is the total area of the ramp
So im guessing i need to do area of the square - quarter of an area of the cylinder.. right?
crimson sedge
Yeah.. I'll try this out and show to my teacher. Thank you very much.
crimson sedge
yo
cosmic steppe
crimson sedge
nope
paper sun
crimson sedge
i ain need help with that but in like 10 min i do
cedar kilnBOT
@crimson sedge Has your question been resolved?
shell echo
is multiplying first row with (-3) the best way to do it?
cedar kilnBOT
is multiplying first row with (-3) the best way to do it?
shell echo
or is that even legal / allowed
tawdry salmon
or is that even legal / allowed
simultaneous equations?
shell echo
sorry but i'm not sure what that means, english isn't my first language
i have to find the value of x and y
tawdry salmon
solving for x and y?
tawdry salmon
equate y
tawdry salmon
you see there is a y without a coefficient
shell echo
so whenever its just a single letter i should try to equate it?
tawdry salmon
kinda yeah
shell echo
thanks 👍
tawdry salmon
np
shell echo
.close
hoary vessel
Does anyone know what the name for this is:
It's a point inside a geometrical shape, where if a ray is casted from there to any other point on the edge of the shape, it would not intersect with any wall.
or in other words, the point where all the edges of the shape are visible by just turning around
cedar kilnBOT
Does anyone know what the name for this is:
It's a point inside a geometrical sh...
@hoary vessel Has your question been resolved?
fair geyser
there would be multiple points, likely each point wouldn't have to be called anything, just part of [this set name]
cedar kilnBOT
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@hoary vessel Has your question been resolved?
marsh nymph
cedar kilnBOT
marsh nymph
bit unsure how this is supposed to work
dawn junco
what are you unsure of here ? @marsh nymph
marsh nymph
what are you unsure of here ? <@461316203437096970>
like it becomes a first order problem right
but how do u actually solve it?
dawn junco
if you set w = y'
the equation becomes w' + 3/x w = 12x^2
that's how it becomes first-order
marsh nymph
isnt that what w(x)=y'(x) the same thing as that?
marsh nymph
so w(x)=y'(x) isnt the same as w=y'
dawn junco
it's the same
marsh nymph
gotcha so just take integrating factor and solve
cedar kilnBOT
@marsh nymph Has your question been resolved?
marsh nymph
how do i do the y part afterwards? i ended up getting to the point where $w=2x^3+c$ where $w=y'$ and $y'(1)=3$ and $3=2+c \implies c=1$
wraith daggerBOT
Gremlin
marsh nymph
im just stuck at $y'(x)=2x^3+1$
wraith daggerBOT
Gremlin
marsh nymph
whoops meant to update this yea so after i did that how do i express the solution?
ive found both C's
dawn junco
well if you have both c's you've won I guess
marsh nymph
but how do i write out the answer like actually lmao
dawn junco
what have you got then?
marsh nymph
$y(x)=1/2 * x^4 +x+7/2$ and y'(x)=2x^3+1
wraith daggerBOT
Gremlin
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marsh nymph
thank u so much
dawn junco
,w solve y'' + 3/x y = 12x^2, y(1) = 5, y'(1)=3
wraith daggerBOT
dawn junco
wtf is going on with wolfram today
marsh nymph
💀
wraith daggerBOT
marsh nymph
shit
dawn junco
recheck your integrating factor stuff I guess
marsh nymph
$e^{3/x}=x^3$ $\integral w'x^3+3x^2w=\integral 12x^5$ $wx^3=2x^6$ aint that right?
wraith daggerBOT
Gremlin
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marsh nymph
jesus that came out sloppy does that make sense tho
marsh nymph
$\int w'x^3+3x^2w=\int 12x^5$
dawn junco
$\int$
marsh nymph
$wx^3=2x^6$
wraith daggerBOT
Gremlin
dawn junco
so what's your integrating factor then ?
wraith daggerBOT
Gremlin
wraith daggerBOT
Gremlin
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marsh nymph
yes lol
dawn junco
$wx^3=2x^6$
you forgot the constant of integration here tho
wx^3 = 2x^6 + c
w = 2x^3 + c/x^3
ok that's where that 1/x^2 comes from in the wolfram answer
marsh nymph
wx^3 = 2x^6 + c
ahjhhh
ngl i started a league game right after i thought i finished lmao
marsh nymph
nope that all makes sense and i got the same as wolfram i appreciate it
cedar kilnBOT
cedar kilnBOT
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indigo marsh
For part e im assigning values for the non basic columns but my solution for x1 in terms of lambda and mu is wrong
cedar kilnBOT
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vernal shell
Do i need to change A or something
There are 4 unknowns and the rank is 2, so you need 4-2=2 free variables for the answer
k4 has definite value
cedar kilnBOT
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inland dragon
Is this good work to show convergence?
cedar kilnBOT
Is this good work to show convergence?
inland dragon
I had to rework a problem from an assignment
So I need to show the professor tomorrow this
I don't know why even do the limit test here really..
cedar kilnBOT
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@inland dragon Has your question been resolved?
ebon juniper
the result makes sense
inland dragon
.close
prime root
Hello
dire thorn
Substitute s(x) and t(x) using the given equations
prime root
Substitute s(x) and t(x) using the given equations
So it would be s(-10)=t(15), s(-10)>t(15), and s(-10)=<t(15)?
I found the x's btw
dire thorn
No s(x)=1/5 x^2-2x+10
Just replace s(x) with the right hand side if that
You do not need anything from part a
prime root
My teacher told me it was the X's from the domain tho
and it helps u get the answer for those
dire thorn
Oh
prime root
i was just double checking if i did it right'
dire thorn
So it would be s(-10)=t(15), s(-10)>t(15), and s(-10)=<t(15)?
Well I guess first one is correct
But the other ones cant be true
prime root
Hmm
dire thorn
Cuz you are saying 0>0 and 0<0 which isnt true
Or not 0 but whatever the y value of the intersection is
dire thorn
So how would i solve the other 2?
Im not sure seems like your teacher wants it a certain way im not familiar with
buoyant latch
Does Rolle's theorem require C1 continuity?
cedar kilnBOT
Does Rolle's theorem require C1 continuity?
buoyant latch
on wikipedia it says that we need a differentiable function
im trying to think if i could make a function that is differentiable on an interval, but since its not continuously differentiable, have its derivative jump from negative to positive
so in a sense it can bypass the 0 point
royal loom
@buoyant latch
buoyant latch
hi
royal loom
how u doin
royal loom
Does Rolle's theorem require C1 continuity?
I feel like this has to be true otherwise we can like jump over what is supposed to be garuanteed
buoyant latch
exactly, that's what i feel too
but then requiring differentiability is not a strong enough condition
since differentiable =/=> C1 continuity
royal loom
just go through the proof
and see if it requires continuously differentiable anywhere
buoyant latch
aha
i think i've found something
the top function indeed has points a, b where f(a) = f(b)
oh but i guess it then depends on how you define the derivative of |x|
if you do d/dx |x| = x/|x| then you lose the 0 point
well actually this isn't differentiable on a, b then
since it'll have the wack 0 point
buoyant latch
but i dont want that
i want a function whose derivative is never 0 but is itself continuous and differentiable not injective
untold torrent
Sorry I don't understand what you mean
What exactly do you want an example of?
I'm pretty sure you don't need the derivative to be continuous for Rolle's theorem to apply
buoyant latch
well if the derivative isn't continuous then it's capable of jumping from negative to positive
meaning no such c exists for f'(c) = 0
untold torrent
Well the derivative can't do that
Look up Darboux theorem (I think that's what it's called)
The derivative always satisfies the intermediate value theorem
Also, just looked at the proof (at least one that I found) and it uses the fact that there is a min/max that isn't on the endpoints, and proves the derivative there is 0.
buoyant latch
ah darboux theorem
untold torrent
To prove the derivative there is 0, you look at the one sided limits (f(x + h) - f(x) over h) and you bound f' by 0 from both sides (at that point)
buoyant latch
that's what i was missing
untold torrent
Yeah so that tells you it's not possible for such a jump
untold torrent
At 0, yes
Rolles theorem doesn't tell you where you will find a derivative of 0
In this case, it's just not at x = 0
And that's true no matter what interval you look at that contains 0
buoyant latch
yeah that's what i was thinking too
buoyant latch
but for the classical x^2sin(1/x) and 0 case
for this
no matter what interval i choose as long as it includes 0 it will also include infinitely many min/max points
untold torrent
Yup
buoyant latch
the same thing breaking the derivative is also saving rolle's theorem
untold torrent
It's a trade-off lol
buoyant latch
if you dont mind i could use some help really understanding darboux's theorem
how is it different to rolle's theorem
buoyant latch
it sounds like a version of MVT
for a < b, every y between f'(a) and f'(b) has a c such that f('c) = y
does that not sound like MVT
untold torrent
But yes, it is similar
Normally, the conditions for IVT is for the function to be continuous
But darboux theorem says that if you are talking about a function which is a derivative, you don't even need for it to be continuous and it will STILL have the same properties
buoyant latch
but a function with a derivative is always continuous
or is it saying the IVT for the derivative
so get a function, be differentiable, its derivative satisfies IVT
untold torrent
I'm saying that if you have f(x)
And you take the derivative, you get f'(x)
Then you are like "oh frick I can't use IVT since I don't know if f' is continuous"
But darboux is like "no worries I gotcha"
buoyant latch
ah
untold torrent
Lol
buoyant latch
since f' is a derivative of a differentiable function
untold torrent
Yes
buoyant latch
IVT will apply even if it's not continuous
untold torrent
👍
buoyant latch
ah but IVT doesn't guarantee continuity
that's so scuffed lmao
it just applies, it doesn't tell you about it's continuity either
untold torrent
ah but IVT doesn't guarantee continuity
That's the wrong way around
Yes, IVT doesn't imply continuity
Continuity implies what IVT guarantees, which is taking every y value between 2 other y values.
buoyant latch
Then you are like "oh frick I can't use IVT since I don't know if f' is continuo...
so at this stage darboux just goes "yes you can use IVT, dont worry about the continuity part"
untold torrent
Yeah
buoyant latch
that's wack
untold torrent
In other words, f' can't have a jump discontinuity (and not even a removable one)
Since that would mean it can skip over values
buoyant latch
and that breaks IVT
untold torrent
Or more specifically, darboux's theorem, yes
buoyant latch
and we cant break IVT here by darboux'x theorem
royal loom
:o
buoyant latch
thus requiring differentiability is enough for rolle's theorem
since you can't actually skip the 0
buoyant latch
i know they shouldn't break because if it could break someone else would've found it already
i just didn't know why it doesn't break
alright thanks very much!
.close
olive briar
cedar kilnBOT
olive briar
is this correct?
g is linear
i did a matrix mapping
like here
b9
how is b9 not linear
i showed it is
cedar kilnBOT
@olive briar Has your question been resolved?
cedar kilnBOT
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Available help channel!
mortal patrol
cedar kilnBOT
mortal patrol
why does t only take values from within [a,a+π]?
oh because it represents the set of unique values doesn't it
wait
tjen how are they getting the circular shapes
as t increases
stiff totem
it's saying that, within each such interval, it makes a full revolution
mortal patrol
yeah how
stiff totem
not that t can only take values within those intervals
because x and y are functions of cos and sin, respectively
they're the circular trigonometric functions
stiff totem
their whole thing in parametrisations is tracing out circles
do you know what the parametrisation (cos(t),sin(t)) looks like
mortal patrol
a circle righjt?/
stiff totem
yes
mortal patrol
from combining them into cartesian
stiff totem
so scaling the x or y coordinate turns it into an ellipse
mortal patrol
that's how I make sense of it
x^2 + y^2 = 1
but that requires them to have the same constant factors
but here cost has multiple 2
while sin(t) has multiple 1
stiff totem
yes, so that's a scaling in the x direction
stiff totem
so it's an ellipse that is twice as wide as it is tall
mortal patrol
via cos^2 + sin^1 = 1
mortal patrol
so it's an ellipse that is twice as wide as it is tall
where can I read up more on this visualising parametisation thing?
becuase I have no sense of it rn
I can only make sense of it by converitng to cartesian
stiff totem
I can't find anything specific right now, but any vector calculus textbook should have an introduction chapter covering parametrisations
it depends on what you're doing, but looking at parametrisations in different coordinate systems might also help break your dependence on converting to cartesian
mortal patrol
ok thanks!
do you have any tips/general intuitive ways of thinking about it that I could use?
like carestian is nice and intuitive to me
2d/3d space
points in uniformly arranged space
etc
stiff totem
qualitatively, sine and cosine just oscillate up and down
if they're scaled evenly, you get a circle
mortal patrol
"evenly"?
stiff totem
if you scale the cosine on the outside, you're just stretching how far it oscillates in that direction
as in with the same coefficient
mortal patrol
perhaps unit circle interpretation is supposed to help more?
mortal patrol
if they're scaled evenly, you get a circle
but here they're clearly not scaled evenly
stiff totem
I was continuing but you interrupted
mortal patrol
sorry
stiff totem
if they are scaled evenly, you get a circle
mortal patrol
Pardon my einstellung, but this does not seem very "circular" to me
stiff totem
(2cos(t), sin(t)) just stretches the x direction by a factor of 2
mortal patrol
does seem very "wavy"
mortal patrol
(2cos(t), sin(t)) just stretches the x direction by a factor of 2
yes I get tat
but how does that tuple translate to a circle
mortal patrol
something circular*
stiff totem
as I said, it stretches in the x-direction by 2
they still oscillate at the same rate, so the closed loop itself doesn't change beyond scaling by 2 in that direction
mortal patrol
how does this encapsulate the "wavy/oscillating" nature of sin/cos
is basically what I mean
stiff totem
mortal patrol
oh right unit circle interpretation
stiff totem
but it's the exact same thing for the ellipse
when you follow the point tracing out the ellipse, the cosine wave you get is stretched out by a factor of 2
mortal patrol
when you follow the point tracing out the ellipse, the cosine wave you get is st...
Thanks!
cedar kilnBOT
@mortal patrol Has your question been resolved?
wispy iris
.reopen
fair copper
cedar kilnBOT
fair copper
this working out is confusing me so much
how do u even derive that
and what happens to the -2
idk how to derive (x/2)
foggy merlin
chain rule
foggy merlin
you dont know chain rule ?
fair copper
ive never done it with a fraction
fair copper
-sin(2x)*2
foggy merlin
where is x
fair copper
i edited it
fair copper
so -sin(1/2x)*1/2
foggy merlin
yes
fair copper
ye thats wwhat i meant
foggy merlin
now u know how to differentiate sin(ax)
fair copper
ok that makes sense but why can u just move the negative from cos to the 1/2 on the other side
its 3rd line of wokring out
foggy merlin
wdym
fair copper
like right here
the negative on cos just gets moved to thee other side of the equation
foggy merlin
do u have any background in algebra ? 🤔
foggy merlin
except you have a multiplication here
fair copper
how can u multiply both sides my -1 when its already multiplication
wouldnt u have to divide
foggy merlin
it is of course 1/(-1)
but that thing is just equal to -1 so whatever
foggy merlin
?
cedar kilnBOT
@fair copper Has your question been resolved?
cedar kilnBOT
cedar kilnBOT
Available help channel!
autumn dew
quadttic
autumn dew
.close
cedar kilnBOT
cedar kilnBOT
Available help channel!
plain hazel
why is 1/x-1 involved in proving the base case?
cedar kilnBOT
why is 1/x-1 involved in proving the base case?
tropic oxide
it's...always involved?
exotic furnace
ngl the 1 superscript with the period looks like ! and it triggers me
cedar kilnBOT
Available help channel!
last trench
cedar kilnBOT
wraith daggerBOT
cedar kilnBOT
@last trench Has your question been resolved?
tropic otter
orchid island
This question can be solved using vector analysis
orchid island
First assume unit vector I and j to North and South
Then use velocity of wind = acctual velocity of wind - acctual velocity of man
In first case man's velocity = 4i and wind's = -Pj (assuming)
In second case mans velocity =8i and wind's= -q(i+j)
And in both case acctual velocity of wind = xi+yj
Wre i and j are unit vectors
mortal patrol
cedar kilnBOT
mortal patrol
why is the largest value here $\sqrt{2r}$
wraith daggerBOT
Kalgar
mortal patrol
how did they get the sqrt
they said above that the largest value is when cosθ =sinθ
so we have r|2sinθ|
but largest value of sinθ = 1
so shouldn't this be 2r
spice kraken
no
sin theta and cos theta never equal 1 at the same time
when does sine theta = cos theta?
mortal patrol
but like algebraically
why can I not replace the sinθ in the equation
to simplify it to r|2sinθ|
mortal patrol
when does sine theta = cos theta?
π/4
spice kraken
the maximum of cos theta and sin theta doesn't equal maximum of 2sin theta
the maximum occur when cos theta = sin theta
spice kraken
the maximum occur when cos theta = sin theta
this tells you theta =pi/4, not that you should replace cosine with sine
mortal patrol
the maximum of cos theta and sin theta doesn't equal maximum of 2sin theta
can this claim be formalised somehow?
spice kraken
I don't see how it's not formal
mortal patrol
like with derivatives
mortal patrol
I don't see how it's not formal
I am having trouble seeing why we can't replace cosθ with sinθ
spice kraken
their maximum is different
mortal patrol
ok yeah I think I get what yo umean
I'm pretty much considering a completely different functoin l;ol
right?
cedar kilnBOT
@mortal patrol Has your question been resolved?
mortal patrol
their maximum is different
.close
pseudo merlin
hi
pseudo merlin
for graphing g
um do i use null factor law
and do
-2x = 0
x = 0
or can u not do that
cus i remember with null factor law
u just ignore the one not in the brackets
slate lintel
this is for g? yeah you can do that
you can also be a bit silly and say that -2x = 0 so either -2 = 0 or x = 0, and the first one is clearly false
livid hound
you can ignore non-zero constants being multiplied when checking for zeroes
livid hound
a number like 5
a constant that isn't 0
in g)
-2x(x-1)(x+3)
is the product of
-2
x
x-1
x+3
determine when each of these are 0
pseudo merlin
um
but i swear
with quadratics
we just ignore the thingy in front
when finding the x intercepts with null factor law
slate lintel
you can ignore non-zero constants being multiplied when checking for zeroes
that's what ram was talking about here
pseudo merlin
can u simplify that pls
pseudo merlin
you can ignore non-zero constants being multiplied when checking for zeroes
oh
i see
nvm
i didnt see the edited haha
slate lintel
if you have 5x(x+3) = 0 you can ignore the 5 [or just divide both sides by 5]
pseudo merlin
if you have 5x(x+3) = 0 you can ignore the 5 [or just divide both sides by 5]
yaaa i remember dat
okok i see
thx :D
.close
thorn tangle
cedar kilnBOT
thorn tangle
how did they get this result?
shut tapir
What's cis(theta)
cedar kilnBOT
shut tapir
Yup
tawdry salmon
never got to use this notation though
shut tapir
Same
cedar kilnBOT
Available help channel!
crimson sedge
cedar kilnBOT
tawdry salmon
of the 5 numbers you need to choose 3 of them to make the 3 digit number
then you need to permute the different numbers in the 3 digit number iirc
crimson sedge
i times 3 of them?
fair geyser
there's one step, you do 5×4×3
tawdry salmon
actually it depends if its with replacement or without right
fair geyser
it specifies different numbers
merry nacelle
cedar kilnBOT
merry nacelle
C)
The unit vector is that of (2, -1) normalized or is it of the gradient vector direction of the point at (2, -1) which i calculated to <9, 9>
I don't know if this makes more sense?
cedar kilnBOT
@merry nacelle Has your question been resolved?
cedar kilnBOT
@merry nacelle Has your question been resolved?