whats radical?
#help-0
clear stump
chrome plank
whats radical?
I don't know, try with wolfram
high rapids
,w define radical
clear stump
its the thing above the numbers in that image
square root?
ocean sealBOT
real gazelle
square root?
radical means the same thing as root basically
(the cut-off word is radicand)
lone heartBOT
@alpine sable Has your question been resolved?
lone heartBOT
lone heartBOT
Available help channel!
half osprey
lone heartBOT
placid zinc
Have you learned the dot product?
half osprey
nah
modern orchid
$a^2+b^2=c^2$ so $c=\sqrt{a^2+b^2}$
ocean sealBOT
π=√g
half osprey
what
lone heartBOT
@half osprey Has your question been resolved?
hot bison
I need to find the vertical stretch of this function
lone heartBOT
I need to find the vertical stretch of this function
hot bison
I initially thought it was 8 but it isnt
real gazelle
wait it is 8 though
if you mean the amplitude of the function
what's the full question? @hot bison
hot bison
thats the full question
the vertical stretch is supposed to multiply al the y coordinates by 8
right?
real gazelle
8 is correct then yea
hot bison
wait one sec
how do I prove that it is 8 then?
this gives this
while this gives this
oh wait a second
it is 8 what
thats so weird I thought I just did it and it failed
thanks for the help, it was really helpful
.close
inland vessel
Is this proof correct? Or did I do it backwards.
lone heartBOT
Is this proof correct? Or did I do it backwards.
inland vessel
And also, can you derive proofs by starting from what you intially wanted to prove?
tame shell
It's not clear from what you wrote down what you're trying to prove at all
rancid blade
if you read woog upside down and backwards it is boom
real gazelle
Is this proof correct? Or did I do it backwards.
refer to the proof I posted earlier btw
if you want a clean example
inland vessel
if you want a clean example
I tried perfecting it but
I didn't really understand your proof that well
inland vessel
alright
real gazelle
okay so the first thing is that the thing we want to prove goes at the bottom of the proof
which is the lim n->inf v^(1/n) = 1 part
let's start from the very beginning
inland vessel
Okay.
real gazelle
do you agree with me that lim n->inf of 1/n is 0?
inland vessel
yes
inland vessel
do you agree with me that lim n->inf of 1/n is 0?
sure
real gazelle
alright so now we can multiply both sides by the log of v
and the limit is still zero right
inland vessel
yes, multiplying log(v)*0 results 0
inland vessel
log(0)=1
real gazelle
1/n * log v is the same thing as log (v ^ (1/n)) right
inland vessel
i wish i could use latex on this one
inland vessel
1/n * log v is the same thing as log (v ^ (1/n)) right
yeah
real gazelle
awesome
real gazelle
i wish i could use latex on this one
what do you mean
real gazelle
the next step is the part that requires understanding of some calculus, do you know what I mean when I say the log function is "continuous"?
inland vessel
the next step is the part that requires understanding of some calculus, do you k...
uh no
real gazelle
okay so basically it means I can draw the log function without lifting up my pencil
it doesn't have any breaks or anything
compare it to a function like this
,w graph step function
inland vessel
doesn't that mean it's infinite?
ocean sealBOT
real gazelle
doesn't that mean it's infinite?
basically it means there's no breaks
real gazelle
like there is in this graph at x=0 ^
inland vessel
basically it means there's no breaks
uh
so you mean, the line is uniform or it's just there are 0 breaks if it's continous
real gazelle
like do you see how that graph jumps from y=0 to y=1
inland vessel
like do you see how that graph jumps from y=0 to y=1
yes
real gazelle
yep, there's 0 breaks if it's continuous
inland vessel
that's a non continous?
inland vessel
Aight
real gazelle
which is why we're able to "move the log out of the limit"
if that makes sense
so like the limit of the log is the same thing as the log of the limit
inland vessel
uh ight?
inland vessel
Multiplying by log v there doesn't follow.
he didn't include it in his description but he did it
vocal quiver
Oh wait nvm
Got confused between log v (which is a constant) and log n
inland vessel
if that makes sense
And that gives us lim n->inf v^(1/n) = 1 after doing logs?
real gazelle
that gives us log(lim v^1/n) = 0
but then we can undo the log on both sides
to get lim n->inf v^(1/n) = 1
yep!
so you see I did the same sort of "taking the log to undo the exponent" thing that you did
inland vessel
you forgot to include "log(0)" there
real gazelle
what do you mean?
inland vessel
like taking the log of both sides
real gazelle
there is no log(0), log(0) is undefined
real gazelle
oh we're not taking the log of both sides
inland vessel
log(1)?
inland vessel
real gazelle
so you can think of it, rather, as taking the exponential of both sides
vocal quiver
It's applying exp to both sides
real gazelle
yep
inland vessel
ah realised there was a log
nvm
anyway
i guess I have some understanding of the proof
inland vessel
uh no really
ocean sealBOT
nesymerp1
inland vessel
i replicated it
inland vessel
do you have any questions about any step?
^
real gazelle
alright cool lemme see
yep, you skipped a step in between these two lines but otherwise it looks good
also, keep in mind that proofs usually have words explaining what you're doing
inland vessel
oh right
well like i said, in HS we didn't really get thought this stuff of proofs
im in the first year tho
will be second year in like uhm, weeks or 1 month from now
real gazelle
yeah, you won't go through stuff like this in the first year of high school
but that's okay!
you can still learn about it on your own
it's really interesting
inland vessel
It could be but...
I've also learnt other things like
basic set theory language, sets and elements
unions
set building
basic integrals, anti-derivatives (that are very straightforward and basic) and some derivatives.
now im doing revisions on the really critical stuff, like logarithms, equation systems and also trig
real gazelle
wait you did derivatives and integrals?
real gazelle
Lol on your own or in school?
real gazelle
that's awesome tho :)
inland vessel
i know what integrals do, vaguely. And also reimann integrals.
real gazelle
keep it up
real gazelle
ah cool!
inland vessel
the fucked up thing is, I learned integrals before derivatives and anti-derivatives.
real gazelle
hahahaha
inland vessel
💀
real gazelle
integrals and derivatives are super useful tho
inland vessel
well yeah
real gazelle
so it's really cool that you're learning about them now
before you even take a calculus class
inland vessel
Currently I've pressed the stop buttom on it.
Because I wont overextend myself.
The latest thing I learned was reccurance relations, (sequences), basic ones like fibbonaci sequence and other sequences.
real gazelle
that's okay, don't do anything that burns yourself out
just keep forging ahead whenever you can :)
real gazelle
oooh awesome
inland vessel
but only with 2d functions, because otherwise...
uhm
you need a closed loop integral for 3d functions right?
i would need greenes thereom to continue
real gazelle
hahaha things get complicated when you get to multivariable calculus 😅
inland vessel
hahaha things get complicated when you get to multivariable calculus 😅
säs
i guess
real gazelle
I wouldn't recommend going there until you're comfortable with single variable calculus
inland vessel
eh yeah
like taking the double integral of
2x+2y with respect to x and y.
isn't hard or anything
but wtf, idk man
real gazelle
yea it's cool
and also super useful
if you want to go into a STEM field later in life
real gazelle
then calculus will definitely be super useful :)
both of those fields are really cool
inland vessel
ya
but all i need to do is fix stupid mistakes I do
like taking the sqrt of
4 and then forgetting to remove the square root of the answer.
like, i accidently write sqrt in the wrong places 💀
but that happens when im tired or something
real gazelle
hahahaha that's alright that just comes with practice
inland vessel
i guess
real gazelle
trust me, as long as you keep doing it, the mistakes keep going down and down
inland vessel
i hope so, eh
real gazelle
it definitely will
carmine bronze
how exactly do I get it to the form f(x)/g(x)?
lone heartBOT
how exactly do I get it to the form f(x)/g(x)?
wary stream
how exactly do I get it to the form f(x)/g(x)?
Factor out the LCM
So then it's a product
raven dagger
2x( e^1/x -1)
F(x) = e^1/x -1
G(x) = 1/2x
carmine bronze
ohh that makes sense
wary stream
2x( e^1/x -1)
F(x) = e^1/x -1
G(x) = 1/2x
Don't do the work for people
It doesn't help them learn
I stated how they should initially start and they can work based on that
carmine bronze
.close
lone heartBOT
lone heartBOT
Available help channel!
coarse wedge
lone heartBOT
coarse wedge
Im not sure how to do this one
lone heartBOT
@coarse wedge Has your question been resolved?
coarse wedge
.close
final crag
hello
final crag
Can someone explain how this answer makes sense?
i think i get the first line of the given statement
but then how does it become the 2nd given statement
lone heartBOT
@final crag Has your question been resolved?
remote heron
but then how does it become the 2nd given statement
looks like contrapositive 
but why the exists
lone heartBOT
@final crag Has your question been resolved?
real gazelle
what is the question here?
are you asking how "for a particular number x, neither (2x+1) nor (x-7) equals 0" is translated into "there exists x such that not Q(x)"
real gazelle
looks like contrapositive <:thonk:406575732563902485>
wha what does this have to do with contrapositive
remote heron
the second and 3rd line is contrapositive right
real gazelle
oh I see what you mean, yea
in this context it's called universal modus tollens tho
real gazelle
are you asking how "for a particular number x, neither (2x+1) nor (x-7) equals 0...
if you're asking this, it's because of de morgan's law
Q(x) is that either (2x+1) = 0 or (x-7) = 0
negation of that is that (2x+1) ≠ 0 AND (x-7) ≠ 0
hope this helps
wind bloom
to find x do i do 7x-1=4x+2
lone heartBOT
to find x do i do 7x-1=4x+2
abstract fractal
What makes you think so?
abstract fractal
You can't just blindly set things equal to eachother
How do the two triangles relate to each other?
wind bloom
they r similar
random void
similar is not always equivalent
abstract fractal
How do the sides of similar triangles relate?
wind bloom
How do the sides of similar triangles relate?
they have to be proportional?
abstract fractal
its 8/4x+2=12/7x-1 right
Yes
north rivet
I need to show that the von Neumann algebra $L(\mathbb Z)$ generated by the discrete icc group $\mathbb Z$ is isomorphic to $L^\infty(\mathbb T, m)$ where $m$ is the arc length measure
ocean sealBOT
Max.
north rivet
And T is the unit circle in C
And Z id not icc this was a mistake but still i need to show this isomorphy
lone heartBOT
@north rivet Has your question been resolved?
glacial hedge
I am doing a proof on n^2<25 implies that n > -5 and ive got to (n^2<25) => (n+5>0 and n-5<0) How do i do the final step to finihs my proof (I mean like how do i justify it)
glacial hedge
Right
placid zinc
👍
glacial hedge
.close
livid raven
Does anyone have a simpler way of doing this?
lone heartBOT
Does anyone have a simpler way of doing this?
tight gulch
livid raven
I have
But not when finding the anti derivative
Or are those two things separate
You’re talking abt $$sin(x)cos(y) = 1/2 (sin(x + y)- sin(x - y))?$$
ocean sealBOT
Cheese Grater
livid raven
<@&286206848099549185>
lone heartBOT
@livid raven Has your question been resolved?
livid raven
This is depressing fr
Does no one hear my desperate cries for help
…
I should of been an English major 😔
I hate English
lone heartBOT
@livid raven Has your question been resolved?
livid raven
Legit ugly cryin rn
novel night
You’re talking abt $$sin(x)cos(y) = 1/2 (sin(x + y)- sin(x - y))?$$
Yeah, that leads to a quicker way to do this problem, and ones like it. Those kinds of formulas apply any time you use trig functions, including inside an integral.
obsidian crane
You’re talking abt $$sin(x)cos(y) = 1/2 (sin(x + y)- sin(x - y))?$$
you can use those formulae in integration
livid raven
Oh
lone heartBOT
@livid raven Has your question been resolved?
stone glen
Oh
There are other formulae as well btw, to make this easier, have a look at your old notes once
alpine sable
If I want to create a graph where every time x goes up by 1, y goes up by 10% of its value, how do I do that?
mortal trellis
$y=y_0 \cdot 1.1^x$
ocean sealBOT
Denascite
mortal trellis
where y_0 is your starting value
lament basalt
It's about logarithms
lone heartBOT
It's about logarithms
lament basalt
What's the logarithm of:
lament basalt
there is no question
No I mean
what's the logarithm of 25x with a base of 5?
I can't seem to know whether is it:
2
1
Or 2,1 for both the square of 5 and the exponent of x
vale wigeon
it's neither 2, nor 1, nor is it 2.1
if you're asking what $\log_5(25x)$ can be \textbf{simplified} to, one possible simplification is $2 + \log_5(x)$.
ocean sealBOT
Ann
alpine sable
how do you take off the decimal of numbers in desmos? like if there is 1232.7583 how do you turn it into 1232? like is there a function?
ripe rain
You mean greatest integer function?
alpine sable
floor?
lost coral
any other way to factor other than this?
lone heartBOT
any other way to factor other than this?
lone heartBOT
lone heartBOT
Available help channel!
lyric gust
what is the formula to to scale an array proportionate to the mean of another array, i would like to make a much smaller dataset look proportionate to a bigger set
lyric gust
wait i gotta idea :3
north rivet
.reopen
native cloud
...?
lyric gust
.close
alpine sable
I'm not sure if I'm just silly, but I've tried this question a couple of times and I just can't seem to get any of the answers shown
alpine sable
I'd really really appreciate any help
lilac nest
How did you calculate the area of the washer?
alpine sable
5π²
analog robin
take the area of the entire circle and then subtract the area of the hole to get the area of the washing machine
lilac nest
You have to subtract the area of the smaller circle from the bigger circle
alpine sable
Oh wow, yea it is pretty simple actually
Thankyou both very much for your time. I hope you both have a great rest of your day 
analog robin
np
lilac nest
Np, have a great day too
alpine sable
.close
mortal trellis
doing exactly what you described on a rubiks cube does give you a different result than just applying A directly
azure trout
lone heartBOT
keen pasture
discriminant < 0
You should learn those 3 cases. In an exam you're not allowed to open a help channel
azure trout
I knoww
azure trout
discriminant < 0
So it’s 0? Because there’s no x intercepts?
keen pasture
What?
keen pasture
discriminant < 0
b^2 -4ac < 0
keen pasture
In the 3rd case there is the c missing
azure trout
Shoot
keen pasture
What do you think it is?
azure trout
I thought it was 0?
keen pasture
So you have this table
Which is almost correct
Do you know how to read out what a,b,c is if you have a quadratic?
keen pasture
Nah
keen pasture
a is just the number before x^2. Here it is a 1
b is also the number before x. So it is the k here
azure trout
Then c is just 9
keen pasture
Yeah
azure trout
Now what?
keen pasture
2nd case in the table, which you need to learn by heart
azure trout
I just plug the numbers in?
keen pasture
Yeah
azure trout
k^2-4(1)(9)=0
keen pasture
Yeah
keen pasture
Simplify
keen pasture
Do the same approach
azure trout
okok
keen pasture
Maybe you should ask your teacher where the c is in the 3rd case
azure trout
Okay - why?
keen pasture
Do the same approach
But now we're in the 3rd case. b^2-4ac <0
keen pasture
Okay - why?
Because the c is missing
azure trout
4^2-4(3)(k)<0
keen pasture
Yeah
azure trout
k>4/3
keen pasture
✅
azure trout
lone heartBOT
azure trout
Whereee would I start?
pliant yew
you can use the vertex form of a qudratic
y=a(x-h)+k
you have h and k
then you can find a using the other point
azure trout
and then just convert to ax^2+bx+c?
pliant yew
yeah
pliant yew
np
azure trout
b would be -4 right
pliant yew
uh
lemme do it rq
you have to multiply by 3
cuz
you got y=a(x-2)^2+4
and a is 3
when you put in the other point
so you get
y=3(x-2)^2+4
y=3(x^2-4x+4)+4
azure trout
why do you have to multiply by 3 what
pliant yew
y=3x^2-12x+16
cuz
the form you need is y=ax^2+bx+c
so you gotta distribute everything
azure trout
it would just be x^2-4x+8
azure trout
??
pliant yew
how do i explain this
ok
let's just look at x^2
x^2
isn't the end
cuz theres the 3
3x^2 is the whole thing
azure trout
what 3 are you talking about
pliant yew
the a from the vertex form
pliant yew
you got y=a(x-2)^2+4
here
azure trout
there is no 3 in there?
pliant yew
ohhhhh
ok
i see what's confusing
ok
so
let's start again
vertex form is y=a(x-h)^2+k
using the vertex
you get
y=a(x-2)^2+4
azure trout
yes
azure trout
why would you have to solve for a
pliant yew
cuz you can't just leave a, as a, that's an important coeffiient
just forget about the standard for a sec
it may seem like a here is the same as a there
azure trout
kk
pliant yew
but they're two different things
so yeha
you gotta solve for a
16=4a+4
a=3
y=3(x-2)^2+4
azure trout
ok so
pliant yew
alr maybe i just suck at explaining
pliant yew
y=3(x^2-4x+4)+4
this is the vertex form
azure trout
NAH WAIT UR RIGHT NVM
pliant yew
ok there's now another form that you would need to know
i think it's called interept form
y = a ( x − p ) ( x − q )
yuh
azure trout
yes
azure trout
kk
pliant yew
alr
im back
ok
so
right
y = a ( x − p ) ( x − q )
so
plug in the points that we have
y=a(x-2)(x-6)
solve for a by pluging in the other point
12=a(3-2)(3-6)
12=-3a
a=-4
so
y=-4(x-2)(x-6)
expand
y=-4(x^2-8x+12)
y=-4x^2+32x-48
pliant yew
b is just 32
azure trout
oh ok
pliant yew
cuz -4*-8 is 32
tawdry radish
dont know if im going about this word problem the right way
lone heartBOT
dont know if im going about this word problem the right way
tawdry radish
just a warning, its kind of wordy. but I dont think the actual math is alot
alpine sable
please send a photo, not a pdf
edgy flare
My dude that is 25 pages of Information.
tawdry radish
its not all
edgy flare
Probably more suitable for a discussion chat.
edgy flare
I read the first 3 because I play RuneScape but then seen 3/25
I was like I don’t got time😂
wind yoke
dont know if im going about this word problem the right way
what qn do u wan t help?
tawdry radish
its on page 6, but it relates to using a fraction to influence rates i.e units of currency /second
wind yoke
its on page 6, but it relates to using a fraction to influence rates i.e units o...
ok lemme look
tawdry radish
ok, skip 1st to pages and i think the question still makes sense so like 3-6
thank you
tawdry radish
first 2 pages is setup of question really, without math
tawdry radish
Although various factors are the cause of the eventual time requirements of the task, I would like to use my experimental value of $20/1800seconds, and for the sake of the argument (math) use those figures as if it were done by the power drill alone, which is a 2 / 3 “multiplier” as compared to the screwdriver method, or .66 times faster. and then use the $20/1800 figure as if it were done by the screwdriver method.
Because I have decided to get 2 different data points on 2 different tools/methods, I can now form a framework of how the tools, and ultimately the efficiency increase they bring, compare to one another, and also the actual value that the efficiency brings.
i.e. I am reasonably-and-justifyably able to attribute a “stat” to the tools, as they relate to one another, in the scope of a specific task. consider the below example :
If you were to do the same task, using a screwdriver, you can know:
$20/1800 will earn rate of 1.1 pennies/second
same task with a power drill, you can know:
if you have a drill : $20/ 1188 (1800 x .66) = 1.7 pennies/second
therefore, I can conclude :
Every second that I am using a power drill, it is netting me .6 pennies/second — and this applies to all tasks outside the scope of where I ran my initial experiment (taking apart the toaster oven)
I need to apply the drill “multiplier” in the form of .6 pennies/second , im assuming, as opposed to simply taking the first time (1800) and multiplying by ( .66 ) as i showed in the first time when deriving the figure.
because ULTIMATELY i DID take actual 1800 seconds to do it. and the reason that is super significant is because that 1800 of actual time needs to be used in the accountance, because again ULTIMATELY, it serves as a concrete and static
fraction of my available FREE-TIME.
1800 / 10,512,000 == 1 / 5840
- am I right that I should use the 2nd method of calculation, as the first was only necessary to serve as a starting point, and
- Since with each completion of a task that is over my current baseline rate of .22 pennies/second, that number that is “worth-per-second” should hypothetically rise,
because why would i STILL use my time at a calculated worth of .22 pennies / second after having just establishing a tremendous pattern of using my time to gain 1.1 pennies/second?? I should be averaging each result of every task as to effect the input parameter of “worth-per-second” of the subsequent task. HOW do I do it in this context. How does that change for someone who has a lesser number of FREE TIME than the 10,512,000 that I have, since all of the above is contingent on the idea that the task i did was 1/5840 of available time. e.i.
How does it change the math if someone wanted to repeat the process for another set of tools, using a 1800 second task, but having half of that available time, because generally, as time becomes lesser and lesser, in theory the rate should go up as a result. at face value, this is relative and different between person to person, how much there
lone heartBOT
@tawdry radish Has your question been resolved?
lone heartBOT
@tawdry radish Has your question been resolved?
tawdry radish
everyone gave up huh? :S
lone heartBOT
@tawdry radish Has your question been resolved?
livid flame
is a reflection over y=x just a rotation of 90 degrees?
lone heartBOT
is a reflection over y=x just a rotation of 90 degrees?
pliant dune
no
pliant dune
for example the line y=x isn't rotated when reflected
livid flame
just looks like it
fresh parcel
thats a special case
livid flame
is there an easy way to visualise the inverse of a graph?
pliant dune
reflecting in the line y=x
ebon fractal
is there an easy way to visualise the inverse of a graph?
As you would visualize doing the non-inverse.
Where the input is now the y, and the result is the x.
livid flame
hmm ok ig I thought there was an easier way to do it
like in terms of just rotating the page
For C, how can we use present value to calculate balance outstanding because isn't the n in the present value annuity supposed to be representative of the time left not the time that has gone by?
livid flame
For C, how can we use present value to calculate balance outstanding because isn...
they use 120 in the present value annuity which is the time that has gone by
lone heartBOT
@livid flame Has your question been resolved?
slender gull
Find $a < 0$ for which the inequalities\
$2\sqrt{ax} < 3a - x$\
$x - \sqrt{\f{x}{a}} > \f{6}{a}$\
have solutions in common.
ocean sealBOT
What the hell am I doing here?
slender gull
I know x < 0 as well. Then I would probably consider squaring heavily.
After a little bit of work
I get,
(x-9a)(x-a) > 0
vale wigeon
desmos this shit 
slender gull
(x-9a)(x-a) > 0
For the first one.
And, (ax-4)(ax-9) > 0
For the second.
After this I am unable to make respective cases to get a complete solution set.
vale wigeon
maybe write t := -x and b := -a and rewrite everything in terms of known-positive numbers before proceeding
and then just solve both inequalities for sqrt(x)?
slender gull
and then just solve both inequalities for sqrt(x)?
x is negative, why and how do I solve an inequality for negative number in a sqrt?
weary wyvern
Just solve each inequality separately
slender gull
Yes, I did.
slender gull
(x-9a)(x-a) > 0
...
vale wigeon
i have a feeling you've screwed it up at some point but w/o seeing your work i could not say where
slender gull
I can show my work, no issues.
weary wyvern
Assuming ur right, just graph each quadratic to get solution sets
slender gull
That gives me the wrong answer haha, tha is why I am here, let me post my work as ann says.
I might have found it already.
$2\sqrt{ax} < 3a - x$\
$4ax < 9a^2 - 6ax + x^2$\
$x^2 - 10ax + 9a^2 > 0$\
$(x-9a)(x-a) > 0$\
Now the second one,\
$x - \f{-6}{a} > \sqrt{\f{x}{a}}$\
$x^2 - 12\f{x}{a} + \f{36}{a^2} > \f{x}{a}$\
$a^2x^2 - 13ax + 36 > 0$\
$(ax-4)(ax-9) > 0$
Geez.
ocean sealBOT
What the hell am I doing here?
slender gull
🥲
slender gull
**What the hell am I doing here?**
This is it, I hope.
lone heartBOT
@slender gull Has your question been resolved?
lone heartBOT
@slender gull Has your question been resolved?
lone heartBOT
@slender gull Has your question been resolved?
lone heartBOT
@slender gull Has your question been resolved?
real gazelle
Geez.
Hey! for your second one it should be x - 6/a, not x - -6/a
That's probably the issue
oh wait I see you fixed that on the next line
hmmm
@slender gull the problem is that you can't necessarily square both sides of an inequality and have it be the same
For example, 6^2 < (-9)^2, but 6 > -9
You need to guarantee that both sides of the inequality are positive first before you can do that
the problem in the first equation is that 3a-x can be negative, which screws the inequality up if you square it
If you start with $$2\sqrt{ax} < 3a - x,$$ then the correct way to proceed is to insist that $$4ax < 9a^2 - 6ax + x^2$$ AND $$3a - x > 0$$
ocean sealBOT
Eric Tao (he/him)
real gazelle
and then the solution set to the original equation is the intersection of the two solution sets you get
hope this helps
Similarly with the second equation, you need to insist that x - 6/a is positive
That will get you the solution you want
Right now you have extraneous solutions
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livid raven
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livid raven
How did they take out (x+1)/2^(3/2) to start the u sub?
By just taking out 1/sqrt(8)
Idk how to phrase my question
placid zinc
If u = x+1 / 2^(3/2)
Then du = [1 / 2^(3/2)] dx
Or, dx = 2^(3/2) du
placid zinc
You can technically do what they did exactly by abusing linearity of d.
But, eh.
ivory seal
ivory seal
<@&286206848099549185>
spare compass
How to post a image with @ocean seal bot
lone heartBOT
How to post a image with <@510789298321096704> bot
@spare compass Has your question been resolved?
kindred anchor
This should go in #latex-help
raven rover
How to post a image with <@510789298321096704> bot
Wym?
You can just share an image as usual
mossy zodiac
Hi again amigos! I have a concept question today
lone heartBOT
Hi again amigos! I have a concept question today
median oar
it's easier
mossy zodiac
in other words, why am I allowed to break u+12 apart?
median oar
well $\frac{a+b}{a}=\frac{a}{a}+\frac{b}{a}$
ocean sealBOT
Frosst
mossy zodiac
i think the integral symbol being there was making it difficult for me to see that
median oar
👍
mossy zodiac
my instinct is to multiply them
median oar
just gotta get used to it
mossy zodiac
a+b(a)
mossy zodiac
so the u^-1/2 is being multiplied there at the end but they have skipped the step in the example?
median oar
are you asking $\frac{12}{\sqrt{u}}=12\cdot u^{-\frac{1}{2}}$
ocean sealBOT
Frosst
mossy zodiac
i think so
i know to push the denom. to the top and change the sign
to get rid of the radical
but i'm struggling with how they tacked it on to the 12
median oar
do you understand the line i just wrote?
median oar
$\int \frac{12}{\sqrt{u}} du = \int 12\cdot u^{-\frac{1}{2}} du$
ocean sealBOT
Frosst
median oar
then we know that $\int a\cdot x dx = a\cdot \int x dx$
ocean sealBOT
Frosst
median oar
if a is a constant not dependent on x
median oar
you can still write it as 1 * ∫ f(u) du
or are you asking if there's no 12 in the original question
cos then it'd just say $\int \frac{u}{\sqrt{u}}du = \int \sqrt{u} du$
ocean sealBOT
Frosst
mossy zodiac
thanks for the patience, my teacher did not post a lecture for this section so this helps a lot
median oar
all g that's what we're here to do :D
mossy zodiac
now that i understand why we can attach the term to the 12, where did the first exponent come from? the positive one
the negative came from the radical
raven rover
Rules of exponents
wet veldt
Did you try reducing it yourself?
median oar
$\frac{u}{\sqrt{u}}=\frac{\sqrt{u}\cdot\sqrt{u}}{\sqrt{u}}=\sqrt{u}$
ocean sealBOT
Frosst
raven rover
u / √u = √u
mossy zodiac
haha, now i get it
i thought the denom. was just being tacked on to the 12
i think this one is solved for now, thanks a ton guys
mossy zodiac
.close
woeful wasp
Hey guys, so I am doing Calculus 1 Homework, and I am double checking my answers with a derivative calculator, but I was just wondering how 2 ended up as the numerator in the derivative of f(x)=(x-1)/(x+1)
woeful wasp
I highlighted the steps on Symbolab, it feels like I am looking at it, but I am missing something
WAIT
nevermind I am bozo
another moment where i realized as soon as I asked xd
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steady fiber
lone heartBOT
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@steady fiber Has your question been resolved?
wet veldt
Interesting. How did you get that dV?
steady fiber
in the first line of my work?
wet veldt
Yeah
steady fiber
alternative form of torricelli's law. A(y)=pi(r^2)
wet veldt
For some reason, I'm getting pi(9^2 - y^2) dy/dt for dV/dt
Though I'm not really using a law, I'm using calculus to do it
steady fiber
hmmm yea if u sketch it out what i'm getting is pi[81-(9-y)^2]
this is kinda the diagram i'm basing it off of, just instead of 4 using 9
lone heartBOT
@steady fiber Has your question been resolved?
wet veldt
It's really strange. Do they want the first 30 minutes not to count to the answer?
lone heartBOT
steady fiber
.reopen
lone heartBOT
✅
lone heartBOT
wet veldt
I see, well you could try after subtracting 1800 to see if it works, but the work you posted looked correct to me.
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Available help channel!
native cloud
Can anyone help me with this?
lone heartBOT
Can anyone help me with this?
naive valley
what's the value of the integrand at the upper endpoint of the interval of integration?
lone heartBOT
@native cloud Has your question been resolved?
native cloud
what's the value of the integrand at the upper endpoint of the interval of integ...
Wdym by integrand?
3 and 0?
<@&286206848099549185>
lilac nest
Integrand is the function you are integrating
naive valley
e^3 - e^0
ok, and only one of the four answers gives you e^3 when you plug in i=n, which is it?
native cloud
D?
naive valley
aye
you can also check that 3/n is the correct rectangle width for n rectangles spanning [0,3]
native cloud
Where is 'i' coming from btw
naive valley
the sum index, i=1 to n
native cloud
Oh right
I was thinking of the imaginary number 'i' for some reason lol
Thanks!
.close
unborn meteor
lone heartBOT
unborn meteor
Can someone help me with sketching f(x)?
I literally don’t understand how I’m meant to sketch the inverse of this let alone the original function
edgy spoke
Have you tried https://www.desmos.com/calculator
Its an amazing site where you can put in a fucntion, and it will draw it for you
unborn meteor
yeah, im just trying to figure out how to draw this myself though
because in an exam i wont be able to just whip out desmos 😂
edgy spoke
Oh 🫤 oops then 😂
unborn meteor
wats the as
ocean sealBOT
Bananach
alpine sable
I didn't flip anything
writing it that way
will let you separate 1
and it will be simpler
unborn meteor
um
how does that help though
i dont get it
i get that u added a 1 -1
at the top
but then whats the enxt step
split the -1 out?
alpine sable
it's $\frac{e^{-x}}{1 +e^{-x}}$
ocean sealBOT
Bananach
alpine sable
correct?
unborn meteor
yes
alpine sable
now write it as $\frac{e^{-x}+1-1}{1 +e^{-x}}$
ocean sealBOT
Bananach
unborn meteor
yep
alpine sable
can you see the next step
unborn meteor
would it be to bring out the -1 into = -1/1+e^-x
?
and then change the e^-x+1/e^-x + 1 into 1/1
oh wait but that literally just gets u back
to the same position
nope ive got no clue
alpine sable
now write it as $\frac{-1}{1 +e^{-x}}+1$
ocean sealBOT
Bananach
unborn meteor
oh wait so u do do that
oh right
yeah my bad
i thought it was the same as the original
alpine sable
do you see how it got simplified?
unborn meteor
yes
i see that
so then would u now just sketch the reciprocal of an exponential graph
with the reflection and shifts
unborn meteor
ahhaah
thats alright bro
thanks for trying to help
anyone know how to sketch this
.close
lone heartBOT
unborn meteor
wait i just realised they give me the graph already
when they ask me to sketch the ivnerse of this
@alpine sable
lmfao
so sorry bro
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alpine sable
lol
unborn meteor
yeah was thinking something was up
dont think its possible
or atleast really hard
alpine sable
I suspected it was the inverse function
lone heartBOT
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Available help channel!
violet canopy
lone heartBOT
violet canopy
can i get some help with this please
unborn meteor
are there solutions for that?
pretty sure w=18
and x = -9
expanding the -3 power into the brackets
we get 2^(3)(-3) * 3^(-6)(-3) = 2^-9 * 2^18
and then using the index laws
we can flip the 2^-9 to 1/2^9
so therefore 1/2^9 multiplied by 3^18
so c = 2, x = 9, and w =18
violet canopy
thx
unborn meteor
nws
lone heartBOT
Available help channel!
native cloud
Help plz
unborn meteor
take derivative
mortal trellis
Fundamental theorem of calculus
unborn meteor
convert the sqrt into index form
so (2t+7)^1/2
and then chain rule
oh wait actually
there are boundries
mortal trellis
Completely unnecessary to calculate any integral or derivative here
native cloud
native cloud
Completely unnecessary to calculate any integral or derivative here
What should I do then?
mortal trellis
And can you tell me what the derivative of F(x) is?
native cloud
This?
mortal trellis
How are f(x) and F(x) related
native cloud
How do I convert f(x) to F(x)