<@&268886789983436800>
#help-23
junior smelt
thats like 4 in 5 minutes
They tend to come in packs 
(only for us to have to put them in packs, of course 
)
safe radishBOT
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median latch
Where is my mistake here? My answer (written note) vs the hint answer
median latch
My concern is that the hint answer doesn't have 1/6 while I do (and I forgot to write + C, i know)
calm cloak
My concern is that the hint answer doesn't have 1/6 while I do (and I forgot to ...
notice that the coeeficient of x while integrating in the final step is 1/6
safe radishBOT
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lean otter
I need the given and to prove
safe radishBOT
I need the given and to prove
lean otter
Gimme a minute
8, 9 and 10
Theorem 6.2. is converse if basic proportionality theorem btw
safe radishBOT
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 was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
I just need the given and to prove
So
lean otter
```
What step are you on?
1. I don't know where to begin.
2. I have begun but go...
1
???
lean otter
you need the *what*?
Given and to prove
warm warren
aint 'given' just the entire question
warm warren
- given traingle ABC, D is a mifpoint of AB, E is a midpoint of AC
- to prove DE || BC
lean otter
And for Q9?
tiny veldt
LOL ISNT THAT NCERT 9TH TEXT BOOK
warm warren
And for Q9?
why dont you try?
lean otter
LOL ISNT THAT NCERT 9TH TEXT BOOK
10th
warm warren
tell me the given and to prove for q9 yourself
lean otter
why dont you try?
Given : Ab || dc
Diagonals intersect at O
Prove : Ao equal to CO
Bo DO
Correct? 😭
tiny veldt
if u need help u can ping me im not gonna interfere with her explanation
safe radishBOT
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woven fjord
Im solving an ode but this looks very wrong to me can someone check please...
safe radishBOT
Im solving an ode but this looks very wrong to me can someone check please...
@woven fjord Has your question been resolved?
woven fjord
Im solving an ode but this looks very wrong to me can someone check please...
<@&286206848099549185>
plucky elk
the original question is blurry
safe radishBOT
✅ Original question: #help-23 message
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@woven fjord Has your question been resolved?
tardy mango
Im solving an ode but this looks very wrong to me can someone check please...
sign error
,w differentiate -C*(b-u)^(-1) wrt u
flat frigateBOT
woven fjord
sign error
its integrate
tardy mango
its integrate
My point is that if
$$\int C(b-u)^{-2} \dd{u}=-C(-u)^{-1},$$
then the derivative of $-C(b-u)^{-1}$ with respect to $u$ should be $C(b-u)^{-2}$. But it isnt.
flat frigateBOT
Civil Service Pigeon
tardy mango
Recall the relationship between differentiation as integration as "opposites".
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glass egret
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@glass egret Has your question been resolved?
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@glass egret Has your question been resolved?
severe pilot
systems of DEs using matrix. i think i did it well, but I'm confused on how to use the initial condition? do i need to at all?
safe radishBOT
systems of DEs using matrix. i think i did it well, but I'm confused on how to u...
median vigil
confirm whether it has that initial value
safe radishBOT
@severe pilot Has your question been resolved?
icy slate
can anyone help with precalc homework here i might be dumb and im very confused
safe radishBOT
can anyone help with precalc homework here i might be dumb and im very confused
Asking the actual question right away is more likely to get responses.
Asking "Can I ask...?" or "Does anyone know about...?" doesn't give people enough information to decide whether they can help, and answering can feel like a promise to help with the actual question, which they might find themselves unable to.
icy slate
Suppose h(x) is as graphed below. On the same plot, graph 1
2
h(x) + 3.
improbably doing this wrong
really wrong
thin bridge
you have the correct function for h(x)
but that isn't what they asked for and isn't needed
its doable using just the coordinates from the graph
consider using a few key points
icy slate
explain this like im 5 casue i get easily confused
thin bridge
looking at the vertex,
when x=-2, h(x) = h(-2) = 2 right?
icy slate
im trying to undertsand rn holdup
ok yes
the red line is the transformation of the function
is the genereal transfomration just a slight cpompression?
thin bridge
now what will
h(-2)/2 + 3
be
thin bridge
the red line is the transformation of the function
not sure what you're doing with those red lines on the graph
this will be a vertical compression followed by vertical shift
icy slate
not sure what you're doing with those red lines on the graph
this will be a vert...
the lines are just what im thinkingthetransfomration will be
icy slate
you have the correct function for h(x)
but that isn't what they asked for and is...
what is it asking for ive ben stuck on this for a while
thin bridge
they want you to graph
1/2 * h(x) + 3
when x=-2, h(x) = h(-2) = 2 right?
now what will
h(-2)/2 + 3
be
thin bridge
why are you using a different function now
icy slate
im not gonna lie iforgot alot ofthe math i learned before thisquarter so im struggling 😭
thin bridge
don't overthink what i'm asking you
when x=-2, h(x) = h(-2) = 2 right?
now what will
h(-2)/2 + 3
be
icy slate
holdup im thinking
icy slate
why are you using a different function now
how you mean the curved functions
thso are fromother proplems isolved earlier
thin bridge
ok, forget those for now
thin bridge
how are you getting 2
thin bridge
$\red{h(-2)} = 2$
$$\frac{\red{h(-2)}}{2} + 3 = \what$$
icy slate
were both texting this rn
flat frigateBOT
ραμOmeganato5
thin bridge
can you draw a cleaner line
icy slate
beleive it or not were both artst drawing shitty lines
icy slate
he said at -2 the y value is 7.5
thin bridge
that's very different from what's shown on his graph
also wrong
icy slate
damn
thin bridge
at x=-2, he had y value of 4 which was correct
where's 7.5 coming from
icy slate
he said at -2 the y value is 7.5
my mistake
i heard him wrong
at -2 y val is 4 at x=5 its 7.5
thin bridge
ok, that's fine
icy slate
hold up let me draw this on my own graph now
like that
this is it right for thisone
thin bridge
yes
icy slate
thanks alot
me and bro were tweking out over thiscause we were confused
might come back for help on another one but i need to go eat right now
rich jay
Please help, I understand Piecewise but I don't understand this 3-Constraint and fraction-function shenanigans happening in this.
safe radishBOT
Please help, I understand Piecewise but I don't understand this 3-Constraint and...
outer pollen
,rccw
flat frigateBOT
outer pollen
<@&268886789983436800>
stoic torrent
Please help, I understand Piecewise but I don't understand this 3-Constraint and...
Can you normally plot 5/4 x +63 = y?
rich jay
what is happening
stoic torrent
With not constraints
rich jay
Can you normally plot 5/4 x +63 = y?
I dont know dude
I don't understand these fraction functions
stoic torrent
Can you draw 5x - 2 = y
rich jay
I only get like the coefficient and non-coeffecients, do I need to do rise over run on both the numerator and denominator?
stoic torrent
Okay heres a trick for drawing lines
rich jay
Do I need to put the constraint numbers as the input for the x value?
stoic torrent
Put x = 0 and find value of y1
This will get you a point on line (0,y1)
Repeat by putting y= 0 and find x
This will get you a second point (x1,0)
Plot these two points and join them using a ruler please
This would get you ur line
stoic torrent
Put x = 0 and find value of y1
This will get you a point on line (0,y1)
Repeat ...
Was this understandable?
rich jay
my brain is not functionging
stoic torrent
Right okay lets do this again
If you have two point of a line, we can draw the line by connecting those two right?
rich jay
Can you draw 5x - 2 = y
5x we would do rise over run 5/1 and the y value is -2 so we begin from there
right???
stoic torrent
Okay so lets do it that way then
stoic torrent
Okay no hold on
outer pollen
maybe it would be helpful to remind yourself of what the general linear equation looks like, OP.
rich jay
maybe it would be helpful to remind yourself of what the general linear equation...
y = mx+b
outer pollen
and what is m? what is b?
rich jay
should I stop trying to use rise over run???
stoic torrent
m is the rise over run
outer pollen
m is the gradient.
stoic torrent
m is called the slope
outer pollen
anyway I'll step back for Yajant.
rich jay
Ah
stoic torrent
anyway I'll step back for Yajant.
No no please go on
tardy mango
rich jay
Put x = 0 and find value of y1
This will get you a point on line (0,y1)
Repeat ...
the value for y is 5, x is 1, because that's rise over run, then the denominator is the same thing being 4/1 so 4 is y and x is 1, 63/4 is not possible because the graph doesn't stretch that far to 63, so I'm definitely doing this wrong, and both of these are all on the -7 y axis from my understanding, so -7 is the region
when I do 5/1 and 4/1, there are literally 2 points beneath each other, without any correlation because I'm dumb.
stoic torrent
Okay, see it would be better here to find the point where the first line cuts the x axis
generally we find where it cuts the y axis right?
rich jay
Then there's the using the constraints as the x input, which gives us 5/4(-7) + 63/4 equating to 7.
and if we switch to -3 as the x input, we get 12.
rich jay
Can you draw 5x - 2 = y
I did this, but I needed to literally add negative values to the Y value
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@rich jay Has your question been resolved?
rich jay
.close
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lavish ermine
hey I'm stuck on this problem
safe radishBOT
hey I'm stuck on this problem
plucky elk
integration by parts 
burnt notch
Yep perfect
delicate shore
,, int_{textstyle 0}^{textstyle 1} x^{textstyle 2} e^{textstyle 2x}, dx
flat frigateBOT
delicate shore
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cunning pasture
Let B = A U (A - B).
Prove that A = B.
safe radishBOT
Let B = A U (A - B).
Prove that A = B.
flat frigateBOT
Annie Maqionde
warm warren
For the first assume assume an element $x \in B$ and work your way from the RHS of the given statement
flat frigateBOT
Annie Maqionde
warm warren
for the second assume $x \in A$ and continue likewise, i assume
flat frigateBOT
Annie Maqionde
vague phoenix
Are A and B disjointed sets?
If they are not then it fairly basic to show by using $\forall x \in \mathbb{U}$
flat frigateBOT
Minλ
safe radishBOT
@cunning pasture Has your question been resolved?
flat frigateBOT
Meolve
cunning pasture
How to prove that $B \subseteq A$?
flat frigateBOT
Meolve
cunning pasture
Hi.
opaque fern
Support your claim
restive niche
Support your claim for option D
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@cunning pasture Has your question been resolved?
buoyant shadow
i don;t understand what it's saying, phi is true or false
main mural
i think it's empty set
main mural
B is true
restive niche
If you're having trouble proving option D, I suggest you look at a scenario like this
buoyant shadow
this is python sybntax lmao
buoyant shadow
they all seem true
cunning pasture
Mhm.
buoyant shadow
ok probably D
cunning pasture
wth
The C just changed to C' in the last option
lmao
anyway
thanks everyone for help
.close
safe radishBOT
azure harbor
!da2a
safe radishBOT
Asking the actual question right away is more likely to get responses.
Asking "Can I ask...?" or "Does anyone know about...?" doesn't give people enough information to decide whether they can help, and answering can feel like a promise to help with the actual question, which they might find themselves unable to.
azure harbor
USE GOOGLE
hard crest
not worth the time
azure harbor
not worth the time
hayley can you close please
hard crest
.close
safe radishBOT
azure harbor
to the original message being deleted
this like the second time today
normal moss
Interesting
hard crest
hmm, the bot didn't say anything
it does that sometimes idk when
normal moss
it does that sometimes idk when
Probably grumpy
hard crest
yeah
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jaunty crag
What is a cube
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What is a cube
jaunty crag
.help
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jaunty crag
.help what is a cube
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No command called "what" found.
frozen marlin
jaunty crag
Ohh thanks
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jaunty crag
Wait I have one more question
A goat is tied to the edge of a 100 square foot circular field with a rope. How long must the rope be for the goat to graze on exactly 50 square feet?
.help
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vague phoenix
A goat is tied to the edge of a 100 square foot circular field with a rope. How ...
you got the area then can find radius
normal moss
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waxen hound
need help transforming this into x^3
safe radishBOT
need help transforming this into x^3
waxen hound
am i correct in saying:
- dilation 1/5 from x axis
- dilation 2 from y axis
- translation 2 down
- translation 2 left
azure harbor
am i correct in saying:
- dilation 1/5 from x axis
- dilation 2 from y axis
- tr...
dilation is scaling?
waxen hound
dilation is scaling?
yes
azure harbor
yes
wait. so you want to switch only the right hand side to x^3 or what?
like do transformation to put all the cofficients and stuff on the right hand side?
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azure harbor
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brazen nimbus
if i want to calculate the are between these 2 functions
safe radishBOT
if i want to calculate the are between these 2 functions
brazen nimbus
can i just do f - g or would i have to transform these functions first
plucky elk
can i just do f - g or would i have to transform these functions first
Not always. You need to integrate the larger function minus the smaller function
brazen nimbus
Not always. You need to integrate the larger function minus the smaller function
yes of course
like integrate of f-g
wait
plucky elk
They might have intersection points so the larger one may switch
plucky elk
How do you know f is larger than g
plucky elk
You need to solve it using algebra
brazen nimbus
You need to solve it using algebra
also, how do i know what the limits are?
like the x values
plucky elk
You need to solve it using algebra
.
Intersection points tell you
Do you know what intersection means for the functions
brazen nimbus
Do you know what intersection means for the functions
yes
where they cross
so i gotta do that too?!
thats much work
i guess
austere panther
can someone explain the solution from the step after the closed form sum multiplcation (the introduction of the k index)
safe radishBOT
can someone explain the solution from the step after the closed form sum multipl...
karmic merlin
can someone explain the solution from the step after the closed form sum multipl...
Try something simpler first, find the coefficient of x^n in ( sum of x^n)(sum of x^n)
karmic merlin
also how did we get these
Substituting n=0,1,2 into the equation
safe radishBOT
@austere panther Has your question been resolved?
austere panther
Try something simpler first, find the coefficient of x^n in ( sum of x^n)(sum of...
oki
thanks
safe radishBOT
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 was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
@bleak ridge Has your question been resolved?
bleak ridge
No
ocean crypt
someone help me derive the sum-to-product identities?
safe radishBOT
someone help me derive the sum-to-product identities?
warm warren
Of what?
ocean crypt
i mean there is a way to remember this?
plucky elk
do a lot of problems using it
ocean crypt
youre right
karmic merlin
i mean there is a way to remember this?
Understanding how to derive it might help
warm warren
hm solving problems only way out
ocean crypt
Understanding how to derive it might help
yes but i also have to remember the formula of addition
warm warren
Well mnemonics(but bad idea)
ocean crypt
cos(a+x) = cos(a) cos(x) - sin(a) sin(x) ?
cos(b + x) = cos(b) cos(x) - sin(b) sin(x)
cos(a) + cos(b)
mm
cos(a + x) + cos(b + x) = cos(a) cos(x) - sin(a) sin(x) + cos(b) cos(x) - sin(b) sin(x)
cos(a+x)+cos(b+x)= cos(x) (cos(a) + cos(b)) - sin(x)(sin(a) + sin(b))
terse sand
yes but i also have to remember the formula of addition
You could also remember the approach to derive them
ocean crypt
can you help pls
terse sand
can you help pls
Me ?
terse sand
I can try
ocean crypt
You could also remember the approach to derive them
here?
ocean crypt
ok so we have to prove cos(x+y)
terse sand
You need to know how we derive those formulas, that way you can just derive it everytime until you memorise it
OR
You do enough problems that it sticks with you
ocean crypt
i found a "trick" like when there is + i have to put cos cos - , and where there is - i have to put cos cos +
ocean crypt
You need to know how we derive those formulas, that way you can just derive it e...
first
terse sand
i found a "trick" like when there is + i have to put cos cos - , and where there...
You can do that as well if you're sure it'll stick with you forever
ocean crypt
i dont think so
terse sand
For multiplication you can just observe cos(x + y) & cos(x - y)
ocean crypt
i need to remember them also when i dont use them so much
ocean crypt
For multiplication you can just observe cos(x + y) & cos(x - y)
can we derive this^
terse sand
can we derive this^
The addition part or multiplication one
ocean crypt
actually we just need one
like we can use cos(x+y) and then cos(x+(-y))
so we just need cos(x+y)
terse sand
cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
cos(x - y) = cos(x)cos(y) + sin(x)sin(y)
We can both of these to get
cos(x + y) + cos(x - y) = 2cos(x)cos(y)
Tadaa we have sum-to-product identity for cos
ocean crypt
so you are saying that
cos(x+y) = cos(x) cos(y) - sin(x) sin(y)
cos(x-y) = cos(x) cos(y) + sin(x) sin(y)
now we sum
cos(x+y) + cos(x-y) = 2cos(x)cos(y)
terse sand
Multiply by -1 on both sides now
ocean crypt
2sin(x)sin(y) = cos(x-y) - cos(x+y)
terse sand
We have the sum-to-product for sin
You could change the function on RHS by subtracting or adding 90° & taking care of sign to get the whole equation in terms of sin function
terse sand
cos(x+y) + cos(x-y) = 2cos(x)cos(y)
Or you could do that here as well directly to obtain the equation for sin
ocean crypt
ok x-y = b => x = b+y
terse sand
no
ocean crypt
why
ocean crypt
yes i have to sub x into x+y = a
terse sand
Yes you can do that to get y
ocean crypt
b+y+y = a so i get 2y+b = a
terse sand
yes
ocean crypt
2y = a-b => y = (a-b)/2
terse sand
Yes
terse sand
Yes
W
You are getting the idea, I am not focusing much on calculations right now just to tell you
ocean crypt
ok
terse sand
That is the derivation of multiplication identities given that one knows about addition ones
Addition ones could be proven by taking a unit circle
ocean crypt
i think i will memorize cos(a+b) and sin(a+b)
just for now
cos(a+b) = cos(a)cos(b) - sin(a) sin(b)
terse sand
i think i will memorize cos(a+b) and sin(a+b)
Yes you can do that as well, just remember that cos is an even function & sin is an odd function, it helps in sin( x +- y )
ocean crypt
sin(a+b) = sin(a) cos(b) + sin(b) cos(a) ... ?
ocean crypt
ok nice
terse sand
Sin one is easier actually, you just have to use both pairs
a , b with sin , cos
b , a with sin , cos
Combination not permutation If I am Correct idk
terse sand
ok now suppose i remember cos(a +- b) and sin(a +- b)
Good
terse sand
can i get all the 4 idetites
Yes, cos into cos
Sin into sin
Cos into sin
Sin into cos
ocean crypt
ok we got sin into sin right?
terse sand
You could just remember one, and change one function into other function ig but we will get the term of pi in that, so it wont look much nice
It should be possible too
ocean crypt
cos(b) - cos(a) = 2sin((b+a)/2)) sin((a-b)/2) i mean we got this
terse sand
Yeah
terse sand
ok we got sin into sin right?
We got cos into cos as well
ocean crypt
what if its cos(b) + cos(a)
terse sand
cos(x+y) + cos(x-y) = 2cos(x)cos(y)
Here
ocean crypt
how can i switch quickly
cos(x+y) + cos(x-y) = 2cos(x)cos(y)
a = x+y
b = x-y
x = a-y
b = (a-y) - y
b = a-y-y = a - 2y
-2y = b-a
2y = a-b
y= (a-b)/2
terse sand
a = x+y
b = x-y
( you could add both of these to get result directly )
terse sand
Yes
terse sand
Yes
ocean crypt
ok now the one with the sines are left
terse sand
You could derive sin into cos by using sin(x +- y
Exactly same thing
Just for different addition identities
ocean crypt
you mean cos(x + pi/2) = sin(x) ?
terse sand
Yes
ocean crypt
can you remember me
terse sand
like why this is wrong
Do you know about graphical transformation
terse sand
There are two ways to verify this,
-
graphical transformation
-
using the expansion of cos(x + y)
terse sand
I think itll be too long to tell you whole about graphical transformation
It is just by drawing graph of LHS & RHS to see if it'll be same
If it is, then there's equality
terse sand
Okay
ocean crypt
so what where you saying
do i have to use cos(x-pi/2) = sin(x) ?
to get the formulas for the sines ?
ocean crypt
ok whats the cleanest way
terse sand
You could use sin(x + y) & sin(x - y)
ocean crypt
ok
sin(x + y) = sin(x) cos(y) + sin(y) cos(x)
sin(x-y) = sin(x) cos(y) - sin(y) cos(x)
i add them
sin(x+y) + sin(x-y) = 2 sin(x) cos(y) (1)
ops
terse sand
Yes
ocean crypt
and now ?
sin(x+y) - sin(x-y) = 2 sin(y) cos(x) (2)
a = x+y
b = x-y
sin(a)+sin(b) = 2 sin((a+b)/2) cos((a-b)/2) (1)
terse sand
sin(x+y) - sin(x-y) = 2 sin(y) cos(x) (2)
How did you write this
ocean crypt
i substracted
terse sand
Okay
ocean crypt
sin(a) - sin(b) = 2 sin((a-b)/2) cos((a+b)/2) (2)
ocean crypt
Nice!
terse sand
This is the way
ocean crypt
Thanks so much
terse sand
I hope you did the calculations well
ocean crypt
i think so
terse sand
Just in case, check the result from some trusted source
Remember that cos is even function and sin is odd
Okay ?
ocean crypt
yes
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azure harbor
wild cape
already b& but fsr it didnt auto delete
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normal moss
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merry lintel
Hello
I was given the function $$(lnX)^2 - 2ln{X}$$
In section A, I was asked to find the domain of the function, which I found was $$X > 0$$
After that,
B. Find extreme points of the function (if there are any) and define them as either Min or Max
C. Find the x-intercepts of f(x)
D. Draw the function.
E. Find a domain where both f(x) and f'(x) are positive.
F. $$g'(x) = f(x)$$
Within the domain $$X > 0$$
Find the extreme points of g(x), and define them as either Min or Max.
As I explained above, I got through A, but I'm unsure as to how I'm supposed to continue.
Isnt $$(lnX)^2 = 2lnX$$? Or did I misremember? I dont see how I'm supposed to find an extreme point when $$f(x) = 0$$
safe radishBOT
Hello
I was given the function $$(lnX)^2 - 2ln{X}$$
In section A, I was asked to...
flat frigateBOT
Kaz
molten acorn
$lnX^2 = 2lnX$, not $(lnX)^2=2lnX$
merry lintel
$lnX^2 = 2lnX$, not $(lnX)^2=2lnX$
is $$lnX^2$$ where the X is squared, not the entire thing?
flat frigateBOT
Kaz
quiet plume
is $$lnX^2$$ where the X is squared, not the entire thing?
There is a difference between $\ln(x^2) = 2\ln(x)$ and $(\ln(x))^2$, which is just $\ln(x) \cdot \ln(x)$.
flat frigateBOT
Azyrashacorki
molten acorn
is $$lnX^2$$ where the X is squared, not the entire thing?
yep
flat frigateBOT
~ 阿莲 <• a.lien •>
merry lintel
I see
so um how would I find the derivative of that? Would it be $$f'(x) = lnX - (1/x)$$
flat frigateBOT
Kaz
flat frigateBOT
Kaz
merry lintel
?
merry lintel
I encountered that the other day but I still dont really understand what that is
quiet plume
Or the product rule if you interpret (ln(x))^2 as (ln(x)) * (ln(x))
merry lintel
I dont understand
quiet plume
The chain rule is important to know, but you can use the product rule if you're more familiar with that
$(f(x) \cdot g(x))' = f'(x)g(x) + f(x)g'(x)$
flat frigateBOT
Azyrashacorki
quiet plume
Here you have $\ln(x) \cdot \ln(x)$
flat frigateBOT
Azyrashacorki
merry lintel
wait is that product rule or chain rule
quiet plume
This is the product rule
merry lintel
I see
merry lintel
**Azyrashacorki**
im familiar with this
what is the chain rule though
Since its come up a few times I should probably learn it
quiet plume
When you have functions $f(x)$ and $g(x)$ that are composed together into, say, $f(g(x))$, you can compute the derivative of $f(g(x))$ via the chain rule, namely $$(f(g(x)))' = f'(g(x)) \cdot g'(x).$$
flat frigateBOT
Azyrashacorki
quiet plume
In this case, you can interpret $(\ln(x))^2$ as the functions $f(x) = x^2$ and $g(x) = \ln(x)$ being composed together, giving you $f(g(x)) = (\ln(x))^2$.
flat frigateBOT
Azyrashacorki
merry lintel
why is f(x) $$x^2$$
flat frigateBOT
Kaz
winged cloak
why is f(x) $$x^2$$
if f(x) = x² then f(ln(x)) = ln(x)²
merry lintel
I dont get it
merry lintel
$$y = x^2$$ ?
flat frigateBOT
Kaz
merry lintel
?
quiet plume
If you agree that $f(g(x)) = (\ln(x))^2$ when $f(x) =x^2$ and $g(x) = \ln(x)$, then that's pretty much it.
flat frigateBOT
Azyrashacorki
quiet plume
They're just two functions which, when composed, give (ln(x))^2.
f(x) = x^2 is a function
g(x) = ln(x) is a function
f(g(x)) = (ln(x))^2 is a function, the composition of f by g.
You could write f(u) = u^2 if having x in there confuses you.
It doesn't change the definition of the function sending (something) to (something)^2
merry lintel
I have in my formula sheet $$[f(u(x))]' = f'(u)*u'(x)$$ Is that it?
flat frigateBOT
Kaz
quiet plume
yes
merry lintel
I see
Unfortunately I have to go but I think I might understand it now maybe? I'll give the question another try later.
Thank you for the help :)
.close
safe radishBOT
safe radishBOT
Available help channel!
normal moss
safe radishBOT
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torn umbra
Hello im back now I dont know what implicit differentiation is :D
worldly lantern
do you know what an implicit function is ?
worldly lantern
its basically when you cant isolate y purely in terms of x
torn umbra
Oouuhhh its like using 2 different equation to single out a y or x function
small sandal
your solving for dy/dx in an equation where you have both x and y as variables
worldly lantern
a simple example would be
y²x + x²y = 1
do you want me to work through this example to show you how its done ?
torn umbra
Yes pls
worldly lantern
ok so
in the equation y²x + x²y = 1
you wanna find the derivative of y wrt x
i.e. dy/dx
so you just differentiate the whole thing
d(y²x) + d(x²y) = d(1)
solving this out using chain rule and product rule looks something like
2yxdy + y²dx + 2xydx + x²dy = 0
the same thing couldve been done wrt x
which would give us
2yx(dy/dx) + y² + 2xy + x²(dy/dx) = 0
solve for dy/dx
the idea to differentiate an equation , by treating the variables as functions
is implicit differentiation
small sandal
Another way to find $\frac{dy}{dx}$ \ for a multivariable function $f(x, y)$ is to set $f(x, y) = 0$ and use $\frac{dy}{dx} = -\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}}$
1sec lemme fix this
torn umbra
Ook
small sandal
there we go
flat frigateBOT
KB
worldly lantern
proof ?
small sandal
idk the proof of it, but it is a way you can find dy/dx of a function of this form
torn umbra
:0
small sandal
take your example $y^2 x + x^2 y = 1$ Let $f(x, y) = y^2 x + x^2 y -1$ and $\frac{df}{dx} = -\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}}$ We now solve these partial derivatives, starting with f with respect to x yields $\frac{\partial f}{\partial x} = y^2 + 2xy$ and for f with respect to y yields $\frac{\partial f}{\partial y} = 2yx + x^2$ putting this into our formula we get $\frac{dy}{dx} = -\frac{y^2+2xy}{2yx+x^2}$ and then from there we simplify
flat frigateBOT
KB
torn umbra
Oouuhhh so that wut it means
small sandal
but unless you have been tought this method follow Smith's way
safe radishBOT
safe radishBOT
Available help channel!
echo gazelle
how do I find the integral when they’re multiplied together
safe radishBOT
how do I find the integral when they’re multiplied together
echo gazelle
did I do this wrong
left folio
did I do this wrong
U = x^2
echo gazelle
Y
left folio
Dv = cosx dx
median vigil
well i don't think the integration by parts is wrong per se but it doesn't seem very helpful
median vigil
ideally you would like to reduce the power of x so that evendually it would be reduced down to x^0
echo gazelle
only when its rlly obvious like when theres lnx
median vigil
there are various acronyms out there that you can memorize if you want, e.g. LIATE
median vigil
log, inverse trig, algebraic (powers of x), trig, exponential. you pick the first one that applies to be u
safe radishBOT
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Available help channel!
echo gazelle
how do i know when to use usub and when to use integration by parts
stoic torrent
Depends on the integrand
echo gazelle
how do i know based on the integrand
mortal sandal
if integration by parts works, then use integration by parts
echo gazelle
do i use usub if something can cancel out
mortal sandal
well, it's probably easier to check u-sub first
echo gazelle
like the x
mortal sandal
if you have a clear du then usub is a good idea
echo gazelle
do i use usub if something can cancel out
wait no
small sandal
after the first few steps of either you'll realise your either making progress or not
mortal sandal
if not it may still be possible
stoic torrent
see in integration by parts u have to see whether the two functions that you take give a good second term, as in after differentiating u and integrating v, the product is easy to integrate or not
As for subs, normally there is a weird or long denominator/ creepy random term in numerator
echo gazelle
is this u sub or ibp
stoic torrent
Then sub generally works as you most of the time it gets cancelled out
stoic torrent
is this u sub or ibp
Sub
echo gazelle
:C
stoic torrent
See numerator has a creepy term
echo gazelle
hows it creepy its just a chil term
stoic torrent
If you differentiate the (1+ lnx) do you get 1/x right?
stoic torrent
If you differentiate the (1+ lnx) do you get 1/x right?
Thats one way of thinking
echo gazelle
yah
stoic torrent
so we can say that sub will work out beautifully here
stoic torrent
hows it creepy its just a chil term
I mean if we try ibp, the integration wouldnt be so nice for (lnx)^5
echo gazelle
o
stoic torrent
If you differentiate the (1+ lnx) do you get 1/x right?
Try to observe and pick up on this thinking logic asw
helps to identify quickly
but sometimes when there is a big mess of trigo functions, dividing by sec or tan helps out too
echo gazelle
If you differentiate the (1+ lnx) do you get 1/x right?
is that just so it cancels
it cancels easy
but I had this case where it didn’t cancel but I could still use usub
burnt notch
In fact, there's no rule about cancellation 🤔
echo gazelle
o
burnt notch
I don't know how you've been taught u-sub
stoic torrent
Oh right i forgot about this case!
Sometimes the other part in x can be simply rewritten as the substitued new var
But again this comes with pratice
its one of those things you pick up while solving qsns
burnt notch
$\int (x+1) \cos(3x²+6x) dx$
flat frigateBOT
Alberto Z.
burnt notch
**Alberto Z.**
Another example, how would you integrate this? @echo gazelle
stoic torrent
-# i got confused why u pinged me😭😭
stoic torrent
do you see that it would be painful to integrate cos(3x^2 +6x)?
if you need more hints ask Alberto directly
echo gazelle
echo gazelle
Another example, how would you integrate this? <@1280172065118228531>
isnt thsi the cancelling...
safe radishBOT
@echo gazelle Has your question been resolved?
safe radishBOT
@echo gazelle Has your question been resolved?
ionic blaze
how do i know when to use usub and when to use integration by parts
both work
usually usub is limited tho, u try to use usub first and if that doesn't work go IBP and if that doesn't work
ionic blaze
erm
cursed but it's right...
you could just do du=6x+6 dx => du/6=(x+1)dx and remove x+1 dx entirely-
echo gazelle
iksnt that what i did .............................................................................
ionic blaze
iksnt that what i did .............................................................
I have no idea what iksnt means but
it's
I don't think it's wrong
it ain't wrong I guess
ionic blaze
iksnt that what i did .............................................................
u got another question or u g
safe radishBOT
@echo gazelle Has your question been resolved?
primal bone
I have no idea what iksnt means but
Context clues - it's a typoed "isn't"
primal bone
iksnt that what i did .............................................................
It is technically what you did, but inside an integral it's generally bad form to have both u and x involved at the same time
Which is why it's often considered better to deduce all the things you need to substitute in before the integral-rewriting
primal bone
erm
cursed but it's _right_...
you could just do du=6x+6 dx => du/6=(x+1)dx and ...
@echo gazelle so see here - there is a (x+1) (times) dx term in the integral, so we can replace those in a fell swoop with du/6
Instead of having to cancel out terms that involve x in an integral that's "talking" about u (by "talking", I'm referring to the fact that there's a du at the end, so we're integrating with respect to u)
fringe dock
safe radishBOT
fringe dock
can someone please help me on this question
ill break down my knowledge for part a
i know that sin^2x can be written as
well
i know that
Cos2A = cos^2x - sin^2x
sin ^ 2x = cos^2x + cos2A
but now im in a dead end...
royal kiln
sin ^ 2x = cos^2x + cos2A
Convert cos²x into sin²x now
royal kiln
I meant sin²x=1 -sin²x +cos2x
U get this right ? After converting cos²x into sin²x
By shifting i meant taking the sin²x from right to left
fringe dock
I meant sin²x=1 -sin²x +cos2x
U get this right ? After converting cos²x into s...
yeah
royal kiln
By shifting i meant taking the sin²x from right to left
And did u understand this?
fringe dock
yes i do
wait
ill shift to the left
to get
2Sin^2x = 1 + cos2x
right
yes
i think so
royal kiln
Yes then divide both sides by 2
royal kiln
sin ^ 2x = cos^2x + cos2A
Sorry there is an issue
royal kiln
i forgot 1
U did?
fringe dock
i didnt
royal kiln
Right,yeah
royal kiln
Yes
fringe dock
whats next
fringe dock
but wait
its not done
wait i think it is
omg i misread the question lool
i thought we had to
turn it allll into sin2x
thats why i was confused and found it hard
royal kiln
Why sin2x?
fringe dock
i mean
cos2x
i just misread the question
it said in terms of cos2x
not into cos2x
okay so
royal kiln
Yeah
fringe dock
is part a helping me for part b?
royal kiln
Do u need help with b too?
fringe dock
no i dont think so
royal kiln
is part a helping me for part b?
It can ,if u use what u found
There r other alternatives too
fringe dock
It can ,if u use what u found
why cant i integrate sin^2x directly?
reverse chain rule
royal kiln
why cant i integrate sin^2x directly?
Did u try?
queen parcel
reverse chain rule
reverse chain rule is essentially u-sub.
power rule doesn't work since it's not x by itself to a power
fringe dock
reverse chain rule is essentially u-sub.
power rule doesn't work since it's not...
whats the difference between u sub and reverse chain rule i always get confused which to use
fringe dock
Then show it
U can easily do it without any sub honestly, if u use what u found
yeah its easy to do with what i know
but
queen parcel
whats the difference between u sub and reverse chain rule i always get confused ...
they're basically the same thing: when you say reverse chain rule, what do you think in your mind?
what does it mean to you?
fringe dock
what does it mean to you?
nothing really i just needed clarification
queen parcel
oh yeah no worries
yeah, u-sub is the calc 2 method of undoing chain rule
chain rule -> derivatives, u-sub -> integrals
fringe dock
but
the main issue is rewriting it into a term i can do easily
like here we rewrote it
but its hard to notice
if there was no part a and it was just one question i couldnt have done it
like this,,,
queen parcel
oh, how should i know that i need to rewrite it instead of doing u-sub or something else?
royal kiln
if there was no part a and it was just one question i couldnt have done it
U could hv since they didnt specify that u need to use u-sub
U still could hv done u sub ,it's practice that matters
safe radishBOT
@fringe dock Has your question been resolved?
late tree
why when working with intergrals and doing u sub we have du equal the derivative of u
safe radishBOT
why when working with intergrals and doing u sub we have du equal the derivative...
primal bone
<@&268886789983436800> do ya thing
broken forum
lets say i have y = 3x
what is dy/dx
late tree
huh
broken forum
i could solve for dy in terms of dx
dy/dx is a fraction
if i have that dy/dx = 3
i could say dy=3dx
muted sapphire
a small change in y
is equal to 3 * the small change in x
aka y = 3x
dy = 3 dx
dy / dx = 3
late tree
i still dont undertsand ignoring the fact your just using it as a fraction wich ig works but i still dont know how that makes du the derivative of u
late tree
ok
muted sapphire
how about you try an exercise and tell us what part u don't get
late tree
u=x^2?
muted sapphire
yes!, and now?
late tree
du/dx=2x
muted sapphire
good, and now?
late tree
idk
muted sapphire
du / dx = 2x
In actual maths, you can't really just multiply by dx
would you like to know how it really works?
late tree
yea
muted sapphire
well do you knwo the fundimental theorem of calculus?
late tree
umm yeah i know it but i dont know why its true
muted sapphire
well it states this:
primal bone
(...where F is any function that satisfies F'(x) = f(x) )
late tree
obv
primal bone
(still should be said - I've had students not be able to tell the difference between F and f somehow)
late tree
yeah idk notation in calc hits a wall in readability compared to algebra
safe radishBOT
@late tree Has your question been resolved?
safe radishBOT
@late tree Has your question been resolved?
royal kiln
<@&268886789983436800>
loud wren
<@&268886789983436800>
upbeat drum
Smh i was gonna use this channel
safe radishBOT
safe radishBOT
Available help channel!
opaque shard
need help with this
safe radishBOT
need help with this
royal kiln
What have you tried?
royal kiln
If u can find DE ,it means u can find the coordinates of D and E right?
opaque shard
yeah
royal kiln
What do you get?
opaque shard
uh let me think
royal kiln
x =k
And y is 0 so the equation would be x=k for DF
royal kiln
Do u k the equation of a straight line which is parallel to y axis?
royal kiln
ohh i get it
Yea?
opaque shard
cause x of D is k and its just x=k
royal kiln
Yes