To go from f to F you integrate
#help-0
native cloud
a^(n-1)/n-1
karmic rapids
a^(n-1)/n-1
what's the "inverse" of integration
ocean sealBOT
illuminator3
karmic rapids
why did you pull that formula out your ass
that's if you want to integrate a polynomial
native cloud
I saw it in the book
karmic rapids
you have $\int f(x)dx$, how do you get $f(x)$
ocean sealBOT
illuminator3
native cloud
f'(x) + C?
ocean sealBOT
illuminator3
native cloud
Change y and x
karmic rapids
no, you just apply $f$ to it
ocean sealBOT
illuminator3
karmic rapids
$f(f^{-1}(x)) = x$
ocean sealBOT
illuminator3
native cloud
I see
karmic rapids
do you understand that
native cloud
Yes
karmic rapids
in our case f is differentiation
and f^{-1} is integration
if we have $\int f(x)dx$
ocean sealBOT
illuminator3
karmic rapids
how do we get $f(x)$
ocean sealBOT
illuminator3
native cloud
d/dx ∫f(x)dx ?
karmic rapids
yes
native cloud
Oooo
karmic rapids
differentiation is the "inverse" of the integration
native cloud
Okie
karmic rapids
in your question you have $g(x) = \int \limits_1^x f(t)dt = F(x) - F(1)$ where $f(t) = \sqrt{2t+7}$
you want to find $g'(x)$ so you can just differentiate $F(1) - F(x)$
ocean sealBOT
illuminator3
mortal trellis
You messed up the order of F(1) and F(x)
karmic rapids
fuck this is the third time this has happened this week
ocean sealBOT
illuminator3
karmic rapids
remember that $F(x) = \int f(x)dx$
ocean sealBOT
illuminator3
karmic rapids
you don't need to differentiate f(t)
native cloud
So I just substitute?
mortal trellis
g(x)=F(x)-F(1). How do you find g'(x)
native cloud
$\int \sqrt{2\left(9\right)+7}-\int \sqrt{2\left(1\right)+7}$
ocean sealBOT
AirToastie
mortal trellis
What do you want to sub
karmic rapids
$\int \sqrt{2\left(9\right)+7}-\int \sqrt{2\left(1\right)+7}$
don't substitute yet
native cloud
Okay
karmic rapids
just find $g'(x)$ for now
ocean sealBOT
illuminator3
karmic rapids
g(x)=F(x)-F(1). How do you find g'(x)
@native cloud
native cloud
F'(x) - 0
karmic rapids
F(1) is a constant
yes
and what's F'(x)?
the prime signals differentation
and the uppercase letter integration
$F'(x) = \frac{d}{dx} \int f(x)dx$
ocean sealBOT
illuminator3
native cloud
But how do I calculate that?
native cloud
d/dx ∫(d/dx ∫f(x)dx) dx ?
karmic rapids
no
what happens if we differentiate something we've just integrated
remember that differentiation is the inverse of integration
native cloud
what happens if we differentiate something we've just integrated
f(x)?
ocean sealBOT
illuminator3
karmic rapids
do you understand that ?
native cloud
Yes
ocean sealBOT
illuminator3
native cloud
G(x)
ocean sealBOT
illuminator3
ocean sealBOT
illuminator3
native cloud
f(x)
ocean sealBOT
illuminator3
native cloud
0?
karmic rapids
why
native cloud
It's a constant..?
mortal trellis
Are you just guessing stuff?
g'(x)=f(x). Then what is g'(9)
karmic rapids
yes
native cloud
Woah this is confusing
karmic rapids
what part
native cloud
I don't know what I just calculated
karmic rapids
you calculated g'(9)
karmic rapids
But why am I just substituting?
given f(x) = 2x, how do you calculate f(2)
native cloud
Substitute 2 in x
karmic rapids
Yes
native cloud
Oh so this problem is teaching me how to calculate
karmic rapids
given g'(x) = sqrt(2*x+7), how do you calculate g'(2)
native cloud
given g'(x) = sqrt(2*x+7), how do you calculate g'(2)
1/2(2(2) + 7)^-1/2 * (2)
mortal trellis
This exercise is teaching you that differentiation is the inverse of integration
native cloud
Oh
native cloud
1/2(2(2) + 7)^-1/2 * (2)
Is this right btw?
mortal trellis
And that you can compute the derivative of an integral without having to calculate the integral and the differentiate it
No
Why are you differentiating that
native cloud
It's a prime?
mortal trellis
g'(x) is already given. You just need to substitute
mortal trellis
Yes
next flax
lone heartBOT
next flax
versed crater
Think about what's happened to the graph
How has the x axis been affected
Has the y axis been affected at all
next flax
hmmm
versed crater
And remember whatever you do in the brackets is opposite
next flax
no the y axis has not been affected
versed crater
I.e a stretch is a fraction
versed crater
Firstly 1/2 would stretch the curve by a scale factor of 2
So yes
That looks right
next flax
Firstly 1/2 would stretch the curve by a scale factor of 2
there is no f(1/2x)
next flax
there is only f(-1/2x)
next flax
erm
versed crater
In the y axis
next flax
afaik there's not much of a discernment of a reflection
it looks identical in terms of shape
versed crater
Reflection in the y axis not x
next flax
but uhh
it's not in the positive axis?
its literally no different
in terms of y axis to me
versed crater
Ok forget the 1/2 for a second
Imagine you have the curve f(X)
And f(-x)
For X= 2 f(X) gives f(2) but for F(-x) we get f(-2)
So it is a reflection in the y axis
Let's say f(X) = x
Or the line y= x
f(-x) = y=-x
A reflection in the y axis
next flax
i still don't understand i really apologize
maybe you can tell me the difference between -1/2f(x) and f(-1/2x)
next flax
Imagine you have the curve f(X)
that's also something i'm a bit stumped on
versed crater
So
If it's outside the bracket
It's a stretch in the y axis
If it's inside
It's a stretch in the x axis
versed crater
i appreciate your help man
Nw
next flax
if i divided 30 / 0.75, it becomes 40
if i multiply 30 * 3/4, it becomes 22.5
which one?
the first one right?
next flax
Nw
because we need to divide, opposite operation of what is provided for that numbe
.close
ocean sealBOT
AimaneSN
toxic wigeon
So I have my mathematics exams coming up in some days and I have no clue of what to do , and like the syllabus is 4 chapters
So I’m starting off with the chapter expansions
And then I realised I do not remember learning anything from the past grades
I am currently in 9th grade
So like I asked my friend about it and all and he sent me this
And the problem is
I absolutely do not get what any of these mean unless someone explains me
So I went to trustworthy google and YouTube
And I might be so dumb that I couldn’t find the videos even if I was continuously searched for one hour
So my question was , are there any good videos of people explaining the formulas so I’ll remember them?
Sorry if I made any typos or you can’t understand some parts of my question , English isn’t my first language
keen bear
So I have my mathematics exams coming up in some days and I have no clue of what...
these look like memorization tables for common expansions of binomials. Have you learned the FOIL method?
toxic wigeon
I have not learned the foil method
keen bear
ok, it looks like there are some trinomials in here as well, so you may be better off learning the table method, but the basic principle is that when multiplying two polynomials, you need to multiply each part of the first one by each part of the second one. If you can understand this principle, you'll be able to absorb much of this without having to memorize
using the first one as an example:
$(a+b)^2 \
=(a+b)(a+b) \
=a(a+b)+b(a+b) \
=a^2+ab+ab+b^2 \
=a^2+2ab+b^2$
ocean sealBOT
GoldenPhoenix
keen bear
notice how for lines three and four I distributed the entire first binomial into the second one.
This is often done through the FOIL method, where FOIL stands for First, Outside, Inside, Last, as a reminder to multiply each part of the first set of parentheses with each part of the second set.
toxic wigeon
I am very confused still , that’s the thing that’s stopping me from understanding
I feel like I have neglected middle school algebra and this is why I’m failing to understand the ways it’s being done
But thanks for mentioning the methods , I’ll try to look for videos that explain it with these examples , if you know any please send the links for them
keen bear
can you name the specific part that's confusing you, or is it just a general fuzziness?
toxic wigeon
I guess general fuzziness because i haven’t done math in a long while
Like I understand what some sentences try to mean but when I imagine putting them into practice I just don’t get it and then I reflect back it turns out I don’t get what the sentence means from the start
keen bear
that's fair. I recommend looking up some more basic concepts like distribution for multiplication, which is the foundation of how this works
fervent mountain
hello
fervent mountain
is anyone now
marsh rapids
fervent mountain
keen bear
,rotate
ocean sealBOT
fervent mountain
why it is not minus 1
keen bear
you gotta be careful, you can't take a cube root of a binomial like that
untold ore
princeton lifesaver page69 lol
keen bear
when you have a plus sign under a radical, you need to make sure that the entire thing is an expanded power before taking the root, you can't just do it term by term
keen bear
why it is not minus 1
where would the -1 be?
fervent mountain
Just leading power of that
thx bro
say...I only do that step by step?
I see now
it is not a standard-form of poly
limpid ingot
where do i start
lone heartBOT
where do i start
languid bolt
wait nvm
limpid ingot
so dab is 95?
languid bolt
nvm forget what i said
limpid ingot
ok
languid bolt
my brain lagging, i was seeing a trapezium idk how
but yes it's the same
dab is 85
limpid ingot
k
untold ore
would angle da b and dcb be the same
yes, since DAB + CBA=180, CBA+A=180
untold ore
just use exterior angle theorem x=A+ABD
lone heartBOT
lone heartBOT
Available help channel!
bronze hazel
lone heartBOT
clear stump
vertices?
oak perch
Just calculate those three dot products
lone heartBOT
@bronze hazel Has your question been resolved?
keen socket
<PQR
so
basically find vectors PQ, QR, PR
use the dot product rule such that it equals 0
then you know which angle is 90degrees
which is angle PQR
azure trout
lone heartBOT
azure trout
How would I express answers in rational form?
languid bolt
do you know what a rational form is?
heady pollen
i would just calculate it tbh
2^6=64
1^4=1
so you get (16*64)^1/2
and note that 16=4^2 and 64=8^2
so sqrt(1664)=48=24
heady pollen
or just say that sqrt(16*x^6y^4)=4x^3y^2
heady pollen
i think so
hard mountain
Doesnt look right to me
languid bolt
it is
languid bolt
Doesnt look right to me
wdym
hard mountain
Question asked specifically to express in rational form with positive exponents
hard mountain
one second writing it up
indigo lotus
and?
ig he meant to say that you can write it in the form (54)^-1
but representing the answer as 1/54 is correct
as its just the correct representation of the answer that the question is asking for and 1/54 is rational and doesn't have any negative exponent
indigo lotus
so sqrt(16*64)=4*8=24
and also we can write 24 as (1/24)^-1
so would you say that 24 is incorrect answer?
heady pollen
i think so
hard mountain
and also we can write 24 as (1/24)^-1
yes, (16 * 2^6)^(1/2) = (2^4 * 2^6) ^(1/2) = (2^10)^1/2 = 2^(10/2) = 2^5
for a)
languid bolt
yeah it's 32
azure trout
so did I get mine right?
languid bolt
so did I get mine right?
yes you did
hard mountain
yes, thats incorrect. Where did the (2^2) in the first line come from?
hard mountain
yes so if you rewrite the numbers inside the paranthesis into rational form you should get
(2^4 * 2^6 * 1) ^(1/2)
seems like you wrote 2^2 instead of 2^6
keen socket
@hard mountain very long way of explain
@azure trout 7a) (16x^6y^4) = 4x^3y^2, sub x=2 y=1 then u get 4(2)^3 which is 2^5
azure trout
2^5 isnt rational form
azure trout
oh
azure trout
is this right
hard mountain
going to check it now
7b) can also be written in rational form with positive exponents
keen socket
is this right
this is right
hard mountain
1/54 is the right value, but isn't what the question was really asking for. To write it in rational form: 1/54 = 1/(2 * 27) = 1/(2 * 3^3)
azure trout
nvm it right
wdym
azure trout
Same with 8a)
but is it the right value
languid bolt
1/54 is the right value, but isn't what the question was really asking for. To w...
random question, are you saying that 1/54 is the wrong form?
hard mountain
yes
languid bolt
how?
hard mountain
think it should be 1/(2 * 3^3)
languid bolt
reason?
hard mountain
question asked to express it in rational form with positive exponents
hard mountain
you're right, it does technically meet the requirements
hard mountain
yeah forgot everything i said. 8a) is correct
1/54 is a technically a rational form with a positive exponent. A rational exponent form, however, would have been 1/(2 * 3^3)
hard mountain
1/54 is irrational
rational number = a/b for some natural numbers a, b ?
keen socket
it wants it in exponential form
languid bolt
rational number = a/b for some natural numbers a, b ?
rational number = a/b for every real number where b not equal to 0
languid bolt
it wants it in exponential form
it doesnt say that it wants it in exponential form
it says that it needs to have a positive exponent
keen socket
lol ok bro write that as the answer
hard mountain
hhh, its always an interpretation issue
hard mountain
1^1 / 54 ^ 1
languid bolt
but it's exponent is positive
keen socket
like 7a) was 2^5
hard mountain
like 7a) was 2^5
thats assuming they wanted "rational exponent form", but the question actually says "rational form with positive exponents"
languid bolt
writing 1^1 makes no sense
but it's still a positive exponent
keen socket
bruh fo sho
hard mountain
writing 1^1 makes no sense
thats what i thought too. Question is should we answer what we think they want, or what the question actually asks for ?
languid bolt
depends on the teacher ig
lone heartBOT
@azure trout Has your question been resolved?
lone heartBOT
lone heartBOT
Available help channel!
keen python
I've solved this but got - 4th root of 2
where as my prof got it positive
keen python
why did he do it like this ?
i instead got this
sqrt( x * ( 1/6x - 1)) => sqrt(x) * sqrt(1/6x - 1)
what is correct now ?
whole belfry
keen python
wow ok
the 2nd step is already too much for me
and then the x = -n
never have seen this before
keen python
ok this makes sense
now why di the prof all of the sudden took the absolut value ?
whats the reason for that
whole belfry
you mean the part marked in yellow?
keen python
yes
lone heartBOT
@keen python Has your question been resolved?
keen python
but look sqrt(6-x) = sqrt(x * (6/x -1) ... why all of the sudden -x * (1-6/x)
ahhh by in my case....
sqrt(x * (6/x -1) -> sqrt(x) * sqrt(6/x -1)
sqrt(x) would be a complex number right ?
whole belfry
yes ,sqrt(x) is a complex number because x<0
keen python
but still we could just do
sqrt(|x| * (6/x -1))
whole belfry
but look sqrt(6-x) = sqrt(x * (6/x -1) ... why all of the sudden -x * (1-6/x)
the order of 1 and 6/x is reversed
keen python
ye but why
whole belfry
and -x is positive
whole belfry
sqrt(-x) is OK
keen python
so again its not complex number
I see I get it, since x < 0...
6/x - 1 would be an negative number, and that under the root, wouldnt work out
thats why we tricked around
whole belfry
correct
you should ensure the number under square root is always positive
the imaginary number is not considered
keen python
and -6/x does not need to be absolut
bc x is <0 thefor -6 / some number < 0 => positive
whole belfry
yes
whole belfry
hahaha actually no, you can directly get the answer from the second formula
whole belfry
but it's limited to this question because the largest power is an even number
change the variable is a safer method
my solution is simpler i think
whole belfry
the -∞ goes to +∞
whole belfry
only when x goes to -∞
keen python
other than going from -inf to inf, i dont see what it changes
but if we look at the professors approach
then going to +inf
would have been easier
so i wouldnt need to take care of -negatve value in root right ?
whole belfry
yes , but +∞ means you donnot need to consider the change of +and-
keen python
ye
thats the advantage
i will keep this in mind and also your root trick is also nice
thanks for the help !
lone heartBOT
Available help channel!
dawn quail
lone heartBOT
dawn quail
Is there a type-o with this last x value? -3 should be 3?
if I graph this it looks like this
vale wigeon
why is the graph a loop?
high rapids
Obvious typo lol
dawn quail
this MHF4U course is so buggy
high rapids
The thing you drew also fails the vertical line test hence not even a function
dawn quail
instructor is just reading off paper not explaining things, like throwing someone into the ocean with sharks
lol
yeah this instructor is a real a-hole 😦
.close
snow crystal
Can someone help me with this question please ?
lone heartBOT
Can someone help me with this question please ?
lone heartBOT
@snow crystal Has your question been resolved?
lone heartBOT
@snow crystal Has your question been resolved?
lone heartBOT
@snow crystal Has your question been resolved?
austere smelt
austere smelt
https://cdn.discordapp.com/attachments/983371916423356436/997258395386118225/Tri...
I missed a bracket there but sinx and tanx are being multiplied on the right hand side
lone heartBOT
@austere smelt Has your question been resolved?
alpine sable
https://cdn.discordapp.com/attachments/983371916423356436/997258395386118225/Tri...
I can't help, other than by telling you that it's not clear what the question is
Please state what the question is
austere smelt
Please state what the question is
it's a trigonometric identity where you need to make either side equal to the other
to prove that the statement is true
here I'm trying to make the left hand side equal to the right hand side ( sinx * tanx )
alpine sable
https://cdn.discordapp.com/attachments/983371916423356436/997258395872661534/Tri...
(1-cosx)(cosx+1)
=cosx+1-cos^2x-cosx
=1-cos^2x
Yet you wrote down cos^2x-1
austere smelt
(1-cosx)(cosx+1)
=cosx+1-cos^2x-cosx
=1-cos^2x
Yet you wrote down cos^2x-1
ooh
I am a professional dumbass
alpine sable
Welp in the end it's just a matter of adding a negative sign
alpine sable
I am a professional dumbass
Mood
worn fox
(1,2) and (2,1)?
yes i have given a counter example
(1,2) is not in R but....
im not seeing where the miscommunication here is
yes you want a counter example to show that the second relation is not symmetric
which means you needs to find (x,y) in R such that (y,x) is not in R
I gave you two ordered pairs
you tell me exactly what it means for a relation to be symmetric
(1,2) is in R but (2,1) is not, so not symmetric
azure trout
lone heartBOT
azure trout
How would I transform an appropriate base function for this?
chrome plank
What do you think is the base function?
azure trout
f(x)=x^2?
chrome plank
why did you choose that one?
chrome plank
You have a function which has been transformed
azure trout
Yes
chrome plank
Your job is to understand what transformations have been used and to obtain that function, but with the least amount of transformations possible
What transformations can you see in this one?
azure trout
Compression by a factor of 4
Translated down 7 units
Ummmm
That’s all I can tell so far lol
What do I do with the exponent? 😅
chrome plank
$2^{x-5}$
ocean sealBOT
Nonna
chrome plank
That looks like a translation on the x axis to me
azure trout
I’m not that familiar lol
chrome plank
Ok, let's say you have a transtation vector
which says how much you are translating a function on the x and y axis
ocean sealBOT
Nonna
chrome plank
When you have a function, to translate it by that translation vector you subtract each component to the variable you want to translate
Let's make an example
v(3, 2)
f(x) = y = 3x²
that becomes
(y - 2) = 3(x - 3)²
or
y = 3(x - 3)² + 2
azure trout
I’m so confused
chrome plank
Ok
let's say you want to translate f(x) = x² (totally unrelated function) and you want it to go 2 upwards and 5 on the right
it becomes
y - 2 = (x - 5)²
and usually is written like
y = (x - 5)² + 2
chrome plank
So how would we do that for 2^x-5?
2^(x - 5) probably means that they translated the function 5 steps on the right
azure trout
It just becomes 2x
chrome plank
do you mean 2^x?
azure trout
Yes oops 😭
azure trout
So how would I get my points from@the base function
chrome plank
What do you mean?
azure trout
Look at b)
chrome plank
Oh
What do they mean by "in mapping the base function onto the given one"? Do you just have to say which transformations took place?
azure trout
Also is it a vertical or horizontal compression
azure trout
What do they mean by "in mapping the base function onto the given one"? Do you j...
I think so
chrome plank
Also is it a vertical or horizontal compression
Stretches: g(x) = f(c · x) [c>1]
Compressions: g(x) = f(c · x) [0<c<1]
azure trout
I thought it was the other way around?
chrome plank
I thought it was the other way around?
Ayy, my bad ahah
azure trout
lol no problem
So what kind of compression is it? Horizontal or vertical
It looks horizontal
chrome plank
I'm pretty sure it's vertical
azure trout
Have a fabulous day!! 
chrome plank
nice
lone heartBOT
<#490557019623915520>
ocean sealBOT
dawn ridge
I i Donot got idea even to start in standard format.. Someone check out first few step?
lone heartBOT
@dawn ridge Has your question been resolved?
dawn ridge
<@&286206848099549185>
real gazelle
do you know how to row reduce a matrix
lone heartBOT
@dawn ridge Has your question been resolved?
lone heartBOT
real gazelle
lol what did you figure it out
lone heartBOT
Available help channel!
stable night
lone heartBOT
stable night
For question 4, how do i start with finding y = 1/f(x)?
Like do i find the intercepts first or
little drum
For question 4, how do i start with finding y = 1/f(x)?
Like do i find the inter...
for starters, write what 1/f(x) is... derive the asymptotes
little drum
then you may use the derivatives to figure the highs and the lows, pick out some points of the curve for your convenience and sketch it
little drum
so the veritcal asym is when ln(2x + 1) + 5 = 0 right?
hmm
stable night
then you may use the derivatives to figure the highs and the lows, pick out some...
ohh
if u sub the bottom equation for 0 the x is rly weird tho..
-.49663
little drum
btw, just so you know, you can think about f(x) as the graph of ln(x) moved 1/2 to the left and ln 2 + 5 upwards
it's just $y = \ln (x + 1/2) + (\ln 2 + 5)$
ocean sealBOT
Arya
stable night
oh right that makes sense
wait but do i need that when i wanna sketch the reciprocal?
stable night
so the veritcal asym is when ln(2x + 1) + 5 = 0 right?
wait also how do i find the vertical asym lol, is it this?
little drum
yeah
stable night
the answer is really weird thou, alot of decimal places
little drum
no worries? you just need to sketch a rough graph-
stable night
oh lol
okay so ive found that
now ill find the y intercept?
when do i know if i have to use differenciation to find the turning points?
little drum
now ill find the y intercept?
yes... points on the curve like: 'as x tends to infty', 'when y = 0', etc. helps understand how the graph roughly looks like
little drum
the icing on the cake is when you take derivatives, as the first derivative helps you check when the graph starts falling and when it starts rising
stable night
Oh so do i have to find it everytime i draw a graph?
Or only when the question explicitly asks for the rising/falling points
little drum
the second derivative helps you derive whether the graph is convex or concave in an interval
little drum
Or only when the question explicitly asks for the rising/falling points
I mean, it's suggested to do so whenever you're plotting a graph
little drum
looks correct
except, you could literally have used $\frac{e^{-5}-1}{2}$ instead of -0.4966
ocean sealBOT
Arya
little drum
Also
wait lmao
what's with the left end of your graph...
you need to make it so it looks like it's descending along the asymptote and not like as if it's about to intersect it-
little drum
stretch it just a little more so the steep descent is actually visible and you should be fine
stable night
little drum
great
little drum
,w plot y = 1/[ln(2x+1) + 5]
ocean sealBOT
little drum
,w ln e
ocean sealBOT
little drum
:p the 'ln' for wolfram is log as the natural base is e
if it's any other, it'd simply mention the base
stable night
ohh makes sense
all is good yes
Actually do u mind helping me check this part as well ◉‿◉
lone heartBOT
@stable night Has your question been resolved?
young patio
Hi,
I came across this question and I can't seem to figure out how to even start solving it. I don't even know how to search for methods for these types of questions.
Could someone possibly explain how I could solve these types of questions?
Please ping me if you reply, thanks!
shell patio
Hi,
I came across this question and I can't seem to figure out how to even start...
Let 'x' be the total quantity (in ml)
Sarah drank x/3 amy drank x/4
Given that (x/3) - ( x/4) = 100
Solving the equation, you get x=1200
So the answer is 1200-300-400 = 500
young patio
Let 'x' be the total quantity (in ml)
Sarah drank x/3 amy drank x/4
Given that ...
Wow, I never thought of it like that. Thanks for your help!
.close
turbid sigil
lone heartBOT
turbid sigil
Is this super easy? As in saying $f(x,y)=u(x,y)$ then $u_x=v_y=0$ and $v_x=0$ so $f^\prime(z)=0$?
ocean sealBOT
cali5nia
oak perch
Yeah u is constant therefore f is constant
turbid sigil
@oak perch thanks
lone heartBOT
Available help channel!
turbid sigil
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tribal oxide
lone heartBOT
rigid grail
segs
tribal oxide
how do i think about this?
last ether
Use exponent identity:
$$a^b = e^{b\ln{(a)}}$$
ocean sealBOT
Umbraleviathan
oak perch
$=5(1+(\frac{4}{5})^{n})^{\frac{1}{n}}$
ocean sealBOT
Cogwheels of the mind
tribal oxide
**Cogwheels of the mind**
(4/5)^n tends to zero?
oak perch
Yeah, but you need to be more rigorous: 1<=1+(4/5)^n<=3/2 for n large enough, so 1<=(1+(4/5)^n)^(1/n)<=(3/2)^(1/n)
Take limit on left and right hand side you will see the limit of the middle term which is what you are looking for
tribal oxide
Yeah, but you need to be more rigorous: 1<=1+(4/5)^n<=3/2 for n large enough, so...
where is the 3/2 coming from
tribal oxide
Take limit on left and right hand side you will see the limit of the middle term...
left and the right side of the inequality?
hello
oak perch
Or just
Use Hopital :
That limit equals the limit of
$e^{\frac{In(\frac{4}{5})(\frac{4}{5})^{n}}{1+(\frac{4}{5})^{n}}}$
ocean sealBOT
Cogwheels of the mind
oak perch
Therefore =e^0=1
oak perch
Take limit on left and right hand side you will see the limit of the middle term...
Never mind, I thought it was wrong, but it’s correct on second thought
tribal oxide
Therefore =e^0=1
the answer is 5
oak perch
left and the right side of the inequality?
Yeah , limit of 1 and limit of (3/2)^(1/n) are both 1
tribal oxide
i will get 5 but i have one thing im stuck on
oak perch
You forgot 5 in the front
oak perch
**Cogwheels of the mind**
5 multiplied by the thing whose limit is 1
tribal oxide
Yeah , limit of 1 and limit of (3/2)^(1/n) are both 1
ok so the middle limit is also 1?
oak perch
I took 3/2 cause it’s a number greater than 1, can be anything you like, 3/2, 2,7,100…
oak perch
ok so the middle limit is also 1?
Yeah
tribal oxide
**Cogwheels of the mind**
the l hopital method looks a bit confusing cuz we have not learnt much of derivatives yet, only some basics.. we applied it on this step?
oak perch
Yeah I brought up hopital cause I thought my first thought was wrong, but my first thought is actually correct so never mind that hopital thing
tribal oxide
Yeah I brought up hopital cause I thought my first thought was wrong, but my fir...
how would i find the limit of (1/n)(4/5)^n
n tend to infinity
oak perch
rigid grail
hey segss
oak perch
Actually it’s true for any a>0, people who answered that guy explained in their answers
tribal oxide
Actually it’s true for any a>0, people who answered that guy explained in their ...
ok but
tribal oxide
how would i find the limit of (1/n)(4/5)^n
what abt this
tribal oxide
https://math.stackexchange.com/questions/861197/if-a1-prove-that-lim-a1-over-n-1...
here what does n tend to?
oak perch
Infinity
tribal oxide
okay i see
tribal oxide
how would i find the limit of (1/n)(4/5)^n
ok i found this
by using l hopital💀 , this is a part of the basic differentiation i know
it comes 0
and i get 5
tribal oxide
by using l hopital💀 , this is a part of the basic differentiation i know
but i remember our teaher saying theres many places i cant use it, and i also doont know how to check when to use it
oak perch
0/0 or infinity/infinity
f/g should be one of those two kinds
And f’/g’ should also be of those kinds or it has limit
tribal oxide
not those
oak perch
What do you mean
tribal oxide
f and g differentiable in neighbourhood of a except possibly at a and find interval around n where g' is never zero except possibly at a
oak perch
I expand them to be real functions
f(x)=In(1+(4/5)^x)
g(x)=x
When limit f(x)/g(x) exists then limit f(n)/g(n) also exists and it equals limit f(x)/g(x)
tribal oxide
I expand them to be real functions
im not sure what this mean💀
oak perch
f(x)=In(1+(4/5)^x)
g(x)=x
Here
tribal oxide
as in what is a real function
oak perch
I mean I expand them from a function N—>R to (0,infinity)—>R
tribal oxide
f(x)=In(1+(4/5)^x)
g(x)=x
what does x tend to here?
oak perch
Positive Infinity
tribal oxide
I mean I expand them from a function N—>R to (0,infinity)—>R
is N set of natural numbers?
oak perch
Yeah
tribal oxide
we can change the domain?
oak perch
Why not I just did
oak perch
When limit f(x)/g(x) exists then limit f(n)/g(n) also exists and it equals limit...
Because of this
tribal oxide
When limit f(x)/g(x) exists then limit f(n)/g(n) also exists and it equals limit...
if x tend to positive infinity is this limit equal 0?
tribal oxide
When limit f(x)/g(x) exists then limit f(n)/g(n) also exists and it equals limit...
what is the benefit of replacing x by n
tribal oxide
Benefit? So that we can use hopital
why cudnt we use it before
tribal oxide
okay so when we use n, we talking abt the function N-->R, and when were talking abt x we mean function (0,inf)-->R
oak perch
Depends, this case it is functions from (0,infinity) to R, or [0, infinity) to R,
Can be an another interval , just depend on questions
We don’t ask why we do something, you can do whatever you want, as long as you make no mistake in your steps
tribal oxide
Depends, this case it is functions from (0,infinity) to R, or [0, infinity) to R...
are we talking about both f and g?
or just f
oak perch
Both
oak perch
Yeah
oak perch
Whatever you want, (1 million,infinity), (7π,infinity)
We only need the limit when x approaches infinity
(-1375926, infinity), whatever
Or just R
tribal oxide
💀 but that might be wrong still cuz the domain of f/g is R-{0}, i can see it wont have any effect on finding the limit
but thank you for your help, i have to go now
.close
median oar
does this logic work,
for all A, B is true (given)
if B is true, there is a chance for A to be true (conclusion)
karmic rapids
if B is true for all A then A must be true for a subset of B, yes
every cat is an animal but not every animal is a cat
median oar
how would i write this with logic symbols?
karmic rapids
$$\forall a \in A : B(a)$$
$$\not \forall b \in B : A(b)$$
ocean sealBOT
illuminator3
median oar
the first line is about for all a in the set A where B is true
second is there is some b in B where the set A contains it?
karmic rapids
second is there is some b in B where the set A contains it?
not for all b in B, A is true
=> there is some b in B for which A is true
median oar
in other words for some b in B?
median oar
and the A(b) part means for some b in B, A of this is true
this being b
so the b is true under condition A
in the animal example it would be there is some animal that is a cat
karmic rapids
and the A(b) part means for some b in B, A of this is true
yes
karmic rapids
in the animal example it would be there is some animal that is a cat
yes
median oar
ok, what's the difference between the small a and the big A
small one just means any one u select
karmic rapids
small a is one big A is all
median oar
ok ok 👍
thank you very much for the help lost so much of this from 2years ago haha
.close
lone heartBOT
lone heartBOT
Available help channel!
ebon bloom
According to textbook, the set S of all positive integers which are perfect squares is countably infinite with f:N->S: f(n)=n^2 as bijective function. Now, why is this function bijective? It isn't surjective. For instance, -2 can't be mapped to S.
sly mantle
@ebon bloom -2 is not in S
naive valley
N and S both contain only positive numbers, no?
lone heartBOT
lone heartBOT
Available help channel!
brave junco
Why does the integral of 1/e^x+8^x from 0 to infinity diverge?
vale wigeon
$\int_0^{\infty} \paren{\frac{1}{e^x} + 8^x} \dd{x}$?
ocean sealBOT
Ann
brave junco
Ope srry, nah. both e^x and 8^x in denominator
ocean sealBOT
tepid wagon
If I make 17 cents every 8 hours and 45 minutes how much would I make in a month? I am stuck on the entire problem because I don't have a brain.
lone heartBOT
If I make 17 cents every 8 hours and 45 minutes how much would I make in a month...
@tepid wagon Has your question been resolved?
kindred anchor
$17$ cents $\rightarrow 8\frac{3}{4}$ hours
ocean sealBOT
QuantumBee
kindred anchor
$x$ cents $\rightarrow 720$ hours
ocean sealBOT
QuantumBee
fair ingot
lone heartBOT
chrome plank
$x^{\frac a b} = \sqrt[b]{x^a}$
ocean sealBOT
Nonna
lone heartBOT
@fair ingot Has your question been resolved?
alpine sable
When you have a variable raised to a power then raised to another power you just multiply the exponents
nova widget
ok in general, $\left(a^b\right)^c = a^{bc}$
ocean sealBOT
geometric
nova widget
so you can rearrange the expression to ${4}^{\frac{5}{3}\cdot\frac{6}{5}} \cdot {a}^{\frac{2}{3}\cdot\frac{6}{5}}$
ocean sealBOT
geometric
lone heartBOT
lone heartBOT
Available help channel!
austere compass
Let $A \in K^{n,n}$ be a matrix.
If $\lambda \in K$, then the set forms $VA(λ) = {v \in K^{n,1}
\mid Av = \lambda v}$ a subspace of $K^{n,1}$.
When exactly is $VA(\lambda) = {0}$?
ocean sealBOT
Levens
weary wyvern
Let $A \in K^{n,n}$ be a matrix.
If $\lambda \in K$, then the set forms $VA(λ) ...
When lambda is not an eigenvalue
lone heartBOT
@austere compass Has your question been resolved?
austere compass
When lambda is not an eigenvalue
why?
lone heartBOT
austere compass
.reopen
lone heartBOT
✅
mortal trellis
per definition of what an eigenvalue is
lambda is an eigenvalue if and only if there is a nonzero x with Ax=lambda x if and only if VA(lambda) is not {0}
austere compass
so any other value would always give {0}?
marsh rapids
lambda is an eigenvalue of A <=> there exists v != 0 such that Av = lambda v
hence lambda not eigenvalue <=> only v such that Av = lambda v is v = 0
potent path
now this :)
lone heartBOT
now this :)
potent path
do i have to actually put in values? I don't see a way to do it analytically
gray isle
sin(x) = sin(pi - x)
gray isle
consider factorising the denominator as well
and applying limit identites
you can also choose to do a sub to get something more recognisable
potent path
im confused
gray isle
if you cant immediately spot what to apply
what do you have after applying
sin(x) = sin(pi - x)
and factorising the denominator
gray isle
best you can do is look it up and find a translation
unique lichen
Hmm u forgot to replace the x with t+pi if u take t= x-pi
potent path
what is t
gray isle
did you look up "substitution"?
potent path
yep
gray isle
t is another variable
potent path
im substituting x for pi
gray isle
no
that's not what i'm recommending you do
do a substitution to change the variable to get something nicer
we already know that plugging in x=pi will get you 0/0 (an indeterminate form)
so what we currently want is to do some manipulation, apply some identities and cancel the components responsible for that 0/0 indeterminate form
which will allow you to evaluate the limit
potent path
ok, what identities would allow me to do this?
gray isle
the one involving
$$\lim_{k\to 0}\frac{\sin(k)}{k}$$
ocean sealBOT
ℝamonov
unique lichen
Do u know l'hospital rule?
gray isle
yes
potent path
ok that's where I got to
@gray isle I know im difficult lol. If it were up to me I wouldn't take this course :)
but if you can walk me through it it would be greatly appriciated
gray isle
ideally your substitution would've directly led to you having sin(t)
anyway since you did this, you can apply sin(-t) = -sin(t)
potent path
ok. now what do I do with the (t+pi)
gray isle
leave it as it is for now
after applying the odd property of the sine function you'll have
$$\lim_{t \to 0} \frac{-\red{\sin(t)}}{(t+\pi)\cdot {\red{t}}}$$
ocean sealBOT
ℝamonov
gray isle
you can then consider more limit properties to get
$$\br{\lim_{t \to 0} -\frac{1}{t + \pi}} \cdot \underbrace{\br{\lim_{t \to 0} \frac{\sin(t)}{t}}}_{\text{you correctly identified this to be 1 earlier}}$$
ocean sealBOT
ℝamonov
potent path
which is just -1/pi
gray isle
yes
potent path
and that's the final answer right
potent garnet
lone heartBOT
potent garnet
I need to use p-adic valuation.
vale wigeon
try the 2-adic valuation more specifically
azure trout
lone heartBOT
lone heartBOT
languid bolt
yes
lone heartBOT
Available help channel!
flat saddle
lone heartBOT
potent garnet
What do you need?
ocean sealBOT
potent garnet
Yes. If you can show that two sets are identical, then they are subsets of each other.
flat saddle
Is what I did showing that they are identical?