oh okay ty
#help-26
unreal stream
sonic bone
ok ty
topaz sinewBOT
@sonic bone Has your question been resolved?
cyan creek
In an integral such as integral of 1/sqrtx dx where you use a substitution x=u^2, what happens when you replace the u’s with x’s if there is a singular u term?
topaz sinewBOT
In an integral such as integral of 1/sqrtx dx where you use a substitution x=u^2...
cyan creek
For example if the answer is 5u, then solving for u, we get u=+-sqrtx
Do we take the positive or negative
Actually here, the domain for x would be positive so it’s positive, but would there be a case when the domain isn’t an issue and there is a choice of plus and minus?
topaz sinewBOT
@cyan creek Has your question been resolved?
whole nacelle
i know we need to use spherical coordinates and ive found that rho limits are 3 to 0 and the theta limits are 0 to 2 pi
topaz sinewBOT
i know we need to use spherical coordinates and ive found that rho limits are 3 ...
whole nacelle
not sure about the phi limits tho
i drew a cross section out and have trouble determining the shaded region
topaz sinewBOT
@whole nacelle Has your question been resolved?
whole nacelle
i drew a cross section out and have trouble determining the shaded region
<@&286206848099549185>
topaz sinewBOT
@whole nacelle Has your question been resolved?
whole nacelle
anyone please
worldly nimbus
The shaded region would be the part of sphere bounded by x^2+y^2=z^2 because everything that is inside x^2+y^2=z^2 will also be bounded by x^2+y^2=3z^2
whole nacelle
The shaded region would be the part of sphere bounded by x^2+y^2=z^2 because eve...
so i assume its the innermost cone?
worldly nimbus
Yea
whole nacelle
then what would the phi limits be?
worldly nimbus
The maximum angle it can make with z axis is 45 degree
whole nacelle
still need help pls ping me
topaz sinewBOT
@whole nacelle Has your question been resolved?
whole nacelle
.close
neon iron
So I am looking at example of telescoping an recurrence relation. And I am wondering how do they the two last lines from?
topaz sinewBOT
So I am looking at example of telescoping an recurrence relation. And I am wonde...
neon iron
So why are we plugging values in telescoping?
supple mist
neon iron
ohhh, but is the point of telescoping not to show from n-2 to the nth term and then cancel?
topaz sinewBOT
@neon iron Has your question been resolved?
grave vigil
The point of telescoping is to apply a recurrence relation multiple times to find T(N) in simpler terms. In this problem, you can find that T(N) = T(1) + 2c(N-1)
neon iron
The point of telescoping is to apply a recurrence relation multiple times to fin...
I see where does N=4 come into play by applying this reccurence multiple times'?
topaz sinewBOT
@neon iron Has your question been resolved?
topaz sinewBOT
@neon iron Has your question been resolved?
lime steeple
topaz sinewBOT
lime steeple
sorry for the fact that it is difficult to understand this task due to the different language
topaz sinewBOT
@lime steeple Has your question been resolved?
lime steeple
.close
leaden tinsel
I really dont understand it pls someone help me I aint got a clue how to find out a suitable scale
topaz sinewBOT
I really dont understand it pls someone help me I aint got a clue how to find ou...
leaden tinsel
.close
oblique venture
can someone help me with this question pls
topaz sinewBOT
can someone help me with this question pls
@oblique venture Has your question been resolved?
subtle jetty
how do i differentiate this?
topaz sinewBOT
how do i differentiate this?
worldly nimbus
By chain rule
restive inlet
linearity and power rule
subtle jetty
idk how to diffentere the 1/3
hasty smelt
u dont
subtle jetty
?
restive inlet
1/3 is just a constant being multiplied
you can consider constant multiply rule
restive inlet
no
hasty smelt
(C * f(x))´ = C * f'(x)
subtle jetty
yes
hasty smelt
indeed
subtle jetty
c is 1/3 right?
hasty smelt
yeah
subtle jetty
f(x)is -3
topaz sinewBOT
@subtle jetty Has your question been resolved?
topaz sinewBOT
ruby mural
topaz sinewBOT
@neon iron Has your question been resolved?
topaz sinewBOT
@neon iron Has your question been resolved?
topaz sinewBOT
topaz sinewBOT
Available help channel!
green gulch
topaz sinewBOT
green gulch
what should be my strat here
boreal leaf
Try replacing x first
vale furnace
basically writing tan as sin/cos and go from there
boreal leaf
Answer is -1?
green gulch
basically writing tan as sin/cos and go from there
idk how to go from there :(
oh
nvm
vale furnace
Answer is -1?
no
thorny flameBOT
Pure
green gulch
yes
boreal leaf
💀
vale furnace
,, 1\cdot\lim_{x \to 0} \frac{(5^x -1)}{\sin{(1-3^x)}}
thorny flameBOT
Pure
boreal leaf
Like i did
vale furnace
second limit use lhopital 
vale furnace
bruh okay let me go eat dinner first and ill come back
green gulch
lmao okay np
vale furnace
@boreal leaf maybe he can help
green gulch
okay
boreal leaf
Apply loptial
boreal leaf
Don't forget chain rule
boreal leaf
Oh shit
green gulch
i just want to know how to do this without
i am quite bad with limits at the moment
am i allowed to use the sinx/x limit here?
green gulch
i think so
and then
the limit would go to 1
leaving with 5^x-1/1-3^x?
is that how it works lol
boreal leaf
How did you get the x tho
green gulch
what x
boreal leaf
For sinx/x
vale furnace
oke ill type this rq
green gulch
i dont know if it works that way i just guessed
vale furnace
,, \text{rewrite this as} \ \frac{5^x -1}{1-3^x} \cdot \frac{1-3^x}{\sin{(1-3^x)}}
and use common limit to eval
green gulch
yes that is what i did
thorny flameBOT
Pure
vale furnace
yes
vale furnace
vale furnace
green gulch
i used lhopital and got 16/3 but how to do without 🥲
green gulch
thanks
topaz sinewBOT
@green gulch Has your question been resolved?
vale furnace
okay here we can use 3 standards limits namely,
\begin{alignat*}{2}
&\lim_{x \to 0} \frac{\tan{x}}{x}& =& \ 0 \
&\lim_{x \to 0} \frac{\ln{(1+x)}}{x}& =& \ 0 \
&\lim_{x \to 0} \frac{e^x -1}{x}& =& \ 0
\end{alignat*}
green gulch
okay
vale furnace
hm
thorny flameBOT
Pure
vale furnace
sorry i was just fixxing the latex
green gulch
yep
vale furnace
you can express each part of your limit in terms of these standard lims
see if you can see that
vale furnace
ah wait i just realised i mistyped all that lol
[
\lim_{x \to 0} \frac{\tan{x}}{x} = 1,
]
[
\lim_{x \to 0} \frac{\ln{(1+x)}}{x} = 1,
]
[
\lim_{x \to 0} \frac{e^x -1}{x} = 1.
]
thorny flameBOT
Pure
vale furnace
not 0
green gulch
wow okay i was a bit confused lol
so ive gotten to
$(ln(1+16x^2))/3x^2
$(ln(1+16x^2))/3x^2$
thorny flameBOT
tim
green gulch
i suck at latex
is this correct
@vale furnace
i got it
thanks a lot
i should memorise the standard limits
.close
topaz sinewBOT
topaz sinewBOT
Available help channel!
kindred knoll
Montrer que |(1-2x)^(3)-(1-6x)|=x^(2)|12-8x|
topaz sinewBOT
Montrer que |(1-2x)^(3)-(1-6x)|=x^(2)|12-8x|
sweet shard
<@&286206848099549185> (Sorry for ping, in a hurry)
!15m
topaz sinewBOT
<@&286206848099549185> (Sorry for ping, in a hurry)
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
@kindred knoll Has your question been resolved?
kindred knoll
@prisma shore
waxen flame
if you're in a hurry, the least you could do while waiting is translate the problem to english lol
kindred knoll
Show that [(1 − 2x)3 − (1 − 6x)| = x2|12 - 8x, for all x belongs to R.
smoky sparrow
Show that [(1 − 2x)3 − (1 − 6x)| = x2|12 - 8x, for all x belongs to R.
can you simplify $(1 - 2x)^3 - (1 - 6x)$?
thorny flameBOT
south
smoky sparrow
once you simplify that
remember that |ab| = |a| |b|
so |x^2| = x^2 as x^2 is always nonnegative
smoky sparrow
can you simplify $(1 - 2x)^3 - (1 - 6x)$?
taking the absolute value of this will give you the right-hand side
oblique bolt
topaz sinewBOT
oblique bolt
forgot this stuff
oblique bolt
so if I do bottom one, I do 5 by 2
oblique bolt
so what would be the other rectangle's size?
21
oblique bolt
10 for the bottom and 21 for other
35?
why that
where does the 3 come in?
do I do the usual multply those by the .5 (1/2*bh)
got it so the area formula for rectangle is l*w i remember now
is this right?? I tried breaking it into two objects liek yall said
it said my asnwer was wrong when I added the triangle and rectangles areas together
how??
22 isnt the height of the entire shape?
thats confusing but ill try to plug in the number
so is the rectangle is 864?
the are for it
oh mb i forgot to clear
and the the top is 48*7
thank you!
.close
crimson iron
neon iron
How do I find the horizontal/vertical asymptote of $y=3(0.5^x)-4$
thorny flameBOT
Matt
formal moat
How do I find the horizontal/vertical asymptote of $y=3(0.5^x)-4$
so when we want to look for asymptotes we are using a few tricks,
-
are there any values for x that makes the function undefined
example: dividing with zero, 3/(x-1) in this case if x is 1 we would have an undefined function. this usally finds the vertical -
we look what happens for extremly large and extremly small values of x. to see what actually happens to the curve.
example: y=e^x+2 we can see that for very small values of x the e^x part will dissapear leaving us with y=+2 which will be the asymptot
Hope this help
neon iron
so when we want to look for asymptotes we are using a few tricks,
1. are there...
thank you 
.close
dusty pelican
is it possible for the sum of a constructible number and an inconstructible number to be a constructible number?
topaz sinewBOT
is it possible for the sum of a constructible number and an inconstructible numb...
dusty pelican
rightttt okay that makes sense. is it the same idea for the product and quotient of a constructible and inconstructible number?
Do you happen to know what the case is for two inconstructible numbers? are they closed under addition, multiplication, and division provided they are distinct inconstructible numbers, or is that not even true?
wait nevermind, im dumb
of course that cant be true
Thank you for your help!!
.close
strange coral
How do I factor secx(sec^2 x)-secxtanx (tanx-1)
topaz sinewBOT
How do I factor secx(sec^2 x)-secxtanx (tanx-1)
haughty oxide
Hey
half edge
whats it w ppl just giving out answers lol
topaz sinewBOT
Hey
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
half edge
@haughty oxide try to guide them instead of posting the solution lol
half edge
u good, just know for future reference
noble laurel
Instead, ask guiding questions and help them through the problem, explain along the way where necessary if they have questions. Not the biggest deal ever 👍🏼
haughty oxide
Thanks so much!
haughty oxide
How do I factor secx(sec^2 x)-secxtanx (tanx-1)
Yeah, hi. Let me walk you through it
strange coral
Okay thank you
haughty oxide
Oh lol, I made a mistake anyways
strange coral
lmaooo
haughty oxide
Okay so, that solution doesn't work. Let's try again
So the expression contains secx and tanx, so we probably need to use the identity 1+tan^2x=sec^2x
Our main objective will be to get to a point where that will help us
strange coral
Alright
haughty oxide
So firstly, how could we start the factoring? Is there anything in common?
strange coral
the sec x?
haughty oxide
Perfect. So let's factor out a secx to start
strange coral
Thats where I got lost didnt know how to take it out\
haughty oxide
Oh okay. So secx is just a term, right. When we factor it out we get something like this
secx*(sec^2x-tanx(tanx-1))
Can you see why?
strange coral
No can you explain please?
haughty oxide
Okay, so we have 2 terms, secx*(sec^2x) and secxtanx(tanx-1), correct?
strange coral
yes
haughty oxide
So since secx is in common, we're going to factor that out from both terms
What that means is we're going to remove 1 secx from each term and multiply it on the outside. Does that make sense?
haughty oxide
Welcome!
strange coral
factor out tan?
haughty oxide
Hmmm. Is tan in common?
strange coral
only in the latter part
but you been in the full equation?
mean*
Sorry im exhausted.
haughty oxide
but you been in the full equation?
So in the full equation, can we factor out tan?
strange coral
probably not?
haughty oxide
Yeah, we can't, since tan isn't in common
Instead, maybe try something else
Sometimes, in order to factor properly, you need to distribute first
strange coral
so distrube tan?
strange coral
sec x(sec^2x-tan^2x+tanx)?
strange coral
Was thinking a trig identities but dont think any would help at this point yet?
haughty oxide
That's a great instinct. Actually, they're perfect to use now
strange coral
Really?
haughty oxide
Which identity can we use?
haughty oxide
Perfect. What do we get after that?
strange coral
Thank you so much!
strange coral
.close
noble laurel
(x-2)/(x-2) is equivalent to 1 ONLY if x≠2
Otherwise it is undefined
That is why
And a function can’t be continuous where it isn’t defined
So there will be a hole at the point x=2
sweet shard
it does change the graph
yes?
so?
it just does?
why don't you think it's possible?
follows from dividing by 0 being undefined
or you could learn to define functions piecewise
toxic stirrup
im pretty sure i need to do mod 4 here, but the -7y is a bit of a problem
vale furnace
do mod 7
toxic stirrup
mod 7?
hazy pumice
If there was a solution in the integers, then 7|(x^2 - 10)
(The expression is a multiple of 7)
vale furnace
mod 7?
similar to how you were doing mod 4 but mod 4 doesnt work here because x^2 = 0,1 mod 4 and 7x = 0,1,2,3 mod 4
why would you do mod 3 when mod 7 is a lot more straightforward
in fact mod 3 doesnt work
hazy pumice
Please show your work once you have it because I am interested in seeing how this plays out 
vale furnace
topaz sinewBOT
@toxic stirrup Has your question been resolved?
toxic stirrup
is there something nice fro squares mod 7
@vale furnace
vale furnace
…
So you just went for 20+ mins
Wtevet
Just try
1^2 mod 7, 2^2 mod 7 … and see
barren parcel
im pretty sure i need to do mod 4 here, but the -7y is a bit of a problem
To prove that the equation x^2 - 7y = 10 has no integer solutions, consider the equation modulo 7. If x^2 - 7y is congruent to 10 modulo 7, then x^2 is congruent to 3 modulo 7. Checking all possible residues of x^2 modulo 7 (0^2 ≡ 0, 1^2 ≡ 1, 2^2 ≡ 4, 3^2 ≡ 2, 4^2 ≡ 2, 5^2 ≡ 4, 6^2 ≡ 1), we find that none is congruent to 3. Therefore, the equation has no integer solutions, as there is no integer value of x that satisfies x^2 ≡ 3 (mod 7). This proof relies on the fact that the quadratic residues modulo 7 are 0, 1, 2, and 4, and none of them is congruent to 3 modulo 7.
topaz sinewBOT
@toxic stirrup Has your question been resolved?
toxic stirrup
.close
deep crow
How do i do this
topaz sinewBOT
How do i do this
winter temple
Where are you stuck on?
deep crow
derivative of f
deep crow
Is this a multiple choice question
It is
i have the ansewr but i want to know the process of getting to it
winter temple
For now you can think of f’(a) as the slope of tangent to f(x) at x=a
The question told you the slope of the tangent should be -1
So you should check the slopes of all the options
The one with slope -1 is the answer
topaz sinewBOT
@deep crow Has your question been resolved?
winter temple
Hmm I guess
just use calculator to find which value of x satisfies f’(x)=-1
deep crow
Hmm I guess
just use calculator to find which value of x satisfies f’(x)=-1
i got like .934
winter temple
(.934,f(.934)) is a point the tangent passes through
The tangent has the following point slope form:
y - f(.934) = -1 (x - .934)
Tangent: y = -x + (.934 + f(.934))
deep crow
i got .836 now tho
deep crow
Hmm I guess
just use calculator to find which value of x satisfies f’(x)=-1
also is there a way to find 6x^5 -4x^3 = 1 by hand
winter temple
i got .836 now tho
You probably got the wrong value
deep crow
How do i find the value of x?
winter temple
It should be -.934
winter temple
also is there a way to find 6x^5 -4x^3 = 1 by hand
idk about quintic
just stick to your calculator
deep crow
idk about quintic
just stick to your calculator
Are quintic functions complicated
winter temple
Well yes
deep crow
(.934,f(.934)) is a point the tangent passes through
Wait can you explain how you know this part
Im kinda confused on how you knew that that was the point the tangent passes
winter temple
f’(a) is the slope of tangent to f(x) at x=a right?
deep crow
yea
winter temple
would this imply that the tangent pass through (a,f(a))
boreal leaf
I have a matrix AB
topaz sinewBOT
I have a matrix AB
boreal leaf
Is if I want to find (AB)²
Do I just square the components of AB
Nvm
I'm stupid
.close
restive bear
topaz sinewBOT
restive bear
i have no clue where to start on this
thorny flameBOT
gianky.e
restive bear
Yeah thats what i figured, but wouldn’t you need a calculator from there?
fathom loom
now pick a random interval lets call it [a,b] and make sure with f(a) > 0 and f(b) <0 or f(a) <0 and f(b) >0. then there exists a point call it c such that f(c) = 0
restive bear
so you just apply random intervals till you find something that works?
fathom loom
then f(a) > 0 and f(b) <0
then all u gotta say is that there exists a point c such that sin(2c) + 5 - 2^c = 0
restive bear
wait so it’d be like this?
fathom loom
There are many intervals you can put
restive bear
restive bear
so you could just put any two intervals that apply and that’d be it
no matter how broad it is
fathom loom
yeah as long as the condition meets
fathom loom
honestly you can pick any interval but pick easy numbers so you wont have to use a calculator
anytime 🙂
normal umbra
hey
normal umbra
i have a question to use my line of best fit to estimate y when x = 2.5
keen matrix
- what's your data set?
- what's your line of best fit?
normal umbra
1. what's your data set?
2. what's your line of best fit?
wait ill take a picture
keen matrix
,rccw
thorny flameBOT
keen matrix
so simply just plug in 2.5 into your line of best fit, such an extrapolation is well within the range of your data set, no? :)
normal umbra
so simply just plug in 2.5 into your line of best fit, such an extrapolation is ...
Hmmm
keen matrix
you have the equation for your line of best fit?
normal umbra
you have the equation for your line of best fit?
i got it dont worry
thanks
wait i got another question
if i have an equation y = 3 over 5x + 380
keen matrix
$y=\frac{3}{5}x+380$
thorny flameBOT
PajamaMamaLlama
normal umbra
yeh
but how would i use that to "use ur equation to estimate the cost of construction of a house that is 300 metres squared"
keen matrix
$\widehat{\text{constructioncost}}=\frac{3}{5}\cdot300+380$, no? :)
thorny flameBOT
PajamaMamaLlama
normal umbra
what does that dot mean\
keen matrix
multiplication
keen matrix
you can plug in 300 into your equation
and that tells you the estimate for the construction cost given 300 meters squared
normal umbra
how did u know that x would be the metres squared tho?
keen matrix
well that's what your equation is in terms of, right?
normal umbra
yea
keen matrix
so then yes, x is in meters squared
normal umbra
i still dont get it ngl
why couldnt metres squared be y?
if ur plugging it into y = mx + c
normal umbra
so then yes, x is in meters squared
u there?
keen matrix
y is construction cost, x is house size in meters squared
normal umbra
y is construction cost, x is house size in meters squared
yea but why is y construction cost and x is house size
idk how they r chosen
topaz sinewBOT
Available help channel!
jade wharf
trying to solve for all the angles in the triangle using law of cosines. this is the process i used for one of the angles and i wanna see if what i did was right
topaz sinewBOT
trying to solve for all the angles in the triangle using law of cosines. this is...
@jade wharf Has your question been resolved?
Available help channel!
neon iron
Optimise the following Boolean function $F$ together with the don't care conditions $d$. Find a simplified boolean expression for $F$ in sum-of-products using K-map
\env{align*}{
\m F{x,y,z,t} &= \m\Pi{1,5,7,8,9,11,13} \\
\m d{x,y,z,t} &= \m\Sigma{0,4,6,12}
}
neon iron
im a bit confused here
the function is given in terms of its maxterms
converting to its minterms gives [
\m\Pi{1,5,7,8,9,11,13} = \m\Sigma{0,2,3,4,6,10,12,14,15}
]
thorny flameBOT
neon iron
but there is some overlap between the don't care conditions and the minterms in here, so what is going on?
hollow drum
converting to its minterms gives \[
\m\Pi{1,5,7,8,9,11,13} = \m\Sigma{0,2,3,4,6,...
I guess converting it is fine but I don't think you need to. The maxterms are 0 in a k map
So with your 4 term function, the kmap is a 4 x 4, you can fill in the don't cares and the max terms as 0s then the rest are 1s
thorny flameBOT
\let\bar\conj
\begin{tabular}{c|c|c|c|c|}
\multicolumn{1}{c}{}& \multicolumn{1}{c}{$\bar z\bar t$}& \multicolumn{1}{c}{$\bar z t$}& \multicolumn{1}{c}{$zt$}& \multicolumn{1}{c}{$z\bar t$} \\
\cline{2-5}
$\bar x \bar y$ &d &0 &1 &1 \\
\cline{2-5}
$\bar xy$ &d &0 &0 & d\\
\cline{2-5}
$xy$ &d &0 &1 & 1 \\
\cline{2-5}
$x\bar y$& 0& 0& 0&1 \\
\cline{2-5}
\end{tabular}
hollow drum
Yeah, you can find the function from that now
neon iron
Yeah, you can find the function from that now
okay so like the first row we have t changing
so we get x'y'z out of that
third row xyz similarly
and then lastly the fourth row xy'zt'
hollow drum
and then lastly the fourth row xy'zt'
Why the last row?
neon iron
i mean there is a 1 there
hollow drum
That is true, but a don't care bit can be in a grouping
neon iron
yeah thats 2 and 4 cells respectively
hollow drum
Powers of two that form rectangles, I'm assuming that orange line, you indicate is a group of 4 but it's a T shape
hollow drum
If you want a hint, you should have 3 groupings
neon iron
If you want a hint, you should have 3 groupings
sorry cant seem to find those 
hollow drum
Well, as mentioned, the don't cares can be included
So you have the red line one
The one in the third row, the zt/z!t is a grouping
Then the entire 4th column is a grouping
neon iron
its fine to have overlap?
thorny flameBOT
hollow drum
That would the SOP for that kmap
neon iron
great
another question i have
"Write the Boolean expression for the output of a 2-to-1 Multiplexer and draw the corresponding logic diagram. What is the propagation delay of a 2-to-1 Multiplexer?"
\let\1\conj
im pretty sure its simply [
Y = \1S I_0 + SI_1
]
right?
thorny flameBOT
topaz sinewBOT
hollow drum
"Write the Boolean expression for the output of a 2-to-1 Multiplexer and draw th...
I got nothing on MUX
neon iron
thats fair
topaz sinewBOT
Available help channel!
edgy quail
topaz sinewBOT
edgy quail
!status
topaz sinewBOT
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
2
source: ||2023 thailand MO, P3||
i’ve only gotten to ||f(0)=0 by P(0, 0) and f(-c)=-f(c) by P(y, x)||
i’ve tried subbing different stuffs like ||(x, -y)|| and ||(-x, -y)|| to try to abuse the fact that ||x^2=(-x)^2|| but that didnt help a lot
i need a hint
pulsar sun
And obviously if u have noticed f(0) = 0 then u must have seen f(x)=0 is a solution too
pulsar sun
no its not
Doesn't matter tbh
edgy quail
And obviously if u have noticed f(0) = 0 then u must have seen f(x)=0 is a solut...
yes
but i cant just claim that
pulsar sun
Wdym?
edgy quail
currently my guesses rn is f = 0 or x
edgy quail
but i cant just claim that
like
that's not the end
rn my idea is
assume f isnt 0 (because we know f=0 worls) and try going on from there
pulsar sun
that's not the end
Ofc u need to see if there are more solutions
edgy quail
Ofc u need to see if there are more solutions
there are
f(x) = x works
rn i am tryijg to use the fact that f(-x) = -f(x) and the fact that x^2 = (-x)^2 to do stuff rn
pulsar sun
Also constant functions i.e. f(x) = c is only satisfied when c=0
edgy quail
but i've tried subbing (x, -y) ans f(-x, -y) and that was not very helpful
pulsar sun
Lemme see if I can work something out
edgy quail
so can i have a hint
wait maybe injectivity/surjectivity
lemme see ah
assuming f =/= 0, take x_0 st f(x_0)=/=0, now plug y_1 and t_2
y_2*
such tgat f(y_1)=f(y_2)
pulsar sun
Notice that since f(-x) = -f(x) therefore if f(x) is a polynomial it must be of odd degree
pulsar sun
you can just say "f is odd"
Yk that term? i explicitly made the choice to omit it
edgy quail
such tgat f(y_1)=f(y\_2)
anyway
pulsar sun
Do u have the answer?
edgy quail
let's try proving injectivity
pulsar sun
To the question
edgy quail
Do u have the answer?
easily findable online but i dont wanna get spoiled
i'm pretty sure it's f = 0 or identity
don't spoil me
ok no
edgy quail
such tgat f(y_1)=f(y\_2)
anyway lets prove injectivity
$f(x_0)f(y_1)f(x_0-y_1)=(x_0)^2f(y_1)-(y_1)^2f(x_0)\
f(x_0)f(y_2)f(x_0-y_2)=(x_0)^2f(y_2)-(y_2)^2f(x_0)$
thorny flameBOT
candies
edgy quail
ok lets see if i can 抵銷 stuff
lemme 相減
$f(x_0)f(y_1)[f(x_0-y_1)-f(x_0-y_2)]=f(x_0)(y_2^2-y_1^2)$
thorny flameBOT
candies
edgy quail
so $f(y_1)[f(x_0-y_1)-f(x_0-y_2)]=(y_2^2-y_1^2)$
thorny flameBOT
candies
topaz sinewBOT
@edgy quail Has your question been resolved?
south osprey
What do the values in P represent and what do the values of x represent in this question?
topaz sinewBOT
What do the values in P represent and what do the values of x represent in this ...
strange whale
P is a partition of [0, 1], i.e. you break [0, 1] into multiple parts to approximate the integral of f
that's what you should be seeing
and the x* are a choice of points inside each of these subdivisions (and you use the values of f(x*) to approx the value of the function on the whole interval containing x*)
@south osprey
topaz sinewBOT
@south osprey Has your question been resolved?
south osprey
and the x* are a choice of points inside each of these subdivisions (and you use...
so then when im breaking up [0,1] is this what i should be doing?
Because it seems a little lengthy to me im not sure if its right
.close
odd pagoda
why should it be discontinuous at uncountably many points?
its discontinuous on Q
rationals are countable
yes but its continuous at the irrationals
I mean exactly what I said
if x is irrational then f is continuous at x
yes
which is completely fine
you are completely mixing up the statements here
if that were true, then also f(x)=x wouldnt be intregrable
topaz sinewBOT
topaz sinewBOT
Available help channel!
neon iron
since the limit of (1+1/n)^n is e, can i just say the limit of (1-1/n)^(-n) is also e because plugging in (-n) gives us a subsequence and thus must converge to the same limit?
topaz sinewBOT
since the limit of (1+1/n)^n is e, can i just say the limit of (1-1/n)^(-n) is a...
neon iron
or since the indices are negative it doesnt count as a subsequence?
topaz sinewBOT
@neon iron Has your question been resolved?
neon iron
my bad, its -n
orchid brook
Well, it will be e
It is just as if n tends to -infinity
I did not get the reasoning however
neon iron
any subsequence will converge to the same limit, so if the original sequence is (1+1/n)^n, plugging in (-n) gives us (1-1/n)^(-n) which also converges to e because it is a subsequence
is that valid?
vernal matrix
because it is a subsequence
no it isn't
neon iron
okay, thats where i had doubts. is it because were not skipping any indices, just making them negative?
vernal matrix
Pretty much yeah - subsequences basically have to "skip indices" (basically being created from a subsequence of the indices) and so you can't e.g. "go backwards"
neon iron
i see. so what would the reasoning be instead?
neon iron
It is just as if n tends to -infinity
this?
orchid brook
Do you know the proof why it is e?
For the infinity limit.
neon iron
we are just supposed to use the fact that (1+1/n)^n has limit e, we have not studied l'hopitals rule yet
orchid brook
Okay
topaz sinewBOT
@neon iron Has your question been resolved?
subtle jetty
For a number of years, the number of pups that gray seals gave birth to on Sable Island has been counted every year. In a model, the development in the number of gray seal pups on Sable Island can be described by
N(t)=787*1.14',
where N(t) denotes the number of gray seal pups at the time † (measured in number of years after 1970).
■a) Determine N'(t).
topaz sinewBOT
For a number of years, the number of pups that gray seals gave birth to on Sable...
subtle jetty
ye?
thorny flameBOT
Mycobacterium
subtle jetty
ye
karmic dove
**Mycobacterium**
yes
hollow drum
i get it to 0
Because this function doesn't have a variable t
subtle jetty
?
subtle jetty
oh
hollow drum
Because you had two constants originally and the derivative of constants is 0
hollow drum
It's suppose to be n(t) := 787 * 1.14^t
.
topaz sinewBOT
@subtle jetty Has your question been resolved?
wide bane
topaz sinewBOT
wide bane
When doing question b and wanting to show that it counts for n+1, should I then use the recursive function ?
Should I set it up as 2*s_n+2 = 2^n+1 - 2 ?
I have a few questions:
A) Is the base case just putting 1 on n's place since we have Z >_ 1?
B) Whats is the step after? THe inductive or hypothesis steP?
wide bane
base case can be s_1 = 0
I finished it now and ill write it out
We first set our base case up to show that s_n = 2^n - 2 holds for n = 1, since that is the smallest value.
s_1 = 2^1 - 2 = 2 - 2 = 0. Since this is true, which we know from the recursive function, we assume that s_n = 2^n - 2 for alle n in Z larger or equal to 1. Therefore it must also hold for n+1. We will now prove this by induction:
S_n+1 = 2 * S_n + 2 = 2 * (2^n - 2) + 2 = 2 * 2^n - 4 + 2 = 2^n+1 - 2
We first set our base case up to show that $s_n = 2^n - 2$ holds for $n = 1$, since that is the smallest value.
[ s_1 = 2^1 - 2 = 2 - 2 = 0. ]
Since this is true, which we know from the recursive function, we assume that $s_n = 2^n - 2$ for all $n \in \mathbb{Z}$ larger or equal to 1. Therefore it must also hold for $n+1$. We will now prove this by induction:
[ S_{n+1} = 2 \cdot S_n + 2 = 2 \cdot (2^n - 2) + 2 = 2^{n+1} - 4 + 2 = 2^{n+1} - 2 ]
thorny flameBOT
bed
odd jacinth
for all n >= 1
then you're already assuming it for n+1 :c
wide bane
> for all n >= 1
then you're already assuming it for n+1 :c
oh? isnt that what im supposed to do?
odd jacinth
no you assume it's true for an integer n (that s_n = 2^n - 2), and then you show it's true for s_(n+1)
if you're already assuming the result there's no reason to prove it
wide bane
So I should instead write: "Since this is true, which we know from the recursive function, we assume that s_n = 2^n - 2 for n. We know wish to prove inductively that it holds for n + 1" ?
odd jacinth
well we don't know it's true for all n before doing the induction
we assume [it's true] for all n >= 1
then it's what the question asked, nothing left to prove, question done
the point is to not assume the result, it's showing that if it's true for some n it's true for all the integers after
odd jacinth
Base step: n = 1 checked
Induction step: We assume there exists n in N which satisfies the property s_n = 2^n-2. Then s_(n+1) = 2s_n+2 = 2(2^n-2)+2 = 2^(n+1)-2, therefore n+1 satisfies it too.
In conclusion, the said property is true for all integers n >= 1
The whole point of a proof by induction is that you have a property P(x), P(k) true for some k
and then you show P(n) implies P(n+1)
so P(k) implies P(k+1) => P(k+2) => P(k+3) => ... for any integer m >= k you want
wide bane
you mean n in Z?
odd jacinth
doesn't matter since s_n isn't defined for n < 1
wide bane
so its not wrong to write n in Z?
odd jacinth
it's not wrong bc you'll write n >= 1 anyway
exactly bc you need a base step
for a proof by induction to make sense
wide bane
okay
wide bane
Base step: n = 1 checked
Induction step: We assume there exists n in N which sat...
the hypothesis part here is just "We assume there exists n in N which satisfies the property s_n = 2^n-2." right?
odd jacinth
yes
wide bane
We assume there exists n in the natural numbers which satisfies the property s_n = 2^n - 2
odd jacinth
in terms of math logic, we're showing P(n) => P(n+1)
Base step of an induction: P(k) is true for some k
Induction step: we show P(n) implies P(n+1)
Conclusion: P(k)=>P(k+1)=>...=>P(m) for all m >= k, so P(m) is true for all m >= k
that's what an induction does in math logic
and to show P(n) implies P(n+1), we assume P(n) and deduce P(n+1)
that's why induction is like it is
wide bane
So for this one the first steps would be:
"We first set up our base case and put 0 in n's place to see if it holds for the smallest value:
(1+h)^0 >= 1 + 0* h.
1 >= 1
We assume that there is n in Z >= 0 which holds that (1+h)^n >= 1+nh"
@odd jacinth
odd jacinth
Z >=0 yes
wide bane
yea
odd jacinth
The property we will prove is P(n): (1+h)^n >= 1+nh for all n >= 0
Base step: n = 0 checked
Induction step: we assume there exists n >= 0 such that P(n) true, ie (1+h)^n >= 1+nh.
From that, we want to deduce P(n+1): (1+h)^(n+1) >= 1+(n+1)h is true.
wide bane
right
wide bane
The property we will prove is P(n): (1+h)^n >= 1+nh for all n >= 0
Base step: n ...
how would you start something like this ?
we want n+1 on one side I guess
odd jacinth
I would multiply by (1+h) and see how things turn out
(1+h)^n >= 1+nh so (1+h)^(n+1) >= (1+h)(1+nh)
= 1 + h + nh + nh² > 1+(n+1)h
topaz sinewBOT
@wide bane Has your question been resolved?
static jacinth
Let $(X_n){n \in \mathbb{N}}$ be a sequence of independent random variables such that:\
$X_1 \equiv 0$\
$\mathbb{P}(X_n=k)=\left{\begin{matrix} {1\over n^3}\ \mathrm{if}\ k=\pm 1,\pm2,\cdots ,\pm n \ 1 -{2 \over n^2}\ \mathrm{if}\ k=0\ \ \ \ \ \ \ \ \ \ \ \ \
\end{matrix}\right.$\
Show that for $\alpha > {1 \over 2}$:\
$\sum{i=1}^{n} {X_i \over n^{\alpha}}\xrightarrow{\mathbb{P}}0$.
topaz sinewBOT
Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of independent random variables suc...
thorny flameBOT
Casiel368
static jacinth
The thing I want to show is this: $\mathbb{P}({\omega \in \Omega / |\sum_{i=1}^{n} {X_i(\omega) \over n^{\alpha}}|> \varepsilon})\rightarrow 0$ when $n\rightarrow \infty$.
thorny flameBOT
Casiel368
static jacinth
<@&286206848099549185>
topaz sinewBOT
@static jacinth Has your question been resolved?
static jacinth
.close
timber sphinx
topaz sinewBOT
timber sphinx
What is the inverse function?
And if there is not, how do i prove it/check it?
i know that in order for function to have an inverse, it must be bijective
rigid ivy
What is the inverse function?
!xy
topaz sinewBOT
What is the inverse function?
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
timber sphinx
It was sent to my in exactly this format
timber sphinx
.
pastel salmon
Po pierwsze podaj dziedzine tej funkcji, oraz podaj granice na koncach okresłonosci, anstepnie, uduwopdnj ze jest rosnaca na swej dziedzinie wiec jest róznowartosciowa, a wtedy bedzie odzworowaniem "na" zir wartsic co da 1-1 i na, czyli bijkecja, natomaist nie podasz wwzrou janwego na fucikje odrtoan bo jest to rownanie przestepne, i nie podasz t, czyli tylok stqierdzisz ze f. odrotna istniej ale jest okreloslna postacia uwiklaną
to wszystko w sensie wskazówej
k*
🙂
vital gulch
suppose we had the function f: reals --> reals, where f(x) = x^2
can someone explain how f(x) = x^2 becomes invertible if we restrict its domain to x =>0
i thought that only bijective functions can be invertible?
if we restrict the domain to x => 0 doesn't f only become an injective function? thus by definition it's not invertible?
or do we assume that our codomain becomes restricted to positive reals, thus making f a bijective function?
manic swan
vital gulch
yea i understand that
manic swan
Like this it's doesn't work
But if you because the inverse function have 2 vlaues
vital gulch
but the problem is i thought a function is only invertible if it's bijective
manic swan
+sqrt(x) and also -sqrt(x)
vital gulch
if we restrict the domain of x it's still not bijective
assuming our codomain is all reals
manic swan
But if you take only one part which x² from x>= 0 then you will get the inverse is sqrt(x)
If you take x<0 then you will get the inverse as -sqrt(x)
loud oasis
the range of x² is the positive reals regardless of any restriction
vital gulch
the range of x² is the positive reals regardless of any restriction
yea but what about the codomain?
loud oasis
the codomain must also be the positive reals
manic swan
Think about if like this
vital gulch
the codomain must also be the positive reals
yea so then that would mean we have to restrict it then?
for x^2 to be a bijective function
loud oasis
yes, which is why we restrict it to x ≥ 0
vital gulch
ok so both the codomain and the domain are restricted to x => 0
for the function to become bijective
and therefore invertible
vital gulch
ok so both the codomain and the domain are restricted to x => 0
*positive reals
loud oasis
the codomain is only the positive reals regardless of whether the domain is restricted, so for it to be bijective we must restrict the domain to match
vital gulch
but the codomain can be anything right?
for x^2 the range is only positive reals
but the codomain can be reals
.close
ivory wave
,, f(x)= 2 \cdot 3 ^{-(x+4)}-8
topaz sinewBOT
,, f(x)= 2 \cdot 3 ^{-(x+4)}-8
thorny flameBOT
Akira (fumo)
ivory wave
I've began but I'd like to get corrected
the parent function is 3^x
horizontal asymptoe is y=-8
reef fjord
thorny flameBOT
Akira (fumo)
reef fjord
y int is just plugging in 0?
ivory wave
now somewhat stuck at transformations
ivory wave
y int is just plugging in 0?
yup
reef fjord
wydm
like what transformations does this function represent of 3^x?
is that what you're asking?
ivory wave
no like can i transform 3^x as well?
ivory wave
I know this
here's a shortcut for it and the table dw
can i transform 2 and 3^x?
I know HT 4 by left and VT 8 down
what about 2 and 3^x?
reef fjord
2 could be interpreted as various different things with exponent properties, but most clearly as a vertical stretch
i'm a bit confused by what you're asking
3^x is the parent function
f(x)
ivory wave
ye I know but do i need to transform it or no?
only 2?
since 3^x is a parent function i get that
reef fjord
are you asking if you can do this with only two transformations, if so, i believe you can
because 2 can be seen as 3^log3(2)
and you can add it to the exponent
and with a bit of manipulation of terms, it can be viewed as a shifting term
so only a vertical shift, and a horizontial shift
:)
@ivory wave
ivory wave
so only a vertical shift, and a horizontial shift
- VS by 2
- RYA
- HT 4 left
- VT by 8 down
correct?
reef fjord
you can combine the VS and HT
but yes, that's right
i should've said 3 transformations of 3^x
i keep forgetting the negative
lmao
VS and HT are the same-ish with exponential functions
i.e. you can view a VS as a certain HT
and vice versa
reef fjord
why is there 2 ks
ivory wave
reflection in y-axis
ivory wave
why is there 2 ks
k is -1
can i just say RYA since there's a negative?
<@&286206848099549185> meow
reef fjord
i'm fine with what you have as RYA, i'm just confused where you're confused lmao
what do you need corrected exactly
ivory wave
- VS by 2
- RYA
- HT 4 left
- VT by 8 down
this
you can't transform 3^x since it's a parent function
right?
reef fjord
are you trying to transform 3^x to match the function you have now?
ivory wave
no im asking if i can transform it typically?
even it's a parent function
like for example this one
the parent function is 2^x
reef fjord
yes
ivory wave
it did not transform it
(my teacher did that example)
so i can't do same thing for 3^x?
reef fjord
I'm still so confused what you mean by 'not transform it';
reef fjord
????
ivory wave
how did you type dot in desmos
reef fjord
*
reef fjord
can you transform 3^x into that function?
ivory wave
$f(x)= 2 \cdot \textcolor{red}{3} ^{-(x+4)}-8$
thorny flameBOT
Akira (fumo)
reef fjord
you want to reverse f(x) to make 3^x?
ivory wave
reef fjord
brain
ivory wave
- VS by 2
- RYA
- HT 4 left
- VT by 8 down
....
ivory wave
yes
ivory wave
like for example this one
my teacher didn't transform 2^x in this example
can i also not do it for 3^x since it's a parent function?
reef fjord
you only apply those sorts of transformationns when theres a HT, HS and HC
cause those are the 'inside' of the parent function
sonic bone
help
sonic bone
i watched the video
but his problem a lil different
and i think im doing this wrong
sweet shard
Where did pi/180 come from