ill confirm whether it is D or not
#help-39
nimble socket
pearl pondBOT
@buoyant crater Has your question been resolved?
woven matrix
it's D
compact palm
for the last step how do we turn the limit into 1?
pearl pondBOT
for the last step how do we turn the limit into 1?
half tendon
L hopital
wet osprey
You can use Taylor series probably
half tendon
You can use Taylor series probably
I don’t think Taylor series is introduced in conjunction with limits and deravatives 
wet osprey
Well once you have limits you know what Taylor series are
half tendon
Mhm that’s true
meager trellis
well like, how did you define e^x
chrome plank
it's not circular, you'd just be evaluating more derivatives than necessary
half tendon
you'd be applying one of the definitions of e
Are you talking about the exponent properties where some n not equal to zero n^0=1?
wet osprey
L hopital
I just don’t think you should be using such heavy machinery that involves derivatives when proving derivatives by first principle
half tendon
I just don’t think you should be using such heavy machinery that involves deriva...
Mhm
Idk how to prove it with that definition
If i remember, substitution or conjugated or something weird idk
wet osprey
You shouldn’t hit everything with lhopital
When you hold a hammer everything looks like a nail
light helm
because that underlined part is the derivative of e^x at x=0
which means that you'd already know the derivative of e^x
versed mica
using lhopital literally assumes you know the derivative of e^x lmao
half tendon
using lhopital literally assumes you know the derivative of e^x lmao
Yeah 
chrome plank
because that underlined part is the derivative of e^x at x=0
which means that yo...
good point - l'hopital is circular reasoning then
chrome plank
By Taylor series
also circular reasoning because you're using the derivative of e^x to prove the derivative of e^x
half tendon
Yeah lmao
meager trellis
well if you defined e^x as that series then it's fine
chrome plank
you'd need to use the epsilon-delta definition of a derivative to prove this
half tendon
you'd need to use the epsilon-delta definition of a derivative to prove this
Real analysis moment
half tendon
well if you defined e^x as that series then it's fine
You would just differentiate the entire series right 
I forgot how to apply series in deravatives
chrome plank
no you need to give your definition of e^x
- define the derivative of e^x is e^x
- let y = e^x
- prove the derivative of y is e^x (the question)
circular XD
half tendon
This is humor at its maximum
versed mica
1. define the derivative of e^x is e^x
2. let y = e^x
3. prove the derivative of...
yea that's why this question would be meaningless if you defined it that way which seems to be what they did
when they say y'(0) = 1
half tendon
How would you prove it using precalc terms
versed mica
what is your definition of e^x
pearl pondBOT
Available help channel!
wary sigil
just question e) guys. on the back the answer is y -2z -4 = 0 but i don't know how it's getting that
wary sigil
i got 2 direction vectors from the yz plane
(0,4,2) and (0,4,0)
and cross product'ed them
got (-8,0,0)
so my cartesian equation is: -8x + 0y + 0z + D = 0
but then after subbing in the y or z intercept (0,4,0) or (0,0,-2)
i just get d = 0, so the cartesian equation is just -8x = 0
i have no idea how to get y - 2z - 4 = 0
help would be appreciated, thanks
summer imp
and cross product'ed them
The plane is perpendicular to the yz plane, so that means that the normal vector of the plane you’re looking for is inside the yz plane, not perpendicular to the yz plane.
That’s why what you did didn’t work. In fact the plane you get from your equation is exactly the yz plane.
pulsar lark
So it's perpendicular or not?
pearl pondBOT
@wary sigil Has your question been resolved?
summer imp
To work it out, consider the two points you get from the y and z intercepts.
This should give you a line where the plane intercepts the yz plane.
Get the vector v which generates that line.
Then you can use the fact that the plane is perpendicular to the yz plane to find another direction vector. Call it w.
Now, v and w are linearly independent and they both lie in the plane you want, so taking their cross product will yield the normal to the plane you’re seeking.
wary sigil
The plane is perpendicular to the yz plane, so that means that the normal vector...
OHH
wary sigil
To work it out, consider the two points you get from the y and z intercepts.
Th...
yeah i get one vector, (0,4,2)
but how do i get the other one?
what is w?
how do i get that
summer imp
Well the plane is perpendicular to x=0, so some vector like (1,0,0) should also be in it
wary sigil
Well the plane is perpendicular to x=0, so some vector like (1,0,0) should also ...
so i cross (1,0,0) and (0,4,2)?
next ivy
pearl pondBOT
next ivy
Can someone help me with a please
next ivy
Acceleration is asking for the second derivative
orchid heath
,rccw
jolly parrotBOT
orchid heath
u need to calcul a
orchid heath
Which is second derivative of s(t)
orchid heath
$v(t) = 2kt +6k^2 -10k$
jolly parrotBOT
SELVATOR
orchid heath
Ur variable is t
Not k
K is constant
Isn't it?
K could be any number unless 0
orchid heath
Means 2kt + 6k² - 10k = 0
Do u know how to solve this?
a' = 0
(ax)' = a
derivative of constante is 0
orchid heath
Exactly
orchid heath
Means 2kt + 6k² - 10k = 0
Do u know how to solve this?
next ivy
t = 5 - 3k (when k =\ 0)?
the =\ is the equal sign with a line through it
Forgot the name
orchid heath
He needs time when V = 0
orchid heath
Read the question
It says k is constant
$2kt +6k^2 -10k = 0$ so: $ 2kt = 10k - 6k^2$
jolly parrotBOT
SELVATOR
orchid heath
Apparently the 2k isnt a constant, my friend said its attached to a variable
Probably ur friend meant 2kt not 2k
next ivy
**SELVATOR**
Does this equal t = 5 - 3k?
orchid heath
Nice
next ivy
So thats the answer?
next ivy
I dont but my friend does, explain as if i do
Ill forward the message
So he can understand
orchid heath
$a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$
jolly parrotBOT
SELVATOR
orchid heath
v(t) = s' (t)
(Derivative)
Afer derivating s(t) ( as time t is the variable) we got: v(t) = 2kt + 6k² - 10k
Wanna explain it by integrals?
That's physics
next ivy
No keep it in calculus
next ivy
No, im trying to figure it out
next ivy
t = 5 - 3k?
Do u think
U could js
Explain the steps
For a
So i could tell my friend
Cuz my understanding is very limited
next ivy
Did u learn primitives?
I didnt, dont know if he did
orchid heath
next ivy
2x - 1
next ivy
2xt - x^2
orchid heath
U are good
orchid heath
2t + 12t - 10?
How did u mistake right there
s(t) mean t is variable
Any other letter or number is a constant
next ivy
So what would the correct answer be
next ivy
2k?
orchid heath
Yes
next ivy
Is that the proof?
next ivy
(2t - 4){\prime} = 2
orchid heath
!done
pearl pondBOT
If you are done with this channel, please mark your problem as solved by typing .close
@next ivy Has your question been resolved?
pearl pondBOT
Available help channel!
rich vector
I need help on all of these problems please! ;_;
pearl pondBOT
I need help on all of these problems please! ;_;
rich vector
More specifically, the ones with the x
verbal wyvern
I need help on all of these problems please! ;_;
what was your thought process for the local minima/maxima question
because you chose some points where f isn't defined
rich vector
Well I didn't remember exactly what is was to be honest, so I just chose the points where it changed from a negative number to the highest possible point for max
similar process for the local minumum
verbal wyvern
for the local maximum you need to look for a point where the functions value (the y value) is larger than all other y values on the graph at points "near" that point
similar idea for the local minimum but the y-value is smaller
verbal wyvern
I need help on all of these problems please! ;_;
for the question where it asked you to sketch the graph of f, you graphed f with positive slope when it has slope of -3, it also has a y intercept of 7 so keep that in mind as well
rich vector
for the local maximum you need to look for a point where the functions value (th...
Ahh okay thank you!! Yeah i'm starting to remember that now that you pointed it out
rich vector
for the question where it asked you to sketch the graph of f, you graphed f with...
Alrighty! I totally forgot about that, thanks for pointing it out
gray lantern
(small note remember for first question the y axis is scaled by 3)
verbal wyvern
I need help on all of these problems please! ;_;
i think you have the right idea with the (f o f)(x) question, you might have accidentally switched the values around in the intervals or forgot a negative sign so just double check your work
what did you get when you plugged f(x) into f to find (f o f)(x)?
rich vector
what did you get when you plugged f(x) into f to find (f o f)(x)?
I was wondering what I did wrong too ;; So I got essentially this!
verbal wyvern
ok that looks good
so there would be two conditions you have to satisfy to find the domain:
x != 2 and f(x) != 2
you had the right numbers in your original answer, when you solve both conditions you get x != -2 and x != -4/3
so the domain for (f o f)(x) is all real numbers except for when x= -2 and x=-4/3
that should help you write it out in interval notation
rich vector
Huh.. It looks like I wrote it correctly then, i'm not sure why my answer is marked as wrong
OH WAIT I JUST REALIZED I THINK THE PARENTHESES ARE MESSED UP
pearl pondBOT
@rich vector Has your question been resolved?
kind thicket
can anyone help me with this? i tried using some subs but somehow it doesnt work, i dont know how to start
wild fable
Prolly some kings rule type shiiii
distant narwhal
maybe it's a little cheating
but the function is periodic
and each 2pi integrate to 0
so overall is also 0
wild fable
,w integrate sin(sin(x)-x) from 0 to 2022pi
jovial kiln
it is 0
jolly parrotBOT
jovial kiln
first off, we only need to check from 0 to 2π
because it is periodic
then make the sub u = 2π-x
you get the equation I = -I where I is the initial integral after sub
the only solution to that is 0
kind thicket
first off, we only need to check from 0 to 2π
im sorry but check the periodicity?
kind thicket
you get the equation I = -I where I is the initial integral after sub
the only s...
bcs 2I = 0, so I =0 ?
jovial kiln
im sorry but check the periodicity?
integral from 0 to 2022π is the same as 1011 complete revolutions of 0 to 2π
pearl pondBOT
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Available help channel!
quiet reef
for part b
pearl pondBOT
for part b
quiet reef
why switch to cos in the middle in awnser sheet
do we take the derivative one time to get rid of d/dx
?
summer imp
Yes, you take the derivative, which turns the sin into a cos, and then you use part a) to write it back into a sin
summer imp
a pops out due to the chain rule, since the derivative of what is inside the sine functions is ax + kpi/2, which has derivative a.
The pi/2 appears by using the result from part a)
compact ridge
if you try applying the chain rule $\frac{d}{dx} \left(ax + \frac{k \pi}{2} \right) = a$
jolly parrotBOT
south
compact ridge
yes
compact ridge
then you should call ax + kpi/2 = u
cos(u) = sin(u + pi/2)
in other words, if you shift the sin function pi/2 units to the left, you get cos x
quiet reef
ya that does make sense
quiet reef
cos(u) = sin(u + pi/2)
rthis was very good, makes it simpler to see the resemblence from part a
aight thanks brothers
compact ridge
yeah so whenever trig and calculus appear like this
compact ridge
cos(u) = sin(u + pi/2)
also you get cos(u - pi/2) = sin(u) for free from this
keep an eye out but there's no need to memorise them
the other big one is cos(x) = sin(pi/2 - x) and sin(x) = cos(pi/2 - x)
quiet reef
but i feel like one should atleast be used to using the trig identities, even if not remembering. potherwise it is hard to see what to use
compact ridge
mhm that's the first step first
you get cos after you differentiate so you need a suitable horizontal translation to get back sin
quiet reef
ya, gotta work more on thoose co function identities
quiet reef
thanks again man , lifesaver
compact ridge
and the ones before are phase shift identities
compact ridge
thanks again man , lifesaver
np!!
quiet reef
and the ones before are phase shift identities
YES i was looking for that too
compact ridge
IB server maths channel is definitely not as active as here
quiet reef
nah ik, but man everyone solves theese IB math questions so eaisily
But i know i will fail some questions on test due to time stress, but theese are not any rocket science. you just gotta see clearly and know your stuff
compact ridge
But i know i will fail some questions on test due to time stress, but theese are...
if I can be honest with you, induction is free marks
people always complain about (3D) vectors though
also differential equations are also fairly free if you've studied those (you haven't yet I know)
quiet reef
does the other proofs ever show up at test?
compact ridge
does the other proofs ever show up at test?
yes they can definitely ask you direct and contradiction
quiet reef
also differential equations are also fairly free if you've studied those (you ha...
I have, i did a speedrun during a break couple of weeks.
it was p[ainful
its the 100 integrals, 100 limits, 100 derivatives, 100 calc 2 lol
compact ridge
bruhh
quiet reef
like 40 h of my 140 h break was looking and writing down math. LOL wtf im down bad
aight gtg, btu thanks for thrid time
fiery sable
I need help finding the equation of the curve. it could be a single curve or two curves composed at different intervals. the specific use case of the function is interpolation between colors on a heatmap. I manually sampled few points from the heatmap and plotted the graph using spline.
compact ridge
I need help finding the equation of the curve. it could be a single curve or two...
it's definitely two curves
it's not a sigmoid function cause it would be too uneven: the slope for 2 < y < 4 is noticeably steeper than that for 0 < y < 2
try two exponential functions joined at around x = 0.6
analog girder
Can someone help me with this question? (Ignore my working out idek what i was doing) The correct answer is A but for point P, should it be (2+1 (moving to right), 4x2+2)= (3,10) and point Q is (-3+1, -4x2+2=-6)? Idk
wild fable
u really only need to look at what happens to the y coordinate here
analog girder
oh but why?
wild fable
because only one option here has the pair of transformed y coords
so u dont have to calculate the transformed x
analog girder
Oh but in general I shoudl still be looking at the X value right? Just in this case for this question I don’t right?
wild fable
ya
chilly karma
Help please I did this question and apparently I got the wrong answer? Idk where i went wrong???
flat cedar
Help please I did this question and apparently I got the wrong answer? Idk where...
you have split the quadratic wrong
toxic lichen
would be (2y+5)(y-6), no?
oak ivy
right
toxic lichen
you put the +5 and -6 in the wrong place
oak ivy
huh
flat cedar
would be (2y+5)(y-6), no?
yes you are riht, i am sorry
pearl pondBOT
pearl pondBOT
Available help channel!
pearl pondBOT
guys is this an acceptable proof
pearl pondBOT
like does it make sense
plush sundial
ye
unkempt yacht
guys is this an acceptable proof
is this a test?
brave sky
nah its a hw application task
unkempt yacht
ok
unkempt yacht
guys is this an acceptable proof
it is mathematically correct, but personally if i were grading this, i would only give it 3/4 points maximum
unkempt yacht
you should explain better on what you are going to do
imagine i don't know anything about induction proof. i would not know what the hell are you doing here
outline your steps properly, i.e. base case, inductive hypothesis and induction step
brave sky
so n is the base case
unkempt yacht
no it's not
unkempt yacht
what do you think P here represents?
unkempt yacht
no
brave sky
proposition
unkempt yacht
correct
unkempt yacht
which brings me to my second point: it is better to write P(1): 1*2*3 ... rather than P(1) = 1*2*3 ...
you are proving propositions. you first prove P(1) is correct, assume P(k) is true, then prove P(k + 1) is true
unkempt yacht
P(k) is the inductive hypothesis
brave sky
and k is any integer
unkempt yacht
any integer greater than 1, yes
unkempt yacht
yes
brave sky
wait so whay is the base case
unkempt yacht
P(1) is the base case
brave sky
what if it said n> or equal ton2
unkempt yacht
in general, your induction proof should look like this:
- Base case:
P(1): 1 * 2 * 3 = ... = ... (true) - Inductive hypothesis:
Suppose P(k) is true; that is, 1 * 2 * ... * (n + 1) * (n + 2) = .... We shall prove P(k + 1) - Induction step:
P(k + 1): .... = ..... = ......
brave sky
then rhe base case woild be p(2)
unkempt yacht
then rhe base case woild be p(2)
do you know how an induction proof works?
brave sky
yeah
at the end woild i say since p(k) is true then p(k+1) is true fir every natural number of n
i mean k
unkempt yacht
for every k > 1, yes
this means that P(1) is true implies P(2) is true, which implies P(3) is true, so on and so forth
brave sky
but why is it k> 1 and not > or equal to 1
unkempt yacht
which has the same effect of proving P is true for all integer n
unkempt yacht
but why is it k> 1 and not > or equal to 1
because you already proved P(1) is true, no need to assume P(1) is true again
unkempt yacht
no problem. close if you are done with this channel
brave sky
.close
pearl pondBOT
pearl pondBOT
Available help channel!
storm verge
pearl pondBOT
storm verge
acd=dob=a, why is dcb=a/2?
also are we just supposed to assume that cda=cdb=90?
.close
tulip cradle
I don’t get how they’ve done the derivative still
pearl pondBOT
I don’t get how they’ve done the derivative still
tulip cradle
pearl pondBOT
@tulip cradle Has your question been resolved?
merry carbon
Chain rule ⛓️ derivative of the inside of the first term wrt M is pi * (theta - 1)v’(W - (theta - 1)M), of the second is -(1 - pi) * v’(W - M)
Of course evaluate both at M = 0, and you get both the insides of the v’ thingys being v’(W), then a matter of algebra and factoring 
hazy escarp
Hey so I’m solving this and I got in a problem while drawing the given to get the nature of the triangle since I got 2 sides and I’m not sure how to get the third since I can’t use Pythagorean theorem, the sides I got are equal and seeing it can’t be proved to be right do I just consider it isosceles or do I get the third side to see if it might be equilateral and how
green dew
Hey so I’m solving this and I got in a problem while drawing the given to get th...
which question r u on
cause im lost in what u js said
green dew
nature of triangles?
hazy escarp
The figure the nature of AEC
hazy escarp
nature of triangles?
Yes
green dew
u have AE
hazy escarp
Yes
green dew
and its length
hazy escarp
And AC
green dew
AC and AE?
hazy escarp
Since 6 is given and the other is 4+2(C mid point and AO radius)
green dew
ok
hazy escarp
I have those but shouldn’t I get the third side as well
Which is CE
It can be either isosceles or equilateral
So I’m not sure
green dew
equilateral triangles are special cases of isoceles triangles
i dont think it will matter for the rest exercice
but u can indeed find CE using trigonometry
hazy escarp
If I wanna look at it it’s an isosceles triangle but how do I prove it
hazy escarp
but u can indeed find CE using trigonometry
I never took that
green dew
I never took that
then u cant prove its equilateral
even so, its not equilateral
using trig
hazy escarp
I know I’m just asking how to prove it’s isosceles with only 2 sides
Shouldn’t I get the 3rd
green dew
no
js two equal sides is enough
as i said, equilateral triangles are js a special case
of isoceles triangles
hazy escarp
I see
hazy escarp
I know that
green dew
then two sides is all u need
hazy escarp
Hey so I’m solving this and I got in a problem while drawing the given to get th...
In number 3
Do I use Thales theorem
Wait nvm
pearl pondBOT
@hazy escarp Has your question been resolved?
hazy escarp
,close
pearl pondBOT
pearl pondBOT
Available help channel!
mystic bison
this is physics but might as well ask anyway
how is the current 0.6mA? they are doing 9/15000 im guessing but would it not just be 9/10000 aka 0.9mA maybe im just confused on how the cap and inductor interact with the stuff
pearl pondBOT
this is physics but might as well ask anyway
how is the current 0.6mA? they ar...
@mystic bison Has your question been resolved?
flat cedar
this is physics but might as well ask anyway
how is the current 0.6mA? they ar...
When a switch is just closed:
Current flows through the capacitor for a brief time and no current flows through inductor.
mystic bison
When a switch is just closed:
Current flows through the capacitor for a brief ti...
I see so thats what makes the current flow through the 10 and 5 res instead of just 10?
wouldnt it be a different current
for the 5k resistor
like I2 since its a junction point
flat cedar
I see so thats what makes the current flow through the 10 and 5 res instead of j...
Yes it just flows through 10k and 5k
flat cedar
wouldnt it be a different current
No.
When we say current won't flow in inductor, it means the whole thing (2k+inductor) system is kind of of no use
mystic bison
and thats cause the inductor intially fully resists the current right
flat cedar
Yup that's right
mystic bison
ic that makes more sense ty
ripe pagoda
If all the real solutions of the equation 4^x - (a - 3)2^x + (a - 4) = 0 are non positive, then
a) 4 < a <= 5
b) 0 < a < 4
c) a > 4
d) a < 3
what i did was sub 2^x = t
so i get a quadratic in t
im assuming the roots to be x and y
so x + y = -b/a = (a - 3)/(a - 4)
now x and y are both negative, so (a - 3)/(a - 4) should be negative
and xy = c/a = (a - 4)
xy should be positve as both are negative, so (a - 4) should be positive
which means (a - 3) is negative, so i can infer that a should be greater than 4 and less than 3
so obviously this doesnt work, where does this approach fail?
i know the actual solution but its kinda long, this was my first approach and i thought itll be over quick but it didnt
was gonna go for option elimination but yeah
rose robin
If all the real solutions of the equation 4^x - (a - 3)2^x + (a - 4) = 0 are no...
is x and y the root for the quadratic in terms of t or the roots to the original equation?
ripe pagoda
well i thought that theyll both be the same as im just subbing in t right
or is this wrong
tender ingot
If all the real solutions of the equation 4^x - (a - 3)2^x + (a - 4) = 0 are no...
-b/a = (a-3)/(a-4)?
rose robin
well i thought that theyll both be the same as im just subbing in t right
well if for example, t=0.5 is a root, then t=2^-1, so x=-1 is the root of the original equation
so the roots of t doesn't have to be negative
north ruin
can anyone helpo me with this im pretty close but missing some key points?
jolly parrotBOT
Xetrov
pearl pondBOT
can anyone helpo me with this im pretty close but missing some key points?
Show your work, and if possible, explain where you are stuck.
north ruin
okay
so well i used the definition of continuity to write out de epsilon neigbourhood of the funtion but im stuck on how to use the inreasing and intermdeiate value property how to use it and complete the proof
pearl pondBOT
@north ruin Has your question been resolved?
pearl pondBOT
@north ruin Has your question been resolved?
pearl pondBOT
@north ruin Has your question been resolved?
sour hearth
If all the real solutions of the equation 4^x - (a - 3)2^x + (a - 4) = 0 are no...
Your mistake was in concluding a - 3 is negative. Instead, check the actual root conditions:
1. Substituting 2^x = t, we get the quadratic:
t^2 - (a-3)t + (a-4) = 0
with roots t_1, t_2.
2. Since t = 2^x > 0 and all solutions must be \leq 1:
• t_1 + t_2 = a - 3 \leq 2 \Rightarrow a \leq 5.
• t_1 t_2 = a - 4 > 0 \Rightarrow a > 4.
Thus, 4 < a \leq 5 (Option A).
analog ore
A square with side length ( (a+b) ) (where ( a ) and ( b ) are integers, with ( a > b )) is divided into four equal pieces as shown in the figure:
Express ( c ) in terms of ( a) and ( b ).
tardy reef
You can solve it with simply the pythagoras theorem
analog ore
how ? i dont see any appropriate triangle
tardy reef
You can break the pieces into triangles
cursive wraith
how ? i dont see any appropriate triangle
can you see that the skewered lines inside the square are perpendicular to each other?
analog ore
can you see that the skewered lines inside the square are perpendicular to each ...
i can see it but its never explicitly stated in the exercise
cursive wraith
i can see it but its never explicitly stated in the exercise
well yeah, you have to prove it
cursive wraith
so
the square is naturally cut into 4 quadrilaterals
notice that they have equal lengths
analog ore
um yea
cursive wraith
and so they are all similar
analog ore
yup
cursive wraith
so this means
that all four angles at the middle "crossing" are equal
4 equal angles summing up to 360°...
cursive wraith
uh
analog ore
nevermind
so
we proved that the crossing is perpendicular
dont we have to draw a hypothenus ?
cursive wraith
Well now that we have the right angle(s), we just draw more lines to create right triangles
cursive wraith
A square with side length ( (a+b) ) (where ( a ) and ( b ) are integers, with ( ...
See for example, in the top quadrilateral
analog ore
like this ?
cursive wraith
Join the bottom left and top right edges by a line
cursive wraith
like this ?
Exactly
analog ore
then we will have 2c² = (a² + b² )²
analog ore
yea my bad
alrightt tysm
cant believe im a second year engeneer and struggling on this basic stuff
bye !
jolly parrotBOT
Midofey
tardy reef
$c = \frac{(a+b)(a-b)}{2}$
The diagram is correct, but this makes no sense. You still gotta apply pythag to relate the sides with c.
lethal flame
The diagram is correct, but this makes no sense. You still gotta apply pythag to...
I did do pythagoras
maybe i messed up
pearl pondBOT
@analog ore Has your question been resolved?
lethal flame
$c = \sqrt{\frac{a^2 + b^2}{2}}$
jolly parrotBOT
Midofey
lethal flame
is this correct? @tardy reef
tardy reef
is this correct? <@617550509477330944>
. Yes it is. as you can cross-check from what the OP and rafilou calculated earlier
flint zenith
pearl pondBOT
flint zenith
How can I determine these?
I have a guess but I don't know if it's logically sound
flint zenith
for onto -> since the mapping goes from R4 to R4, it is onto
for one-to-one each input variable has an output variable
flint zenith
same for this but the one-to-one confuses me because I feel like I may be missing something
unkempt yacht
for onto -> since the mapping goes from R4 to R4, it is onto
not necessarily
flint zenith
alr
flint zenith
it is not
unkempt yacht
why?
flint zenith
because both x and y are mapped to x
unkempt yacht
mmmm, i'm looking for a particular keyword
the ..... of the map T does not equal to R^2
flint zenith
range ?
unkempt yacht
exactly
flint zenith
consider the map T: R^2 -> R^2, T(x, y) = (x, x)
so would this be mapped onto a line in a 2-dimensional plane
unkempt yacht
similarly, can you justify the range of this map is also four-dimensional?
flint zenith
each element is mapped to a unique element
unkempt yacht
i'm assuming we are talking about the map 3
unkempt yacht
you are essentially saying T(x, y, z, w) = (y, x, w, z) is both one-to-one and onto
flint zenith
yes
flint zenith
each element is mapped to a unique element
that's what Im saying by saying this?
paper wharf
hi
unkempt yacht
technically that only means injectivity, but you also said it's surjective
flint zenith
yea
unkempt yacht
i'm not saying you are incorrect, but i want you to explain why do you think so
flint zenith
each of the variables are used in the transformation
unkempt yacht
have you heard of the term "kernel"?
flint zenith
I have but I'm not very familiar with it
unkempt yacht
or "null space"
paper wharf
yea
flint zenith
I understand null space
but I don't understand how to apply kernel to linear transformations
and ik they're basically the same thing
unkempt yacht
right. then you should also know that T is injective if and only if ker(T) = {0}
flint zenith
each element is mapped to a unique element
how can I tell that ker(T) = {0}? essentially this?
unkempt yacht
well of course T(0, 0, 0, 0) = (0, 0, 0, 0)
but since all T does is swapping the first and second entries, and the third and fourth entries, of course no other element maps to (0, 0, 0, 0) except for (0, 0, 0, 0) itself
thus ker(T) = {0}, thus it is injective
likewise, T is onto if and only if im(T) = R^4. it's obvious the subspace of R^4 generated from (y, x, w, z) is four-dimensional and thus is R^4 itself
flint zenith
I see
unkempt yacht
is it?
flint zenith
it is not
unkempt yacht
why?
unkempt yacht
and you are done
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muted coyote
hello can someone help me, i am unsure of my answer i solved it in 2 ways and i got $(6,2)$ and $(7,3)$
jolly parrotBOT
counterclokwise
north talon
is there anything else you're given?
unkempt yacht
hello can someone help me, i am unsure of my answer i solved it in 2 ways and i ...
not enough info unfortunately
you would need to know either the length of the upper small side, or the area of the shape
muted coyote
not enough info unfortunately
i forgot to mention this sorry
it is a rectangle
i made a new rectangle inside it
i mean sq
pearl pondBOT
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muted coyote
<@&286206848099549185>
compact ridge
I don't get how the two shapes show a rectangle?
oak ivy
Like I think it's a rectangle with those cut off pieces also being a rectangle
pine temple
Hmm
jolly parrotBOT
Bammmyouuu
pine temple
Ngl, not bad for my first time using LaTex
pine temple
**Bammmyouuu**
@muted coyote But yeah ^
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little hound
pearl pondBOT
little hound
Comparison test..
I have to get them correct in one go which is why I am stuck, unsure which ones are incorrect
.close
pearl pondBOT
bright rose
someone help me with v)
I have found that w = cos(2pi / 7) + isin(2pi/7)
and from iv)
alpha = w+w^2+w^4
also -cos(pi/7) = cos(6pi/7)
so we have that w+w^2+w^3 = cis(2pi/7)+cis(4pi/7)+cis(6pi/7)
but after that i am stuck
bright rose
i know you would have to do some finnicky stuff with equating real and imaginary parts from alpha
pearl pondBOT
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compact ridge
someone help me with v)
I have found that w = cos(2pi / 7) + isin(2pi/7)
and fro...
ah so directly from part iv, you have Re(w^4) = cos(8pi/7) = -cos(pi/7) and so on
spark parcel
compact ridge
?
you need to be familiar with these sorts of identities
try using the unit circle and marking the angles theta = 8pi/7 and pi/7
then compare their x-coordinates
so in general yes cos(x + pi) = -cos(x)
then the ones for Re(w) and Re(w^2) match directly
bright rose
ohhh
so Re(w+w^2+w^4) = - 1/2
=> cos(4pi/7)+cos(2pi/7)+cos(8 * pi/7) = -1/2
because -cos(pi/7) = -1 * -cos(pi+pi/7)
which equals cos(8pi/7)
thank you
i get it now
compact ridge
so Re(w+w^2+w^4) = - 1/2
=> cos(4pi/7)+cos(2pi/7)+cos(8 * pi/7) = -1/2
because ...
np!
very similar for the imaginary part, e.g Im(w^6) = -Im(w)
alright no worries
bright rose
pearl pondBOT
bright rose
im assuming you would let w be cos(2pi/9), and you would have to prove that:
Re(w+w^2+w^3+w^4)
= - 1/2
bright rose
oh wait
i figured it out
after a lot of trial and error
so you have that:
w+w^2+w^3+...+w^8+w^9 = 0
=> w+w^2+2^3+w^4+w^5+w^6+w^7+w^8 = -1
=> w+w^2+w^3+w^4+w^4(w+w^2+w^3+w^4) = -1
=> (w+w^2+w^3+w^4)(1+w^4) = -1
=> (w+w^2+w^3+w^4) = -1 / (1+w^4)
and we need to prove that Re(w+w^2+w^3+w^4) = - 1/2
so thus we need to prove that:
-1 / (1+w^4)
= - 1/2
which is:
-1/(1+cos(8pi/9)+isin(8pi/9))
dividing the fraction gives:
-(1 + cos(8pi/9) - isin(8pi/9)) / 2(1+cos(8pi/9))
which simplifies to -1/2
yay
.close
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✅
compact ridge
cos(2pi/9) = cos(2pi - 2pi/9) = cos(16pi/9)
cos(4pi/9) = cos(14pi/9)
cos(6pi/9) = cos(12pi/9)
cos(8pi/9) = cos(10pi/9)
but then cos(2pi/9) + ... + cos(16pi/9)
is just the real part of w + .... + w^8 = -1, by the geometric series
(if you practice enough you'll automatically know 1 + w + .... w^n = 0, well there's a nice geometric reason too when you add up all the vectors tip to tail)
so each half equals -1/2
placid geyser
dividing the fraction gives:
-(1 + cos(8pi/9) - isin(8pi/9)) / 2(1+cos(8pi/9))
How'd u get that step?
bright rose
How'd u get that step?
Times the top and bottom by the conjugate
placid geyser
Oh, ok thanks
bright rose
is just the real part of w + .... + w^8 = -1, by the geometric series
(if you p...
Could you tell me more about this vector tip proof?
compact ridge
Could you tell me more about this vector tip proof?
observe
you just end up recreating the original regular polygon
(it's regular cause all n points are the same distance to the origin, and eachcentral angle is equal at 2pi/n)
bright rose
Interesting
So for a nth root of unity, you can form a nth sided polygon as the sum of all those roots which always equals zero
compact ridge
you just end up recreating the original regular polygon
(by moving the vectors around in that order)
compact ridge
So for a nth root of unity, you can form a nth sided polygon as the sum of all t...
yep!!
the resultant is the zero vector
bright rose
And the angles in that polygon works be (n-2)*pi
compact ridge
so the sum must be (the complex number) 0
compact ridge
And the angles in that polygon works be (n-2)*pi
unfortunately that fact isn't useful
for complex numbers
bright rose
Actually it is because it means that the angle between the vector sum of two roots of unity is always (n-2)*pi
bright rose
Oh then it would be the angle of the vector sum of two consecutive roots of units
bright rose
That angle would be:
(5-2)*pi which is 3 pi
Oh
Yeah
That's wrong
I gtg but cya man
Thank you
This was very interesting
.close
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woven totem
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quick venture
Can somebody explain why my second x value is wrong it should be 5.6992rad
quick venture
Thanks
keen quail
Can somebody explain why my second x value is wrong it should be 5.6992rad
youve written 4.19.. instead of 4.91
in the last line
pearl pondBOT
@quick venture Has your question been resolved?
torpid schooner
How do I determine the bounds for this blue curve?
pearl pondBOT
How do I determine the bounds for this blue curve?
pearl pondBOT
@torpid schooner Has your question been resolved?
rough forge
How do I determine the bounds for this blue curve?
Make a table
The period is 10π
For each value of theta, draw the radius
pearl pondBOT
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sour sluice
why didn’t he reduce the common expressions? is it because there were terms separated by subtraction in the numerator?
sour sluice
that’s the only reason? and it would be the same if there was addition?
sour sluice
like if it were on the right side with those mentioned terms?
oak ivy
yes
sour sluice
wtf
dapper spruce
why didn’t he reduce the common expressions? is it because there were terms sepa...
you can multiplié both the sides by e^-x
toxic lichen
why didn’t he reduce the common expressions? is it because there were terms sepa...
can you highlight which common expressions you thought could get reduced
dapper spruce
this will make it simple maybe
sour sluice
oak ivy
But it is not present in xe^xe^x
sour sluice
if it were present, how would the cancelation play out?
oak ivy
so 1+e^x could be taken common
sour sluice
if it were on both sides of the numerator, would you slash both of them and then slash the denominator down one power?
oak ivy
1+e^x[xe^x+e^x-xe^xe^x] divided by (1+e^x)^2
Remove the 1+e^x from numerator and remove the square from denominator
sour sluice
ah that makes sense
what if for some reason (1+e^x) was on the right side of the numerator, between Xe^x and e^x?
oak ivy
Uh same thing right since it's being multiplied
oak ivy
what if for some reason (1+e^x) was on the right side of the numerator, between ...
So it is not present on left side?
sour sluice
So it is not present on left side?
no it is, sorry forgot to say that
oak ivy
what if for some reason (1+e^x) was on the right side of the numerator, between ...
It doesn't chang
celest cedar
I thout that point as +
oak ivy
Ssame thing
oak ivy
^
Take it out common
Since it's being multiplied
You mean (1+e^x)(xe^x+e^x)-xe^x(1+e^x)e^x right
sour sluice
so you would have (1*e^x)(all the other terms + 1)
sour sluice
You mean (1+e^x)(xe^x+e^x)-xe^x(1+e^x)e^x right
yes
sour sluice
wouldn’t a 1 be in there to account for the second term of (1+e^x) ?
if we’re talking about the term being on both the left and right
sour sluice
hold on rq i’m gonna draw something out
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dense zinc
assume a(x) and b(x) are polynomials
degree of a(x) = degree of b(x) + 1 (the degree of a(x) is higher by 1 then the degree of b(x))
would this mean that the division of b(x) by a(x) would always result in a reminder that is not 0?
mental vine
do you mean b(x)/a(x)?
dense zinc
yep
supple mural
``assume a(x) and b(x) are polynomials``
``degree of a(x) = degree of b(x) + 1`...
yeah
mental vine
alright
not only is the remainder not 0
unless b(x)/a(x) is simplify-able
the remainder will be b(x)
pearl pondBOT
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Available help channel!
pearl pondBOT
flat cedar
Wtf?
frozen lantern
yeah
pearl pondBOT
oak ivy
Troll?
open rivet
spam message
frozen lantern
some guy pinged everyone, dropped a random server invite, occupied three channels.
main oriole
some guy sent a scrupulous link
flat cedar
Yeah
frozen lantern
no you cant lmao
oak ivy
Everyone doesn't work
latent quail
Banned
oak ivy
Why do these idiots come here
frozen lantern
eh yeah
open rivet
they prob got hacked
main oriole
Why do these idiots come here
big server = big publicity for their ip grabber
frozen lantern
dont feed him
latent quail
Don’t click on any suspicious link
open rivet
let it go people, move on
oak ivy
Ok
main oriole
!original
pearl pondBOT
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.
wild fable
let it go people, move on
lettt it gooo let it goooo cant hold it back anymoreee
pearl pondBOT
Available help channel!
cobalt hinge
pearl pondBOT
cobalt hinge
!status
pearl pondBOT
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
1
cobalt hinge
What
light helm
look it up
uneven radish
state the angles u know for sure first
for example, a medium triangle can be created using the angles 45 and 30
using angle sum of triangle the remaining angle is 180-45-30=105
but then theres a straight line
straight lines add up to 180
so the angle right next to it is 180-105=75
cobalt hinge
Yeah I know that
uneven radish
what i feel like u need to do
is treat a few unknown angles as letters
like angle something has y degrees
angle something has z degrees
i feel like that substitution would be better to calculate areas sum of triangles
light helm
conventional angle chasing doesn't work here
uneven radish
i assume the whole big triangle is isoceles
if it is it saves us loads of time
because
if its isoceles would u agree the bottom 2 angles are the same
but they all add up to 180
so 180-45-135, 135/2 = 67.5
the bottom left angle 67.5-45=22.5, and the bottom right angle 67.5-30=37.5
cobalt hinge
Yes
uneven radish
say we sub in a few values
so 45+30+y+z=135
what we know now
is the bottom left small triangle
one of the angles is 75 degrees
now lets think the bigger triangle on the bottom
top angle 75 degrees, + y + 45 + z = 180
y+z should add up to 60 agreed?
ok lemme draw a diagram first
uneven radish
essentially yes
notice that 135-y angle
because its corresponding angles
the angle right above it should also equal to 135-y
same with its left and right
so we have y+z=60
now lets think here
bottom right big triangle
the top angle should be 180-45-30-y aka 105-y
now focus on small right triangle
top angle is 105-y
however with corresponding angles in z
it also equals to 45-z
something like this
what we can try is a quadrilateral
we know polygons with 4 sides add up to 360
,rotate
jolly parrotBOT
uneven radish
we know y+z=60
pearl pondBOT
@cobalt hinge Has your question been resolved?
pearl pondBOT
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Available help channel!
scarlet trellis
pearl pondBOT
scarlet trellis
My initial approach was to write a program on my calculator to solve it (you are allowed any calculator with any apps) but it was just taking too long to write that out on a calculator keyboard (this is a timed test)
other than that, i have no idea where to even start
1000 numbers is way to difficult to write out manually
maybe i could make use of the table of values from the grapher app? but no idea how to even get a function here to graph
toxic lichen
there ain't that many trios of primes adding up to less than 13
the three lowest ones already add to 10
actually is it just (2,3,5) and (2,3,7)
i think thats it
uneven radish
ye
our smallest positive integer is 30, 2(3)(5)
with powers having 1
then multiply 2 for each, 60, 120, 240, 480, 960
we already have 6 here
now mutiplt by 3 starting from 30, disregarding copies
90, 270, 810
then 5
150, 750
now lets see if a and b = 2
(4)(9)(5)=180