how many ways for that?
#help-39
cursive wraith
cursive wraith
wait so why wouldnt we do it for 4, and others i dont get it
because in all the ways we counted, x_1 + x_2 + x_3 + t = 21
so there's no need to get rid of stuff we haven't counted at all
north talon
ohh
cursive wraith
we just overcounted when x_1 + x_2 + x_3 + t = 21 and t >= 16
north talon
okay
cursive wraith
yep correct answer
north talon
yay
cinder flower
as a generating function shill i will also mention....
,w expand (\sum_{i=3}^{24}x^i)^3 * (\sum_{i=0}^{15}x^i)
jolly parrotBOT
cinder flower
the coefficient of x^30 is 1968
north talon
how did you get that?
cursive wraith
how did you get that?
wolfram alpha computed this
cinder flower
yea i did not do much
north talon
no i meant like the sum
cursive wraith
no i meant like the sum
you allow the teacher to answer between 0 and 15 questions
and you force each of the 3 students to answer at least 3 questions
north talon
whys the top bound 24
cursive wraith
the upper bound is 24, because you know the other two students answer at least 6 questions together
cinder flower
also could be anything >= 24 and it would work
cursive wraith
and yes anything above 24 would also work
because it won't contribute to the x^30 coefficient
north talon
oh
cursive wraith
(they would answer more than 30 questions in total)
north talon
thats cool
north talon
okay haha
cinder flower
unless you had a way to specifically find the x^30 coefficient without multiplying everything out
north talon
if i get bored in an exam i would do this
cinder flower
but you would probably just do something like what rafilou walked you through for that
pearl pondBOT
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Available help channel!
rotund osprey
pearl pondBOT
rotund osprey
im so lost
pretty much i ended up w like
x(2x+6y) = 3.6 and y(2x+6y)=4.8
idek if thats right
but
what do i do
idk how to solve that
rough forge
idk how to solve that
Do you know the formula for a vector projection?
jolly parrotBOT
bacc the sigmašš¤
rough forge
You are given |b|
Just do the dot product between a and b
then equate the components
solve for x and y
rotund osprey
x(2x+6y) = 3.6 and y(2x+6y)=4.8
this is what i get
when i equate componetns
but the numbers are like
super annoying to work w
rough forge
(2x+6y)x = 3.6
(2x+6y)y = 4.8
rotund osprey
yeah
how do i solve that
without making it stupidly convolute
maybe my brain is not working rn
rough forge
You solve one equation for one variable, say eq. (1) for y
rough forge
Then plug that into eq. (2)
rotund osprey
idk if theres a better way
cos i got
y = (3.6-2x^2)/6
and then plugigng that into the other thing
i get rly
clunky
hard to work w numbers
rotund osprey
huh
rough forge
weird
they have good solutions
just doing something wrong
I think it's easier to solve do (2)-(1)
(2x+6y)(y-x) = (4.8-3.6) = 1.2
No this doesn't help
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dull hemlock
This isnt related to a specific question but I was wondering if anyone knows of any resources to help me learn about metric spaces
red hedge
pearl pondBOT
red hedge
Need help with d and e
light helm
did you do parts a,b,c?
red hedge
Yeah
Hereās how I did it
And then I input negative 1 and 1 in place of h to find whether itās increasing or decreasing at the point before and after the min/max
light helm
i don't follow your reasoning for the last lines of each part
red hedge
Well everything cancels out except for the h^2
light helm
not everything else should cancel
double check your signs
red hedge
For a b or c
light helm
cancellation issues for b and c
part a is missing things
red hedge
Alr one sec
light helm
for part d) and e)
you can use stuff like compound angle identities
light helm
do you know how to expand
sin( p + q) =
cos(p + q) =
red hedge
We didnāt learn this stuff yet š¦
We do this in like 4 more units
But isnāt is sin(p)sin(q) + cos(p)cos(q)
For the first one
And idk abt cos
light helm
look up compound angle identity, double angle identity
trig is usually taught before these types of questions
red hedge
Ig Iāll ask my teacher but I might be fucked
light helm
have you not been officially taught derivatives yet?
red hedge
No
I know power rule but weāre supposed to learn that in the next c purse
Course
Thatās about it tho
red hedge
Nope
light helm
i suppose you could just graph around the indicated point
red hedge
Alr
light helm
or make a table of values
dreamy shoal
How do I do this one
pearl pondBOT
How do I do this one
dreamy shoal
Is the next step
3(-csc^2(pi-t)^2?
Is the next step
3(-csc^2(pi-t)^2(pi-t)?
Actually ^
jolly parrotBOT
riemann
plush bramble
you're using this right?
the "outer" function is x^3 and "inner" function is cot(pi - t)
pearl pondBOT
@dreamy shoal Has your question been resolved?
plush bramble
no
dreamy shoal
How
plush bramble
just get this part right first
dreamy shoal
Which one is that I only know them by name and idk what langrange mean
Is that newton
Or the other one
plush bramble
you're using this right?
.
dreamy shoal
How did you know that you need to apply that formula out of all of these
plush bramble
by looking at the problem and decomposing y = f(g(x))
dreamy shoal
So f outer g inner?
So the derivative of f
3()^2
With the original g inside
Ohh so itās 3 functions
Thatās my mistake
I again didnāt identify that itās 3 chainsš„²
plush bramble
,w diff cot^3(pi - x)
jolly parrotBOT
dreamy shoal
Why? You multiply by the inner function no?
If it was just cot(pi-t
Then itās -csc^2(pi-t)*(pi-t)
plush bramble
$\frac{d}{dt}(\pi - t) = ?$
jolly parrotBOT
riemann
dreamy shoal
Ohhh
Cuz pi is just a number
Right
So -1
Letās gooo
Thank youš
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ornate garden
occupied
ornate garden
how is
3/5 * 5/8
how are the fives "allowed" to cancel eachother out?
i just need to understand the mechanics of it please
sharp vigil
,, \frac 35 \cdot \frac 58 = \ab(\frac 31 \cdot \frac 15) \cdot \ab(\frac 51 \cdot \frac 18) = \frac 31 \ab(\frac 15 \cdot \frac 51)\cdot \frac 18
jolly parrotBOT
cloud
compact veldt
3/5 is 3 * 1/5
5/8 is 5 * 1/8
3 * (1/5 * 5) * 1/8
= 3 * 1 * 1/8 = 3/8
you're basically multiplying 5 by 1/5
which is 1
ornate garden
so any time two numbers are the same, when X:ing fractions, i can just cancel them out?
4/9 * 9/8?
=4 / 8 ?
compact veldt
4/9 * 9/8?
yes
compact veldt
so any time two numbers are the same, when X:ing fractions, i can just cancel th...
$\frac{a}{b} \cdot \frac{b}{c} = \frac{a}{c}$
jolly parrotBOT
w
ornate garden
now where talking
great! does that rule have math name? or just "its just maths, stupid" ? š
compact veldt
i mean, its just cancelling terms
not sure if it has a name, i wouldnt specifically call it anything
ornate garden
yeah. feels so general. is there a case it wouldnt be correct?
compact veldt
if i were writing a proof, i wouldnt say, this is true because...
compact veldt
yeah. feels so general. is there a case it wouldnt be correct?
uh well i guess if b were 0??
u wouldnt even write something over 0
other wise as long as it follows this form
ornate garden
oh! feels like road is open! thanks alot!
compact veldt
when a, b, c are all complex
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whole carbon
I have 700 analyses and their conclusions.
I have 2 humans analysing the conclusions.
They either say "Correct", "Incorrect", or "I don't know".
Human one analysed let's say 500 of them. Human two analysed 400.
I want to see if the difference in the percentage of "Incorrect" conclusions they give is significant (maybe human 1 says 50% incorrect and human 2 says 55% incorrect). Is the difference significative?
whole carbon
Should I use a z-test?
hot stone
Sounds like a 2 proportion z test
But not completely sure without knowing your class, the exact problem etc
kindred mantle
y = (x^5) + x
pearl pondBOT
y = (x^5) + x
kindred mantle
can this function be inverted
I dont want a numerical solution
well i suppose it can becuase its bijective
but how would i go about it
sharp vigil
if it was a polynomial of lower degree (e.g. a quadratic or cubic) then you could use the (quadratic, cubic, etc.) formulas, but there is no such formula for quintics
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abstract sentinel
Hi, I'm trying to do an example problem with finding f = O(g).
I've put it in the form of lim x -> inf, f(x)/g(x).
I've been able to rewrite it to the form in the pictures. However I'm not really sure where to go from here.
My understanding is I should be aiming to put it in the form so that there are only constants in either the numerator or denominator.
abstract sentinel
f = (logn)^logn, g = n/logn
I don't really know what to make of e^log n * ln(log n)
Picture shows log(log n), but it should be ln(log n).
untold dawn
you guys study log in which grde
abstract sentinel
If I use L'Hopitals rule, I just kinda get stuck in a loop where I constantly get derivatives of logs, which gives me 1/n which fails to eliminate the term in the denominator
abstract sentinel
Thanks
plucky python
Hi, I'm trying to do an example problem with finding f = O(g).
I've put it in th...
do u mean proving f is O(g)?
untold dawn
i tried my best
abstract sentinel
do u mean proving f is O(g)?
Yes. But I'm just stuck specifically on rewriting the fraction rn
Such that the denominator or numerator only has constants
plucky python
Yes. But I'm just stuck specifically on rewriting the fraction rn
are you sure? cus wolframalpha suggests that f is not O(g)?
abstract sentinel
I don't know how to compare e^logs to linear or logarithmic time
Proving or disproving
plucky python
Proving or disproving
ah kk
abstract sentinel
I'm trying to figure out the logic
I assumed e^logs is exponential time, but apperantly not.
plucky python
If I use L'Hopitals rule, I just kinda get stuck in a loop where I constantly ge...
how exactly?
f/g simplifies to
$\frac{\log(n)^{1+\log(n)}}{n}$
jolly parrotBOT
LY
abstract sentinel
Yes
plucky python
so if u diff. w.r.t. n, bottom becomes 1
abstract sentinel
Yes, but when you diff the top, it gives 1/x
Let me double check my math to make sure that's true
plucky python
Yes, but when you diff the top, it gives 1/x
ah right ic
abstract sentinel
(1+logn)(logn)^logn * 1/(x * ln(10))
abstract sentinel
I think I might have to use the property of multiplying the equivalent of 1/1
Like logn/logn
Or some other form
plucky python
$\frac{e^{x\ln(x)+\ln(x)}}{e^x} = e^{x(\ln(x)-1)+\ln(x)}$ which obviously blows up to infinity
jolly parrotBOT
LY
plucky python
And how e^x got in the denominator
u set x = ln(n)
and rewrite everything in terms of x
abstract sentinel
$\frac{x*e^{x*ln(x)}}n$
jolly parrotBOT
Thelonious
plucky python
n = e^x
abstract sentinel
But it doesn't? if x = log(n), then 10^x = n, not e^x.
The bases in those logs differ
Or am I missing something?
Chatgpt gave me something similar. Ill work through it. I have to practice substitution techniques. Thank you for the help.
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abstract sentinel
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timid imp
how does 33 1/3% = 0.6?
pearl pondBOT
how does 33 1/3% = 0.6?
timid imp
it is literally just 0.333333333 etc but my awnser sheets say 0.6?
plush bramble
how does 33 1/3% = 0.6?
It doesn't I already explained it to you
plush bramble
Just assume it's another error then
timid imp
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fierce schooner
Why is my answer wrong
pearl pondBOT
Why is my answer wrong
plush bramble
Show your work
pearl pondBOT
@fierce schooner Has your question been resolved?
rare edge
pearl pondBOT
acoustic pivot
correct
acoustic pivot
they represent how many times you multiply that number by itself, like on A, you would multiply -2x by itself 5 times. -2x time -2x times -2x times -2x times -2x
rare edge
they represent how many times you multiply that number by itself, like on A, you...
And is that the slope?
acoustic pivot
not exactly, but it makes the x input grow exponentially as it comes out as a y output
rare edge
So why isnāt it D?
acoustic pivot
well, try taking numbers and inserting them as x, just try a small number first like 2 or 3
rare edge
well, try taking numbers and inserting them as x, just try a small number first ...
So 3-2(2)^2
acoustic pivot
exactly
rare edge
That would give us 3-2(4)
acoustic pivot
mhm
acoustic pivot
i was just affirming, like saying yes
acoustic pivot
well you should calculate it further, with would give you -5
rare edge
Okay so -5
acoustic pivot
and then try it with a different number, which can give you a good idea of the direction the graph is taking
rare edge
What do I do with that
rare edge
and then try it with a different number, which can give you a good idea of the d...
So if I try 3
3-2(3)^2
3-2(9)
3-18
-15
acoustic pivot
right, so now you can look on the graph and see if those points are on the graph, it doesnt go down to -15 but it shows -5
if the answer was D then it would show the line passing through the coordinate (2,-5)
rare edge
if the answer was D then it would show the line passing through the coordinate (...
Well that isnāt on the graph
rare edge
and then I would just try it with the others
And I should try 2
For every number
Or 1
acoustic pivot
2, because no matter what you put 1 to the power of, it will always equal 1
rare edge
if the answer was D then it would show the line passing through the coordinate (...
2 being the X value and -5 being the y right
acoustic pivot
o well because any number to the power of 5 or 6is gonna be pretty big, but thats why you use 2
rare edge
o well because any number to the power of 5 or 6is gonna be pretty big, but that...
So far we know
It isnāt C or D
For A I got -61
But that aināt on the graph
acoustic pivot
wait, sorry you should have used -2, which is my bad
which is still -61, but if you input -2 into B it should come up with a positive answer
acoustic pivot
if you put a negative number to the power of an even number, than it comes out positive, but if you put it to the power of an odd number, it will come out negative
look at it like this, -2x-2 is 4, which is positive, but -2x-2x-2 is -8, which is negative again
bitter trail
are you trying to solve for this
rare edge
Correct which is the 3
bitter trail
You see how from intervals of roughly [-0.5,0.5], it looks like a straight line
rare edge
You see how from intervals of roughly [-0.5,0.5], it looks like a straight line
On the X coordinate
Yes I see it
bitter trail
basically thereās a multiplicity
rare edge
basically thereās a multiplicity
Go on
bitter trail
the straighter the line on the vertex point, the greater the multiplicity
i forgot what u call (h,k) for more generalized equations
rare edge
So 0.5 is our multiplicity
bitter trail
multiplicity is based on the exponent
for example, (x-3)^3 tells us that there are 3 x-intercepts. but on the graph, it wont show that. thatās why on (3,0), it looks like a straight line
rare edge
Ok so
bitter trail
So how does the 0.5 equate to the exponent
explain how u got to that conclusion
rare edge
explain how u got to that conclusion
U said the straight line on the interval -0.5,0.5
Iām just lost
bitter trail
U said the straight line on the interval -0.5,0.5
yes the x-values
rare edge
On where the exponent comes into this
bitter trail
from x= -0.5 to x=0.5
bitter trail
straighter the line -> bigger the multiplicity -> bigger the exponent
rare edge
So since itās 0.5
It should be the smallest option
Therefore itās D?
?
The line is small
bitter trail
for example, g(x)= (x-2)^2 has multiplicity of 2
h(x)= (x-6)^9 has multiplicity of 9
the graph of h(x) at its vertex will appear as a straighter line than g(x) since the exponent is bigger
sorry my phone froze
bitter trail
bitter trail
Since itās the biggest exponent
yessssss
rare edge
yessssss
This was the reasoning to
It didnāt mention a line
Is it 5?
Hence the name
rare edge
Ahhhh
rare edge
Is the answer 5
bitter trail
to find how many turning points there are
you take the degree and subtract 1
so 5-1
rare edge
Got it
bitter trail
Itās 4 or less then 4
uhhhhh
bitter trail
yeah ik
rare edge
So itās any number under 5
rare edge
This has to be 3
bitter trail
i dont see how it can have one point of inflection
bitter trail
let me count hold on
rare edge
bitter trail
no itās two
bitter trail
rare edge
Huh
bitter trail
Whatās a turning point
but itās when the graph changes concavity
bitter trail
letās take the middle point from the relative maximum and origin
from the relative maximum to the middle point, the rate of change is still decreasing bc the graph is concave down
but from the middle point to the origin, the rate of change is now increasing bc the graph is roughly reaching concave up
rare edge
Ohhhh
bitter trail
and then the same for the middle point in between the origin and the relative minimum
rare edge
This is a positive
So itās either B or D
And since it isnāt a straight line
The Exponet is low
So is it H
B*
rare edge
Uhhh what
bitter trail
refer to this
ignore fhis
rare edge
just count how many point of inflections there are
5
bitter trail
Since its 4-1
n-1=4
solve for n
n-1 determines how many points of inflection there are on the graph
n is the degree
bitter trail
yeahhhhhh
rare edge
So how do Ik if itās C or D
bitter trail
Holy moly hold on
bitter trail
refer to this
refer to this
this basically tells us what the second biggest exponent value is
find out what function that specific shape looks like
bitter trail
itās D bc the graph i circled looks like a x^3 graph
WIST
NO NO
No no its not D
rare edge
C?
bitter trail
yea
rare edge
Got it
bitter trail
was it right or no
rare edge
Yuppp
bitter trail
yay
pearl pondBOT
@rare edge Has your question been resolved?
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true bluff
I understand how they got the first 2 lines
pearl pondBOT
I understand how they got the first 2 lines
true bluff
but I don't understand how the second line proves that the statement is true
like how can we just subtract the number and say ok this is true because we subtracted this number from that
plush bramble
like how can we just subtract the number and say ok this is true because we subt...
Inequalities work like that
vital estuary
(x^2-2y)^2 >=0
plush bramble
If x > 3, then x-1 > 2
true bluff
ok that makes a lot more sense
vital estuary
it is true that for all real numbers a^2 >= 0
true bluff
I'm not very good with understanding inequalities
vital estuary
this is known as the trivial inequality
true bluff
it is true that for all real numbers a^2 >= 0
yes
vital estuary
by rearranging the inequality, we get a perfect square
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rare edge
If I have 2(2)^2
pearl pondBOT
If I have 2(2)^2
sacred crest
Exponentiation comes before multiplication, yes. So if your function is 2x^2 you would calculate 2*(2^2)
rare edge
Exponentiation comes before multiplication, yes. So if your function is 2x^2 you...
So this is right
sacred crest
Seems so
jolly parrotBOT
sacred crest
Idk tbh, I donāt remember the quadratic formula
rare edge
<@&286206848099549185> check it out
sacred crest
I suppose though you can insert into calculator and see if it works
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jade mist
In a bag, there are 3 red marbles and "B" blue marbles. Two marbles are randomly selected from the bag without replacement. The probability that the two marbles are the same color is 0.5. Calculate the sum of all possible values of B
pearl pondBOT
In a bag, there are 3 red marbles and "B" blue marbles. Two marbles are randomly...
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midnight haven
Could someone explain the solution simply to me? I don't understand the 'counting' thing
pearl pondBOT
Could someone explain the solution simply to me? I don't understand the 'countin...
@midnight haven Has your question been resolved?
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coarse osprey
Hello, why do we have to multiply by the absolute value of x instead of x when trying to evaluate the limit using squeeze in this problem? I think my foundational knowledge of absolute values is lacking.
plucky python
Hello, why do we have to multiply by the absolute value of x instead of x when t...
you don't have to multiply by |x|, they've basically done that to be safe for if x is negative
plucky python
Hello, why do we have to multiply by the absolute value of x instead of x when t...
(basically notice that if x > 0, then)
$-x \leq x \cos(\frac{3}{x}) \leq x$
jolly parrotBOT
LY
plucky python
but if x is negative, notice that you need to flip ur x's around
so $x \leq x \cos(\frac{3}{x}) \leq -x$ for negative $x$
jolly parrotBOT
LY
plucky python
they've just used absolute values so they don't have to do this
plucky python
so multiplying by absolute value of x can always be done ?
i'm not entirely sure i understand ur question? but yeah since abs(x) is non-negative, u can always multiply by it and not have to worry about ur signs flipping etc.
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midnight haven
is there a product expansion formula something like the taylor expansion?
rigid lava
midnight haven
theres an infinite product for sine but also an infinite series (taylor series)
is there a formula to get those expansions like the taylor one?
u can use taylor for exp and then u have sin x = sh (ix)/2i
tropic saddle
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fluid dome
We have to prove that c is bigger or equal to 2h
pearl pondBOT
We have to prove that c is bigger or equal to 2h
tulip matrix
We have to prove that c is bigger or equal to 2h
,rotate
jolly parrotBOT
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fluid dome
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ā
fluid dome
<@&286206848099549185>
fluid dome
Canāt u use similar triangles
I tried what i could , i js gave up
midnight haven
gyat
Conditions:
\begin{align}
h^2 + c_2^2 &= b^2 \
h^2 + c_1^2 &= a^2 \
b+c_1 &> h\
a+c_2 &> h\
\end{align}
Adding the Pythagorean equations:
\begin{align*}
a^2 + b^2 &= 2h^2 + c_2^2 + c_1^2
\end{align*}
Squaring the inequalities:
\begin{align*}
a^2 + b^2 + 2(bc_1 + ac_2) + c_1^2 + c_2^2 &> 4h^2\
2h^2 + 2c_2^2 + 2c_1^2 + 2(bc_1 + ac_2) &> 4h^2\
2(bc_1 + ac_2) &> 2h^2 - 2(c_2^2 + 2c_1^2)\
bc_1 + ac_2 &> h^2 - c_2^2 + 2c_1^2\
&> h^2 - c^2\
\end{align*}
shit
i made a mistake
jolly parrotBOT
icannotdoanymorecauchy
pearl pondBOT
@fluid dome Has your question been resolved?
midnight haven
im trying 
Conditions:
\begin{align}
h^2 + c_2^2 &= b^2 \
h^2 + c_1^2 &= a^2 \
b+c_1 &> h\
a+c_2 &> h\
\end{align}
Adding the Pythagorean equations:
\begin{align*}
a^2 + b^2 &= 2h^2 + c_2^2 + c_1^2
\end{align*}
Squaring the inequalities:
\begin{align*}
a^2 + b^2 + 2(bc_1 + ac_2) + c_1^2 + c_2^2 &> 4h^2\
2h^2 + 2c_2^2 + 2c_1^2 + 2(bc_1 + ac_2) &> 4h^2\
2(bc_1 + ac_2) &> 2h^2 - 2(c_2^2 + c_1^2)\
bc_1 + ac_2 &> h^2 - (c_2^2 + c_1^2)\
&> h^2 - (c_2^2 + c_1^2)\
c_1\sqrt{c_2^2 + h^2} + c_2\sqrt{c_1^2 + h^2} &> h^2 - (c_2^2 + c_1^2)\
(c_2^2 + c_1^2) + c_1\sqrt{c_2^2 + h^2} + c_2\sqrt{c_1^2 + h^2} &> h^2
\end{align*}
jolly parrotBOT
icannotdoanymorecauchy
fluid dome
This doesnt solve it
fluid dome
Thanks for trying
midnight haven
${\frac{h}{\sqrt{c_1^2 + h^2}} = \frac{\sqrt{c_2^2 + h^2}}{c}}$
jolly parrotBOT
icannotdoanymorecauchy
midnight haven
i really have to do it on paper, dont it
stoic imp
pearl pondBOT
midnight haven
have u graphed it
stoic imp
yes
midnight haven
may i see
stoic imp
stoic imp
hi spider man
midnight haven
but i found an importnat piece
stoic imp
is that u venom
midnight haven
proving that c_1^2 + c_2^2 >= 2h^2
fluid dome
Imma open another channel and ask u
midnight haven
okie
midnight haven
the area of the rectangle
is (4-x)sqrt(x)
stoic imp
is (4-x)sqrt(x)
how
midnight haven
why do u need y =x?
midnight haven
rename the points
stoic imp
?
midnight haven
like
stoic imp
is (4-x)sqrt(x)
what about minimum area
stoic imp
you mean find critical points
pearl pondBOT
lemme check smth
Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).
midnight haven
):
stoic imp
JK
midnight haven
well
stoic imp
do whatever you want
midnight haven
theres only one critical point
stoic imp
,w solve d/dx((4-x)sqrt(x)) = 0
jolly parrotBOT
stoic imp
theres only one critical point
which is
stoic imp
interval?
jolly parrotBOT
midnight haven
stoic imp
wdym
midnight haven
.
the second derivative on that interval is always negative, so there's only maximum
midnight haven
that answers the second part
stoic imp
ok
dapper nova
I'm having trouble solving with with the Direct Comparison Test. Can anyone help? I've shown that 1/x^x < 2 for all x
dapper nova
Yeahh
Lol
But after I've shown that
1/n^n < 2, you have the n! which makes it impossible
midnight haven
n+1 wont work definitely
dapper nova
im in bc calc lmao
midnight haven
i got a 5
dapper nova
i have no clue of this stirling approximation
midnight haven
and i cant even solve that
midnight haven
I'm having trouble solving with with the Direct Comparison Test. Can anyone help...
u have to prove it converges ?
dapper nova
yeah its convergent
midnight haven
so u want to compute it or ?
@midnight haven what is the goal here !
to prove
convergence
ah ok
midnight haven
what is dct ?
dapper nova
direct comparison test
midnight haven
why dont u use it then ?
dapper nova
because this worksheet is for the direct comparison test
so we cannot use knowledge that is in the future
plucky python
I'm having trouble solving with with the Direct Comparison Test. Can anyone help...
hint: ||use the first 2 terms when u expand n!/n^n||
dapper nova
I have already expanded it, but I'm not sure how this is helpful
plucky python
well when u expand that, we get
plucky python
n(n-1)(n-2)...3 * 2 * 1 / n^(n-1) * n
ah that's not quite what i meant
n * (n-1) * ... * 2 * 1 / n * n * ... * n
now think about what happens when we take the last 2 terms of the num. and denom.
in latex
$\big ( \frac{1}{n} \big )\big ( \frac{2}{n} \big )\big ( \frac{3}{n} \big ) \cdots \big ( \frac{n}{n} \big )$
jolly parrotBOT
LY
tiny jay
how would i approach this?
pearl pondBOT
how would i approach this?
tiny jay
wouldnt it be the transformation thing but as a matrix since ur jusy putting the standard matrix in it
coral rover
wouldnt it be the transformation thing but as a matrix since ur jusy putting the...
yeah
cool cool so the size of A
tiny jay
4x3
coral rover
hmm
coral rover
okayy so the last row is just -27
that's right
coral rover
ur chilling
coral rover
love the about me though
tiny jay
ty bro
coral rover
ofc
coral rover
how do i close this
".close"
tiny jay
.close
pearl pondBOT
coral rover
what if i multiplied
it by 1/2
[ 1 0]
mhm
yes
bc 1/2 is not an integer
and 1/2 is a suitable scalar
pearl pondBOT
pearl pondBOT
Available help channel!
worthy island
whats the half angle for sin^2?
pearl pondBOT
whats the half angle for sin^2?
worthy island
so sin^2(x/2)
normally its sqrt1-cosx/2
so for sinx i mean
but what if its squared
acoustic path
so sin^2(x/2)
consider how you get from sin(x/2) to sin(x)
acoustic path
consider how you get from sin(x/2) to sin(x)
you replace x with 2x
pearl pondBOT
pearl pondBOT
Available help channel!
ocean lily
craggy ridge
hello!
craggy ridge
ive got a few questions about this.
- why isnt the formula of the cross product and the vector projection the same and how do they differ
- what does (Ahat * B) / Ahat mean as an answer in the final questions down at the bottom
- what is the difference between scalar values and vector values
sharp vigil
the cross product is a different operation from vector projection, why would they be similar?
craggy ridge
i thought they were similar because their formulas look very close together. i believe that A & B are both vectors as well
so i looked up vector multiplication and cross product is what came up
sharp vigil
the projection is based on the dot product, which is [ \vb A \cdot \vb B = AB\cos \theta ] as shown at the top of the page
jolly parrotBOT
cloud
craggy ridge
right, so whats the difference between the two formulas?
sharp vigil
one is based on sin and the other is based on cos, so the dot product is bigger the closer they are to parallel and the cross product is bigger the closer they are to perpendicular
craggy ridge
but the outputs are different too, no? iirc dot product is supposed to find the value of the difference in the angle between two vectors, but im not sure what the cross product is about or if it relates to this at all
last i checked it was about finding the magnitude of the component in the z direction
sharp vigil
the dot product outputs a scalar, and the cross product outputs a vector
craggy ridge
so the final values for the second formula in the first picture there has nothing to do with multiplying 2 vectors together at all?
and its actually about taking the components of those 2 vectors?
sharp vigil
there are two completely different ways of multiplying two vectors together. one of those ways is the dot product, and the other is the cross product
all projection formulas are based on the dot product
craggy ridge
thats good to know. so what about the second question? how would i go about interpreting that?
sharp vigil
i don't see the formula (Ahat * Bhat)/Ahat anywhere (not that it would make sense, we can't divide by vectors)
sharp vigil
so $\hat{\vb A}$ is a unit vector (length 1) in the direction of $\vb A$. $ (\hat{\vb A} \cdot \vb B )$ is the dot product of the unit vector of \vb A with the vector \vb B (which is a scalar)
jolly parrotBOT
cloud
sharp vigil
so then $(\hat{\vb A} \cdot \vb B)\hat{\vb A}$ is the scalar multiplication of that dot product with $\hat{\vb A}$
jolly parrotBOT
cloud
sharp vigil
meaning $(\hat{\vb A} \cdot \vb B)\hat{\vb A}$ is a vector in the direction of \vb A with a magnitude of $\abs{\hat{\vb A} \cdot \vb B}$
jolly parrotBOT
cloud
craggy ridge
so $\hat{\vb A}$ is a unit vector (length 1) in the direction of $\vb A$. $ (\ha...
unit vector in the direction of A meaning the component of A parallel with B?
sharp vigil
no, $\hat{\vb A}$ is a vector which is parallel to \vb A and has length 1
jolly parrotBOT
cloud
craggy ridge
so its used to denote direction instead of magnitude?
sharp vigil
yes
craggy ridge
cool
i guess i misunderstood a different part of my textbook then
oh yeah i did
yeah that makes sense
so when we look at $(\hat{\vb A} \cdot \vb B)\hat{\vb A}$ again, we see that B multiplied in the direction of A then multiplied by the direction of A again, is the vector component of B onto A?
jolly parrotBOT
Jack of all Spades
sharp vigil
the dot product of Ahat and B is the component of B in the A direction
scalar multiplying it by Ahat gives a vector with that magnitude in the direction of A
craggy ridge
alright i think im good with that now
last one, difference between scalar values and vector values
craggy ridge
also in reference to this image
actually
we already solved this i think
because the scalar value of the projection of B onto A is just the magnitude not yet multiplied by the direction of A
is that right?
sharp vigil
yes
gusty prism
mol involved with light š
pearl pondBOT
mol involved with light š
gusty prism
like
id imagin e i need to convert?
kj and mol someway
I show work of my idea and we'll see i need 5 mins
sry
midnight haven
pearl pondBOT
long gorge
how can you change (\sum_{n=1}^\infty n x^{n - 1}) into (\sum_{n=1}^\infty n x^n)?
jolly parrotBOT
Invariance
long gorge
yeah, just multiply by x lol
long gorge
It's understandable that you missed it! The x^n term as a whole is dependent on n, so it's not something you'd necessarily think about partially bringing in or out of the sum
sure
midnight haven
yes let me send it here
i have a "solution" but it is wrong
i am definitely making a solly error
long gorge
how are you taking the (-4x + 5)^n out of the limit?
and what method are you using to find the radius?
long gorge
I think you have another typo in 5 n + 12
midnight haven
yes i do
its (5n+7)
but still it does not really change the fact that the limit is 0 when it should not be
long gorge
its (5n+7)
oh wait sorry, it isn't a typo, and I think that's your problem: it's + 12 because 5(n + 1) + 7 = 5 n + 12
(5 n + 12)! / (5 n + 7)! = ?
also you're doing the ratio test a bit differently than I learnt (including the (-4 x + 5)^n term), but be careful with the "-4" bit because that will affect the radius
pearl pondBOT
@midnight haven Has your question been resolved?
midnight haven
oh wait sorry, it *isn't* a typo, and I think that's your problem: it's + 12 bec...
yes but its not that its the fact that the limit equals 0
long gorge
yes but its not that its the fact that the limit equals 0
what is (5 n + 12)! / (5 n + 7)! ?
you're missing terms in the numerator
midnight haven
5n+7
pearl pondBOT
no, not quite
long gorge
try expanding the first few terms of 5n + 12
that should solve your main issue
.close
pearl pondBOT
midnight haven
try expanding the first few terms of 5n + 12
factorial?
long gorge
yes
long gorge
got it?
long gorge
np!
midnight haven
.close
pearl pondBOT
Available help channel!
rare edge
pearl pondBOT
cosmic charm
Donāt know where I went wrong
14 - 28 is not -42 is it