just do the formula
#help-38
nimble stone
fringe rover
Ngl I’m sorta stuck
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charred marsh
The exponential function e^{x} cannot be extended to [the projectively extended real line]
https://en.wikipedia.org/wiki/Projectively_extended_real_line
charred marsh
> The exponential function e^{x} cannot be extended to [the projectively extende...
why is that?
silver furnace
think about what happens as x -> infinity and x -> -infinity, do they wrap around to the same "point"?
charred marsh
is that a problem
maiden hare
What would be e^{inf}?
solid kilnBOT
what is an 🍓
maiden hare
Why?
maiden hare
Because e^x tends to +inf when x tends to +inf?
charred marsh
yeah, i guess
maiden hare
What about when x tends to -inf?
maiden hare
Most functions, yes
short lantern
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short lantern
why can’t we divide it further
sharp vine
what
short lantern
um
sharp vine
when you get a number lower than the divisor and you don't have anything further, you leave it as the remainder
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latent meteor
How do u do q9?
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How do u do q9?
wraith hinge
,rccw
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wraith hinge
do you know about partial fractions decomposition
latent meteor
i do, i forgot
i normally do a weird method and it works
but not for this
so no
latent meteor
do you know about partial fractions decomposition
so no.
wraith hinge
look it up then tbh
lots of tutorials and explanations online
these channels are kinda more so for specific questions you have
latent meteor
can u send a good one pld
wraith hinge
uh sure
latent meteor
idk how to do it if p(x) has a greater degree than g(x)
where p(x) is the numerator
latent meteor
ty
latent meteor
https://tutorial.math.lamar.edu/classes/calcii/partialfractions.aspx
still dont know how to do 9d 😭
oh wait
do i just do long division to bring the degree of the numerator down?
welp...
if someone can show working for like 10c that would be great
so ik whats going on
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flint lichen
I dont know how to do. I have to find N. Examples:
log2 8 = N
logN 4 = 2
logN = 2
lnN = 1
these are just some examples. I dont understand the connections in logarithms. I know how to do log2 N = 4. I know that N = 4 squared but i cant understand the ones i put above. How am I supposed to consistantly find N in all of those scenarios even if i get harder ones like logN 1/25 = -2. Is there like a formula to do these? I know its supposed to be similar to dividing and multiplication like a * b = c -> b = c/a but i just dont get the connections. Have tried to do them but mostly its through guessing not any actual formula or something similar
edgy cedar
$\log_b(a) = c \iff b^c = a$
solid kilnBOT
$Pure$
edgy cedar
So $\log_b(a)$ is asking b to the what power is a
solid kilnBOT
$Pure$
flint lichen
So with the first example i gave Log2 8 = N its 2^N = 8?
mellow violet
yep
mellow violet
what is base of logN?
mellow violet
it's not an integer btw
tepid rock
It doesn't have to be 10, however if you're told to assume bases when not given, then that's that.
flint lichen
I thought if its not said or shown then its 10. Like you dont write the power 1 to any number (8^1 and so on)
flint lichen
it's not an integer btw
what does this mean?
tepid rock
I thought if its not said or shown then its 10. Like you dont write the power 1 ...
Those two sentences are unrelated.
The base being 10, and 8^1 = 8
Absolutely unrelated.
edgy cedar
Depends on what you’re doing some take log to be base e some base 2. it’s just what’s more convenient I guess
mellow violet
what does this mean?
n is not an integer
flint lichen
But if its for example logN 1/25 = -2 is it N^-2 = 1/25. How would finding the N work there. As with 2^N = 8 I can see that 2^3 = 8 so that N is 3. But with logN 1/25 = -2 I cant just guess it if i put it in the formula
edgy cedar
It just means $N^{-2} = \frac{1}{25}$
solid kilnBOT
$Pure$
edgy cedar
If you don’t see what N is try writing 25 as something squared
flint lichen
Can it be 1/25 = 1 / N^2 --> N^2 = 25 and then squareroot and N = 5?
Or is that wrong
If i do N^-2 = 1/N^2
flint lichen
is there a simpler way to do it? since you said to make 25 something squared?
N^-2 = 1/5^2
edgy cedar
25 is 5^2
flint lichen
Am I supposed to just "see" it from there?
edgy cedar
Then it’s pretty obvious that N is 5
edgy cedar
Well I suppose if you’re new to exponent rules it might take some practice
No I’m just more experienced
flint lichen
So If I just follow the formula you showed me earlier I can do all of these?
edgy cedar
Yep $\log_b(a) = c \iff b^c = a$
solid kilnBOT
$Pure$
solid kilnBOT
$Pure$
edgy cedar
You’d need a calculator
flint lichen
Ok
edgy cedar
Should just be perfect squares cubes etc
tepid pumice
hi, i solved for this question and got these answers but it says on of the answers are wrong
tepid pumice
so like i have this
dense breach
You left out the negative sign in front of 1/12 at the beginning
deep oxide
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deep oxide
Can someone tell me what is c supposed to mean
Do I add 5 to every transport expenses
<@&286206848099549185> plz help
remote thistle
<@&286206848099549185> plz help
The wording seems vague since it doesn't say if each executive's expense individually increases by 5 or if it's one executive increasing by a lot. I think your interpretation is the safest one where each executive increases by $5. So you'd get the same histogram but shifted to the right by 5 (also assuming we changed the buckets with this increase)
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barren sun
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barren sun
find the derivative of the function
barren sun
How do you get it?
woven nova
notice that e^(ln(x))=x
so if we write e^ln(f(x)) we get f(x)
so in our case
3^sqrt(x)=e^ln(3^sqrt(x))
from there on we can simplify a bit
$3^{\sqrt{x}}=e^{\ln 3^{\sqrt{x}}}$
solid kilnBOT
Martin
woven nova
$\ln (a^b)=b\cdot\ln a$
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Martin
barren sun
can you dumb it down a little
woven nova
we are using:
e^(ln(x))=x
this is because e^x and ln(x) are inverse to each other usually
which is why we can just write f(x) as e^ln(f(x))
woven nova
if we want to prove the other property:
log_b(x^k) = k * log_b(x)
m = log_b(x)
=> x = b^m
x^k = (b^m)^k=b^mk
log_b(x^k)=log_b(b^mk)=mk
remember the value of m
log_b(x^k)=k * log_b(x)
edgy cedar
log(x^k) = log(x • x • … •x) k times = logx + logx +…+logx k times 
woven nova
oh yeah that is nicer
but then you have to prove that property as well haha
but lets say that is a given
good point tho, thx
edgy cedar
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timid python
How do I go about doing this? I don’t think I started with the right idea
timid python
lapis aspen
when solving efficiency, how do i know what is output or input in a word problem? im kinda confused,
fathom ivy
without any context the only advice i can give is “output is the smaller number” because having higher than 100% efficiency violates physics
lethal anvil
without any context the only advice i can give is “output is the smaller number”...
I sure which it didn't. Wouldn't that be neat
lapis aspen
can you post the word problem?
generally, im confused in any context,
lapis aspen
without any context the only advice i can give is “output is the smaller number”...
oh ok,,
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lethal anvil
generally, im confused in any context,
Seems like you got it there 👍
stoic flare
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stoic flare
Are they telling us to draw it again
vapid lynx
looks like it? are you learning protractors or something
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little granite
Could someone help me me prove this trig question using the double angle formulae
flint sluice
can you use the trig circle
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@little granite Has your question been resolved?
little granite
No, I need to solve it using the trig identity
Tan2A=(2tanA)/(1-tan^2A)
Need to solve algebraically
winged hinge
what's tan(180 - x)
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oblique walrus
.close
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wintry mantle
for a question like this when do we know to apply l'hopitals rule? I keep having issues to when to apply it
wintry mantle
i have to do l'hopitals here by making it sin(1/k)(1/k) but how do I know I will have to do that?
zinc ginkgo
Do x = 1/k
keen compass
you dont have to apply lhopital for sin(x)/x
it should be common knowledge knowing the limits of sin(x)/x
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patent hound
What if it's "parallel" instead of 'perpendicular'?
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What if it's "parallel" instead of 'perpendicular'?
patent hound
Would the working still be the same?
tepid rock
Parallel to the plane?
patent hound
Ye
patent hound
Wld u say I'd learn it in year 12?
tepid rock
Sure, perhaps before that.
patent hound
Not to mention, there'd be infinite lines.
How do I approach the question if so ?
tepid rock
There'd be an entire plane parallel to your plane passing through A.
And that plane would have infinite such lines passing through A.
So I suppose just write the equation of plane, and then write the line equations from there.
wraith hinge
Hey, how do I solve 16?
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Hey, how do I solve 16?
wraith hinge
This isn't a geometric series
And the sequence inside the series diverges
So I'm confused
Oh, sum of geometric series?
Is that it?
serene briar
you can honestly say $\frac{1+3^n}{2^n}$ is strictly increasing
solid kilnBOT
candies
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wraith hinge
you can honestly say $\frac{1+3^n}{2^n}$ is strictly increasing
I'm not sure how that helps prove divergence
The way I did it is fine though, right?
Using geometric series
serene briar
I'm not sure how that helps prove divergence
cuz then it shoots off the infinity
ig u can say
[original expression] > summation 3^n/2^n which diverges, so [orig expression] must be divergent
wraith hinge
ig u can say
[original expression] > summation 3^n/2^n which diverges, so [orig ...
Oh, I get it
Comparison test
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summer meteor
e^(x+y) = 6
e^x + e^y = 5
trim joltBOT
e^(x+y) = 6
e^x + e^y = 5
serene briar
hint 1: ||try writing e^(x+y) as e^x * e^y||
hint 2: ||the first equation becomes ab=6, second one becomes a+b=5||
hint 3: ||solve by quadratic||
click each hint after at least 5 minutes trying ig
shrewd ridge
5 minutes at least? 
summer meteor
how do I put them on the same line though 
(5 - b)b = 6
b^2 - 5b + 6
(b + 1)(b - 6)
so replace b with e^y then I..... (then you jump in here)
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@summer meteor Has your question been resolved?
maiden hare
(b + 1)(b - 6)
That's not a correct factorization
Just use the quadratic formula if you can't directly factorize this
Find the possible values for b and a
(honestly it's not hard to find two integers that add up to 5 and multiply into 6)
summer meteor
3 and 2 sorry
maiden hare
Yeah, so e^x and e^y are 2 and 3 (or 3 and 2)
summer meteor
so just ln(2) and ln(3)?
maiden hare
Yes
summer meteor
thank you
cursive arrow
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cursive arrow
how do i compute the inner product of this
stark bison
Consider using integration by parts twice
cursive arrow
sure?
actually nvm yh i see it
i know that is some notation that is | with the 0,1
a bit like the intergral, but just a straight line. What is that?
stark bison
The evalutation bar?
cursive arrow
Perhaps?
stark bison
[ \eval_0^1 ]
solid kilnBOT
A Lonely Bean
cursive arrow
yes
Am i supposed to write that out for when i take x^2*e^x outside of the intergral?
stark bison
Yeah but it's kind of too early, you still need to integrate xe^x
stark bison
Right, now you add that to $2xe^x\eval_0^1$
solid kilnBOT
A Lonely Bean
stark bison
Btw you should have + in front of the integral when applying product rule
cursive arrow
where?
stark bison
**A Lonely Bean**
My bad I meant x^2 e^x
solid kilnBOT
// mav
stark bison
Not really
stark bison
Right, now you just simplify
[ \left[x^2e^x - 2xe^x\right]\eval_0^1 + 2\int_0^1e^x\dd x ]
solid kilnBOT
A Lonely Bean
stark bison
That will yield the overall answer
cursive arrow
hy is the a minus inbetween
stark bison
The product rule is
[ \int u \dd v = uv - \int v \dd u ]
That's where the minus sign comes from
solid kilnBOT
A Lonely Bean
stark bison
**A Lonely Bean**
And the sign in front of the integral is positive because we applied product rule twice, so its sign changed twice (minus times minus gives plus)
stark bison
Let's see
cursive arrow
stark bison
**A Lonely Bean**
= e - 2e + 2(e - 1) = -e + 2e - 2 = e - 2
stark bison
I think you should have 2 * 0 * e^0 in the first line
Wait why is it 2 * 0 * e^1
Ah
cursive arrow
2*x? at x=0?
stark bison
Yeah but e^x = e^0 at x = 0 but whatever
cursive arrow
have i misswritten
stark bison
Yeah you should have 2 * 0 * e^0 there
Or 2 * 1 * e^1
Kinda hard to tell in which order you are doing those
stark bison
So x^2 * e^x - 2xe^x evaluated at x = 1 is e - 2e and at x = 0 it's just 0
So you have -e + the integral
And you know that the integral is equal to 2e - 2
Hence it's -e + 2e - 2
cursive arrow
i have to go
but i cant get it to work
even though this should be the simplest part of everything
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copper yarrow
if I know that tan(pi/12)=2-root3
how will I find tan(pi/24)
stark bison
Use the half angle identity for tan (if you don't know it, you can derive it from the double angle identity for tan)
copper yarrow
Use the half angle identity for tan (if you don't know it, you can derive it fro...
how would you derive the half angle identity for tan?
stark bison
Let's start with the double angle identity for tan, can you recall it?
fair raft
There is a wikipedia article for it.
https://en.wikipedia.org/wiki/Tangent_half-angle_formula
When you know the tangent value, you could determine the sine and cosine value by draw a right triangle.
copper yarrow
Let's start with the double angle identity for tan, can you recall it?
Yeah
It's given to us in a formula booklet for our test
stark bison
Alright, so we have
[ \tan 2x = \frac{2\tan x}{1 - \tan^2 x} ]
solid kilnBOT
A Lonely Bean
stark bison
Isolate tan(x) in it
copper yarrow
(tan2x-(tan2x)(tan^2x))/2=tanx
stark bison
I think it's better to not have fractions there, no need to divide by 2
So $\tan 2x - \tan 2x\tan^2 x = 2\tan x$
solid kilnBOT
A Lonely Bean
copper yarrow
right
stark bison
Move all tan(x)'s to one side and every thing else to the other side (even if it's tan(2x))
copper yarrow
Move all tan(x)'s to one side and every thing else to the other side (even if it...
divide B.S by tan^2x
?
stark bison
What's BS?
copper yarrow
both sides
stark bison
No, you will still have a quadratic equation
Okay let's move everything to the right side
[ \tan 2x \tan^2 x + 2\tan x - \tan 2x = 0 ]
solid kilnBOT
A Lonely Bean
stark bison
As weird as it may seem, you can apply quadratic formula to this
copper yarrow
oh
bitter ermine
How to find how many digits will be in decimal form 3^16?
winged hinge
3^2 = 9
cyan zinc
digits of a number is the floor of the base 10 logarithm plus one
so you want
floor(log_10(x)) + 1
naive dirge
digits of a number is the floor of the base 10 logarithm plus one
ok rhats interesting
but makes sense
trim joltBOT
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.
@bitter ermine Has your question been resolved?
winged hinge
I mean just note that up until you've multiplied 9 by itself 8 times, the pattern is 1 more digit
with leading digit going down by 1
bitter ermine
I didn't understand this
9×9×9×9×9×9×9×9
Ohh i got it now
9×9 will not surpass 2 digit
Right?
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bitter ermine
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proud fulcrum
what am i supposed to do after i find the slopes of f(x) and g(x) ? all i know is that im supposed to flip g(x) or something 😭😓 this is ap calc derivatives
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what am i supposed to do after i find the slopes of f(x) and g(x) ? all i know i...
@proud fulcrum Has your question been resolved?
proud fulcrum
<@&286206848099549185>
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@proud fulcrum Has your question been resolved?
proud fulcrum
<@&286206848099549185>
but how😭
like what does that mean and entail
for which one f(x) or g(x)
ohh
ok so i plug it in in the derivative of f or just original one
welll
i could say normal bc i alr used derivative for g but also i can say derivative cuz i mean it is a calc question for derivative😭
umm
i think orginal because the equation says that
it’s 1
wait derivative of x^2 is 2x ?
okay lemme do that
16/4 is 4
so is y = 4
YAAAA
so the coordinates for point P is (4,4)
okayyyy i’ll keep that in mind
thank you!!
.close
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obtuse brook
does anyone know whats the first step i should take when solving equations like this?
obtuse brook
or this
i just get really confused
I know how to do like radical + radical but not something like this from the start
im assuming its quite simple but i just dont know how to begin
solid kilnBOT
JS
obtuse brook
Yeah I can simplify the cube roots
Let me do it rq
how do u do that
like the bot sending
so it shows
i simplifed it to 4 cuberoot 3
pure lark
you can write in Latex and the bot will convert.
obtuse brook
how do i write in latex?
pure lark
Like this: $\sqrt[3]{320}$
solid kilnBOT
JS
pure lark
wow that's huge 😄
obtuse brook
$\sqrt[3]{192}$ simplifies to $4sqrt[3]{3}$
solid kilnBOT
Rahm Bow
obtuse brook
$\sqrt[3]{192}$ simplifies to $4\sqrt[3]{3}$
solid kilnBOT
Rahm Bow
obtuse brook
😄
pure lark
yep you got it 😄
solid kilnBOT
Rahm Bow
pure lark
yep do that one as well and then add them up
obtuse brook
i get $2\sqrt[3]{3}$
solid kilnBOT
Rahm Bow
solid kilnBOT
Rahm Bow
obtuse brook
like that right
pure lark
there's a 5/6 multiplied with the cube root(192) and a 1/8 with the cube root(24) as well....
but yeah you can reduce it to something cube root(3) and that's it I think...
obtuse brook
there's a 5/6 multiplied with the cube root(192) and a 1/8 with the cube root(2...
Yea thats where I get stuck
How does the new equation look now
for this
pure lark
put in the simplified numbers for both the cube roots, and then recalc
pure lark
the second one should be 2 x cube root(3) instead of 6 x cube root(3).
pure lark
well the two got switched 😛
pure lark
np 🙂
pure lark
are you then comfortable factoring out the cuberoot and adding the fractions?
pure lark
ok, well from what you have we can simplify to: $\sqrt[3]{3} x (\frac{5}{6} x 4 + \frac{2}{8}) $
obtuse brook
$\sqrt[3]{3} x (\frac{5}{6} x 4 + \frac{2}{8})$
solid kilnBOT
Rahm Bow
obtuse brook
that?
pure lark
yes you got it.. then just simplify the stuff in brackets and it's done.
obtuse brook
yes you got it.. then just simplify the stuff in brackets and it's done.
but i dont even know how you got from that to that and where did the x's come from?
pure lark
the x is just Latex's rendering - needs a \times instead of x.
obtuse brook
this is how they show me do it
but i dont know what they did for the beginning
idk if im jsut dumb or something but i swear i cant remember
pure lark
it's the same thing
obtuse brook
Im still really confused ngl
pure lark
ok you good till this step?
obtuse brook
Yes
pure lark
ok so, you can write that two ways - let me write them both, gimme 2 mins
obtuse brook
kk
pure lark
so what you have can be written as $\frac{5 \times 4 \times \sqrt[3]{3}}{6} + \frac{2 \times \sqrt[3]{3}}{8}$ which is the same as they have on the second line. Or, $\sqrt[3]{3} (\frac{5 \times 4}{6} + \frac{2}{8}) = \sqrt[3]{3} \frac{43}{12}$
does this make sense?
solid kilnBOT
JS
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brave ermine
<@&286206848099549185>
tropic smelt
Hello. Notice that dx/dt is given in terms of x, not t
So they use the chain rule to find
d(x')/dt
d(x')/dt = d(x')/dx * dx/dt
grim crown
for a) do you need to prove one direction with induction
grim crown
like showing how to represent n!
ie like n! = n x n-1 x p x 2 x 1
or can i just like
n! = n x n-1 x p x 2 x 1 use this fact
like clearly here p is a multiple of n! so it must divide n!
stoic iron
yeah thats totally fine for n >= p implies p | n!
grim crown
yeah thats totally fine for n >= p implies p | n!
and i get that you would have to show what happens if n < p, but what is the other direction
i have one direction but i guess it has multiple cases?
stoic iron
which may be easier to show
grim crown
just by negation right?
yeha i showed that
interesting
so you just show the negation is true
stoic iron
the second two are equivalent by contrapositive
stoic iron
so yeah then it seems like you've done it all
grim crown
so yeah then it seems like you've done it all
do you need to use the binomial theorem for b)
is there another way to do it
we went over the binomial theorem in class but not too much
stoic iron
you do
grim crown
okay
stoic iron
sure
grim crown
i thinmk im messing this up so bad
i dont think im particular im explaining the congruent mod p thing correctly
you can use the binomial coefficient thing to show that (n k) is divisible by n
i think im close but i might need a small hint
because i think i sort of get what its saying
like if you have 1 x .. x p x ... x n
then any prime number will divide it if its < n
which leaves n as the remainder
so it would be mod n?
stoic iron
for part b, in the formula for binomial expansion, the n is now p
because you're doing (x+y)^p
stoic iron
just write out a few terms of (x+y)^p using the formula
you know from the question which terms should die so justify that they do
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stoic iron
yeah i think you got it
you took a lot of care to make sure the denominators were not cancelling the factor of p in the numerator which is good
heavy elbow
okay so
heavy elbow
with this problem, I found out that it had either 2 or 0 positive answers and exactly 1 negative answer
and I plugged in all the possible answers using the formula/method our teach have us
And none of them worked
vagrant prism
,rccw
solid kilnBOT
heavy elbow
here’s the rest of my work (Ik I’m messy af)
,rotate
And this is the method we’re supposed to use
,rotate
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heavy elbow
what did I do wrong?
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heavy elbow
<@&286206848099549185>
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azure pulsar
i think ppl dont know what that method is
heavy elbow
well i dont either T-T
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vapid lynx
its the rational roots factoring method (no idea what step 2 is but its not needed?) but I do not want to check what is wrong since i hate this method
vapid lynx
for what its worth, it does have a rational root but I need to go look up the method again to see why you missed it
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gray jungle
vapid lynx
okay, well the method looks about right, the actual root is -1/2 but I cant decipher your work (if you missed it as a possibility, or just factored wrong when testing it) but I assume you can troubleshoot from here? @heavy elbow
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glass hill
Hey! I'm currently studying the epsilon-delta definition of the limit, and I'm confused by this portion of the textbook (AoPS Calculus).
glass hill
if abs(z) is less than delta, how can the statement on the right side of the underlined portion be true?
calm ginkgo
I'm not sure z is real number or not (because z sometimes represents complex number)
If z is real number
0<|z|<δ→-δ<z<δ→0<z²=|z|²<δ²
So this statement is true
If z is not real number
This statement is fault because z² may be not real number,in this condition we can't compare which is larger.
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glass hill
I'm not sure z is real number or not (because z sometimes represents complex num...
aaaa
sorry!
I forgot to mention that z is a substitution for an expression
so z = x-2
glass hill
I'm not sure z is real number or not (because z sometimes represents complex num...
thanks !!
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real herald
I am trying to solve this system of odes and cannot find my mistake, ive redone it a couple of times can anyone take a look?
zinc urchin
how is the answer -1, like i dont get what d is referring to here, is it the y intercept of the linear equation thats passing thru the parabola?
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how is the answer -1, like i dont get what d is referring to here, is it the y i...
@zinc urchin Has your question been resolved?
zinc urchin
<@&286206848099549185> if anybody could help, that would be great
charred trail
just a suggestion which may or may not work, how about sub in the equation for y into the expression 4y=d, i.e. 12x^2-16x+4=d, and then as there are two points of intersection, the determinant would be positive
this should get you closer to the answer maybe
charred trail
ah you were right with your method
its just that its asking for the least possible integer
you woould have gotten that d>-4/3 right?
zinc urchin
exactly
charred trail
so the least possible integer that satisfies this is -1
and thats why the answer is -1
zinc urchin
so the least possible integer that satisfies this is -1
ok thats where i lose you
how
charred trail
okay no worries, im assuming you know what integers, natural numbers, rationals etc are?
zinc urchin
yes
charred trail
cool
so, if we know that integers are the numbers ....-3, -2, -1, 0, 1, 2, 3.... and so on, but we have that d must be greater than -4/3, what is the smallest integer that we can find which satisfies this?
charred trail
haha great, whenever doing this step, sometimes its useful to visualise a number line
zinc urchin
less goo, my dumbass was still stuck with -1.333 so thats why, didmt pay attention to the word integer 💀
zinc urchin
haha great, whenever doing this step, sometimes its useful to visualise a number...
good tip man thanks a ton
charred trail
haha no worries anytime 🙂
zinc urchin
haha no worries anytime 🙂
i have one more question tho, its simple as well i think, could u help me with that as well?
charred trail
i could try haha
zinc urchin
great
hold up
thats how i am visualizing this
idk if i am right?
i used the arc length formula but i am still nowhere close to ans C
zinc urchin
s=r theta
and the standard one as well
2pi r and angle divided by 360
shouldnt the angle be 180?
charred trail
yes
zinc urchin
i just get 5pi
charred trail
so itll be 180/360 (2 * pi * 5) = pi*5 which is equivalent to 15/3pi as 15/3=5
thats correct
5pi = 15pi/3
zinc urchin
well thanks alot man, appreciate you taking out the time
charred trail
nw not at all
zinc urchin
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vast flint
Let $g(x,y)=(2y-x-y^3)(2x)+(x-2y)(4y)=-2x^2-8y^2-2xy^3+8xy$
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Beous
vast flint
How can I show that there is a neighborhood of $(0,0)$ such that $g(x,y)<0$ for all non-zero elements of this neighborhood.
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Beous
vast flint
The issue is that the hessian of g is not negative definite but just negative semi-definite
the hessian of g is
[[-4, 8 - 6y^2],
[8 - 6y^2, -12xy - 16]]
the 4th order approximation does not help since it is a saddle
if i fix y then i get a quadratic in x
so i can find the maximum of this quadratic since it has negative leading coefficient
i'll see if this works
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tribal dove
n*(n+42) when answer is a prime number
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n*(n+42) when answer is a prime number
charred trail
for n=1
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frozen prism
$f(x) = \frac{(x^2+x-6)}{(x-2)}$
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$f(x) = \frac{(x^2+x-6)}{(x-2)}$
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ahmed349
kindred pier
Limits
frozen prism
$f(x) = \frac{x^2+x-6}{x-2} = \frac{(x-2)(x+3)}{x-2} = x+3$
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ahmed349
kindred pier
**ahmed349**
$f(2)$ is not defined because you cannot divide by zero.
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SWR
kindred pier
But it's limit exists and it is 5
frozen prism
**ahmed349**
but it works here
kindred pier
You are entering the domain of a neat branch of mathematics known as real analysis
frozen prism
ah maybe i should take a step back
and keep that door closed
but like
this is just algebraic manipulation
at what point do i consider the value a limit
kindred pier
It's a cool door
frozen prism
like if i didnt know calculus existed i would just say "ah yes x+3 so any value for x works"
kindred pier
at what point do i consider the value a limit
That's a really hard question to answer without taking a long time.
frozen prism
damn
kindred pier
But you're asking the right questions, which is good
It's awesome actually
You have the mind of a thinker.
Are you taking calculus yet?
What math are you in right now?
frozen prism
You have the mind of a thinker.
damn, i thought i was plain stupid
kindred pier
Nah. You question things. That's a good quality.
frozen prism
What math are you in right now?
college
faculty of artificial intellgince, idk what the course name is though
kindred pier
So you've taken calculus I'm assuming.
frozen prism
yes
kindred pier
Then you've already learned about limits in some sense
frozen prism
but just memorized stuff
yeah
but i literally just now found out that when you approach x thing you look for the value of y
kindred pier
You should also know a bit of set theory too then. Sets, domains, functions on sets.
frozen prism
i just knew you had to solve funny algebra to get the limit
frozen prism
You should also know a bit of set theory too then. Sets, domains, functions on s...
discrete maths yes
kindred pier
Yeah you need limits to do calculus, but you don't need to understand it a deep level
It's funny because analysis is the foundation that calculus is built on, but calculus is so much more of a practical branch on its own that people only learn limits in the most hand wavy way that can get you through it
But you have competencies, so I'll try to explain as best I can
frozen prism
thank you very much
kindred pier
Division is a binary function defined except when the denominator is zero
That's just a fact
kindred pier
**ahmed349**
And here, this is all algebraic manipulation, just like you said
But it does not work at x=2 because then the denominator is 0
kindred pier
**ahmed349**
But if you graph this, it will just look like y=x+3. That's pretty obvious to our eyes
frozen prism
yes
kindred pier
But technically, an open hole exist exactly at x=2 because you would be dividing by zero there.
frozen prism
yes
kindred pier
How do we account for this?
That's where limits come in. We can say f(2) is not defined, but as x approaches 2, we see that f(x) gets closer and closer to 5, without necessarily every needing to get to 5 exactly
kindred pier
**ahmed349**
So someone taking real analysis would say that the domain here is R-{2}. That is f(2) is not defined, but we can find its limit.
This let's us stay consistent with never dividing by zero
frozen prism
But technically, an open hole exist exactly at x=2 because you would be dividing...
this where i get confused though
doesnt this mean $f(x) = x+3 and g(x) = \frac{x^2 + x - 6}{x-2}$ are different
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ahmed349
frozen prism
but $x+3 = \frac{x^2 + x - 6}{x-2}$
kindred pier
f(2) is defined but g(2) is not
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ahmed349
kindred pier
For every $x\ne2$
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SWR
frozen prism
interesting, so this means i have to deal with equations the same way with functions, setting a domain
kindred pier
Yes
It's very meticulous
You can use limits to fill holes though
You can say $g(x)=\frac{x^2+x+6}{x-2}$ when $x\ne2$ and $g(x)=5$ when $x=2$
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SWR
kindred pier
Then $f(x)=g(x)$ everywhere.
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SWR
versed silo
hi, after receiving a 20% mark for this question I'm wondering if my approach was even valid. the marker argued the text in yellow, but that's not what I meant. could i request a remark or am i just wrong?
silver furnace
can you send the whole proof, as it depends on how you argued
versed silo
mkay
silver furnace
the argument looks a bit flawed
you can consider a specific case but you have to generalize like in inductive proofs
I don’t really see that in this proof, you make assumptions and you use those all the way
versed silo
but its not a specific case, im considering a coordinate space where A is the origin and the vertical unit vector is parallel to d
stoic iron
What about a coordinate system where that isn't the case?
versed silo
heres a 2d equivalent of my definitions
A can be anywhere in i,j but is at the origin of i',j'
stoic iron
The point is the result is independent of coordinate system, so you either need to not pick one, or show your argument generalises to all coordinate systems
versed silo
but since ive defined i',j' independently of i,j, thats true right?
silver furnace
yeah that's fine you're allowed to define your own things, but you eventually have to relate it to the overall problem. It looks like you only proved for one case
versed silo
the final sentence says that since its a plane in this new system its also a plane in the general i,j,k
should i be clearer there?
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zinc raft
Is this the less than sign < ?
My equation is
4x + 2y blank 12
It’s hot dogs and water and the total cost is less than 12 so should I put < in the blank?
zinc raft
eternal adder
yes, < is the less than symbol, and that would be an appropriate use of it
zinc raft
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last fossil
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last fossil
well i was thinking of how i can keep the integeral the same while turning that x into -1
i thought of multiplying by -1/x
rough grotto
well i was thinking of how i can keep the integeral the same while turning that ...
couldn't understand this part
your substitution is fine
why r u facing issues?
last fossil
like how i can make du line up with the original statement
cause right now i have -1dx
but in the integeral i have x (sqrt(u)) dx
which dosent sub in directly
rough grotto
u = 1 - x => x = 1 - u
whenever you are substituting you would have to replace x with the substituted terms everywhere
now tell me what will you get when you do that above?
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last fossil
something like this?
edgy cedar
Well you said du=-dx
last fossil
so if du is -dx
mm
like how does that work when i change it to du?
do i multiply everything by -1?
last fossil
$-((1-u)(\sqrt(u)))du$
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Arm
last fossil
kk
rough grotto
you need to simplify and then solve
rough grotto
how would you solve?
rough grotto
**Arm**
from here on?
-((1-u) * sqrt(u)) du
what will you do in the next step?
just show that to me
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Arm
last fossil
got it!
this one was kind weird with the lone x
like usually du perfectly substituted thru
but here we did x = 1-u
rough grotto
yes
last fossil
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obtuse brook
i dont know what to do next from here
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i dont know what to do next from here
last fossil
whats ur goal?
to simplify it further?
well for the first term you can multiply 4 into the numerator
obtuse brook
to simplify it further?
yes
last fossil
yeah
cause -4 technically has a imaginary 1 under neath
$\frac{w}{5}(-4) = \frac{w}{5} \cdot \frac{-4}{1}$
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Arm
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obtuse brook
cause -4 technically has a imaginary 1 under neath
okay so once we do that that 5 stays the same and it becomes $w-4$?
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Rahm Bow
obtuse brook
or -4w*
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wet delta
in a right triangle, the long side is three times the length of the shorter short side. Find the length of all three sides when the area is 24.0 cm2 https://media.discordapp.net/attachments/1172135383753039922/1172135545892245524/IMG_0284.jpeg?ex=655f374b&is=654cc24b&hm=80dbc62cca58e6633f0f182ba81686e060aae5661dcca19609b0e0199121e3f6&=&width=1366&height=1024
stoic iron
If the length of the short side is x, what is an expression for the length of the long side?
stoic iron
Oh I see I misread that's correct
How do you find a missing side in a right triangle when you know two of them?
wet delta
How do you find a missing side in a right triangle when you know two of them?
i dont know any of the sides
but only the area - 24m2
stoic iron
You know the short side is x and the hypotenuse is 3x, find an expression in terms of x for the other side
wet delta
You know the short side is x and the hypotenuse is 3x, find an expression in ter...
24m2 = x * 3x?
stoic iron
no that would be the formula for area if you had both the non hypotenuse sides
there was a famous man how lived very long ago whose name begins with P and he loved triangles and has a theorem named after him
wet delta
no that would be the formula for area if you had both the non hypotenuse sides
but like u need to know the sides
wet delta
there was a famous man how lived very long ago whose name begins with P and he l...
1/2 * a*h
so
like
stoic iron
look up pythagoras' theorem
wet delta
a*h = 48m^2
wet delta
look up pythagoras' theorem
i know it
stoic iron
then it will let you get an expression for the missing side in terms of x
wet delta
x^2 * b^2 = 3x^ then...
@stoic iron
wet delta
i dont understand what this is supposed to be
i put the sides
stoic iron
and b is your missing side
wet delta
that we "know"
stoic iron
$x^2 + b^2 = (3x)^2$
wet delta
but thats kinda not possible
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ΣΑCu
stoic iron
so rearrange this for b
stoic iron
because $(3x)^2 \neq 3x^2$
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ΣΑCu
wet delta
3x^2 - x^2 = b^2
(3x)^2 - 3x * (3x) * (3x)
3x^2 - 3x * 3x???
does brackets like this work like that
stoic iron
(3x)^2 is (3x)*(3x) = 9x^2
wet delta
(3x)^2 is (3x)*(3x) = 9x^2
but 3x^2?
stoic iron
is not equal to 9x^2
wet delta
what is then
stoic iron
just 3 times x^2