use the fact that the particular affine transformations you want can be decomposed into translations followed by a rotation and then a scaling map
#help-41
severe scroll
zinc harbor
a,b are vectors
simple hare
I meant the step where you conclude the integral is equal to ||b-a||
severe scroll
a,b are vectors
a and b are points
zinc harbor
I meant the step where you conclude the integral is equal to \||b-a\||
the integral is equal to b-a
simple hare
Why?
zinc harbor
because of FTC
simple hare
You're dealing with integrating a vector function
zinc harbor
doesn't it work still
simple hare
So what version of FTC are you using
simple hare
Okay so you're doing FTC on components?
zinc harbor
if every step implies the next, that must mean the first step implies the last
yeah but (first step => last step) =/> (step => next step)
simple hare
Thanks smay
severe scroll
I'm just memeing
lol
it seems as though this is just fine no?
we are integrating the magnitude of a vector valued funciton
simple hare
Yeah I think so
severe scroll
|gamma'| is a map R -> R
so we are good
and then you used the triangle inequality
zinc harbor
and then you used the triangle inequality
and then definition of integral, then FTC
severe scroll
wait does this inequality (the only one you wrote) really come from the traingle inequality
zinc harbor
wait does this inequality (the only one you wrote) really come from the traingle...
yeah the one over arbitrarily big sums
wait
forgot to tell a step
i also applied the fact that a_n < B implies lim a_n < B
because integral definition uses limits
severe scroll
i mean idc about the limit step i just care that the weird triangle thing you did works
oh okay i see
yeah everything checks out
i know that you know what a limit is
zinc harbor
i mean idc about the limit step i just care that the weird triangle thing you di...
yeah i'm pretty sure it does (if you're worried about the /n thing use ||v/n|| = ||v||/|n|)
severe scroll
I am worried about the traingle inequality part only
like no other step
okay i think it's fine after staring at it
it's literally just a direct application of the traingle inequality
zinc harbor
yeah but with limits
simple hare
Yeah that is the triangle inequality
simple hare
I think so, each step does seem valid
zinc harbor
alright
simple hare
We discussed the middle steps
amber waspBOT
amber waspBOT
Available help channel!
golden horizon
can someone help me find the answer to this question algebraically? I know there's definitely an intersection between 1 and 2 (shown by graph) but how would I reach that through algebra?
lone bolt
It’s a cubic
Factor the denominator on the rhs
Multiply out, collect terms
And you’ll obtain a cubic
U might be able to do something cheeky with it to solve for what you want
But,
You can use the cubic formula to find x
That’s an algebraic solution
golden horizon
And you’ll obtain a cubic
ahh, it'll be like this right? and ive been using synthetic div to try and solve it but its not working, so would i use something else?
final matrix
have you learnt polynomial long division?
lone bolt
yes
You should first try guess the factor right
But
You if you want an algebraic method for which to solve it, you can use the cubic formula, albeit cumbersome
final matrix
I would recommend substituting factors in until you get an answer of 0 and doing the polynomial long division especially for the small values
personally
but both methods work
golden horizon
ohhh
so would I do (x/x+2) and (5x/x^2-4) both by polynomial long div first and then solving the rest of the eq? or should I combine everything like I did above
golden horizon
ahh, it'll be like this right? and ive been using synthetic div to try and solve...
but then i kinda dont see what I can divide it by here
inner prawn
can someone help me find the answer to this question algebraically? I know there...
isn't missing +x on the black graph?
amber waspBOT
@golden horizon Has your question been resolved?
undone ridge
would these two circles be congruent since they overlap onto each others center?
signal furnace
since they overlap onto each others center?
this implies they have the same radius, so yes
amber waspBOT
amber waspBOT
Available help channel!
slim radish
amber waspBOT
slim radish
Stuck at 3{x}^2=x[x]
nvm i solved
You can realise that [x]=1 from the facts that 3{x}^2 is less than 3 and x is positive
So 3{x}^2=1+{x}
.close
glad matrix
amber waspBOT
amber waspBOT
@glad matrix Has your question been resolved?
rustic portal
how does all of them = 0
did you try it with small positives integers for m
and see what happens?
amber waspBOT
@glad matrix Has your question been resolved?
glad matrix
did you try it with small positives integers for m
and see what happens?
its because they are a multiplicity right
their dervative/slope would be exactly 0
glad matrix
yea
rustic portal
nice, so let's differentiate it
let's say f(x)=(x-1)³
f'(x)=3(x-1)²
f''(x)=6(x-1)
f'''(x)=6
f''''(x)=0
glad matrix
yea
glad matrix
so, your question is?
so right here
how do we get xk-1/2(xk-x^* ) - x *
take the diff and replace x_k+1 with newton meaning like
1/2* (x-x^* ) - x^ * no?
rustic portal
oh, I think it's just iteration for Newton's method
glad matrix
wym
rustic portal
x_n+1 - xn = -f(x)/f`(x)
and f(xk)/f'(xk)=1/2 (xk-x^*)
glad matrix
oh wait yea
x_n+1 is = x_k - -f(x)/f`(x)
so
at the end we get this
what does this even represent
rustic portal
so
that is with each iteration, the error will be narrowed by (around) (m-1)/m times
glad matrix
that is with each iteration, the error will be narrowed by (around) (m-1)/m tim...
and for this part
when m is large it converges slowly because we divide by a huge number?
rustic portal
it's because m-1 and m is so close together
e.g.
1/2 = 0.5
2/3 = 0.66666.…
3/4 = 0.75
4/5 = 0.8
...
999999/1000000=0.999999
which will becomes very close to 1
glad matrix
ooh i see
wb this formula above
we derived that
for what purpose?
1/2(x_k-x^*)
is that just telling us the error will reduce by half each iteration?
its the same formula right m-1/m(x_k-x^*)
rustic portal
is that just telling us the error will reduce by half each iteration?
yea, i think that's the reasoning for that
just with different multiplicity
glad matrix
yea, i think that's the reasoning for that
but then what is the point of this then
for f(x) and f`(x)
rustic portal
hmmm
rustic portal
for f(x) and f`(x)
not sure about this then
glad matrix
hm okat
glad matrix
not sure about this then
i also wanted to clarify this
what does this equation represent
rustic portal
but the whole thing is that higher the multiplicity, weaker Newtow's method work
glad matrix
i didnt understand what the point of this was
rustic portal
you mean the last one? green one?
rustic portal
it's the error for Newton's method with function involving multiplicity m
the efficiency
glad matrix
it's the error for Newton's method with function involving multiplicity m
so like how fast it converges basically?
rustic portal
yea
glad matrix
they derive it form thsi eqn
what does alpha mean?
does it just mean that its the max derivative in the interval [a,b]?
rustic portal
this one, I'm not sure
glad matrix
like the worst case scenario for the convergence rate
rustic portal
reading
glad matrix
reading
yeah but im jus confused on what alpha is
it is the rate of the convergence
but how am i supposed to know that value when doing the problem
like in here they just assume its 1
rustic portal
but how am i supposed to know that value when doing the problem
well, you'll have to calculate when you are really doing a problem about error in methods
glad matrix
ic ok thanks
rustic portal
if it's assuming its 1, it should told you about it.
if it didn't say, you might have to calculate
glad matrix
if it's assuming its 1, it should told you about it.
so when we have a multiplicity we could use this instead
but 1 thing i didnt get was how they reduced teh top eqn to get the 2nd
like the (x-p)^m-1 where did it go
plain coral
factor (x-p)^(m-1) out of the numerator and out of the denominator
glad matrix
ah ic
This would be fine to use right?
using f(x)/f`(x)
instead of the modified
glad matrix
factor (x-p)^(m-1) out of the numerator and out of the denominator
dyk why g`(x^*) = 0?
@rustic portal
rustic portal
dyk why g`(x^*) = 0?
it's because x^* is the root, so f(x^*)=0, hence g'(x^*)=0
glad matrix
it's because x^* is the root, so f(x^\*)=0, hence g'(x^\*)=0
how would u do part a) then
to prove it
rustic portal
wouldn't this be sufficient?
glad matrix
wouldn't this be sufficient?
but i dont understand where i "proved it"
rustic portal
yep
glad matrix
but we dont prove how it is = 0 tho
i still dont understand how we proved this
from what ur saying
rustic portal
since f(x*) is 0
rustic portal
(because x* is a root of f(x)
glad matrix
oooh so f(x^*) = 0
rustic portal
and f'(x) ≠0 therefore the expression is not undetermined
glad matrix
and 0/something
rustic portal
and 0/something
yea
glad matrix
glad matrix
yea
What does this question ask exactly? dyk?
the wording is so fucking stupid it pisses me off
rustic portal
this question might be asking to use the fixed-point method tobfind ³√225?
rustic portal
the wording is so fucking stupid it pisses me off
not sure lol,as you've said it's not that clear
glad matrix
this question might be asking to use the fixed-point method tobfind ³√225?
apprently this is what they meant
sm1 did this and got full marks
But that makes no sense to me
that is just f(x) or wtv
rustic portal
let's see
glad matrix
how is this doing fixed point method on g(x) 💀
rustic portal
i guess you'll have to first find an intial guess
let's say 6³=216, we can try with it?
but seriously, not sure, man
amber waspBOT
@glad matrix Has your question been resolved?
eager lake
sketching on complex plane question:
amber waspBOT
sketching on complex plane question:
eager lake
i dont understand how the equation for the circle changes
amber waspBOT
@eager lake Has your question been resolved?
dusk ferry
how do i find the domain? help mee
amber waspBOT
how do i find the domain? help mee
little dagger
there are methods to find range
keen pawn
Do you know what domain means?
little dagger
which one do yk
keen pawn
there are methods to find range
Domain here!
dusk ferry
there are methods to find range
well i solved the 1-3
little dagger
ok so from 4
dusk ferry
wait i mean 2-4-5 not yet hehe
solemn escarp
how do i find the domain? help mee
square roots are strictly >=0
little dagger
what do you know about square roots?
stoic locust
sqrt 25 = 5
sqrt 100000000000000000000000000000000000000000000000000000000000000000000 = 100000000000000000000000000000000000
dusk ferry
sqrt 25 = 5
sqrt 100000000000000000000000000000000000000000000000000000000000000...
ye
alr so i solved the question 1
is {X | X E R}
fresh ocean
Yes
fresh ocean
for 2, the function can not have a finite value if there is a division by 0
so all such x are not allowed
and so on
just find all the values of x for which the computation is not according to the rules
solemn escarp
but idk the num 2,4,5
for num 2, solve where 2x - 5 = 0, which is where the domain doesnt exist
fresh ocean
and the rest of the numbers is your solution
dusk ferry
for num 2, solve where 2x - 5 = 0, which is where the domain doesnt exist
that one idk dude i sound stupid but yes
idk
how
astral umbra
how do i find the domain? help mee
domain is all the values which give you real outputs
dusk ferry
ok
astral umbra
usually it's all real numbers
solemn escarp
that one idk dude i sound stupid but yes
well, take (x - 3)/x for example
astral umbra
but for square roots and fractions it might change
solemn escarp
well, take (x - 3)/x for example
it is undefined where x = 0 since you cannot divide by 0
astral umbra
for example say $\sqrt{x}$\
we know that square root of negative numbers don't exist\
so it is only defined for x > 0 (that's the domain!)
grizzled pagodaBOT
dusk ferry
the num 2 like this right?
little dagger
yup
solemn escarp
wait whats "Kodomain"?
little dagger
codomain
astral umbra
wait whats "Kodomain"?
codomain?
little dagger
co-domain
solemn escarp
oh sry
solemn escarp
yep
solemn escarp
sqr(x) >= 0 for all real numbers
solemn escarp
sqr(x) >= 0 for all real numbers
make this fit (4)
dusk ferry
ah ok ok
dusk ferry
make this fit (4)
Domain: x ≥ 0 atau [0, ∞). this one?
solemn escarp
you mean answer?
dusk ferry
you mean answer?
num4. {x| , x >/ -3/2, x ∈ R} ?
fresh ocean
num4. {x| , x >/ -3/2, x ∈ R} ?
is that 3 under the square root sign? it doesnt look like that
dusk ferry
is that 3 under the square root sign? it doesnt look like that
2x + 3 ≥ 0
2x ≥ -3
im not sure
fresh ocean
Since the root is for only 2x, you dont need the whole thing to be positive
solemn escarp
2x + 3 ≥ 0
2x ≥ -3
since the 3 is not under the sqrt you just have to solve 2x >=0
dusk ferry
oo
solemn escarp
but for (5) you do
dusk ferry
since the 3 is not under the sqrt you just have to solve 2x >=0
so i should solve only 2x >= 0
dusk ferry
but for (5) you do
{ x | x >= 2, x e r} ? like this?
@fresh ocean yo bacter is that true for num 4
fresh ocean
for 4, x being non negative is sufficient, so x>=0 is a valid domain for it to be a function
since 2x, and x behave similarly for the squareroot function
solemn escarp
so i should solve only 2x >= 0
divide both sides by 2
amber waspBOT
Available help channel!
olive star
I need to find all sequences $(a_n)$ with n a non negative integer and $a_n$ all positive integers such that $a_n$ divides $2^{a_{n+1}}-1$ for all $n$. First i showed that $a_n$ is always odd and with Fermat that if $a_n$ is constant, $a_n$ is never a prime number. I don’t know where to go from here.
grizzled pagodaBOT
Yhelvo
amber waspBOT
@olive star Has your question been resolved?
amber waspBOT
@olive star Has your question been resolved?
amber waspBOT
amber waspBOT
Available help channel!
split sail
ya teezi menak elo
amber waspBOT
ya teezi menak elo
split sail
i did
sterile nymph
generally the angle is named for the side opposite to it.
split sail
i only i have a triangle with a hyp of 52
sterile nymph
so tan W has W opposite, V adjacent, and U hypotenuse.
draw it out and verify though
sterile nymph
dunno, could be either, you don't have either side marked with the length
keen topaz
where is W tho
Doesn't that depend on your preference
sterile nymph
you can reflect the triangle, so either arrangement of the vertices will be possible.
keen topaz
W and T just have to correspond
split sail
wait nvm i understood
so S must be the 90 angle
cuz in the question it says RST
S is in the middle
split sail
but it helps to know the rest
split sail
if it had said SRT, that's the same triangle
same triangle but different points
so R will have the right angle
sterile nymph
no
split sail
what
sterile nymph
S can have the right angle even in triangle SRT
keen topaz
so R will have the right angle
No S
sterile nymph
however,
sterile nymph
the legs are what determines the right angle
split sail
cuz its RST
sterile nymph
in this case RT is the longest edge, so S is the right angle.
split sail
the legs are what determines the right angle
how
sterile nymph
(given that you have a right angle)
sterile nymph
yes
split sail
damn
sterile nymph
so triangle SRT is still a right triangle, and S is the right angle.
because RT is the longest leg
amber waspBOT
amber waspBOT
Available help channel!
foggy drum
amber waspBOT
foggy drum
i don't understand what to do in this at all
foggy drum
Partial fractions?
like write down the difference, multiply divide with it and simplify it?
but that'll work for 2 quadrtics only, no?
there's 3 here
foggy drum
okay I'll try and get back at this
foggy drum
I think this is only what u meant right
nope
foggy drum
I think this is only what u meant right
lemme try this then
warm glacier
Mhm
foggy drum
If it is not possible there is something u can do
is there some other approach to it?
foggy drum
Then u should leave it as quadratic only
but then I won't be able to integrate it
cause it's in product and not in sum
warm glacier
I think this is only what u meant right
..
U are converting it into sum
Just search up for partially fractions
On YouTube
foggy drum
im not understanding your method
warm glacier
Uk get some idea
foggy drum
Just search up for partially fractions
okay that works
warm glacier
Ull*
warm glacier
okay that works
Mhm
It's basically a method to convert it into separate fractions with no multiplication in denominator
amber waspBOT
@foggy drum Has your question been resolved?
amber waspBOT
foggy drum
.close
amber waspBOT
Available help channel!
dawn jewel
yo
pure minnow
Hmmmmm
dawn jewel
i evaluated the expression at the hint
pure minnow
Maybe try rewriting the top part of the first one as 1 - (6x)^2
dawn jewel
why
signal furnace
i evaluated the expression at the hint
you just forgot the chain rule
pure minnow
Cause it might not recognize that as equivalent lol
neat wind
dawn jewel
oof
signal furnace
you just forgot the chain rule
ex. $\frac{d}{dx} \int^{7x}_{0} f(u) \dd{u}=7 f(7x)$
grizzled pagodaBOT
Civil Service Pigeon
dawn jewel
oh
wait
does that happen for all bounds
u just multiply by the derivative
of the bound
signal furnace
I mean that's how chain rule works
dawn jewel
can u illustrate how that applies here exactly
ik chain rule but i dont see where that pops up here
signal furnace
$\int^{7x}{0} f(u) \dd{u}=f(g(x))$ where $f(x)=\int^{x}{0} f(u) \dd{u}$ and $g(x)=7x$
grizzled pagodaBOT
Civil Service Pigeon
signal furnace
It's pretty easy to replicate this logic for the general case, so I'll let you do that
amber waspBOT
amber waspBOT
Available help channel!
low remnant
How do you multiply rational expressions?
amber waspBOT
How do you multiply rational expressions?
cunning birch
like $\frac ab \times \frac cd$?
grizzled pagodaBOT
rafilou2003
cunning birch
multiply numerators between them
low remnant
Ye
grizzled pagodaBOT
rafilou2003
low remnant
Ok but what abt multiplying two complex rational expressions?
Like x^2 - 1 over 4 - x times 5 + x over 2
cunning birch
Like x^2 - 1 over 4 - x times 5 + x over 2
We already told you everything
cunning birch
multiply numerators between them
.
grizzled pagodaBOT
bruhmoment
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
cunning birch
$\frac{(x^2-1)(5+x)}{(4-x)\cdot2}$
grizzled pagodaBOT
rafilou2003
cunning birch
Hey how do I get that Texit app?
on this discord just start your math stuff with $ and end it with $ again
$like this$
grizzled pagodaBOT
rafilou2003
low remnant
$really$
grizzled pagodaBOT
me.dusk
split sail
$\LaTeX$
cunning birch
so I can do $x^2+1$ and this is text
cunning birch
me when $\begin{pmatrix}a&b\c&d\end{pmatrix}$ when I need a matrix
grizzled pagodaBOT
rafilou2003
low remnant
**rafilou2003**
Can we simplify this?
cunning birch
Can we simplify this?
nope
grizzled pagodaBOT
me.dusk
grizzled pagodaBOT
me.dusk
split sail
one easy way to remember is that exp turns + to x, and log turns x to +
split sail
pretty much
low remnant
Ye i messed up
On exp equations u just take the ln of both sides IF the bases arent common or cant be simplified to common bases
How do u turn a log into an exponential expression
?
low remnant
Yep pretty much
cunning birch
well, take the exponential of both sides
low remnant
Wdym
low remnant
Show it in TeXit please
grizzled pagodaBOT
rafilou2003
grizzled pagodaBOT
rafilou2003
low remnant
Oohhhh
split sail
*correctness $\operatorname{log}$
grizzled pagodaBOT
Display name
cunning birch
*correctness $\operatorname{log}$
just $\log$
grizzled pagodaBOT
rafilou2003
cunning birch
but I'm tired
low remnant
Ln is better right?
cunning birch
log is log_10 for you?
low remnant
Yea
grizzled pagodaBOT
rafilou2003
cunning birch
you can raise the equation to any exponent base you want
low remnant
Nice
low remnant
Im really dumb💀
grizzled pagodaBOT
me.dusk
cunning birch
recall
cunning birch
one easy way to remember is that exp turns + to x, and log turns x to +
what was said
so 10^(...+...)
becomes...
10^... times 10^...
grizzled pagodaBOT
me.dusk
cunning birch
the power is not 5-x+1
split sail
10^{-x+1}
low remnant
There u go
cunning birch
the power is not 5-x+1
it's
log(5-x)+1
...
and looks like you didn't apply the rule at all xdd
split sail
$$10^{\log(5-x)+1} = 10^{\log(5-x)}\cdot 10^1 = (5-x)\cdot 10$$
cunning birch
$ $
grizzled pagodaBOT
Display name
low remnant
Ok thx
How do you write the equation of a line perpendicular to a given line through a point?
split sail
in 2D/3D/higher dimension?
low remnant
2D
Algebra 1 stuff
For some reason TOCT puts Algebra 1 content in a precalculus playlist💀
grizzled pagodaBOT
me.dusk
split sail
There are two ways come to my mind, the first one is to use the fact that a line is of the form y=mx+c where m is the slope and c is a constant. The line perpendicular to a line with slope m is 1/m or -1/m, and if you want your new line to intersect a particular point, then substitute the coordinates of the point to the equation of the new line and solve for c.
Another way uses dot product and vectors which I find more conceptual, probably not in your algebra 1 syllabus though. I'm not familiar with the US system.
amber waspBOT
@low remnant Has your question been resolved?
low remnant
How do you write the equation of a line perpendicular to a given line through a point?
storm moth
whats the line
amber waspBOT
@low remnant Has your question been resolved?
torn furnace
How do you *write the equation of a line perpendicular to a given line through a...
don't vanish!
you have a given line? call it $y_1 = m_1 x + b_1$
grizzled pagodaBOT
jan Niku
grizzled pagodaBOT
jan Niku
torn furnace
and it must go through some point, say, $(x_p, y_p)$
grizzled pagodaBOT
jan Niku
torn furnace
so $y_2 (x_c) = y_c = \frac{-1}{m_1} x_c + b$
grizzled pagodaBOT
jan Niku
torn furnace
and $b = y_c + \frac{x_c}{m_1}$
grizzled pagodaBOT
jan Niku
torn furnace
then $y_2 (x) = \frac{-1}{m_1} x + \qty( y_c + \frac{x_c}{m_1})$
grizzled pagodaBOT
jan Niku
sonic musk
can someone explain part c like i am 4 years old
amber waspBOT
can someone explain part c like i am 4 years old
misty compass
Do you know how to differentiate a function?
sonic musk
sort of
misty compass
Your doing the same thing as part a just with the thing you came up with in part b
misty compass
You different f^-1(y)
sonic musk
so y/4 + 9/4?
misty compass
Yeah what is the derivative of that with respect to y
sonic musk
wouldn't it be $1/4$
grizzled pagodaBOT
nothin
misty compass
Yeah
gloomy tide
well... they were asking specifically for the derivative of the inverse
which in most cases happens to be the same (this is a very precise theorem, better to not assume that)
rich elk
Can someone help me with the 3rd question? I’m stuck since I’ve tried using a calculator online and it didn’t work for me.
rich elk
Let me know if either are you aren’t able to read it.
amber waspBOT
@rich elk Has your question been resolved?
rich elk
Can someone help me with the 3rd question? I’m stuck since I’ve tried using a ca...
I meant help with the 4th question. Sorry
fresh ocean
Your matrices E1 to E4 look correct. Are you sure you multiplied them in the correct order?
E4*E3 is first, then multiply that product by E2, and then by E1
amber waspBOT
@rich elk Has your question been resolved?
rich elk
Your matrices E1 to E4 look correct. Are you sure you multiplied them in the cor...
Oh hi, sorry I just noticed your reply.
rich elk
E4*E3 is first, then multiply that product by E2, and then by E1
Talking about these ones?
fresh ocean
yes
I did the product in correct order just to check and the answer did match the inverse
rich elk
?
that's weird
i got something else
Not sure if my multiplication was correct or something
fresh ocean
just try once with online calculators
rich elk
just try once with online calculators
I got this.
Which should not be the correct inverse matrix.
Unless? no
fresh ocean
Aha, ypu have the last transformation down wrong even if you performed it right
it goes r1 -> r1 + r3
which is what you did as well
but in your list you noted it wrong
fresh ocean
And I just made the matrices off of your transforms in the 1st image, so I didnt really bother checking your rest of stuff
fresh ocean
oh wait?
no no, the last step
that is r3 in there which you might have misread as r2 later on
rich elk
That's r1 + r2 -> r3.
fresh ocean
It looks like the operation is r1 becomes r1 + r3 to me
fresh ocean
happens all the time with me too, no worries. Just make sure to check everything properly once you are done
rich elk
Which is painful
That's why row operations are painful.
Wait, even if that was the case.
I still have to deal with the 2 on the center top.
I got it!
@fresh ocean ty
fresh ocean
npnp
rich elk
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Available help channel!
vestal igloo
amber waspBOT
vestal igloo
only need help in ii
am i supposed to prove using angles? kinda confused. see my markings on the diagram
fresh ocean
Do you need to use angles? You can just find the point of intersection and calculate all 3 lengths of the sides and you are done
Or you can claim that the RT and OP are parallel coz the angles POT and OTR are equal
amber waspBOT
@vestal igloo Has your question been resolved?
vestal igloo
Do you need to use angles? You can just find the point of intersection and calcu...
lengths of OR, RT, and OT?
vestal igloo
how do we know which sides are equal?
fresh ocean
equal sides = length of sides is equal
triangle is isosceles if any pair of sides has same length
vestal igloo
triangle is isosceles if any pair of sides has same length
i know
what i mean is which pair of sides are equal?
fresh ocean
that you can find by calculating all 3 lengths
vestal igloo
huh theres no value of y for T
fresh ocean
You know equation for y-axis and equation of line for PQ
tough mica
amber waspBOT
tough mica
,, \lim_{x \to \pm\infty} f (x) = f(x) = 0 \textbf{ if its continuous}
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
tough mica
what i mean
,, \lim_{x \to \pm \infty} \frac{\cos(x) + 5(x+3)^2 - 30x - 46}{x^2} = a \textbf{ at } x_0 = 0 \textbf{ if its continuous}
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
vast spade
**938c2cc0dcc05f2b68c4287040cfcf71**
yes
vast spade
yes what wai said
tough mica
wai?
tough mica
,, \lim_{x \to 0} \frac{\cos(x) + 5(x+3)^2 - 30x - 46}{x^2} = a
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
tough mica
from both sides, 0^+ and 0^- I think
keen pawn
Cool, any ideas?
tough mica
,w cos(0) + 5*9 - 0 -46
grizzled pagodaBOT
tough mica
lhopital
but if we can avoid it just tell me.
,, \frac{d}{dx} \left(\cos(x) + 5(x+3)^2 -30x - 46\right)
keen pawn
uh
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
tough mica
first we differentiate the numerator
keen pawn
why not series expansion instead
tough mica
taylor expansion? when cetntered around x = 0?
okay
of what? cosine or the entire numerator?
tough mica
we still need to differentiate yeah
keen pawn
Do that then!
tough mica
yeah lets hope lhopital and simplifying gets rid of the 0/0 undeterminate form
,, \frac{d}{dx} \left(\cos(x) + 5(x+3)^2 -30x - 46\right) \ = -\sin(x) + \frac{d}{dx} \left(5(x+3)^2\right) - 30
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
tough mica
we need function composition rule for the second term
mmm
,, \frac{d}{dx} \left(\cos(x) + 5(x+3)^2 -30x - 46\right) \ = -\sin(x) + \frac{d}{dx} \left(5(x+3)^2\right) - 30 \ = -\sin(x) + 5\frac{d}{dx} \left[(x+3)^2\right] - 30
solar gust
U^n ==> nu'u^(n-1)
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
tough mica
,, u^n \implies n \times u' \times u^{(n-1)}
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
tough mica
u = x+3
n = 2
,, u^n \implies n \times u' \times u^{(n-1)} \ (x+3)^n \implies 2 \times 1 \times u^{1}
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
tough mica
,, \frac{d}{dx} \left(\cos(x) + 5(x+3)^2 -30x - 46\right) \ = -\sin(x) + \frac{d}{dx} \left(5(x+3)^2\right) - 30 \ = -\sin(x) + 5\frac{d}{dx} \left[(x+3)^2\right] - 30 \ = -\sin(x) + 5\left(2(x+3)\right) - 30
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
proven vapor
**938c2cc0dcc05f2b68c4287040cfcf71**
why no series expansion
tough mica
series expansion of which function
proven vapor
the cos
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
proven vapor
$\lim_{x\to 0}\frac{1-\frac{x^2}{2}+O(x^4)+5x^2+30x+45-30x-46}{x^2}$
grizzled pagodaBOT
kheerii
tough mica
is that the first two terms?
proven vapor
yes
proven vapor
the constant and the x terms cancel and you're just left with $\lim_{x\to 0}\frac{\frac{9}{2}x^2+O(x^4)}{x^2}=\frac{9}{2}$
grizzled pagodaBOT
kheerii
tough mica
I can ignore the O iguess
proven vapor
I can ignore the O iguess
it just represents all the terms in the series which have a term of x^4 or higher
,w limit of (cosx + 5(x+3)^2 - 30x - 46)/(x^2) as x tends to 0
grizzled pagodaBOT
proven vapor
had to check
tough mica
how did u simplify
$\lim_{x\to 0}\frac{1-\frac{x^2}{2}+O(x^4)+5x^2+30x+45-30x-46}{x^2}$
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
tough mica
I got lost here
proven vapor
add up everything
notice that the constant terms are 1+45 - 46 which is 0
and the x terms are 30x - 30x which is also 0
tough mica
**938c2cc0dcc05f2b68c4287040cfcf71**
let me multiply by 2/2 the whole fraction
$\lim_{x\to 0}\frac{1-\frac{x^2}{2}+O(x^4)+5x^2+30x+45-30x-46}{x^2} \ \lim_{x\to 0}\frac{2-x^2+10x^2+60x+90-60x-92 + O(x^4)}{2x^2}$
tough mica
yeah srry
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
tough mica
,w 2 - x^2 + 10x^2 + 60x + 90 -60x -92
grizzled pagodaBOT
tough mica
okay thats 9/2 for sure
yeah we got rid of undeterminate form
so the limit is 9/2
so a should be 9/2 as well for it to be continuous
how do I check my answer?
ok whatever should be right
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tough mica
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tough mica
let $f$ be a function with continuous derivative such that $\lim_{x \to 0} \frac{f(e^{3x})}{x\cos(x)} = 12$ find $f$ and $f'(1)$
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
amber waspBOT
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
proven vapor
f(1) should be easy to spot
for the limit to exist what do you need f(e^3x) to be at x=0?
tough mica
= 12
proven vapor
why?
tough mica
idk
idk
im not sure I just said something random
,, \textbf{we are being told } \lim_{x \to 0} \frac{f(e^{3x})}{x\cos(x)} = 12
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
keen pawn
Hmm, I suggest you use ||LH||
tough mica
you are saying $\lim_{x \to 0} f(e^{3x}) = 12?$
proven vapor
you are saying $\lim_{x \to 0} f(e^{3x}) = 12?$
no
tough mica
if the numerator were not 0 would the limit exist?
I guess not, but I am not 100% about this one
Idk, is complicated for me
I need to put in double the effort compared to other folks
proven vapor
if it were not an indeterminate form can the limit ever exist finitely?
proven vapor
well yeah in this case it's 12
tough mica
implies numerator is approaching 0 aswell
tough mica
well 1/0 = +infty
proven vapor
not exactly, but yeah finite/0 always diverges to + or - infinity
tough mica
but what I mean is that, if denominator would be approaching zero, numerator necessarily needs to apporach zero because it is a finite value, idk if I can say that x/0 = infty for any x but Idk how to explain in it in a more rigurous manner neither
calculus is just a hard topic, and I am a bad student
proven vapor
no what you're saying makes sense
you're basically right
x/0 = infinity is not very precise but you understand the idea
the limit can't exist finitely if the denominator tends to 0 unless the numerator also tends to 0
tough mica
exactly
what do I do with this piece of information though? is there a followup coming because im out of ideas
WAI said to use Lhopi
but we dont know the form of the f(x) function
proven vapor
what do I do with this piece of information though? is there a followup coming b...
well you did the first part
f(1)=0
for the second part yeah LH works
keen pawn
for the second part yeah LH works
I don't think there's any othere "trivial" way anyway
proven vapor
I don't think there's any othere "trivial" way anyway
well you can use the first principle
keen pawn
well you can use the first principle
I mean, you could, yeah
but that wouldn't be pleasent here
proven vapor
using LH is redundant when the denominator is a linear term
tough mica
first priciples? you mean the derivative as a limit?
proven vapor
but that wouldn't be pleasent here
no just separate the cosx
proven vapor
first priciples? you mean the derivative as a limit?
yeah, but don't worry about that
just use LH
proven vapor
doesn't matter, you can still differentiate it
tough mica
,, \lim_{x \to 0} \frac{f(e^{3x})}{x\cos(x)}
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
tough mica
ok top is $3e^{3x}f'(e^{3x})$ ?????????
keen pawn
use chain rule
tough mica
my bad
proven vapor
nope
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
keen pawn
yes
proven vapor
yes
tough mica
my gawd okay
keen pawn
now derive the denominator.
tough mica
,, \frac{d}{dx} \left(x\cos(x)\right) = \cos(x) - x\sin(x)
proven vapor
for the denominator instead of using product rule you can separate the cosx
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
tough mica
,w differentiate xcos(x)
proven vapor
$\lim_{x\to 0}\frac{f\left(e^{3x}\right)}{x\cos x}=\lim_{x\to 0}\frac{f\left(e^{3x}\right)}{x}\cdot\lim_{x\to 0}\frac1{\cos x}$
grizzled pagodaBOT
kheerii
tough mica
cos(0) = 1
proven vapor
yes
tough mica
$\lim_{x\to 0}\frac{f\left(e^{3x}\right)}{x\cos x}=\lim_{x\to 0}\frac{f\left(e^{...
what about it
proven vapor
so it's no longer indeterminate
proven vapor
what about it
now you don't have to use the product rule for the denominator
tough mica
okay
$\lim_{x\to 0}\frac{f\left(e^{3x}\right)}{x\cos x}=\lim_{x\to 0}\frac{f\left(e^{3x}\right)}{x}\cdot\lim_{x\to 0}\frac1{\cos x} = \lim_{x\to 0}\frac{f\left(e^{3x}\right)}{x}$
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
proven vapor
yeah it is
tough mica
if we differentiate denom is 1
proven vapor
but you can only say this here because you KNOW that the limit HAS to exist
in general this fact is not true
proven vapor
if we differentiate denom is 1
yes
tough mica
you mean product rule for limit only applies if both limits exist
proven vapor
yes
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
proven vapor
numerator differentiated is
the same as what you said earlier
tough mica
yeah
tough mica
,, \lim_{x \to 0} \frac{f(e^{3x})}{x\cos(x)} = \lim_{x \to 0} \frac{3e^{3x}f'(e^{3x})}{1} = 3f'(1)
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
tough mica
mmm
idk if this equality is legal
but should be since we had 0/0 undeterminate form at the start so lhopital should me legal
proven vapor
it is
tough mica
anyways, 3f'(1) = 12
proven vapor
why do you think it isnt?
tough mica
why do you think it isnt?
because problem looked hard but after lhopi and a bit of reasoning is actualy normal problem now
which means 3f'(1) = 12 ===> f'(1) = 4 and f(1) = 0
proven vapor
indeed!
proven vapor
because problem looked hard but after lhopi and a bit of reasoning is actualy no...
well, that's the entire point of problem solving
tough mica
.solved
tough mica
amber waspBOT
tough mica
I think I might need to use first principles for this
,, \lim_{h \to 0} \frac{f(x+h) - f(h)}{h}
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
vast spade
first find if the function is continuous
tough mica
true, but differentiability implies continuity
what I mean is that if we found that the function is differentiable at a point, is also continuous at that point
split sail
It seems that the function is not even defined below 1/6
tough mica
okay
thats an error in the domain of the function
I think they meant dom(f) = (-1/6, infty)
proven vapor
what I mean is that if we found that the function is differentiable at a point, ...
you cannot even test for differentiability without checking for continuity
though continuity is not too bad to test for here
tough mica
okay lets check for contiuity and then differentiability
I might be mistaken and taking nonsense
,, \lim_{x \to 0} \frac{-\ln(1+6x)}{x} = -6
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
tough mica
this needs to hold for that piecewise to be continuous
we need to apply lhopi, unless there is other way please let me know
numerator is $\frac{d}{dx}\left(-\ln(1+6x)\right) = -\frac{1}{1+6x} \times 6 = \frac{-6}{1+ 6x}$
grizzled pagodaBOT
938c2cc0dcc05f2b68c4287040cfcf71
proven vapor
we need to apply lhopi, unless there is other way please let me know
this is a pretty famous limit, $\lim_{x\to 0}\frac{\log(1+x)}{x}=1$ which in turn comes from the series expansion of log(1+x) at x=0