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#help-33
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marsh citrusBOT
hybrid plover
Post your work and we can see where you're going wrong
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celest frigate
I have a quick question about calc 3... My professor said gradient will always be normal to the level curve. (Will it also always be orthogonal/normal to the tangent plane of a surface)?
marsh citrusBOT
I have a quick question about calc 3... My professor said gradient will always b...
celest frigate
also gradient does not equal the partial of x y and z right
proud ice
I have a quick question about calc 3... My professor said gradient will always b...
Nah I do not think so
celest frigate
<a , b ,c> if you had something like this
proud ice
gradient vector lies inside the tangent plane
celest frigate
would it be equal to a = partial of x, b = partial of y, c = parital derivative of z?
proud ice
It will not be orthogonal to the plane
celest frigate
oh wait can i share a picture
do you know why this is?
the red vector is the gradient vector btw
yellow plane is the plane and grey surface is the surface
also I noticed there are two ways to solve for a tangent plane
0 = f_x(x1, y1, z1)(x-x1) + f_y(x1,y1,z1)(y-y1) + f_z(x1,y1,z1)(z-z1)
and
z = f(x1,y1) + f_x(x1,y1)(x-x1) + f_y(x1,y1)(y-y1)
in the first one are we treating it as a function with 4 variables?
or 3 inputs
in the 2nd we are treating it like a f(x1,y1) function
but in the first i think we are treating it like f(x1,y1,z1) function
except there is no variable to represent this 3 input function in the 1st formula
@proud ice <@&286206848099549185>
celest frigate
That is, it is a surface of constant value?
hmm i don't think so since its not a plane
it has creases and stuff
dips and folds
proud ice
That's the only scenario where I can think of gradient being orthogonal to tangent plane
Also, level surface does not mean "flat plane"
celest frigate
oh
proud ice
consider the scalar field f(x, y, z)=x²+y²+z²
a level surface would be f(x, y, z)=c for some constant c
proud ice
Then every level surface would be a sphere centered at the origin with radius sqrt(c)
proud ice
yes
celest frigate
hmm I think this one is 3 inputs yes
proud ice
Your level surface is c=x²+y²+z²
In which case, yes, the gradient is orthogonal to this tangent plane
celest frigate
0 = f_x(x1, y1, z1)(x-x1) + f_y(x1,y1,z1)(y-y1) + f_z(x1,y1,z1)(z-z1)
if the formula is this, would this count as a level surface?
partial of x times x values then partial of y etc
because its not explicitly stating an extra variable like
A = x^2 +y^2 +z^2 isn't that also a level surface
where a is a variable
proud ice
if the formula is this, would this count as a level surface?
I'm assuming f_x, f_y, f_z are your partial derivatives of f?
celest frigate
I'm assuming f_x, f_y, f_z are your partial derivatives of f?
yes
proud ice
Then no, that is not a level surface. That is (I believe) the plane tangent to your level surface
celest frigate
and x1,y1,z1 are the x y and z values of the point
celest frigate
Then no, that is not a level surface. That is (I believe) the plane tangent to y...
oh okay
wait so if it was 0 = x^2 + y^2 + z^2
would this be a level surface
and f(x, y, z) = x^2 + y^2 + z^2 do you know what the difference is between these?
proud ice
wait so if it was 0 = x^2 + y^2 + z^2
It would be a level surface of the single point at the origin (a sphere of radius 0)
celest frigate
since the function thinks the 4th variable would be 0? (lets say a = 0)
proud ice
and f(x, y, z) = x^2 + y^2 + z^2 do you know what the difference is between the...
This is the function of your scalar field. It is not level because f(0, 0, 0) != f(0, 0, 1)
celest frigate
oh
proud ice
a level surface requires f(x, y, z)=f(x', y', z') for any two points (x, y, z) and (x', y', z') in the surface
proud ice
a level surface requires f(x, y, z)=f(x', y', z') for any two points (x, y, z) a...
(This is equivalent to requiring f(x, y, z) = some constant)
celest frigate
since derivative of 0 is 0
celest frigate
(This is equivalent to requiring f(x, y, z) = some constant)
ah okay
proud ice
so basically (0,0,0)
Not sure what you mean here
celest frigate
well to be honest, we've been working with 3 variables, and im a bit confused on why f(x,y,z) is coming into the picture when it used to be f(x, y) which is = z
proud ice
yes
celest frigate
not sure what f(x,y,z) means
proud ice
you ask good questions
It's hard to grasp at first
f(x, y) is easier to grasp because it can be represented as a surface in 3D space
f(x, y, z) is technically an upscale of that: A hyper surface in 4D space
Not too easy to visualize...
But there are applications for this
f(x, y, z) assign a scalar value to every point in 3D space
celest frigate
wow so is that equation of tangent plane 0 = fx.... a 4D tangent plane?
celest frigate
0 = f_x(x1, y1, z1)(x-x1) + f_y(x1,y1,z1)(y-y1) + f_z(x1,y1,z1)(z-z1)
0 = f_x(x1, y1, z1)(x-x1) + f_y(x1,y1,z1)(y-y1) + f_z(x1,y1,z1)(z-z1)
this one
proud ice
as a practical application, consider the temperature in a room
proud ice
0 = f_x(x1, y1, z1)(x-x1) + f_y(x1,y1,z1)(y-y1) + f_z(x1,y1,z1)(z-z1)
this one
Not quite, this is still a plane in 3-space.
If it were w = instead of 0 = , then it is a 4D plane
proud ice
as a practical application, consider the temperature in a room
to help understand scalar fields, imagine f(x, y, z) as saying what the temperature in a room is at a point (x, y, z) in the room
That's the best real-world example I can think of
celest frigate
If it were `w = ` instead of `0 = `, then it is a 4D plane
oh i see, sorry can you explain this sorry for all the questions
he puts it in xyz form
of the function
proud ice
oh i see, sorry can you explain this sorry for all the questions
w=Ax+By+Cz+D is a hyperplane
0=Ax+By+Cz+D is a plane
Difference is the introduction of a new free variable; w
celest frigate
oh but doesn't f(x,y,z) = w?
proud ice
w=Ax+By+Cz+D has three variables: x, y, and z. They all affect the dependent variable w
celest frigate
okay
proud ice
0=Ax+By+Cz+D has only two free variables. For example, if x and y are free, then z is dependent on x and y
proud ice
So two free variables and one dependent give you a 2D plane in 3D space
celest frigate
which makes the 4th variable irrelivant even though its in that form
proud ice
And three free variables with one dependent will give you a 3D hyperplane in 4D space
celest frigate
ohh okay
proud ice
Refresh me on your question now. We discussed a lot
celest frigate
basically my question was is the gradient which is < f_x(x1,y1), f_y(x1,y1) > is it orthoganol to the tangent plane of a surface?
proud ice
Right. The question, as it stands now, is ill-formed in this context
There are multiple ways you can define "surface"
If you define "surface" as the set of points f(x, y), then no
The gradient will always point in the direction of maximum increase of f(x, y), and will therefore be inside the tangent plane
celest frigate
oh tbh i don't really know the equation of the surface shown
proud ice
If you defined "surface" as level surface to f(x, y, z), then yes
celest frigate
if its f(x,y) or leveled
celest frigate
If you defined "surface" as _level surface_ to f(x, y, z), then yes
oh okay
i will assume its leveled
proud ice
Based on what you've shown me, you're dealing with level surfaces of f(x, y, z)
You can see an easier example by considering "level curves" to surfaces defined by f(x, y)
These are actually what topographical maps are
Here is an example
celest frigate
oh yeah thats what we did for xy functions
proud ice
The lines defined sections of constant height (i.e. where f(x, y) is unchanged)
level surfaces are the same idea
They are topographic maps in 3D
celest frigate
celest frigate
They are topographic maps in 3D
oh got it
proud ice
In these examples, gradient will always be orthogonal to level surface or level curve
celest frigate
ah okay, but not always to tangent planes since the surface could be a non leveled surface
right
proud ice
ah okay, but not always to tangent planes since the surface could be a non level...
Yes. It's important to know what kinds surfaces you are dealing with
celest frigate
you said a leveled surface is f(x,y,z) and has 3 independant and 1 dependant variable
how come here there is only 1 dependant 2 independant, in that equation yet its labeled as f(x,y,z)?
celest frigate
Yes. It's important to know what kinds surfaces you are dealing with
i was thinking since its = 0 it shouldn't even have f(x,y,z) cause 0 = that equation is already a full equation, why are there three = signs?
Maybe he wrote it wrong
anyways thank you for the help @proud ice u really helped me understand that :)
hope u have a great day
marsh citrusBOT
@celest frigate Has your question been resolved?
gloomy hornet
How solve
tawdry shard
marsh citrusBOT
tawdry shard
how to solve this
marsh citrusBOT
@tawdry shard Has your question been resolved?
tawdry shard
<@&286206848099549185>
tawdry shard
no i dont
any easier way
normally we use this
let a new variable to represent something
tawdry shard
<@&286206848099549185>
main idol
let a new variable to represent something
Try u=(x+8)^3
marsh citrusBOT
marsh citrusBOT
Available help channel!
brave spire
Would someone help me on 10th? I can't understand how x is -2.. my solution is 212.5
marsh citrusBOT
Would someone help me on 10th? I can't understand how x is -2.. my solution is 2...
elfin berryBOT
glass kestrel
Would someone help me on 10th? I can't understand how x is -2.. my solution is 2...
Can you show your steps on solving this?
brave spire
Can you show your steps on solving this?
It's pretty rough, lemme redo it clearly
glass kestrel
Aight take your time
glass kestrel
Lmaooo it's fine
stoic crag
can someone please help me
marsh citrusBOT
can someone please help me
glass kestrel
Have you tried something?
stoic crag
yes
i tried making it into a right angled triangle
with sides of 2km
but that didn't work
glass kestrel
to find the distance from start to A, it would be this, no?
stoic crag
the distance is 2km right?
stoic crag
to find the distance from start to A, it would be this, no?
i think thats what you are supposed to do
glass kestrel
looking at the picture how many blocks is the base triangle? and how many blocks is the height of the triangle?
glass kestrel
i think thats what you are supposed to do
yeah
glass kestrel
for? base or height?
stoic crag
hight
glass kestrel
thats correct
so each grid represents 1km. so the base would be 2km and the height would be 1km
what formula would you use to find the hypotenuse?
stoic crag
a^2 + b^2 = c^2
stoic crag
is it square root 5?
glass kestrel
square root of 5 is approx 2.24
scenic dagger
how many ways can I represent latitude and longitude, say this one of central park?
40.770071, -73.974952
marsh citrusBOT
how many ways can I represent latitude and longitude, say this one of central pa...
sand vigil
how many ways can I represent latitude and longitude, say this one of central pa...
actually this is more of a google question 😂
😏
dim kestrel
i just got it from the internet
There are various ways to represent latitude and longitude coordinates, but some of the most common ones are:
Decimal Degrees: The format you provided, 40.770071, -73.974952, is the decimal degree format, where the latitude and longitude are represented as decimal fractions.
Degrees, Minutes, Seconds (DMS): In this format, the latitude and longitude are expressed as degrees, minutes, and seconds. For example, the coordinates of Central Park in DMS format would be 40°46'12.3"N, 73°58'29.8"W.
Universal Transverse Mercator (UTM): UTM is a coordinate system that divides the earth into a grid and represents locations using easting and northing values. The coordinates of Central Park in UTM format would be 18T 583973mE 4512489mN.
Military Grid Reference System (MGRS): Similar to UTM, MGRS is a coordinate system that divides the earth into a grid and represents locations using a combination of letters and numbers. The coordinates of Central Park in MGRS format would be 18TWL8397301249.
There are also other formats and systems used to represent latitude and longitude coordinates, but these are some of the most common ones.
scenic dagger
my goal is to try to reduce the characters counting without losing too much precision. But I believe if I introduce degrees and minutes and seconds, I'd be just adding more characters
scenic dagger
i just got it from the internet
There are various ways to represent latitude and...
thats a very good answer, thank you!
thanks guys
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torn sphinx
anyone can help me with solving this question ?
marsh citrusBOT
anyone can help me with solving this question ?
glass silo
Which one - there are two visible?
torn sphinx
question 1 thx
glass silo
Cool cool, so note how they said that
The sum of the volumes of two spheres is [324pi cm^3]
and also
The radius of the larger sphere is equal to the diameter of the smaller sphere.
Can you use these to form equations?
torn sphinx
4/3 pi r³ + 4/3 pi 2r³= 324pi?
glass silo
Need to 
that carefully but basically
torn sphinx
then I'll find the radius of the smaller sphere?
glass silo
[and from there you can work out everything else they said]
torn sphinx
can the equation be simplified into 4r³= 324?
my answer is still different to the actual answer
<@&286206848099549185>
daring stump
can the equation be simplified into 4r³= 324?
which equation you want to simplify..?
@torn sphinx
torn sphinx
4/3 pi r³ + 4/3 pi 2r³= 324pi?
this
daring stump
this
yes you can convert
1st take pi common and it`ll cancel out
then you have to add terms r3
you`ll get desired outcome
torn sphinx
4.326....
torn sphinx
anyone can help me with solving this question ?
but still I can't get the desired outcome of the larger spehre's volume which is 288
daring stump
4/3 pi r³ + 4/3 pi 2r³= 324pi?
you trying to find value of r right..?
daring stump
4/3 pi r³ + 4/3 pi 2r³= 324pi?
wait i`m considering that this equation is correct
then r = cuberoot of 81
r=4.32
if equation you wrote here is right..! @torn sphinx
daring stump
ye and it's this if my calculations aren't wrong
what`s the ans
torn sphinx
should be 216
torn sphinx
which is?
daring stump
use r=3 and calculate the things you want to calculate
tell me if this is wrong
ill check once again
daring stump
so the mistake you did was
they said that the radius of lager sphere is double of the smaller right..?
then R=2r
R larger sphere radius
r smaller ------
right..?
till this makes any sense to you...?
torn sphinx
yew
daring stump
so when you write the formula for volume its 4/3 pi r^3
this is right for larger sphere which is 4/3 pi R^3
but for smaller one it`ll become 4/3 pi (2r)^3
got it..?
torn sphinx
ye
daring stump
4/3 pi R³ + 4/3 pi 8r³= 324pi
torn sphinx
ohhhhh
daring stump
so this is the correct equation
you have to take cube of 2 also in consideration
torn sphinx
I get it now
daring stump
glad i could help bro
torn sphinx
thx so much 🙏
daring stump
thx so much 🙏
anytime brother
torn sphinx
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marsh citrusBOT
marsh citrusBOT
Available help channel!
deep gale
marsh citrusBOT
main idol
would I be correct
first one's right. 2nd and 3rd are wrong
deep gale
would it be that for the 2nd 2 that since they go to infinity and -infinity they get closer to 1
main idol
yes
deep gale
gotcha
I just was not sure which one it would be but it makes sense since it just gets closer
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oak void
if
oak void
a quadratic have equation $y = 4(1-x)^{2}-2$
elfin berryBOT
yomiko
oak void
the the turning point would be (1,-2) right?
hazy lion
for which class?
i keep seeing the language turning point
is that calculus? or are you talking about the max?
oak void
what do you mean which class
hazy lion
idk just curious if you can use calculus to discuss it or not
what you can do is you know a square quantity is squared where it is 0, right
the squared on a^2 means that a cant be smaller than 0
so its smallest when its 0
if thats (1-x)^2 then we want 1-x=0, so x=1
thats where it will be smallest
make sense?
@oak void
oak void
yeah
hazy lion
okay
and yea, i think -2
so makes sense to me 
did you check it
,w graph 4(1-x)^2-2
elfin berryBOT
hazy lion
tranquil dove
turning 2a^-1 + 3b^-2 into a single fraction with positive integers
2/a + 3/b^2 is the first step right
marble glacier
mhm
tranquil dove
humble raft
marsh citrusBOT
@tranquil dove Has your question been resolved?
open plume
marsh citrusBOT
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'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
1
marsh citrusBOT
@open plume Has your question been resolved?
tardy mist
marsh citrusBOT
tardy mist
they want me to find the volume between the both functions and i cant find any good variable exchanges that make it work 😦
the last integral doesnt end up with the right answer, i dont have the d( θ) because it is wrong somewhere before
didnt get that far
<@&286206848099549185>
tried another method but didnt work either 😦
the answer should be about 13,8
<@&286206848099549185>
marsh citrusBOT
@tardy mist Has your question been resolved?
tardy mist
anyone?
marsh citrusBOT
@tardy mist Has your question been resolved?
tardy mist
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warm plaza
A triangular pyramid has a base shaped like an equilateral triangle. The legs of the equilateral triangle are all 13 millimetres long, and the height of the equilateral triangle is 11.3 millimetres. The pyramid's slant height is 18 millimetres. What is its surface area?
'
marsh citrusBOT
A triangular pyramid has a base shaped like an equilateral triangle. The legs of...
stark trail
frigid python
Hey, this is more of a history of math problem. The proof that there are arbitrarily many composite numbers between prime numbers is quite trivial. For a gap of the size (N-1), create the sequence N!+2, N!+3, N!4, ..., N!+N - each of the terms in this sequence is composite and therefore we've constructed a "gap" of the size N.
However, I have no idea who said this first. Who was the initiator? Does anyone have any idea?
marsh citrusBOT
Hey, this is more of a history of math problem. The proof that there are arbitra...
@frigid python Has your question been resolved?
main idol
Hey, this is more of a history of math problem. The proof that there are arbitra...
This isn't really well known in math since it's not a big result
frigid python
Ah, okay 😅 So do you think I can include it into my thesis without citing and referencing?
marsh citrusBOT
@frigid python Has your question been resolved?
short jetty
marsh citrusBOT
short jetty
for some reason my brain isn't putting together between the factoring and
right after I get it all to the same side, the factoring by I'm assuming grouping is throwing me off, why can I just have the x-3 raised to the 4 just go to three and completely eliminate the x-3^3
my train of thought is pretty fuzzy on where to go after i isolate everything
wild zodiac
you're not understanding the logic behind this step, in particular?
short jetty
yes
short jetty
yeah
wild zodiac
5x here is the common factor
short jetty
yeah
wild zodiac
in your example, the common factor is (x-3)^3
(x-3)^4 / (x-3)^3 = (x-3)
-6(x-3)^3 / (x-3)^3 = -6
so you are left with (x-3)^3 [ (x-3) + (-6) ]
i can write that out in handwriting if notation is hard to understand
short jetty
nah youre totally fine
so after that i break down the x-3^3 right? but there since its cubed i don't set it to 0 just yet?
wild zodiac
what do you mean by 'break down' the (x-3)^3?
short jetty
split into (x-3)(x-3)(x-3)
wild zodiac
that isn't necessary in order to derive solutions
short jetty
so where would you go from there
wild zodiac
conceptually you need to know this:
if you can rewrite a function as
(x-a)^n * (x-b)^m * (x-c)^k * ... * etc
(stop me if i've lost you already)
short jetty
youre good
wild zodiac
you can derive that there are n-solutions of a, m-solutions of b and k-solutions of c
to ground what i just said in an example
consider f(x) = (x-3)^2 * (x-2) * (x+1)
from this point, we can say for f(x) = 0 we have
x=3 or x = 2 or x = -1
with x = 3 as a double root
short jetty
yeah yeah
wild zodiac
you can derive that there are n-solutions of a, m-solutions of b and k-solutions...
note: for x=0, that is
so back to your original question
you have (x-3)^3 * (x-9) = 0
using that concept
x=3 (triple root) and x=9 are the only solutions
wild zodiac
ah
short jetty
I'm always so unaware of when I can remove parentheses
wild zodiac
that is only because of one condition
(x-3) is (x-3)^1
which just means (x-3)
parenthesis are only used there to signify that we have factorised a term with (x-3)^3 as a common factor
and it has left us with (x-3) by itself
if it was, say
(x-3)^2 - 6
you couldnt just do (x-9)^2
do you understand why?
wild zodiac
yeah
and if instead you had (x-9)^2
(x-9)(x-9) will expand into something a lot different
this is the case for any (x-3)^n where n isn't 1
e.g (x-3)(x-3)(x-3) wouldnt be = to (x-9)(x-9)(x-9) and so on so forth
but with (x-3)^1 specifically
(x-3)^1 = (x-3) = x - 3
there is no operation happening upon the '(x-3)' here
so anything outside it, like the -9, can freely interact
i.e, we can remove the parenthesis
short jetty
yeah thats wild
something in me wants to just set the (x-3) = 0 and then get x = -3
even though its being subtracted from 6
wild zodiac
i can get that since when you see brackets its tempting
short jetty
6 subtracting etc
wild zodiac
you can infact do that for the (x-3)^3 straight away
short jetty
and have them just = 3 triple root?
wild zodiac
if you have f(x) = (x-3)^3 * (literally any function of (x))
like, say
f(x) = (x-3)^3 * (arctan(5/13^(15x)))
you would still have the solution
x=3 (triple root)
(and some other wild solution for whatever is on the other side, there)
short jetty
yeeah yeah
wild zodiac
the important distinction here is the multiplication
short jetty
may i ask another question
wild zodiac
sure thing
short jetty
excuse the mess above lmfao
so when i see this
my first thought is to grab common etc
and i first imagined itd be the x(x-2) ?
to give me this?
short jetty
oh yeah the stuff above the line that i poorly seperated is bad
wild zodiac
well
the solutions to this from step 2 you can see will be
[x^2] = 0
[(x-2)] = 0
[(x-2)^3 - 1] = 0
the first two should be trivial
wild zodiac
[x^2]=0 => x=0 (double root)
[(x-2)]=0 => x=2
the first one is a double root
again thou gh
short jetty
yeah that's what is figured, but they dont give me the option for that
wild zodiac
noticing that things are double roots are only important graphically
do you understand what i mean?
wild zodiac
well rather
short jetty
im not sure
wild zodiac
if your questions is 'What are all the possible solutions for x?'
you would just write 'x=0'
wild zodiac
'double root' just means that, in a graph, the function passes through 0 'twice' at the same spot
short jetty
yeah yeah that part i got
wild zodiac
the last part of the question is to solve [(x-2)^3 - 1] = 0
wild zodiac
here?
short jetty
oh wait
wild zodiac
you were right to factor out a common factor, yes
short jetty
i saw yours, okay so thats the right factor
wild zodiac
for the last brackets, we expand for
expand out for (x-2)(x-2)(x-2) - 1 = 0
(x^3-6x^2+12x-8) - 1 = 0
x^3 -6x^2 + 12x - 9 = 0
using factor thrm notice x=3 is a factor
short jetty
ohhhh i got it, the first part for why thats factored the way it is
short jetty
on the last bit?
wild zodiac
for (x^2 -3x +3)
short jetty
ahhhhh i think im mentally flooded a second
wild zodiac
but tbh, you probably don't want complex answers...
let me just triple check i've not made any mistake
short jetty
yeah nah not yet
wild zodiac
weird that they'd give you this question, then
wild zodiac
this is the answer they want?
short jetty
yeah those are the choices
im surprised double root for zero isnt there
that really messed with me mentally
wild zodiac
right they probably just want you to say:
"ah, this sqrt( ) has a - in it, i should disregard it completely"
wild zodiac
have you done something called polynomial division at your stage?
short jetty
not in depth we're going back over it in a few units
all of the video lessons on this dont have it being done on these
wild zodiac
you can not approach (x-2)(x-2)(x-2)-1=0
without knowing how to compute this
(at least, i dont think so)
short jetty
id have to review that, we did it a while ago but it wasnt used to solve any of these up until this point
mostly factoring by grouping and differences of squares
wild zodiac
i can remind you how you would do it if it'd help
i was only hesitant because what you're doing could be a few grades behind what this is
short jetty
yeah im actually returning to school, way later
wild zodiac
polynomial division is where you know (x-3) * (something) = (x^3 -6x^2 + 12x - 9)
and you want to factor out the (x-3)
to get the (something)
and rewrite it as (x-3) * (something)
short jetty
x^2 + 2x + 4 i think something like that
wild zodiac
wdym?
short jetty
i think is the answer to that, idk
wild zodiac
it's a good guess in that
you know the (something) will likely be a quadratic
because the biggest term of the product function is x^3
so you'd need an x^2 at most to get from x to x^3
but unfortunately, we need to be very exact
and the rest of it (+2x+4) isn't correct
so we can't just guess, we need to follow a division algorithm a little like long division
short jetty
oh yeah i wasn't writing it down or anything, i actually think i found out how they want me to do it
i think they want me to factor an (x-2) out of the (x-2)^3
wild zodiac
this?
short jetty
yeah
short jetty
yeah youre right
wild zodiac
(you would get (0)(0)^2 - 1 = 0, -1=0 which isnt right
wild zodiac
the answers to your question are
x=0,x=2,x=3, & x=3/2 +/- i sqrt(3)/2
you can ignore the 4th and 5th, however
as they are complex
x=0, x=2 and x=3 are the answers
short jetty
no double root?
short jetty
the last test i took marked it wrong because i didn't
short jetty
is it not a usual convention?
wild zodiac
well, i can say that to me it personally isn't as a college student in the field
if your teacher is asking for it though then i would just out of habit
if they are specifically saying, no, x=0 is a single root, not a double root
then they are wrong
it is a double root
short jetty
okay gotha
so for the sake of the answer then, and in general, in this context, x^2 = 0
of this specific problem
wild zodiac
(x^2) * (x-2) * [this last function we're working on] = 0
solution 1:
(x^2) = 0
=> x = 0
solution 2:
(x-2) = 0
=> x=2
solution 3 is what we're working on righ tnow
wild zodiac
solution 3 will be the case:
(x-2)^3 -1 = 0
short jetty
without using polynomial divison or the quadratic formula, how would you approach that
wild zodiac
you couldn't
short jetty
at all
wild zodiac
well
you could find (x-3)
but
you couldn't say 'I am certain that the quadratic that occurs when I factor out (x-3) will give me real solutions'
real meaning 'not-complex'
so really you'd have to approach it in full
short jetty
so i cant solve this without either of those
wild zodiac
which requires polynomial division
short jetty
we havent used that in this unit
wild zodiac
i don't think so, no
short jetty
i think i can use the difference of two cubes here actually
(x-2)^3 - 1
(x-2)^2 + (x-2) + 1
short jetty
yesss
wild zodiac
so from here you have x=3 as solution as required
and you would just show the discriminant test for the quadratic to show it has no solutions in the reals
short jetty
yeah yeah
they didn't want me messing with anything complex in this yet
ahh youre the best, big time thanks
the parentheses getting removed and the double root thing tripped me up majorly
.close
simple briar
hello
simple briar
So,
I calculated the fourth derivative to be 6/x
but then I get n = 100(1000x^3)^1/3 / x
which can't be right
so i'm a little lost
i basically simplified to
1/(3X10^6n^4) < 1/3X10^17
oh
nvm
i'm so slow
i didn't plug in my lower limit
lool
this channel can be closed
glass needle
how would i slove (2x^3y^2z^4)^3 x (2z^2)^3 ??
marsh citrusBOT
how would i slove (2x^3y^2z^4)^3 x (2z^2)^3 ??
nova totem
how would i slove (2x^3y^2z^4)^3 x (2z^2)^3 ??
Bad notation here. Simplify, and is the x in between the multiplication symbol
Use like • or "times"
glass needle
nova totem
Looks like a third term of x
glass needle
so would i dispute the 3?
keen stump
You want to distribute the 3 to every number/variable inside the parentheses
keen stump
try solving it and ill tell you if youre right
glass needle
bc doesnt 2^3 = 6?
keen stump
2^3=8
glass needle
but how?
keen stump
$2^3=222$
elfin berryBOT
CrEpasPmkinPie
keen stump
how?
glass needle
bc i went 2x3, 2x3, 2x3
keen stump
that doesn't even equal 12 lol
keen stump
ohhh
keen stump
$222$
elfin berryBOT
CrEpasPmkinPie
keen stump
$2*2=4$
elfin berryBOT
CrEpasPmkinPie
keen stump
$4*2=8$
elfin berryBOT
CrEpasPmkinPie
keen stump
So what do you get for the question now?
glass needle
lemme see
wait
(2z^2)^3
it also be 8 if i did it with the 2 and 3?
the two exponets
glass needle
;-;
elfin berryBOT
CrEpasPmkinPie
elfin berryBOT
CrEpasPmkinPie
glass needle
uhhuh
elfin berryBOT
CrEpasPmkinPie
glass needle
geez
keen stump
its chonky
keen stump
yes!!
glass needle
keen stump
$8x^9y^6z^{12}*(2z^2)^3$
elfin berryBOT
CrEpasPmkinPie
glass needle
and i still have to simpily ?
keen stump
simplify the right term
glass needle
the one with the ()?
keen stump
yes
glass needle
o ok
keen stump
but here's the rule on the exponents and parentheses
glass needle
what is it?/
keen stump
$(x^a)^b=x^{ab}$
elfin berryBOT
CrEpasPmkinPie
glass needle
thats what
keen stump
?
glass needle
**CrEpasPmkinPie**
what is this?
keen stump
thats the rule for exponentd and parentheses
keen stump
np
glass needle
.close
marsh citrusBOT
marsh citrusBOT
Available help channel!
rancid saffron
How to solve?
marsh citrusBOT
How to solve?
rancid saffron
It requires a decimal number as the answer
nova totem
,w ln(21)
elfin berryBOT
still temple
Hi
still temple
I have a question
mellow isle
I have a question
ask it
stoic saddle
i closed the other one
still temple
How do they get the X values @mellow isle
Or just in general
they just picked them
Cause I assume that they got the y values then multiplied by -1
Yeah I know
It’s a reflection off the X axis correct?
idk what question you are referring to
D
the sheet just looks like its practicing graphing equations
Yea, but the y values are correct, but the X values are incorrect
Why tho
whats f(x)
Wouldn’t the equation be x=2X+4
Cause I found the y values, and that’s correct
The x values are incorrect
can u show what f(x) is
Idk @
what
Idk what you’re asking me
send a picture of the function
or graph associated with it
It’s the very top one
On the left
You just need to translate it
i can barely see it
The points are
(-2,0), 0,-2, 6,1
I understand how to get the Y values
Not the x tho
I dont understand you question, you want to know how to translate the graph ?
Yes
what part are you stuck with
The x values
okay
do you know what f(cx) does to the coords?
like lets say f(x) gives you a point (a, b)
what does f(cx) give you
and similarly what does f(x+d) give you
Wouldn’t it be x= 2X+4
Then you put the points into X
what
still temple
what does f(cx) give you
what does this give you
Not sure @still temple
if f(x) gives some some point (a, b)
Compress
then f(cx) gives you (a/c, b)
f(x+d) gives you (a-d, b)
apply that to the points
How would you write the equation
what equation
Like how would I find the x points
okay so you have the point
(-2, 0)
transform it
Yes
How?
still temple
then f(cx) gives you (a/c, b)
.
How do I transform the x points
Can you explain in simpler terms
I don’t get what you’re telling me
f(x) gives (10, 10)
f(2x) gives (5, 10)
How
still temple
then f(cx) gives you (a/c, b)
.
The equation is 2X+4
okay
2(-2)+4??
what
okay
still temple
then f(cx) gives you (a/c, b)
do you understand this?
No
what part dont you understand
What it means
if you have a function
f(x)
and transform it by some constant c
such that
f(cx) is ur new function
then a point (a,b) will turn into a point (a/c, b)
Yeah I get that, but how do they get the x values transformed
bro
still temple
then a point (a,b) will turn into a point (a/c, b)
.
(a,b) turns into (a/c, b)
c is the constant you transformed by
Can you show me how to get the first x point
okay
lets say
f(x) gives you some point (10, 10)
then f(2x) gives you (10/2, 10)
do you get that ?
10 divided by 2?
yes
still temple
then f(cx) gives you (a/c, b)
.
You divide the x values?
yes
If the original point is (-2,0) then what the transformed point of x for 2X+4
lets say you have f(x)
which gives a point (100, 100)
what does f(10x) give you
what point ?
10
give the whole point
10,100
okay
what about f(x+1)
still temple
f(x+d) gives you (a-d, b)
use this
Idk
f(x+d) gives you (a-d, b)
you have (100, 100)
what does f(x+1) give you
Not sure
-1?
still temple
f(x+d) gives you (a-d, b)
do you know what this means
d is just some constant
so f(x+d) means like f(x + SOME NUMBER)
still temple
what does f(x+1) give you
what is d in the case of f(x+1) ?
Just some number
whats the number specifically
f(x+1)
what is d
100
f(x+d)
you have
f(x+1)
what is d
1
okay
f(x) gives you (a, b)
f(x+d) gives you (a-d, b)
if f(x) gives you (100, 100)
what does f(x+1) give you
101
no, your answer is meant to be a point
and either way its a-d not a+d
okay I'm going to sleep maybe someone else can help
Okay
Can someone else help me please
still temple
.close
raw coral
How do u do this?
marsh citrusBOT
How do u do this?
static quarry
what needs to be true in general for a function to be continuous at a point?
raw coral
left and right sides must equal each other
but there are 2 variables
so doing that wont work
static quarry
no
only x is a variable
a and b are constants that you have to determine
there may be more than one pair of (a,b) that work
which is why they say "find all numbers a and b.."
raw coral
how would $a/x-1$ be continious on 1?
elfin berryBOT
Dracu
static quarry
there's exactly one value of a that will give a/(x-1) a finite limit as x->1
raw coral
woundn't that be 0?
otherwise it's just +-infinity
@static quarry
bruh
wtf
.close
white brook
This is my problem
marsh citrusBOT
This is my problem
white brook
This is what I've tried
In my head it seems correct, so I'm not entirely sure what it's flagging, whether it's a calculation issue, or a syntax issue.
flat raft
you wrote x in there when the variable is clearly t
white brook
Yeah I saw that
I started a new problem but am having a similar issue (I think)
.close
sterile mango
marsh citrusBOT
sterile mango
hi this question is about shoes so the only thing i dont understand here is 1/7 of the GST inclusive price
marsh citrusBOT
@sterile mango Has your question been resolved?
limber star
hi this question is about shoes so the only thing i dont understand here is 1/7 ...
discount of 1/7 of the GST inclusive price means that the price of the shoes including GST is discounted by 1/7
sterile mango
so its just 158.50 x 1/7?
fervent turtle
How would I find inverse tan of -7/5 in degrees
marsh citrusBOT
How would I find inverse tan of -7/5 in degrees
fervent turtle
(Without calculator)
edgy jungle
How would I find inverse tan of -7/5 in degrees
Use right triangle and quadrants
marsh citrusBOT
fervent turtle
.reopen
marsh citrusBOT
✅
fervent turtle
How would I determine if it is Q2 or 4 based on a given tan or will it not matter
stoic saddle
arctan only gives outputs between -90° and +90°, so only Q4 and Q1
marsh citrusBOT
@fervent turtle Has your question been resolved?
hollow pelican
Work on the right. What did I do wrong?
marsh citrusBOT
Work on the right. What did I do wrong?
@hollow pelican Has your question been resolved?