can i send u link?
#help-28
unique valve
cunning vessel
of what?
unique valve
wait
https://docs.google.com/spreadsheets/d/1yTc7CI5EuaX88qn-Dz8-U14TqkXKboYMJrFiurgYLVU/edit?usp=sharing
here
cunning vessel
oh there is raw data
unique valve
i got the mean median mode in every district
cunning vessel
I guess first question asks you to actually calculate the ratio of eyecolors and compare it with summary?
I dont know how far you can use your google sheet command
unique valve
So, the ratio of eye colors is Brown : Hazel : Green : Blue = 2 : 1 : 1 : 1
full forumBOT
@unique valve Has your question been resolved?
cold trail
please
real musk
Can someone help me with this question, how do I continue it to find the global max and min?
real musk
This is the worked solution. How did they work out the end interval????
queen verge
Well, in your question it clearly states that x should be in between -1 and 7, the y values 1 and 57 are calculated by plugging in these x-values in your given function
real musk
Well, in your question it clearly states that x should be in between -1 and 7, t...
Ooh okay, so then whenever the question gives an interval, I’d always have to find the end point of interval? And if the question didn’t give any interval, I just use second derivative??????
onyx glen
This is the worked solution. How did they work out the end interval????
well they just evaluated the function at each endpoint
"find the endpoint of interval" is bad phrasing imo
queen verge
Well, in your question it clearly states that x should be in between -1 and 7, t...
No you can also calculate the second derivative here, they just calculated here so you don't find a maximum that is beyond your interval of itnerest
onyx glen
what they did is that they worked out the value of y (y itself, not any derivative of it) at the endpoints
the reason for this is that the maximum over your interval could occur at either endpoint. so you have to account for them
real musk
No you can also calculate the second derivative here, they just calculated here ...
Is this a more accurate way of doing it? By finding the end point
queen verge
the thing is, if you just would do the ordinary thing: find first derivative, find potential extrema and then check with second derivative which is max or min
then you might find a maximum outside the interval your questions asked you to look in
so that is the only reason why they also evaluated the end points
full forumBOT
full forumBOT
Available help channel!
sharp harbor
full forumBOT
nocturne shard
which one
nocturne shard
$\lim_{x \to 0} \frac{x-\tan x}{1-\cos x}$
glossy valveBOT
TargetVN
nocturne shard
Recall that $1-\cos x \sim \frac{x^2}{2}$
sharp harbor
?
glossy valveBOT
TargetVN
nocturne shard
$\tan x \sim x$
glossy valveBOT
TargetVN
sharp harbor
you don't study about this?
no
nocturne shard
i can still solve this with only the known limits you have learned
no fancy stuff required
sharp harbor
i can still solve this with only the known limits you have learned
i learned 3 methodes
nocturne shard
specify
sharp harbor
limits and derivability and change of variables
hallow vault
trigonometry
sharp harbor
trigonometry
yes
sharp harbor
yes
nocturne shard
nice
lets use them
$\lim_{x \to 0} \frac{x-\tan x}{1-\cos x}=\lim_{x \to 0} \frac{x-\frac{\sin x}{\cos x}}{1-\cos x}=\lim_{x \to 0} \frac{x\cos x-\sin x}{\cos x\left( 1-\cos x \right)}$
glossy valveBOT
TargetVN
sharp harbor
yeah
nocturne shard
ok
$\lim_{x \to 0} \frac{x\cos x-\sin x}{\cos x\left( 1-\cos x \right)}=\lim_{x \to 0} \frac{x\left( \cos x-\frac{\sin x}{x}\right)}{\cos x\left( 1-\cos x \right)}$
glossy valveBOT
TargetVN
nocturne shard
now i rewrite sin x = x * (sin x) / x to factor x out
it also reveals where to use the standard limit
sharp harbor
now i rewrite sin x = x * (sin x) / x to factor x out
yes i see it
nocturne shard
so that term is just 1
$\lim_{x \to 0} \frac{x\left( \cos x-\frac{\sin x}{x}\right)}{\cos x\left( 1-\cos x \right)}=\lim_{x \to 0} \frac{x\left( \cos x-1\right)}{\cos x\left( 1-\cos x \right)}$
glossy valveBOT
TargetVN
nocturne shard
can you continue from here?
sharp harbor
can you continue from here?
no i cant
sharp harbor
Yeah thats just 1 question
yes
sharp harbor
im having dinner now
Enjoy your dinner and thank u agian
hallow vault
how many questions are left
sharp harbor
how many questions are left
2
hallow vault
ok
uh
in ther interval, does it include pi/2
bc the bracket system is different
sharp harbor
in ther interval, does it include pi/2
no its not in
sharp harbor
0<=x <pi/2 ?
exactly
sharp harbor
ok
hallow vault
ok so what is the nature of f(x) = tan(x) - x in that same interval provided
you can find it using derivative test
hallow vault
how do you get minus sign
check your calculation,
define another function, g(x) = tan(x) - (x+x^3/3)
sharp harbor
how do you get minus sign
f'(x ) = 1 -(1+tg^2(x))
sharp harbor
derivative of tanx -x
no of x-tanx
nocturne shard
i mean you can do either of them
hallow vault
tan(x) >= x , you gotta subtract both sides by x
mb
do the same thing with g(x) and confirm its nature
sharp harbor
look at this
hallow vault
f'(x ) = 1 -(1+tg^2(x))
multiply by -1 both sides and the inequality flips
now do the same with g(x)
sharp harbor
I think you found the answer
maybe
sharp harbor
yeaaah
hallow vault
okay you got it, next question
sharp harbor
b) deduce that
hallow vault
2 sin(x) + tan(x) >= 3x
sharp harbor
2 sin(x) + tan(x) >= 3x
yes
sharp harbor
2 sin(x) + tan(x) >= 3x
my mind stoped here
nocturne shard
do the same thing as the previous problem: consider f(x) = 2sin(x) + tan(x) - 3x
sharp harbor
ok
nocturne shard
that is one method, if you're done with that, i'll show you another method
sharp harbor
ok
hallow vault
you can try different value from given interval and come to a conclusion
for small values, you can approximate cos(x) as 1 and sec^2(x) as 1, the inequality holds
for large values sec^2(x) becomes large so it still holds
sharp harbor
i dont understand
hallow vault
you are given a interval, 0<= x < pi/2, for which you have to prove 2(sinx) + tanx >= 3x
after taking the derivative test, you verify your answer
bc you have to know if the inequality holds or not
nocturne shard
just stick to the traditional method
hallow vault
i mean it's just a verification
sharp harbor
what is next and i have to go to shcool right now 😦
can u plese let this room opned to finish our talks
see you soon 👋
nocturne shard
can u plese let this room opned to finish our talks
it'll be closed for timeout
lol
just in case i'll also give a 2nd method
sharp harbor
ok
full forumBOT
@sharp harbor Has your question been resolved?
dense yoke
full forumBOT
dense yoke
is this correct
full forumBOT
@dense yoke Has your question been resolved?
dense yoke
<@&286206848099549185>
full forumBOT
@dense yoke Has your question been resolved?
full forumBOT
@dense yoke Has your question been resolved?
placid vale
I have to find angle KIH
full forumBOT
I have to find angle KIH
placid vale
this is a regular nonagon and AIJ = 45 degrees
oak ruin
steel solar
I have to find angle KIH
Is that a circumcircle? If so, for which point?
placid vale
I,J,G
full forumBOT
@placid vale Has your question been resolved?
placid vale
<@&286206848099549185>
placid vale
<@&286206848099549185>
full forumBOT
@placid vale Has your question been resolved?
placid vale
<@&286206848099549185>
limpid junco
i just want to know the solution
You can draw it in geogebra and measure
limpid junco
I have to find angle KIH
You should at least be able to tell me all the green angles
(Their values)
placid vale
i found the angles but you can't solve it by just having the angles
do you know how to solve it?
limpid junco
do you know how to solve it?
I see the sol
blissful wagon
You should at least be able to tell me all the green angles
interesting
placid vale
@limpid junco
limpid junco
what answer did you get
Idk the answer but i see the sol
limpid junco
?
Correct me if im wrong but this angle should be 65
Angle on the same arc length
placid vale
and whats the answer?
full forumBOT
full forumBOT
Available help channel!
shut umbra
How do I find the increasing and decreasing sections of the following function
high fjord
do you know how to find inflection points
shut umbra
do you know how to find inflection points
No
Is there a different way to find the sections besides inflection points?
high fjord
Is there a different way to find the sections besides inflection points?
well
no
find the second derivative
fathom saddle
It's a pretty simple transformation on log2. If you can graph it, you've got it
high fjord
It's a pretty simple transformation on log2. If you can graph it, you've got it
unless you graph it ofc
fathom saddle
Not like you necessarily need to graph it, I mean
shut umbra
It's a pretty simple transformation on log2. If you can graph it, you've got it
Oh okay
high fjord
i misread the question
nocturne shard
wait, arent we finding when its increasing and decreasing
shut umbra
yes
high fjord
you can just find when the slope is 0
high fjord
wait, arent we finding when its increasing and decreasing
my bad my bad
nocturne shard
It's a pretty simple transformation on log2. If you can graph it, you've got it
though, i think this is easier
log2(x) is strictly increasing in (0,+infinity)
so you do graph transformation to get f(x)
shut umbra
Okay so I know how to graph this
since the domain is (-inf, 3) i just chose numbers within that range
let me graph 5 points and you guys can help me figure out whats next
@nocturne shard
nocturne shard
what i would do is first graph log_2(x)
then graph 1/2log_2(x)
from that, graph 1/2log_2(-x)
then 1/2log_2(3-x)
then 1/2log_2(3-x)-1
yeah this is too much, should have gone with the f'(x)=0 route
shut umbra
well
i didnt want to do that because
i am studying for a test
and we wont have any graphing calculators
full forumBOT
@shut umbra Has your question been resolved?
verbal yoke
To differentiate a logarithmic function, the basic rule is: the derivative of log base "a" of x is equal to 1/(x * ln(a))
Use this to get f'(x) and set it to zero
you get your stationery points
fathom saddle
You may not know about differentiation if you aren't a calculus student. If you are, this does become nicer
verbal yoke
How do I find the increasing and decreasing sections of the following function
However if waht you drew is true, i would assume it is decreasing for x<3 and increasing never
verbal yoke
You may not know about differentiation if you aren't a calculus student. If you ...
I guess you would just have to plot and solve by inspection
shut umbra
You may not know about differentiation if you aren't a calculus student. If you ...
I dont
Alg 2
shut umbra
I guess you would just have to plot and solve by inspection
How would this exactly work?
shut umbra
yes
verbal yoke
just know the asymptotes and general shape
then you can determine where it is decreasing/increasing
shut umbra
then you can determine where it is decreasing/increasing
yeaa but idrk how it works
how can I determine where its doing that
and come up for this case x<3
verbal yoke
and come up for this case x<3
Do u understand how i came to this conclusion in this question?
shut umbra
Do u understand how i came to this conclusion in this question?
Not quite 😦
verbal yoke
Ok, so if a function is decreasing, then as x increases, y will be decreasing
are you following?
shut umbra
yes
shut umbra
i applied what you said and it looks like its true
verbal yoke
In your drawing here, can use see that as we go along the x axis in the positive direction, y decreases?
shut umbra
yes
verbal yoke
DO you understand wher the infintiy comes from?
shut umbra
yes
verbal yoke
So what don't you get?
shut umbra
how did you know it was right at x<3
full forumBOT
verbal yoke
You have noted the asymptote at x=3
And we have determined that y is decreasing as x increases all the way along the curve, right?
shut umbra
ah okay
full forumBOT
Available help channel!
verbal yoke
therefore it is a decreasing function up until x=3
full forumBOT
therefore it is a decreasing function up until x=3
unique valve
https://docs.google.com/spreadsheets/d/191GeuXNGGZk9RrSK-KH7Ii32a2ARw9f0h0S4sGQp_Vk/edit?usp=sharing
slate violet
why do you want us to do your homework for you
slate violet
ah okay I see
slate violet
that's not correct for this task
you want to find the summary statistics for all the dead people combined, and then compare it with the other spreadsheet about the living men
summary statistics would be the mean, median, mode, and range
you can get this information from a table too! there are YT videos on this
(and then the next part asks you to group the data by eye colour and by SEC, so make groups 1 to whatever according to those categories)
unique valve
ok this is the
dead people combined
mode = 1
median = 19.5
mean = 20.34868421
full forumBOT
@unique valve Has your question been resolved?
unique valve
<@&286206848099549185> how do i compare it with the living men
should i just put the summary statistics of the living men to the side of it?
unique valve
(and then the next part asks you to group the data by eye colour and by SEC, so ...
for this i should get the summary statistics by eye colour and by SEC in every district
is this correct?
slate violet
for this i should get the summary statistics by eye colour and by SEC in every d...
no
in fact ignore the districts completely
you want all the data that has green eyes, and that's 1 group
you also want all the data that has brown eyes, and that's another group
slate violet
this is it right?
no you need to sort the data in the dead men sheet yourself
also you should really ask your teacher if you need help
this is a really open-ended task
unique valve
i will ask
@slate violet one last thing
could u check the previous milestone if i did it right or wrong
slate violet
could u check the previous milestone if i did it right or wrong
I have no idea
this is above my pay grade
like your task will take a couple of hours to complete at the very least
slate violet
the purpose of the investigation is not to be right or wrong
it's for you to show your work and your thought process
if I gave you that, I would be doing all your work for you
there's a pyramid of thinking skills
just doing a procedure is understanding and applying
this is an investigation!
you're going to need to analyse and evaluate the data (and your process of sorting and making sense of the data)
notice that "how helpful" and "the best way" are subjective questions
of course you need evidence from the data to justify and support your opinion
but there's many many different interpretations which could be valid
from the same dataset
so even if you mess up, if you can show your steps and your thinking
you'll get a lot of marks
the final answer is just one small step of the entire journey
unique valve
you'll get a lot of marks
really?
unique valve
if i did it all wrong and just show my steps, i'll get graded fairly?
ok i will do that
thank u south
slate violet
ok i will do that
mhm no worries
I have some experience in education at uni, took a course on that
I'm a humanities major and not an education major
there's actually been a huge wave over the past few decades to reform maths eduation, to make it less about memorising formulas and more about understanding concepts
and probably also how they could be used in the real world
a lot of people, statisticians and researchers sure, but also managers, say of a restaurant or a sports team
they need data to run their business and they have to analyse it for a living
small business owners in particular since there's no one you can delegate your job to, no one that has enough time since they're doing the essential work and you're managing the business
forget algebra or trigonometry or calculus
statistics is the one branch that is the most useful outside of school
slate violet
also mean, median, range, and mode is baby statistics
unique valve
what
slate violet
really they're asking you to use your brain on this data
unique valve
there's more harder than the baby statistics?
unique valve
im screwed
slate violet
really they're asking you to use your brain on this data
and not necessarily to know a whole bunch of statistical techniques
if you're not using functions and calculus on statistical distributions, then you're doing baby stats (continuous random variables is what I mean)
(of course there's a lot of discrete distribtions such as binomial and Poisson)
unique valve
do u have any advices so that i can understand math easily?
slate violet
do u have any advices so that i can understand math easily?
uhhh one sec
unique valve
im taking software development
unique valve
you won't be going up too far up in maths then
thank god
slate violet
there's a million takes on this
a lot of people will say to practice, to drill certain skills so that you get the hang of them through mental repetition
but it's also true that a lot of maths comes from understanding, for instance, by drawing a nice picture
there are pictures that show that (a + b)(a + b) = a^2 + 2ab + b^2 for example
really it's subjective
it depends what you need maths for, so Khan Academy has a lot of stuff
unique valve
i will do that
thank u south i will watch some tutorial how to do this
and understand them first
slate violet
no worries!
slate violet
and understand them first
.close
wind horizon
full forumBOT
umbral dome
umbral quail
full forumBOT
umbral quail
my prof told us to turn this into a triangular matrix and then try and solve it
but im so lost
honest hill
did you at least try to turn it into a triangular matrix?
sharp flame
Maybe you should write the determinant out explicitly for a small value like n = 3 and see how you'd reduce it to something doable
And then apply the same concept on this generalised version
umbral quail
did you at least try to turn it into a triangular matrix?
yea but like i literally cant figure out how to
honest hill
okay, like neon said, try to start with a small value
$\begin{pmatrix}1&2&3\3&1&2\2&3&1\end{pmatrix}$
glossy valveBOT
Bonk
honest hill
turn this into a triangular matrix
umbral quail
1 2 3
0 -5 -7
0 0 -18/5
?
honest hill
,w rref {{1,2,3},{3,1,2},{2,3,1}}
glossy valveBOT
honest hill
hmm
umbral quail
oh like that
umbral quail
okk
honest hill
1 2 3
0 -5 -7
0 0 -18/5
?
where does your /5 come from?
umbral quail
but the thing is i cant just get rid of n-1 right?
honest hill
but the thing is i cant just get rid of n-1 right?
why not?
in the 3x3 example the n-1 was 2
and there you could get rid of it just fine
umbral quail
i thought you can only multiply a row but you cant add a number to it
like you can only do addition or multiplication
i mean no addition
honest hill
you can do R2-(n-1)*R1
full forumBOT
@umbral quail Has your question been resolved?
Available help channel!
Available help channel!
ionic ginkgo
full forumBOT
ionic ginkgo
how do we solve B?
onyx glen
onyx glen
no
note that after gluing, arc AB becomes the circumference of the base
find the radius from that
onyx glen
then, note that the radius of the original circle (ie OA) actually becomes the slant height
onyx glen
work things out on paper. i will not be able to validate answers if you just blurt out numbers
ionic ginkgo
work things out on paper. i will not be able to validate answers if you just blu...
alright
well hold on
how is OA
the slant height
if its part of the circumfrence
onyx glen
no
OA is one of the two radii of the original circle
it will be glued to OB
O will be the apex
onyx glen
you fold OA and OB onto each other...
ionic ginkgo
right
onyx glen
idk how else to tell you without making a physical model
(which i can't do at the moment)
ionic ginkgo
give m another
moment ot try to understan
the shape
it creates
tell you what
i think i will just go ask my teacher because i honestly have
no other clue how i could understandit
thank you!
.close
full forumBOT
full forumBOT
Available help channel!
shell axle
hello! when do i know i have to use the reverse chain rule? is it just whenever i see a function in a function
twin wolf
hello! when do i know i have to use the reverse chain rule? is it just whenever ...
its usually pretty easy to tell if u are able to use it or not do u have any other examples
shell axle
its usually pretty easy to tell if u are able to use it or not do u have any oth...
is there like a general rule for when to do it?
liks (2x+1)^anything
or sin(2x)
or e^3x
twin wolf
liks (2x+1)^anything
yeah simple integrals like that
tbh it just comes down to experience
if u do a lot of them u will be able to tell
or like usually
if theres the derivative of some composite function in the integrand
it will work
like xsin(x^2)
shell axle
like xsin(x^2)
what do you do in this situation
full forumBOT
@shell axle Has your question been resolved?
fallen shell
full forumBOT
fallen shell
can someone check which one is correct im not sure
m1 andd m2 are just the two possible answers
sorry edit for m1
royal holly
fallen shell
oh wait i m1 should be correct right
i dont know why i didnt just try and input the numbers in
would this show the line is a tangent to the curve?
because they have the same gradient
full forumBOT
@fallen shell Has your question been resolved?
fringe silo
tah lewandowksy
jagged raptor
Why does graphing triganomic functions graph things the way they do?
glossy valveBOT
gritty flax
I don't really get your question
gritty flax
Why isnt it linear
Periodicity
void nova
Why does graphing a trig function create the parabolas
It's not a parabola, btw
jagged raptor
It's not a parabola, btw
Ik, just the best i could think of rn
jagged raptor
Periodicity
Whats that mean?
void nova
Oh I see
tall tinsel
Китай номер 1
void nova
Whats that mean?
Can a linear function (hence, a straight line) be periodic?
jagged raptor
I dont know what that means
void nova
Well, then you'll have to wait when you will be taught trig graphs
gritty flax
f(x+T) = f(x) for a T>0
jagged raptor
Well, then you'll have to wait when you will be taught trig graphs
Thats in like 3 years for me
tall tinsel
Скоро весь мир захватит Китай
jagged raptor
.close
steel belfry
what am I supposed to do if 8 can't be squared?
This is converting general form of a circle to standard form
onyx glen
what do you mean by "cannot be squared"?
willow geode
radius would be √8 or 2√2
willow geode
what do you mean by "cannot be squared"?
he meant as in general term of circle
where constant left is r²
onyx glen
but 8 is positive, what's the issue
steel belfry
oh okay thanks☺
willow geode
but 8 is positive, what's the issue
idk, he got confused what would be the radius, i guess
golden agate
but why is 121 and 196 positive on rhs?
steel belfry
. close
full forumBOT
@steel belfry Has your question been resolved?
full forumBOT
full forumBOT
Available help channel!
sly trench
I dont understand the first line of the proof
full forumBOT
I dont understand the first line of the proof
sly trench
yes
limpid junco
Because as h approaches 0 , eventtually |h| will be less than delta. So for every |h| < epsilon you have that f(c + h) - f(c) >= 0
limpid junco
Well c + h is in range (c - delta , c + delta)
And we assumed f(c) is minimum in that range.
tranquil lantern
full forumBOT
tranquil lantern
i struggle a lot with
idenitfying what goes on the y axis and x axis
is the any way i cann tell or do i just guess it?
slate violet
idenitfying what goes on the y axis and x axis
x axis is the horizontal one
y-axis is the vertical one
tranquil lantern
ik
slate violet
so a good mnemonic is along the corridor then up the stairs
you go horizontally first for the x before you go vertically for the y
tranquil lantern
im not referring to that, im referring to labelling it
slate violet
oh
tranquil lantern
ik but im just asking how can i know what goes on the y axis and x axis as in the volume and time, what i shown was a worked sol for the first part but im asking how to be able to identify what goes where
slate violet
ik but im just asking how can i know what goes on the y axis and x axis as in th...
what do you mean by what goes on?
the time is (almost) always the x-axis because that's your independent variable
the dependent variable is the y-axis
tranquil lantern
okay ty
tranquil lantern
np!
here theres no time so im kinda stuck now
coz either one in my mind can be the independant variable (the one i change)
slate violet
ah, so the independent variable is the number of miles
the dependent one is the price
the price (C or y) depends on the number of miles (m or x)
tranquil lantern
the price (C or y) depends on the number of miles (m or x)
i see now part a gives it away with the equation
slate violet
yeah it's a skill
I mean when I say independent and dependent I'm borrowing terminology from science
graphs are a really important part of STEM in general, not just maths
wait I switched independent and dependent
SORRY
tranquil lantern
the price (C or y) depends on the number of miles (m or x)
but like what if i decide to go 10 miles wouldnt the price follow my decision? and my decision was travelling that far, hence itll be independent?
slate violet
ok fixed
tranquil lantern
wait I switched independent and dependent
its ok
slate violet
but like what if i decide to go 10 miles wouldnt the price follow my decision? a...
you can decide to go 10 miles directly
yep
so yeah you can't decide to have a fare of 10 pounds directly, that's the difference between the independent and dependent variables
the 10 pounds depends on something else, the distance
trim sun
wait I switched independent and dependent
You didn't
slate violet
You didn't
I fixed everything
tranquil lantern
I fixed everything
yeah 90% of the questions involve time
so its obvious itll be on the x axis
slate violet
yeah that's a common pattern
slate violet
like I thought of an example from physics
where you can change the length of a pendulum, a bit of string with some object hanging off
and then the time period gets affected
so like not 100% but for all your textbook questions here, sure
slate violet
(not involving time)
here you go
T = 2pi sqrt(L/g)
notice how T is now the subject of the equation, haha
slate violet
however im tryna find cases where it isnt
ah I misread
yeah what happens when the x-axis isn't time
yeah so q7 and similar qs are good practice
slate violet
just do a million examples to get the hang of the concept
I feel I can't say much more
slate violet
oooh that's a good one
yes it can be either way for this one
this is something where you have some data, where the two variables don't depend on each other at all
tranquil lantern
yes it can be either way for this one
how about Fahrenheit and Degree Celsius
Pounds and Dollars?
slate violet
they are correlated so there's a trend, which is not the same as being dependent
slate violet
how about Fahrenheit and Degree Celsius
Pounds and Dollars?
yeah, same thing
you can go either way
so 1 pound = 1.2 dollars but 1 dollar = 0.8 pounds approximately
slate violet
no worries!
rain mauve
for part c why would I need to use differentiation?
full forumBOT
for part c why would I need to use differentiation?
tranquil lantern
for part c why would I need to use differentiation?
that looks like an a level paper
😭
rain mauve
that looks like an a level paper
its an AS one actually lol
tranquil lantern
its an AS one actually lol
oh lol
can u show the markscheme?
sly trench
you understand how you do this?
rain mauve
you understand how you do this?
yea but i dont get why i need to use differentiation to solve the question
sly trench
do you know what the graph looks like?
rain mauve
yea lemme take a pic of mine
rain mauve
do you know what the graph looks like?
sly trench
alright
can you draw a line y=k such that the function will have 3 intersections with this line
rain mauve
that part... im not sure what it would look like
rain mauve
alright try the line y=0
ah thats basically the same horizontal line
rain mauve
how many intersections will that have?
3
sly trench
ah thats basically the same horizontal line
exactly
rain mauve
3 as well?
rain mauve
ah 1
sly trench
exactly
so they ask you to find the value for k such that it will always have 3 intersections with the function
so can you maybe try to imagine a certain area where a horizontale line will always give 3 solutions with the function?
rain mauve
between y<10 and y<-10
zenith dome
6root3
sly trench
between y<10 and y<-10
so you are going into the right direction
but we want it to be precise
if you look at the grap
you can see that 10,1 for example will also give 3 solutions
so how can we find the exact values for k?
rain mauve
so how can we find the exact values for k?
apply the discriminant?
rain mauve
what would the discrimant give us?
a value greater than 0
rain mauve
or what do we use that for ?
to see how many roots it has (if any)
sly trench
true but that is not really what we are looking for
do you agree with me that we want to find the x values for the minimum and maximum?
rain mauve
do you agree with me that we want to find the x values for the minimum and maxim...
..why is that?
rain mauve
yup
sly trench
a horizontale line will always give 3 solutions
in otherwords
min f(x) < k < max f(x)
you understand that?
rain mauve
yup
tranquil lantern
you understand that?
ty (i didnt know how to do it too)
sly trench
so we want to find the x values for the minumum and maximum so we can find what?
sly trench
ty (i didnt know how to do it too)
no worries that is why this discord exists 🙂
rain mauve
so we want to find the x values for the minumum and maximum so we can find what?
finding the boundaries where the line intersects the curve at three points
rain mauve
differentiation
okokk i get it
i thought differentiation was only used for stationary points and finding gradients of curves
tysm!
sly trench
noooo
tranquil lantern
is using the discriminant useless for part c?
sly trench
differentation is really important
yes
you can do without it
you can show your work if you want i can look at it
tranquil lantern
i get it now, the problem was trying to identify using differentitation
sly trench
how did u know to use differentiation?
Well honest answer
doing math for much much longer
tranquil lantern
what is you derivative?
+/- 6root3
(cant be equal to)
sly trench
if i was at your level probably notice that the line has to be between those two maximums , so i must find the maximum/minimum so use differentation
tranquil lantern
if i was at your level probably notice that the line has to be between those two...
do u think drawing a graph as soon as u saw what it was asking was the best idea?
i think part b was hinting to it tbh
sly trench
do u think drawing a graph as soon as u saw what it was asking was the best idea...
yes definitely
tranquil lantern
ty tho!
sly trench
+/- 6root3
(cant be equal to)
yes how would you write this down mathematically?
tranquil lantern
-6root3<k<6root3
full forumBOT
full forumBOT
Available help channel!
sly trench
full forumBOT
sly trench
So second order dericitional derivative?
I have no idea what that is
I never got that explained so would like some help understanding it
what is a directional derivative?
honest hill
have you not had directional derivates but theyre asking you to calculate a directional derivative?
sly trench
have you not had directional derivates but theyre asking you to calculate a dire...
yes
gotta love that
honest hill
$D_u f(x,y)=\lim_{h\to 0}\frac{f(x+hu_1,y+hu_2)-f(x,y)}{h}$
glossy valveBOT
Bonk
honest hill
is there nothing in your book about directional derivatives?
sly trench
well we didnt have it as a homework excersise i think only in our group exercises
honest hill
its actually not that complicated, but if you can watch a 5-10 min video about it, thatd still be easier
honest hill
also, $D_uf(x,y)=\nabla f\cdot u$
glossy valveBOT
Bonk
honest hill
sly trench
what in the vector calculus
honest hill
the vector is already normalised, so you can skip step 2 and 3
honest hill
oki
sly trench
i thought this was supposed to be easy ;-;
honest hill
sly trench
it probably is but just looks fucked up
honest hill
ye, that last one
sly trench
alright thanks bonk
honest hill
gl 😄
sly trench
if i sitll dont understand it ill just open one and u will be there alright!!!
alright thanks man
.close
full forumBOT
honest hill
im gonna go have dinner, but i have notifications on 🙂
full forumBOT
Available help channel!
vague moss
I need to use quadratic splines to approximate a curve passing through (0,0), (0.6, 0.04), (0.95,0.16), (1,1).
vague moss
Give me a while while I explain my working.
I set up quadratics:
$$a_1 x^2 + b_1 x + c_1 = 0: 0 \leq x \leq 0.6$$
$$a_2 (x-0.6)^2 + b_2 (x-0.6) + c_2 = 0.04: 0.6 \leq x \leq 0.95$$
$$a_3 (x-0.95)^2 + b_3 (x-0.95) + c_3 = 0.16: 0.95 \leq x \leq 1$$
glossy valveBOT
RadMeerkat62445
vague moss
Substituting x = 0 in the first one, $c_1 = 0$.
glossy valveBOT
RadMeerkat62445
vague moss
Substituting x = 0.6 in the second one, $c_2 = 0.04$.
glossy valveBOT
RadMeerkat62445
vague moss
Substituting x = 0.95 in the third one, $c_3 = 0.16$.
glossy valveBOT
RadMeerkat62445
vague moss
Right so far?
Now, ensuring continuity:
Substituting x=0.6 in equations 1 and 2,
$$a_1 (0.6)^2 + b_1 (0.6) + 0 = 0.04$$
$$c_2 = 0.04$$
Therefore, $b_1 = \frac{0.04-0.6^2 a_1}{(0.6)}$
glossy valveBOT
RadMeerkat62445
vague moss
Substituting x = 0.95 in equations 2 and 3,
$$a_2 (0.35)^2 + b_2 (0.35) + 0.04 = 0.16$$
$$c_3 = 0.16$$
Therefore $$b_2 = \frac{0.12-a_2 (0.35)^2}{0.35}$$
Amending first working message to replace RHS constant values with f(x).
glossy valveBOT
RadMeerkat62445
vague moss
Lastly, substituting x=1 in equation 3,
$$0.05^2 a_3 + 0.05 b_3 + 0.16 = 1$$
Therefore, $$b_3 = \frac{0.84-a_3 (0.05^2)}{(0.05)}$$
glossy valveBOT
RadMeerkat62445
vague moss
Differentiating all equations,
$$\frac{dy}{dx} = 2a_1 x + b_1$$
$$\frac{dy}{dx} = 2a_2 (x-0.6) + b_2$$
$$\frac{dy}{dx} = 2a_3 (x-0.95) + b_3$$
glossy valveBOT
RadMeerkat62445
vague moss
At x=0.6 for the differentiated equations 1 and 2,
$$1.2a_1 + b_1 = b_2$$
Similarly, $$0.7a_2 + b_2 = b_3$$
glossy valveBOT
RadMeerkat62445
vague moss
We have a system of nine unknowns initially.
c values were found constant and hence eliminated. We now have six unknowns.
We have equations:
$$b_1 = \frac{0.04-0.6^2 a_1}{(0.6)}$$
$$b_2 = \frac{0.12-a_2 (0.35)^2}{0.35}$$
$$b_3 = \frac{0.84-a_3 (0.05^2)}{(0.05)}$$
$$1.2a_1 + b_1 = b_2$$
$$0.7a_2 + b_2 = b_3$$
glossy valveBOT
RadMeerkat62445
vague moss
We shall require the curves to be increasing (convex).
Hence,
for $$0 \leq x \leq 0.6: x \geq \frac{-b_1}{2a_1}$$
for $$0.6 \leq x \leq 0.95: x \geq \frac{-b_2}{2a_2} + 0.6$$
for $$0.95 \leq x \leq 1: x \geq \frac{-b_3}{2a_3} + 0.95$$
glossy valveBOT
RadMeerkat62445
vague moss
Therefore, at boundaries and hence taking the more restrictive one:
for $$0 \leq x \leq 0.6: 0 \geq \frac{-b_1}{2a_1}$$
for $$0.6 \leq x \leq 0.95: 0 \geq \frac{-b_2}{2a_2}$$
for $$0.95 \leq x \leq 1: 0 \geq \frac{-b_3}{2a_3}$$
glossy valveBOT
RadMeerkat62445
full forumBOT
@vague moss Has your question been resolved?
full forumBOT
full forumBOT
Available help channel!
cursive grotto
full forumBOT
cursive grotto
Quite lost on this one
full forumBOT
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
@cursive grotto Has your question been resolved?
Available help channel!
dry light
can someone help me with the last part please
full forumBOT
can someone help me with the last part please
dry light
like what do I even do when the boat is moving in the north-east direction
it would have to do something with the velocity right?
is it where the i components and j components of the velocity are equal?
placid vale
is it where the i components and j components of the velocity are equal?
i think so
dry light
dry light
,rotate
glossy valveBOT
dry light
that’s what I got but not sure if it’s right
placid vale
so the position vector is constant?
dry light
no?
because of r = ut + 0.5a(t^2)
when t changes, r would also change so it wouldnt be a constant
full forumBOT
@dry light Has your question been resolved?
full forumBOT
@dry light Has your question been resolved?
dry light
.close
stark wraith
If you were to getting the equation of a surface spanned by a line when it rotates around the z-axis, how would I start this problem?
gritty rose
If you were to getting the equation of a surface spanned by a line when it rotat...
Just show an actual problem
stark wraith
mb
Problem 1. Consider the line (x,y,z) = (1+t, 2t, 2-t). Find the equation of the surface which is spanned as this line is rotated around the z-axis.
I was thinking it would look like a cylinder
so something like x^2 + y^2 = c for some constant c
full forumBOT
@stark wraith Has your question been resolved?
tulip marlin
I don't think it would unless the line is parallel to the z axis
it probably would be more helpful to write the equation of the line in vector form to visualize it
tulip marlin
so something like x^2 + y^2 = c for some constant c
your idea here has some merit.
but wrong shape
tulip marlin
it probably would be more helpful to write the equation of the line in vector fo...
After you've done this and guessed which shape it should be
It would probably be a good to think about this radially I think
full forumBOT
@stark wraith Has your question been resolved?
stark wraith
in vector form it would be <1+t, 2t, 2-t>
is that what you meant?
tulip marlin
l = (1, 0, 2) + (1, 2, -1)t
stark wraith
ah okay
so any point on the surface of the shape would be a point that had previously lay on the line
its hard to visualize what the shape would be
i can only be sure that the cross section is a circle in the xy plane
also what do you mean radially?
oh nvm i have been thinking wrong
when it said rotate i thought it meant like twist the line which sounds weird
but rotating would be a cone
a hyperboloid of one sheet right?
so
x^2 + y^2 - z^2 = c
stark wraith
what does radial mean... like x = rcostheta?
tulip marlin
since you are rotating the line around the z axis
you are interested in the perpendicular distance to the z axis
for each point on the line
full forumBOT
stark wraith
.reopen
full forumBOT
✅
stark wraith
the distance from (0,0,z) to points (x,y,z) which are on the line
sqrt(x^2+y^2+(z-z)^2)
dumb question but i'm assuming the distance between two points is the same formula in 2d or 3d
tulip marlin
well its not
really
its 3d pythag
its the root of the sum of the square of the distances between each coordinate
sqrt((x1 - x2)^2 + (y1 - y2)^2 + (z1 - z2)^2)
tulip marlin
i missed that
its the same as 2d pythag only because its the same z coordinate, yes
tulip marlin
the distance from (0,0,z) to points (x,y,z) which are on the line
sqrt(x^2+y^2+...
yes
stark wraith
`sqrt((x1 - x2)^2 + (y1 - y2)^2 + (z1 - z2)^2)`
i always thought it was the second coord minus the first
wait so it's solved
tulip marlin
(x1, y1, z1) (x2, y2, z2)
what you wrote was correct
it doesnt matter which way round it is because the difference is squared
stark wraith
so the form is x^2 + y^2 - z^2 = c
i haven't learned much about quadric surface but i am guessing the c is the radius
so i can plug in the x,y,z parameters of the line and the formula we got for the radius
tulip marlin
i dont think thats the form
no ok that is the form maybe
i think its more helpful to write it as
x^2 + y^2 = (z + k)^2
for this particular problem
then z + k is the radius for each z
if im not mistaken anyways
====================================
you might not need to know the form anyways
tulip marlin
the distance from (0,0,z) to points (x,y,z) which are on the line
sqrt(x^2+y^2+...
simply take this
and you know r^2 = x^2 + y^2
right
stark wraith
yeah