can someone explain why part b is k=1/10?
#t formulae: part b
lilac wing
green kiteBOT
can someone explain why part b is k=1/10?
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lilac wing
ik chatgpt is sometimes inaccurate but i put it in there and it also gives 0.1 in accordance with the markscheme, i dont understand can someone explain please
is it just asking for what value (k) multiplied by y gives you the second graph
astral swan
is it just asking for what value (k) multiplied by y gives you the second graph
Remember what happens when you transform a function y = f(x) to y = kf(x).
lilac wing
Remember what happens when you transform a function y = f(x) to y = kf(x).
i get that, i jus wanna know like
what does it ask for
lilac wing
is it just asking for what value (k) multiplied by y gives you the second graph
is it just this?
astral swan
is it just this?
Yes. You just need to find the constant that you can multiply your model graph by to get something close to the experimental graph.
lilac wing
gotcha
also, part bii
im not sure, what does it mean?
i have no idea ngl, even looking at the answer how to answer that
astral swan
Well, I'd say because the experimental intensity isn't strictly periodic like our model.
astral swan
The location of peaks kinda coincides, but the intensities are a bit different each time.
lilac wing
gotcha
okay that kinda makes sense
i mean it does make sense but idk if il be able to think of that in an exam lol
well what abt this?
i think if its a peak then it should be
dy/dx = 0
so i get
(3t^2 - 8t - 5)(t^2 +2t-1) = 0
after getting rid of the extra stuff
lilac wing
(3t^2 - 8t - 5)(t^2 +2t-1) = 0
how do i know which to take here?
astral swan
i think if its a peak then it should be
Well, you have high and low peaks. So, just count which experimental peak is the highest, then find the corresponding model peak.
lilac wing
Well, you have high and low peaks. So, just count which experimental peak is the...
the 4th maximum point
where d^2y/dx^2 <0
when we say intense peak
are we talking about every really big spike
that goes up
astral swan
Yes.
astral swan
or are we talking about the ones on the bottom too
No. The peaks are the ones pointing up.
lilac wing
(3t^2 - 8t - 5)(t^2 +2t-1) = 0
this is what im on rn
idk what to do next
cuz i get 4 t values there
but idk which peaks the values pertain to
cuz there are more than 4 peaks and also its infinite ig so idk whats going on there
lilac wing
(3t^2 - 8t - 5)(t^2 +2t-1) = 0
from this
astral swan
cuz i get 4 t values there
Oh, simple: just find the value at each extremum point.
One of them will be the largest, that's what you pick.
lilac wing
okie dokie il find the largest one sec
3.189254788
so from there?
if im not mistaken it looks like this one here
astral swan
No, wait.
lilac wing
First of all, what is this value?
dont have it exactly, cz i used my calculator
astral swan
Also, the most intense experimental peak is the fourth of the highest ones.
astral swan
dont have it exactly, cz i used my calculator
Don't use it.
lilac wing
3.189254788
so tan x/10 = this
kay lemme find it hold on
4/3 +root31/3
tan x/10 = 4/3 +root31/3
x = 10arctan (...)
but that gives
700.. something
yea im so confused
lilac wing
First of all, what is this value?
largest value of t
astral swan
No, hold on. Something isn't right. Let me try.
lilac wing
for dy/dx = 0
okie dokie
this is the answer if you want it
i just dont get it at all
astral swan
In terms of t, the roots of the derivative are t = -1 ± √(2) and t = (4 ± √(31))/3. So, there are four series of extremum points:
x = 10arctan(-1 - √(2)) + 10πm
x = 10arctan(-1 + √(2)) + 10πn
x = 10arctan((4 - √(31))/3) + 10πk
x = 10arctan((4 + √(31))/3) + 10πl
Here m,n, k, l ∈ ℤ.
astral swan
So, now we find out which one corresponds to the maximum value.
astral swan
Extremum points.
astral swan
the max points ofc but which is which?
Minimum, too.
astral swan
how do i find which is which
We find the values of the function at these points.
lilac wing
gotcha
lilac wing
In terms of t, the roots of the derivative are t = -1 ± √(2) and t = (4 ± √(31))...
can i just write n instead of mnkl when im writing this down
since its just an integer anyway
astral swan
We have:
f(x) = (1/2)(sin(x/5) + cos(x/5)) + 2sin(x/5)cos(x/5) + 2
Here you need to remember two identities:
sin(2arctan(x)) = 2x/(1 + x^2)
cos(2arctan(x)) = (1 - x^2)/(1 + x^2)
astral swan
can i just write n instead of mnkl when im writing this down
Those series of points are independent of each other. You need to use different letters.
lilac wing
Those series of points are independent of each other. You need to use different ...
okie dokie
astral swan
You can use n with an index if you want, so n1, n2, n3, n4.
lilac wing
We have:
f(x) = (1/2)(sin(x/5) + cos(x/5)) + 2sin(x/5)cos(x/5) + 2
Here you need...
woah whered this sin(2arctan) come from
never seen that before
is that t formulae
oh it is
astral swan
woah whered this sin(2arctan) come from
Oh, it's a common identity. Often used in integrals, too.
astral swan
You'll need it for evaluationg the function at these points.
lilac wing
so i didnt realise
okay so
what i understand rn
is the smallest positive value of dy/dx gives the closest peak to the y axis
so that smalles tvalue of dy/dx (approx 0.414)
is this first peak
astral swan
Wait, what? Why?
The values of the derivative show how fast the function is growing.
Also, we're looking at the model graph now.
lilac wing
Wait, what? Why?
The values of the derivative show how fast the function is grow...
since we r using tan u add npi + arctan 0.414
astral swan
I'm not following.
lilac wing
The model graph is the first one.
why are we lookin at that one
question part c asks for 2nd graph
astral swan
why are we lookin at that one
Because we have its equation.
lilac wing
the one usedi n the pulsar thing
astral swan
Have you found the values of f(x) at the extremum points?
lilac wing
didnt need em
lilac wing
im tryna figure it out rn
but the negative stationary points
the negative ones we dont use at all, since they wouldnt be the ones we r looking for
we only care abt the positive ones rn so it narrows down to 2
astral swan
There are no negative values of this function.
lilac wing
(3t^2 - 8t - 5)(t^2 +2t-1) = 0
i get 2 negatives and 2 positives
astral swan
Those are just the values of t.
Each of them corresponds to a series of critical points.
lilac wing
hm
astral swan
In terms of t, the roots of the derivative are t = -1 ± √(2) and t = (4 ± √(31))...
We just established this here.
Now you should evaluate f(x) at each series, which allows you to determine which one corresponds to the tallest peaks.
lilac wing
none of them would correspond to the peak near 98 tho
astral swan
Why?
lilac wing
cz when i sub them in i get like
-0.4
0.07
0.13
0.33
0.07 is the smallest so its the first peak, ie when t= 0.414
negative value lies in the other quadrant so we dont need it
astral swan
Don't round.
lilac wing
Don't round.
i have the actual value on my calc, jus takes ages to type em all out
astral swan
Again, you don't need a calculator here.
Well, apart from the very last step where you round to the millisecond.
lilac wing
i cant evaluate 10arctan(-1+root2)+10kpi in my head 😔
astral swan
It's too early to round that. You still haven't found the series corresponding to the tallest peaks.
lilac wing
idk what u mean series
astral swan
In terms of t, the roots of the derivative are t = -1 ± √(2) and t = (4 ± √(31))...
We have four series of critical points.
lilac wing
ya ima b real this question is weird
il come back to it later
ty for helping
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