unions are basically you plopping all the elements together
#help-27
pure flower
pure flower
$n(A \cup B) = n(A) + n(b) -n(A \cap B)$
woven radishBOT
Percy
pure flower
$n(A \cap B)=3$
woven radishBOT
Percy
restive river
I've been doing math for over 10 hours a day for the past week and I'm about losing my ability to remember anything lol
pure flower
you know $n(A)$ and $n(B)$
woven radishBOT
Percy
pure flower
I've been doing math for over 10 hours a day for the past week and I'm about los...
us indeed
pure flower
$n(A \cup B) = n(A) + n(b) -n(A \cap B)$
this is not like fancy or anything btw
restive river
Was I taught it? Probably! Do I remember it? Not at all lol
pure flower
you just plop the sets together and then remove the ones in common because they'd be double counted
pure flower
yeah yeah it's notation
restive river
Okok, I get the first three, now.. how does $n(A \cap B)=22$
woven radishBOT
ari
restive river
And how is the upside down and up cap related
restive river
Okok, I get the first three, now.. how does $n(A \cap B)=22$
Ok so it's 12 + 13 - 3
So n(A n B) is how many cards overlap
pure flower
Okok, I get the first three, now.. how does $n(A \cap B)=22$
It doesn’t no
That’s union
restive river
It doesn’t no
My bad, I meant the cap the other way lol
pure flower
Union
Oh
NO
not the number
just A U B is the Union of a and b
It’s a set
Not a number
devout snowBOT
restive river
.reopen
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✅
restive river
I did 9! to get the 2nd answer, I don't know where they are getting 90,720
<@&286206848099549185>
bright pumice
restive river
I did 9! to get the 2nd answer, I don't know where they are getting 90,720
that should be right
9!
ohhh wait
i misread the right answer
It's okay, I'm sure we're all losing it a bit here
It's order doesn't matter tho right
i got 40,320
for 8!
Aaaaa
So what calculation gets 90720
uhh
362880÷4 =90720
thats divided by 4
why 4
if there's 2 E
that makes 8 selective letters
8! Should be the right answer I don't know how they are getting 90720
restive river
so like for a similar question, that also has 2 of the same letters in the word
cuz ur rearranging the letters
there's stil lgonna be 9 letters
So their wrong
I'm right
1 e doesnt disappear
yeah
9! should be right
thats canvas right
talk to your teacher maybe
wait
pause
Paused
8! Makes sense
it does
cuz no matter which e goes where
but it'd still be 9 letters
9!/4
n!/(n-r)!, where n is the total number of objects and r is the number of objects being arranged.
THERES 2 L'S
Sorry I can't read
ohhh
haha
Okok
So
9! ÷4!
It wouldn't be 4 factorial
no
it wouldnt
Why do they put a factorial in this example I'm looking at
One second
I'll send
ok
i have an example that might be easier
so 9 letters
2 repeat twice
9!/2!*2!
idk why did it in 2 factorials though
for the denominator
Yeah
After they end up using 2
but only 7 of the 9 change for each ig
so 9-2
Not the factorial of 2
well
whats
2!
2*1
2
2!=2
Yeah
OH
True
4! Is like 24
But that wouldn't make sense
lmaoo
yeah
restive river
i have an example that might be easier
they did it in 2 here though
2!*2!
OH
OHHHHH
OHHHHHHHHHH
Ok
Makes sense
I get it now
yep
so 9!/4
Thank you!!
ofcourseee
glad we figured it out
.close
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green sierra
GTO Incorporated is considering an investment costing $210,720 that results in net cash flows of $30,000 annually for 10 years. (PV of $1, FV of $1, PVA of $1, and FVA of $1) (Use appropriate factor(s) from the tables provided.)
(a) What is the internal rate of return of this investment?
(b) The hurdle rate is 8.5%. Should the company invest in this project on the basis of internal rate of return?
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GTO Incorporated is considering an investment costing $210,720 that results in n...
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edgy frost
I know the answer is D. But if x could be equal to 1 and -1 then how would i solve it
devout snowBOT
I know the answer is D. But if x could be equal to 1 and -1 then how would i sol...
edgy frost
Because it's obvious that it can't be equal to +-1 or you are dividing by zero
So why did they say that
waxen merlin
just so what they're saying is mathematically correct
edgy frost
Oh ok
waxen merlin
cus expression wont have any sense otherwise like u said
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torn bobcat
- tan(x)=2; pi <= x <= 2pi find sec(x)
- the points (3.5,-1) and (5.5,-6) are on the graph of a trig function what is the function that models?
- The average depth of water at the end of a dock is 16 ft and it varies 4ft in both up and down directions, caused by the tidal. The tide goes from low to high every 8 hours. The depth of water is at the average depth at 3 AM with respect to time t. Let t=1 corresponding to 1 AM
4)graph y=2tan(x/2-pi/3)+1 and y=1/2csc(2x+pi)+2
devout snowBOT
1) tan(x)=2; pi <= x <= 2pi find sec(x)
2) the points (3.5,-1) and (5.5,-6) are ...
torn bobcat
im confused on how to do all of these
would be helpful if someone showed me the answers and then explained since i need to turn them in soon
<@&286206848099549185>
anyone?
@fervent swallow
<@&286206848099549185>
hollow socket
hey
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round saffron
Ugugucu
solar goblet
?
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robust cape
A radio tower is located 400 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is 30 degress
and that the angle of depression to the bottom of the tower is 24 degress
. How tall is the tower?
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A radio tower is located 400 feet from a building. From a window in the building...
opaque nova
try drawing a picture, might help
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inner mantle
hi
raven nacelle
needed help with this problem
What progress have you made?
long sundial
Are you aware of the Chinese remainder theorem?
inner mantle
What progress have you made?
i found x from the first eqn and substituted it in the 2nd so we get 7a+3 congruence 3(mod5) then from there I found a which is 5b+3 and now im confused what to do next
inner mantle
Are you aware of the Chinese remainder theorem?
yes
should i substitute the value of a in the first eqn ?
eternal tapir
i found x from the first eqn and substituted it in the 2nd so we get 7a+3 congru...
what are you getting for x
inner mantle
7a+3
eternal tapir
you can't just divide by 2
inner mantle
why
inner mantle
np
eternal tapir
anyway it's probably most simple to do this by writing out multiples until you get the lcm of 14,15 60
or directly apply crt
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winged wasp
What is the mistake in this solution
devout snowBOT
What is the mistake in this solution
winter patrol
,rotate
woven radishBOT
topaz beacon
winged wasp
Khan academy
winter patrol
pic pls
winter patrol
your result is equivalent to b from power law for log
option C isn't the same as what you got
winged wasp
option C isn't the same as what you got
I know i just randomly picked it to see answer
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daring blaze
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daring blaze
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patent grove
!15m
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patent grove
,rccw
woven radishBOT
daring blaze
oh sorry
daring blaze
the radius is 5?
daring blaze
is the center of first circle is (8, 14)?
patent grove
is the center of first circle is (8, 14)?
yeah
daring blaze
the radius is 4?
daring blaze
ohhhhh
daring blaze
center of circle number 2 is (4, 10)?
patent grove
center of circle number 2 is (4, 10)?
(4, 9.5)
daring blaze
ohhhh
patent grove
yes
patent grove
(12, 9**.5**)
daring blaze
ohhh okaay
patent grove
The radius is 4 bro…
when you count horizontally that's not the diameter
if you count vertically it's gonna be 9 units
coarse egret
Bro it’s 4
patent grove
no bro
coarse egret
when you count horizontally that's not the diameter
Why does it matter which way u count
Do u know what radius is
patent grove
Do u know what radius is
not this again
coarse egret
The easiest and most visible way to count for that circle is horizontally
When x = 0
Look at x = 0 and x = 8
That’s one full circle
For circle 2
Each of the circles are the same
patent grove
try drawing the diameter when r = 4
the center would not be equidistant from the circumference
it would 4 units and 5 units
if you draw the diameter vertically, you get 9 units
this would mean that the radius is at 4.5 units on the y axis
on plotting the diameter horizontally with y axis as 4.5 we get the diamater as 9 units as well
this would mean that the original diameter of 8 units is not the diamater as its not the longest chord
coarse egret
Circle two starts at x 4 ends at x 12
Length is 8
Radius is 4
How is it that hard to understand
patent grove
thats what i thought before as well
but then when i actually counted it it's not coming as 4 idfk why
coarse egret
The horizontal distance will ALWAYS be the same as the vertical distance if it’s a CIRCLE
patent grove
ik ik
coarse egret
A circle has a definition
patent grove
i thought it was 4 bro
but when i counted it's not coming well anyway
just take it as 4
i think it's probably a problem with how the image is coming on my screen idfk
patent grove
A circle has a definition
yeah ik
if you look at the circle vertically its not actually from y = 10 to y = 18
there's a slight offset
but anyway
daring blaze
uhmmm
lone ravine
The first lampta should be 3/5
Ich denke hier macht es keinen Sinn, da die Funktion keine lokalen Extrema unter der Nebenbedingung haben kann. Angenommen du setzt die Lampta's ein, dann ist die Gleichung nur noch von einer Variablen abhängig
stone stump
The first lampta should be 3/5
lambda btw
lone ravine
Die Gleichung wird also von einer Variablen unabhängig und somit ist die Bedingung, die du ansetzt trivialerweise erfüllt
lone ravine
lambda btw
Danke, du konntest ja auch deutsch?
stone stump
vllt konnte ich das. aber lagrange zeug ist zu lange her
lone ravine
Musst du hier notwendigerweise Lagrenge anwenden?
Wie gesagt, es gibt keine lokalen Extrema. Denke deswegen macht das keinen großen Sinn hier
Sagen wir x1= x und x2 = y
stone stump
was ist überhaupt die frage
lone ravine
Maximum vom 3x+4y unter 5x+7y = 35 nehme ich an?
stone stump
!original
devout snowBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
lone ravine
Können x und y auch negativ sein?
stone stump
ach komm das aufgabenblatt kann dir doch nicht nur so zufällige Symbole wie m=35 geben. da wird doch irgendeine aufgabenstellung draufsein
lone ravine
!original
Danke, für das direkte Nachfragen. Wäre jetzt so schnell nicht drauf gekommen. Das x,y nicht negativ sein kann verändert einiges
stone stump
also ist das hier unter dem Titel lagrange zeug oder lineare optimierung
was du auf jeden fall bisher überhaupt nicht hast einfließen lassen ist x>=0 und y>=0. das sind beides super wichtige bedingungen
aber ich weiß auch nicht mehr wie man das mit lagrange macht, sorry
also in der linearen Optimierung gibt es den simplex algorithmus im allgemeinen. Für R^2 ist der aber vollkommen overkill
hier könnte man das tatsächlich graphisch lösen
aber naja das sollt ihr vermutlich nicht tun
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fervent ledge
are these matrices in echelon form or reduced echelon form or neither?
twilit comet
neither
dense lynx
are these matrices in echelon form or reduced echelon form or neither?
what can you tell me about the properties of matrices in:
- echelon form
- reduced echelon form?
fervent ledge
1 in principal diagona and zeroes below it (echelon)
1 in principal diagonal and zeroes both above and below it (reduced echelon
dense lynx
hm
not quite (in particular, the "principal diagonal" part is false)
here is a good definition of "echelon form" (not reduced)
fervent ledge
so it doesnt need to be one?
dense lynx
principal diagonal does not need to exist in an echelon form matrix
fervent ledge
wdym by that
dense lynx
principal diagonal to my knowledge is the items in a matrix in the same-numbered row and column
which is not relevant when we consider echelon forms
fervent ledge
yea so it doesnt need to be one?
dense lynx
what is your definition of "principal diagonal" ?
fervent ledge
well, consider the matrix as a rectangle, and then just scribble a line which joins the entries and looks the most diagonal-like out of all other possible lines you could have drawn
🤓
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fervent ledge
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languid surge
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languid surge
I'm confused on how to go about solving this problem. I'm confused on the method/logic that I need to use to solve it
fI think t1 =6, t2=8, and t3 =1, but after that I am unsure. Do I just reuse the same numbers or am I suppose to do it in intervals
sullen island
they're hinting at the fact that the concentration of coffee increases only when you're actually drinking coffee
languid surge
it qould be like
g1 0<t≤6
g1 6<t≤6.5
g3 8<t≤8.5
g4 1<t≤1.5
sullen island
well you need to have the times between the coffees also
g1 : 0 to 6
g2 : 6 to 6.5
g3 : 6.5 to 8
...
as for the 1pm, they're expecting you to use 24 hour time here
the final time for the day is 24
languid surge
g1 : 0 to 6
g2 : 6 to 6.5
g3 : 6.5 to 8
...
but I just wrote that?
what would be the time in between the coffee
languid surge
oh
so that means it's
g1 : 0 to 6
g2 : 6 to 6.5
g3 : 6.5 to 8
g4 : 8 to 8.5
g5 : 8.5 to 1
g6 : 1 to 1.5
g7 : 1.5 to 24
is this correct?
sullen island
you still need the 24 hour time here
what does 8.5 to 1 even mean numerically ?
are you going backwards?
languid surge
that 4.5 hrs passed
well isn't it implied that time is going forward meaning it's going from AM to PM
sullen island
there's no AM or PM with real numbers
languid surge
so then what do i do for this
sullen island
what does 8.5 to 1 even mean numerically ?
it's like you went back 7.5 hours somehow
languid surge
you still need the 24 hour time here
and for this
sullen island
1pm = 13 hours past midnight
sullen island
not sure what you're saying here
languid surge
I'm not sure what you're saying eitherabout the 8.5 to 1 thing
languid surge
sullen island
the problem is you end up with two different moments of your day which correspond to the same t-value
you can't distinguish 1am from 1pm like that
languid surge
oh
sullen island
so that's why they're using 24 hour time in this exercise
languid surge
so instead ur saying write 13 and 13.5 to represent them as 1
languid surge
so that's why they're using 24 hour time in this exercise
which makes sense because on a 24 hour clock you can easily represent am and pm without labels
g1 : 0 to 6
g2 : 6 to 6.5
g3 : 6.5 to 8
g4 : 8 to 8.5
g5 : 8.5 to 13
g6 : 13 to 13.5
g7 : 13.5 to 24
So this is how it should be written, correct?
languid surge
was just about to ask that lol
sullen island
well y(t) [the function you're trying to find] is the concentration of caffeine in your body throughout the day
languid surge
so how would i go about that? Do I just enter 30 for every time coffe is consumed and put a 0 when it isn't
sullen island
well 0 when you're not drinking coffee sure
but if you reread the end of the paragraph, they tell you how much caffeine concentration increases while you're drinking coffee
languid surge
0 : 0 to 6
40 : 6 to 6.5
0 : 6.5 to 8
40 : 8 to 8.5
0 : 8.5 to 13
40 : 13 to 13.5
0 : 13.5 to 24
????
that looks really wrong
sullen island
I gtg for a bit
languid surge
okay
sullen island
0 : 0 to 6
40 : 6 to 6.5
0 : 6.5 to 8
40 : 8 to 8.5
0 : 8.5 to 13
40 : 13 to 13....
Yeah 40 when drinking, 0 when not
You gotta correct where you put your 40's tho
languid surge
oh it should be reversed
0 : 0 to 6
40 : 6 to 6.5
0 : 6.5 to 8
40 : 8 to 8.5
0 : 8.5 to 13
40 : 13 to 13.5
0 : 13.5 to 24
So I'm doing the second part rn which states
Assume that the half-life of caffeine in the blood is 4 hours and use the Forward Euler Method to find the concentration of caffeine in the blood over a 24 hour period. Use a step size of dt = 1sec and plot y(t) and g(t) on the same plot.
since it's half life I use:
$y(t) = y_0 e^{-kt}$
right?
woven radishBOT
arrow891
sullen island
It's not like your g(t) is straight up 0
How do you know the solution looks like that
They're telling you to solve the equation numerically anyway
With forward euler
languid surge
and from there I just plug and chug
t = 4 x 3600sec = 14400
y_0/2 = y_0 *e^k (14400)
and then solve for k
languid surge
How do you know the solution looks like that
with half life an atom decays at an exponential rate
sullen island
They told you the equation to find k if you know the half life already at the beginning
sullen island
Yes
languid surge
t is the 40, right?
sullen island
Yeah
languid surge
and that's it?
that seems to simple
also you said I gotta correct my g(t) cuz it's not "straight up 0" lmao
sullen island
No your g(t) is good
languid surge
ah okay
sullen island
You were saying that y(t) should be a bog standard exponential
languid surge
so then k ≈ 0.0753
sullen island
You were saying that y(t) should be a bog standard exponential
But that's only the case if g(t) is always 0
languid surge
oh
languid surge
But that's only the case if g(t) is always 0
so I should solve this $y′ = −ky + g(t)$ which becomes $y=\frac{g\left(kt-1\right)}{k^2}+\frac{c_1}{e^{kt}}$
woven radishBOT
arrow891
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
languid surge
thre that into symbolab lol
sullen island
so I should solve this $y′ = −ky + g(t)$ which becomes $y=\frac{g\left(kt-1\righ...
What's going on with g(t) here
sullen island
Again, they're not asking you to solve the equation by hand
Do you know what the forward euler method is ?
languid surge
code?
languid surge
Do you know what the forward euler method is ?
yea
it's just euler's method formula
sullen island
Yea run it on your computer
sullen island
500 sounds a bit much
You only took 3 coffees which increase the concentration by 40 each
And the concentration only decreases after that
languid surge
import numpy as np
import matplotlib.pyplot as plt
# Define constants
dt = 1 # Step size in seconds
t_half = 4 * 3600 # Half-life of caffeine in seconds
k = np.log(2) / t_half # Decay rate constant
# Define function g(t)
def g(t):
if 0 < t <= 6:
return 40
elif 6 < t <= 6.5:
return 0
elif 6.5 < t <= 8:
return 40
elif 8 < t <= 8.5:
return 0
elif 8.5 < t <= 13:
return 40
elif 13 < t <= 13.5:
return 0
elif 13.5 < t <= 24:
return 0
else:
return 0
# Define Forward Euler Method to solve the differential equation
def forward_euler(y0, g):
y = [y0]
t = np.arange(0, 24 * 3600 + dt, dt)
for i in range(1, len(t)):
y_next = y[-1] + dt * (-k * y[-1] + g(t[i]))
y.append(y_next)
return t, y
# Initial concentration of caffeine in the blood (mg/L)
y0 = 0
# Solve the differential equation
t, y = forward_euler(y0, g)
# Plot the concentration of caffeine in the blood and intake function g(t)
plt.figure(figsize=(10, 6))
plt.plot(t, y, label='Caffeine Concentration (mg/L)', color='blue')
plt.plot(t, [g(ti) for ti in t], label='Intake Rate (mg/L/hr)', linestyle='--', color='red')
plt.xlabel('Time (seconds)')
plt.ylabel('Concentration (mg/L)')
plt.title('Concentration of Caffeine in Blood over 24 Hours')
plt.legend()
plt.grid(True)
plt.show()
You forgot to change the g
languid surge
Are you drinking coffee between 0 and 6am ?
Not unless I have a shift I guess lol
languid surge
import numpy as np
import matplotlib.pyplot as plt
# Define constants
dt = 1 # Step size in seconds
t_half = 4 * 3600 # Half-life of caffeine in seconds
k = np.log(2) / t_half # Decay rate constant
# Define function g(t)
def g(t):
if 0 < t <= 6:
return 0
elif 6 < t <= 6.5:
return 40
elif 6.5 < t <= 8:
return 0
elif 8 < t <= 8.5:
return 40
elif 8.5 < t <= 13:
return 0
elif 13 < t <= 13.5:
return 40
elif 13.5 < t <= 24:
return 0
else:
return 0
# Define Forward Euler Method to solve the differential equation
def forward_euler(y0, g):
y = [y0]
t = np.arange(0, 24 * 3600 + dt, dt)
for i in range(1, len(t)):
y_next = y[-1] + dt * (-k * y[-1] + g(t[i]))
y.append(y_next)
return t, y
# Initial concentration of caffeine in the blood (mg/L)
y0 = 0
# Solve the differential equation
t, y = forward_euler(y0, g)
# Plot the concentration of caffeine in the blood and intake function g(t)
plt.figure(figsize=(10, 6))
plt.plot(t, y, label='Caffeine Concentration (mg/L)', color='blue')
plt.plot(t, [g(ti) for ti in t], label='Intake Rate (mg/L/hr)', linestyle='--', color='red')
plt.xlabel('Time (seconds)')
plt.ylabel('Concentration (mg/L)')
plt.title('Concentration of Caffeine in Blood over 24 Hours')
plt.legend()
plt.grid(True)
plt.show()
And there's one other problem now
languid surge
languid surge
And there's one other problem now
better?
sullen island
The time in g(t) isn't in seconds
You have to be coherent with your units
So you gotta multiply each boundary by 3600 in g(t)
languid surge
on the x axis?
sullen island
You chose t to be in seconds
languid surge
nvm
sullen island
But g(t) expects a time in hours right now
languid surge
import numpy as np
import matplotlib.pyplot as plt
# Define constants
dt = 1 # Step size in seconds
t_half = 4 * 3600 # Half-life of caffeine in seconds
k = np.log(2) / t_half # Decay rate constant
# Define function g(t)
def g(t):
if 0 < t <= 6 * 3600:
return 0
elif 6 * 3600 <= t < 6.5 * 3600:
return 40
elif 6.5 * 3600 <= t < 8 * 3600:
return 0
elif 8 * 3600 <= t < 8.5 * 3600:
return 40
elif 8.5 * 3600 <= t < 13 * 3600:
return 0
elif 13 * 3600 <= t < 13.5 * 3600:
return 40
elif 13.5 * 3600 <= t <= 24 * 3600:
return 0
else:
return 0
# Define Forward Euler Method to solve the differential equation
def forward_euler(y0, g):
y = [y0]
t = np.arange(0, 24 * 3600 + dt, dt)
for i in range(1, len(t)):
y_next = y[-1] + dt * (-k * y[-1] + g(t[i]))
y.append(y_next)
return t, y
# Initial concentration of caffeine in the blood (mg/L)
y0 = 0
# Solve the differential equation
t, y = forward_euler(y0, g)
# Plot the concentration of caffeine in the blood and intake function g(t)
plt.figure(figsize=(10, 6))
plt.plot(t, y, label='Caffeine Concentration (mg/L)', color='blue')
plt.plot(t, [g(ti) for ti in t], label='Intake Rate (mg/L/hr)', linestyle='--', color='red')
plt.xlabel('Time (seconds)')
plt.ylabel('Concentration (mg/L)')
plt.title('Concentration of Caffeine in Blood over 24 Hours')
plt.legend()
plt.grid(True)
plt.show()
the grah is wonky now
sullen island
Now it's fine
It's supposed to be like that yea
Your time scale was all squeezed out in your first graphs
languid surge
oh cool
sullen island
Yea
languid surge
yay
sullen island
At least for that question...
languid surge
languid surge
this is why I don't like coding
sullen island
Yeah the increase in concentration has units of mg/L/hr in the exercise
But now your time is in seconds for g(t)
So in 1 second you increase 3600 times less
So all your 40's should be 40/3600's now
sullen island
this is why I don't like coding
Numerical computing in a nutshell
You fuck up a sign, everything goes wrong
languid surge
reasons why I'm switching to mechanical engineering lmao
less cs stuff
import numpy as np
import matplotlib.pyplot as plt
# Define constants
dt = 1 # Step size in seconds
t_half = 4 * 3600 # Half-life of caffeine in seconds
k = np.log(2) / t_half # Decay rate constant
# Define function g(t)
def g(t):
if 0 < t <= 6 * 3600:
return 0
elif 6 * 3600 <= t < 6.5 * 3600:
return 40 / 3600 # Adjusted for unit conversion
elif 6.5 * 3600 <= t < 8 * 3600:
return 0
elif 8 * 3600 <= t < 8.5 * 3600:
return 40 / 3600 # Adjusted for unit conversion
elif 8.5 * 3600 <= t < 13 * 3600:
return 0
elif 13 * 3600 <= t < 13.5 * 3600:
return 40 / 3600 # Adjusted for unit conversion
elif 13.5 * 3600 <= t <= 24 * 3600:
return 0
else:
return 0
# Define Forward Euler Method to solve the differential equation
def forward_euler(y0, g):
y = [y0]
t = np.arange(0, 24 * 3600 + dt, dt)
for i in range(1, len(t)):
y_next = y[-1] + dt * (-k * y[-1] + g(t[i]))
y.append(y_next)
return t, y
# Initial concentration of caffeine in the blood (mg/L)
y0 = 0
# Solve the differential equation
t, y = forward_euler(y0, g)
# Plot the concentration of caffeine in the blood and intake function g(t)
plt.figure(figsize=(10, 6))
plt.plot(t, y, label='Caffeine Concentration (mg/L)', color='blue')
plt.plot(t, [g(ti) for ti in t], label='Intake Rate (mg/L/hr)', linestyle='--', color='red')
plt.xlabel('Time (seconds)')
plt.ylabel('Concentration (mg/L)')
plt.title('Concentration of Caffeine in Blood over 24 Hours')
plt.legend()
plt.grid(True)
plt.show()
sullen island
Ayyy
languid surge
I got it? Did I win?
sullen island
Now our guy is not dead with 120 grams of caffeine in his body
languid surge
Aight now that I'm done with that I can move onto stats. Yay 🙄
thanks for all the help
sullen island
Gl with mech eng
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shy field
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polar chasm
I wouldn't be too surprised if it didnt have nice algebraic solution
shy field
I wouldn't be too surprised if it didnt have nice algebraic solution
I feel like it has a easy solution but im not seeing it
<@&286206848099549185>
dense lynx
I know x = 3, but how to solve it
you plug in integers until you find one that works
dense lynx
genuinely the fastest way to do this problem
left trench
It seems like youtube problems that says "HARD OLYMPIAD PROBLEM" on title
shy field
you plug in integers until you find one that works
yea but i have to algebricly
dense lynx
you are required to have an algebraic solution?
shy field
yea
left trench
you plug in integers until you find one that works
This is solution to these kind of problems, but if you want real solution like some very complicated algebraic solution, I think you could use simple approaches or functions that are defined on N+
It's not easy though, that's why it's funny.
What exactly are you going to learn from this problem, have you ever solved this kind of problem before "algebraically"?
And problem must say what kind of number is x, it can be complex number, that would be weird with 1 base
devout snowBOT
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shy field
What exactly are you going to learn from this problem, have you ever solved this...
yea but this one is hard as far
shy field
And problem must say what kind of number is x, it can be complex number, that wo...
well x is 3
but idk how to deduce it
left trench
yea but this one is hard as far
What's the difference between others and this one
shy field
mostly bec i cant simplify it or use ln or log
even if i simplify it it gets even more complicated
left trench
well x is 3
You said you want an algebraic solution, you should act like you don't know x yet.
shy field
exactly
left trench
There might be more solutions
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shy field
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crisp anchor
I have learnt that $R[S]$ being the closure of a relation $R$ over $S$, can be computed inductively via $Y_0 = S$, $Y_{n+1} = Y_n \cup R(Y_n)$
devout snowBOT
I have learnt that $R[S]$ being the closure of a relation $R$ over $S$, can be c...
woven radishBOT
Aresiel
crisp anchor
I'm now trying to figure out how to compute the transitive closure of some relation $Q$ by assembling a relation $R$ and initial set $S$, and then using the above method.
woven radishBOT
Aresiel
crisp anchor
I've figured out that $R = {((x_1, y_1), (x_2, y_2)) : x_1\in A, y_1 \in A, ... Q ... }$ where $A = (\operatorname{range} Q) \cup (\operatorname{domain} Q)$
woven radishBOT
Aresiel
crisp anchor
But I can't figure how I further must specify R such that it generates the transitive closure
<@&286206848099549185>
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crisp anchor
I believe I need to find a relation R which is equivalent with composition with Q
crisp anchor
It is such that R[Q] = Q o Q
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restive river
Trying to solve B but stuck at the red part
devout snowBOT
Trying to solve B but stuck at the red part
winter patrol
you are told P(3) = 18
restive river
Yeah
winter patrol
so set that equal to 18
restive river
Wdym..
winter patrol
wait, why do you have +18 on your P(3)
restive river
I threw the 18 in here
winter patrol
you were not
P(3) is the result of plugging 3 into the function ONLY
plugging x=3 in gets you
P(3) = 27 - 9r + 9 + r^2 = r^2 - 9r + 36
and you are told the numerical value of this is 18
yes
winter patrol
first isolate 2^n
restive river
So restart?
winter patrol
no
from what you currently have
isolate 2^n
and it should be clear what the value of n should be
winter patrol
why does it look like there's still a - sign next to the 16
winter patrol
and it should be clear what the value of n should be
restive river
I’m not seeing it
winter patrol
do you know your powers of 2
restive river
Ohhhh
It’s 2 to the power of 4
Thanks
@winter patrol is everything still ok here
Or did it go wrong
restive river
Ok so it should be a^2
winter patrol
yes
restive river
winter patrol
quadratic formula
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regal moat
harsh spruce
@trail eagle
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<@125307464647180288>
harsh spruce
I know it yeah
But our teacher said that we should turn what we have into a series bcz using limit would be challenging
trail eagle
That's rather odd, I don't think you really need a series here.
See, we want gamma to be finite, and the denominator goes to 0. The only way this can happen is if the numerator goes to 0 as well, right? Otherwise, the limit would be c / 0 and that just blows up.
So if you plug in 0 (the value we're approaching) in the numerator, and try to set = to 0, what does that tell you?
honest meadow
im actually curious what series your teacher is thinking about to solve this because i dont see anything
harsh spruce
So if you plug in 0 (the value we're approaching) in the numerator, and try to s...
1+BETA-0=0
BETA=-1?
trail eagle
Not far, e^0 is 1, alpha stays there and (beta + 1)*0 = 0
harsh spruce
im actually curious what series your teacher is thinking about to solve this bec...
I am not sure about this question but we solved something similar to it
harsh spruce
Not far, e^0 is 1, alpha stays there and (beta + 1)*0 = 0
I meant alpha sorry not beta
harsh spruce
Yeah
trail eagle
Now try and use L'Hopital's rule and apply the same idea.
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halcyon yew
Hi
jade oak
Good Job!
wanton thunder
for f(x)=3x 1 and g(x)=x^2-6, find the (f-g)(x)
can someone explain how to solve for this?
devout snowBOT
for f(x)=3x 1 and g(x)=x^2-6, find the (f-g)(x)
can someone explain how to solve...
winter patrol
$(f-g)(x) = f(x) - g(x)$
woven radishBOT
ℝαμΩℕωⅤ
wanton thunder
so
(3x+1)-(x^2-6)
is the answer x^2+3x-5
wanton thunder
is there any part of my answer that was right?
heady plinth
Only +3x lol
winter patrol
yes
wanton thunder
if its asking to fine the (f+g)(x) will it be (f+g)(x)= f(x) + g(x)
honest meadow
if its asking to fine the (f+g)(x) will it be (f+g)(x)= f(x) + g(x)
yes
spring sundial
Why is the rank nullity theorem formulated in terms of columns in particular
spring sundial
closed
spring sundial
Why is the rank nullity theorem formulated in terms of columns in particular
and not rows?
hoary radish
columns correspond to the dimension of the domain for the matrix
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obsidian junco
Stats and Prob, Estimating a Population Mean
devout snowBOT
Stats and Prob, Estimating a Population Mean
obsidian junco
need to know how to do the Za/2 caculation on my ti-84 caculator
zealous basin
radiant anvil
LOL start your own help channel
visual hazel
need to know how to do the Za/2 caculation on my ti-84 caculator
is it from area under distribution curve to z-score?
obsidian junco
is it from area under distribution curve to z-score?
nah I remember now
looked through my notes
InvNorm caculator function
but thanks for willing to help
you may wanna help that newbie
visual hazel
was about to say that but all g
obsidian junco
.close
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primal grove
Trying to solve question c, factoring by grouping, not sure what I'm doing wrong or what to do to after that part
devout snowBOT
Trying to solve question c, factoring by grouping, not sure what I'm doing wrong...
covert root
primal grove
I'm pretty sure I was already doing x is cos(5 pi t/3)
I don't think I'm supposed to use rational root theorem, I don't think we've learnt that
Teacher says factor by grouping then solve resulting trigonometric equation for t
I'm stuck at not being able to factor because the -8 and +17 don't match
<@&286206848099549185>
unborn solstice
I'm stuck at not being able to factor because the -8 and +17 don't match
that cant be factored
unborn solstice
I'm stuck at not being able to factor because the -8 and +17 don't match
wait um it tells you that d^2=9
so you can must put the 9 to the other side
so the 17 becomes 8 and then you can factor it
and also yeah let x=co(5/3pit) to make it visibly easier to factor
unborn solstice
I'm stuck at not being able to factor because the -8 and +17 don't match
did you get it or do you need more help?
primal grove
that cant be factored
Ok, makes sense
Not quite sure what swaps with the 9 tho
Will probably get suck again later
unborn solstice
Ok, makes sense
Not quite sure what swaps with the 9 tho
Will probably get suck...
wdym
what swaps with the 9?
unborn solstice
9 moves across the = , does it become 0?
the question tells you that d=3 so d^2=9
so you have 9= -16cos^3.......+17
you move the 9 to the other side and you get:
-16cos^3......+8=0
and you factor it and solve it
primal grove
and you factor it and solve it
Yeah, so 9 = .... becomes 0 = ....
Was just checking that
unborn solstice
Yeah, so 9 = .... becomes 0 = ....
Was just checking that
well yeah, you basically subtract 9 from both sides
primal grove
well yeah, you basically subtract 9 from both sides
Ok, I'll be alright for a little while now
Thanks
unborn solstice
Ok, I'll be alright for a little while now
Thanks
alr ping me if you need more help
primal grove
alr ping me if you need more help
In dm?
primal grove
wherever you want , dm as well
That'd be awesome, thanks so much
unborn solstice
np
primal grove
.close
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glass perch
help
glass perch
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warm hollow
can someone please check these for me
devout snowBOT
can someone please check these for me
warm hollow
for 25 I got x=pi/2 +- pi(n) , x=0 +- pi(n) , x = 7pi/6 +- 2pi(n) , x= 11pi/5 +- 2pi(n)
<@&286206848099549185>
solar goblet
<@&286206848099549185>
!15min
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Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
solar goblet
for 25 I got x=pi/2 +- pi(n) , x=0 +- pi(n) , x = 7pi/6 +- 2pi(n) , x= 11pi/5 +-...
show your work
steady idol
$x=0, x=-\frac{\pi}{6},x=-\frac{7\pi}{6}$, and $+2k \pi$ for each one
woven radishBOT
democracy_landing
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worldly sonnet
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worldly sonnet
So for the last table they have given a formula for probability X>
1/y (which is basically probability of X>x as 1/y=x) so for the value of x = 2 we need to find P(X>2) which is equal to 1 - P(X<2) and P(X<2) = F(2) and we can see from the second table (on page 178 the left one) F(2) is 5/8
1- 5/8 should be 3/8
Like I don't get that
Please explain
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worldly sonnet
<@&286206848099549185>
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@worldly sonnet Has your question been resolved?
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hot steppe
Show that (p ∨ q) ∧ (¬p ∨ r) → (q ∨ r) is a tautology, by using equivalence laws.
devout snowBOT
Show that (p ∨ q) ∧ (¬p ∨ r) → (q ∨ r) is a tautology, by using equivalence laws...
hot steppe
can someone help me with this because i dont know how to do it
severe orchid
do u know what tautology means?
hot steppe
to prove it is True
severe orchid
that's not a tautology
hot steppe
it means a true statement
severe orchid
ok
well if its F then its vacuously true
so u just need to prove that when its true the right side can only be true
hot steppe
If its F then it is a contradiction
hot steppe
no
severe orchid
ye distribute that
severe orchid
like make (p ∨ q) ∧ (¬p ∨ r) into a single statement
get rid of the and
in the middle
hot steppe
using equivalence laws?
severe orchid
ye
hot steppe
¬(pVq)->¬(¬pVr)?
hot steppe
tilde is not
severe orchid
I get q v rp
severe orchid
-> q v r
hot steppe
what law did u use
severe orchid
ye
hot steppe
ummm
severe orchid
use distributive
hot steppe
(p ∨ q) ∧ (¬p ∨ r) the second p has negation
in the distributive law there is no negation
severe orchid
ok but its irrespective of the sign
like(p ∨ q) ∧ (¬p)
what is this
and then (p ∨ q) ∧ (r)
what is this
and then
(p ∨ q) ∧ (r) v (p ∨ q) ∧ (¬p)
what is this
are u able to send a photo of ur working?
hot steppe
so it doesnt matter if it has negation or not
hot steppe
are u able to send a photo of ur working?
i havent worked on it because im trying to understand how to do it
severe orchid
start here (p ∨ q) ∧ (r) v (p ∨ q) ∧ (¬p)
and try to use equivalence laws along the way
see what u can cancel
hot steppe
just one question
severe orchid
ye
hot steppe
if the equivalence law doesnt have negations but i have can i still use it?
severe orchid
yep
severe orchid
yep
hot steppe
p v (q and r)
that is equivalent to p
severe orchid
i get q v pr
hot steppe
well this says p v (q and r)
severe orchid
can u send ur work?
hot steppe
well i havent done any work
just based on the laws
that is what you get
(p ∨ q) ∧ (¬p ∨ r) is equivalent to p v ( q and r) based ondistributive law
severe orchid
no
hot steppe
why
hot steppe
no
severe orchid
then ur using it wrong
severe orchid
please first tell me how to distribute (x + y)* (z+w)
let x, y, z and w be just random variables
and + is normal addition and * is normal multiplication
u can think of v as + and * as and
its similar enough for this question
hot steppe
wait
hot steppe
please first tell me how to distribute (x + y)* (z+w)
x times z + x times w + y times z + y times w
hot steppe
that would be
p and ¬p V p and r V q and ¬p V q and r
severe orchid
its a lil off
hot steppe
why
severe orchid
x times z + x times w + y times z + y times w
u only had 4 terms here
severe orchid
that would be
p and ¬p V p and r V q and ¬p V q and r
but u have 5 terms here
should be same number of terms
also also v is or and ^ is and
severe orchid
no cuz for the xy thing, u get xz + zx + wy + yw
but for the thing above u get
nvm I'm just stupid
I'm sorry
ye that's right
super sorry
miss read
hot steppe
no problem
hot steppe
that would be
p and ¬p V p and r V q and ¬p V q and r
but there is no law that i can do with this
@severe orchid
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willow ore
Alright, so, another question. Ive simplified an inequality down to this and know the vertical asymptotes are 3 and 0.778, and I also no there are no x intercepts. But I have a very spotty idea on where exactly to go from here. I cant really find a good explanation of how to make an interval table which I need to do to get the answer, nor do I really know what to do without the x intercept, which Im pretty sure I need to make that table. Any help would be appreciated!
frosty mantle
what answer are u looking for exactly
willow ore
Pretty much how to build an interval table from this
and maybe how to build one in general
stark sinew
Best way to do interval tables is to create tables at each root of the denominator AND the numerator, and then in the table just note for each interval whether the num and denom are positive or negative so whether teh overall fraction is
imagine this but with the points being all the roots of the numerator and denominator, and then the signs of the numerator or denominator
stark sinew
then just check the sign of the numerator thats why I said the sign thing
if theres no root the sign should not change across the table!
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pseudo rain
PQ is a tangent at Q and PS//QR. Prove that angle SPQ= angle QSR.
pseudo rain
wasnt given a graph
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weary arch
PQ is a tangent at Q and PS//QR. Prove that angle SPQ= angle QSR.
SPQ is not equa to QSR if its's just random points
Are you sure it's no SPQ = PQR?
Or smth similar, cuz randomly placing the points isnt always gonna give the same result
pseudo rain
im not sure
weary arch
Also "tangent" to what??
pseudo rain
i think the question is too vague
i wasnt given a graph so i dont know where to start
weary arch
Well where'd you get this alleged question
pseudo rain
tutor
weary arch
PQ is a tangent at Q and PS//QR. Prove that angle SPQ= angle QSR.
Well like that it makes little to no sense i think, maybe you missed a part?
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celest sundial
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celest sundial
Relating to question 1, how do I find delta x?
spare crypt
the spaced apart number, 1.5
eager fox
the spaced apart number, 1.5
Dont want to be clog the learning but maybe in a general channel can you tell me how the special roles are earned?
celest sundial
the spaced apart number, 1.5
But it says the in answer, delta x is 3
@spare crypt
spare crypt
oh huh, seems like it's a midpoint rule thing
celest sundial
spare crypt
like you're only allowed to use the data in the middle of rectangles
celest sundial
Wdym?