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#help-27
restive river
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restive river
Can someone help me understand this? I really don't get it
I understand what cartesian equations are, but I don't really understand what I'm supposed to do here
small raptor
Erm
i doubt this is the way to do this, as in theres probably an actual method
but plug in x(t) and y(t) into that equation
and than for that Equation to be true, we know sin^2 + cos^2 =1, so the /4 and /9 will have to cancel
this will get you your a and b
vapid nest
oh bud try x = 2cos(t)
you would get cos(t) = x/2
then try and use the identity cos(t)^2 + sin(t)^2 = 1
to confirm
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soft umbra
(k,l) represents its center
a represent its radius palleral to x axis
b represents its radius palleral to y axis
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lunar kiln
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lunar kiln
,rotate
woven radishBOT
lunar kiln
part b
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tough hatch
can some help me with this integral
devout snowBOT
can some help me with this integral
trail eagle
Split it in 2 and use exponent laws.
trail eagle
,rccw
woven radishBOT
tough hatch
symbolab says that the answer is 4569/80
but what i get is 4569/20
idk where i went wrong
trail eagle
You multiplied by 2 when you should've divided by 2
Look at where you broke it into 2
trail eagle
Or if you'd rather, just take the 1/2 outside the integral right away
lime dew
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lime dew
can someone explain how it simplifies to that?
unique spire
$\frac{2}{3*\sqrt{3}}1=\frac{2}{3\sqrt{3}}\frac{\sqrt{3}}{\sqrt{3}}=\frac{2\sqrt{3}}{3*3}$
late orbit
Just multiply $3sqrt(3)$ to both the numerator and denominator.
lime dew
did they rationalise it?
lime dew
when you rationalise do you take only the surd or the entire denominator
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hard cosmos
Why is (1)/(2/5) = 5/2
devout snowBOT
Why is (1)/(2/5) = 5/2
magic pine
(\frac{1}{x}) is a number such that [\frac{1}{x}\cdot x = 1]
woven radishBOT
maximo
magic pine
by definition
so now say we want to find out what 1/(2/5) is
[\frac{1}{\left(\frac{2}{5}\right)} \cdot \frac{2}{5} = 1]
woven radishBOT
maximo
magic pine
do you see why you would end up with 5/2?
urban harbor
good explanation 👍 : )
hard cosmos
**maximo**
So it’s like (1/5) * (5/1) = 1 because it’s like 5/5, yeah makes sense
But i still don’t understand why (1)/(2/5) = 5/2
hard cosmos
**maximo**
This also makes sense
magic pine
**maximo**
in this equation
magic pine
yes
hard cosmos
Another way that I found was that 1/(2/5) is equal to (1/1) / (2/5)
and you can cross multiply
Idk
Many ways to prove it i guess
hazy sedge
Another way that I found was that 1/(2/5) is equal to (1/1) / (2/5)
thats what i was going to mention to you but i figured thats basically giving you the answer
hazy sedge
Many ways to prove it i guess
at least 2 lol
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autumn quiver
how do we find limit of f(x) (x->inf)?
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how do we find limit of f(x) (x->inf)?
autumn quiver
i cant wrap my head around "..." questions
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autumn quiver
help plx,
supple knot
how do we find limit of f(x) (x->inf)?
f(x) is a rational function, so you just compare the terms with the highest powers
autumn quiver
https://www.youtube.com/watch?v=M2BgBG2Su94&ab_channel=TheOrganicChemistryTutor
so itll be 2?
supple knot
so itll be 2?
right
bright gulch
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faint hearth
Did i get it right?
They want f"(4), you've calculated f'(4) here
bright gulch
F'(4)
Did i get this part right though?
Like i conjoined f'(u) and u'
Is this the right way of doing it
faint hearth
Thats correct, but u' at x=2 is 10
Not 4
u'=3x^2-2, so when x=2, u'=10
So, 17=2(9+12)(10f'(4)+6) should be what you get, not 17=2(9+12)(4f'(4)+6)
faint hearth
Thats correct, but u' at x=2 is 10
@bright gulch
bright gulch
I see
Thanks
You helped me a lot. I havent checked the answer yet but it seems correct so far
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restive river
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restive river
I got 4a^2 + 4b^2 + 8ab, but idk what to do after that
soft umbra
Actually, why don’t you just mix it up?
soft umbra
I got 4a^2 + 4b^2 + 8ab, but idk what to do after that
Just take out 4 and you’ll get
4(a^2+b^2+2ab)
restive river
I got 4a^2 + 4b^2 + 8ab, but idk what to do after that
Oh my God I'm so blind
restive river
Just take out 4 and you’ll get
4(a^2+b^2+2ab)
Yeah I get it now
soft umbra
gl fam
restive river
I'm so tired my brain doesn't work
soft umbra
Type
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To close the channel
restive river
Thank you
soft umbra
I'm so tired my brain doesn't work
understandable
restive river
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soft umbra
What’s your question?
Do you have some questions for reference?
I won’t
yep
No wait
I would get the simpliest equation
2(v+w)=3(v-w)
v=5w
This is my preference
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ember shard
can anyone help me with part b
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can anyone help me with part b
restive river
what is cot(theta) = ?
dawn iris
try multiplying both sides by cot
vernal monolith
try multiplying both sides by cot
After doing this, complete the square
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ember shard
thank you very much
stuck bridge
How to prove that abcd tangent trapezoid's area = this picture:
devout snowBOT
How to prove that abcd tangent trapezoid's area = this picture:
stuck bridge
I probably need some equation from here but I don't know which and how to simplify it prove the equation
https://en.wikipedia.org/wiki/Trapezoid#Area
soft umbra
How to prove that abcd tangent trapezoid's area = this picture:
What does abcd stands for?
Point? side?
stuck bridge
sides
soft umbra
I see
stuck bridge
I learned ABCD are points and abcd are sides
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stuck bridge
<@&286206848099549185>
rose mortar
That area is the isosceles tangential trapezoid
I don't know much but the same equation appears in the section about that figure
And it seems that is given by the inradius
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stuck bridge
hmm could be that the teacher didn't say it was isosceles...
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jade pendant
I am confused I get a different result than what is in the teaching material
timber pebble
you need to write the top as a difference of squares
jade pendant
Huh?
jade pendant
Would it be possible for you to write it down on a sheet step by step so that I can understand it?
timber pebble
sure but im in bed it may be one moment
jade pendant
So u also get a different answer than what is written in the teaching material
restive river
minus square is plus (c+2d)/(c+2d)^2 is 1/(c+2d)
(c^2-d^2)^2 is (c+d)^2(c-d)^2 which divided by c+d is (c-d)^2(c+d)
and final answer is (c-d)^2(c+d)/(c+2d)
which can be simplified as (c^2-d^2)(c-d)/(c+2d)
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timber pebble
So u also get a different answer than what is written in the teaching material
one second here
writing it out now
jade pendant
Thanks
timber pebble
timber pebble
whats :
woven radishBOT
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rough lynx
500 chairs were arranged in rows in the hall. There were equal seats in each row. After the reconstruction of the hall, there were 5 more chairs in each row, and there were 5 fewer rows in the hall than it was at the beginning. Therefore, the total number of seats in the hall decreased by 10 percent. How many rows were there in the hall at first?
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500 chairs were arranged in rows in the hall. There were equal seats in each row...
@rough lynx Has your question been resolved?
twin fjord
what have you done till now?
twin fjord
i see in any case, I will just give a hint: take number of rows to be x and no of columns to be y. Make two equations and solve them to get the answer
idle sun
I've found the zeros as well as some rough end behavior (which quadrants the ends will be in and the direction of the ends) but im not sure how to find out the height of the curves, how to connect them somewhat accurately, and other things. Basically, i need help with the steps after finding the zeros and some end behavior.
devout snowBOT
I've found the zeros as well as some rough end behavior (which quadrants the end...
solemn brook
Why is my y value not as the markscheme
thin fern
I've found the zeros as well as some rough end behavior (which quadrants the end...
finding the height of the curves is hard without calculus
which idk if you know or not
idle sun
finding the height of the curves is hard without calculus
can you help me with the other parts then?
like how would i roughly find the height
the question says a possible graph
so i dont think it needs to be too accurate
i just need help grasping the concept
thin fern
Yes as long as your curves are in the right direction I think you'll probably be fine
queen kernel
sure buddy
Do you have the answer?
thin fern
If you're going for accuracy, maybe create a table of points?
queen kernel
I want to check the result
idle sun
the hoemwork is mostly using like logic and knowledge about graphs to determine it
instead of points
thin fern
okay
thin fern
So in that case, just start from the left (either from the bottom or top based on what you determined end behavior to be)
And then intersect all the zeroes
torpid cairn
You may try to factorise this one to get a more accurate picture
idle sun
ok so far
i factorized it
found the zeros which are 0 and 3
figured out that the ends are in quadrant 3 and 1
thin fern
The 3 is a double root, which is key to graphing it correctly
thin fern
correct
idle sun
yeah i got that
idle sun
this si basically what i have
drew it with a mouse on pc
i have this
i just dont know between 0 and 3 how high it should go
and left of 0 how steep or far it should go
thin fern
i just dont know between 0 and 3 how high it should go
doesn't matter
just make it go up and then back down
idle sun
okay lol
so basically if i want to accurately draw the graph i need calculus
rigt now im limited to making really rough sketches
?
thin fern
so basically if i want to accurately draw the graph i need calculus
or you could do a table of points
thin fern
rigt now im limited to making really rough sketches
pretty much I'd say
idle sun
alright thanks for the help
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echo kraken
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echo kraken
hi, i answered question a, could i get some help on the questions below pls
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echo kraken
<@&286206848099549185>
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orchid wasp
How I do this
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How I do this
lavish nimbus
You're given two points. Use them.
orchid wasp
You're given two points. Use them.
So I just write in the x,y point they gave me
crisp dust
How I do this
Please send a picture with visible points
ionic wigeon
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thorn umbra
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final berry
How can I find the area of the trapezoid?
devout snowBOT
How can I find the area of the trapezoid?
final berry
... what trapezoid?
The trapezoid below the triangle.
shrewd thistle
You have several triangles in the diagram
tawdry wave
How can I find the area of the trapezoid?
Try constructing a triangle here
shrewd thistle
You need to be clearer about the dimension you're looking to solve for
final berry
Try constructing a triangle here
Okay, I used the definition of the midpoint to calculate the side with two ticks to be equal to 11.5, then, I used the Pythagorean theorem to find the measure of the side with one tick to be equal to the square root of 156.75.
I think that's correct.
Then, I can find the measure of the top part of the trapezoid to be 17.
tawdry wave
Okay, I used the definition of the midpoint to calculate the side with two ticks...
The side with one tick is equal to the side with two ticks
shrewd thistle
So you think the side with no ticks = 17?
final berry
So you think the side with no ticks = 17?
No, just the measure of the top part of the trapezoid is 17.
shrewd thistle
45/45/90 = 1^2 + 1^2 = sqrt(2)
shrewd thistle
No, just the measure of the **top** part of the trapezoid is 17.
You're missing some fundamental elements
That IS the hypotenuse of your two triangles
tawdry wave
No, just the measure of the **top** part of the trapezoid is 17.
It isn't
This is not a 45/45/90 triangle
final berry
It isn't
This is not a 45/45/90 triangle
Wait, how is it not?
tawdry wave
Wait, how is it not?
17sqrt(2) is not 23
shrewd thistle
How is it not?
tawdry wave
Not this one
shrewd thistle
The triangle you constructed has to be a 45/45/90, if the lines you drew were perpendicular/parallel
Also, the triangle you constructed is unnecessary to solve the problem
But thanks anyways
final berry
Also, the triangle you constructed is unnecessary to solve the problem
How is it unnecessary ?
shrewd thistle
Where were we?
final berry
Where were we?
Finding the area of the trapezoid.
shrewd thistle
| = ||
By definition
Ergo, the hypotenuse is sqrt(2)
That is the long edge of the rectangle
That is also the opposite side to the hypotenuse of the triangle in the trapezoid
final berry
Doesn't || = 23/2; definition of midpoint?
Meaning | is also 23/2?
Oh wait, the top of the trapezoid should be 4*sqrt(15) according to the Pythagorean theorem.
shrewd thistle
Without measurements, the top of the trapezoid is sqrt(2)
final berry
But I'm still confused about how to find the height and bottom side of the trapezoid.
shrewd thistle
bottom side = top side, by definition of the right trangle in the very bottom right
height = congruent hypotenuse to another 45/45/90 triangle
so if the hypotenuse = 1, then the height =....
final berry
so if the hypotenuse = 1, then the height =....
Is 1/sqrt(2)?
Wait, if you split the trapezoid into one right triangle and one rectangle, the rectangle has an area of (sqrt(23)/2)(4sqrt(15)), and the triangle has an area of (23/2)(sqrt(23)/2). This means that the area should be 2sqrt(345) + sqrt(12167)/4... which is approximately 64.724. I'm not sure if this is right though.
<@&286206848099549185>
Nvm, I figured it out.
.close
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bitter halo
,tex\textbf{} Let ( f: \mathbb{R}^3 \rightarrow \mathbb{R}^2 ) be the linear transformation ( f(x_1, x_2, x_3) = (x_1 - x_2, x_2 + x_3) ) and let ( \bm{v} = (2,3) ); ( \mathbb{S} = \langle (1,2,1) \rangle ); ( \mathbb{T} = { \bm{x} \in \mathbb{R}^2 \mid 3x_1 - 2x_2 = 0 } ).
Find ( f(\mathbb{S}) ), ( f^{-1}(\bm{v}) ) and ( f^{-1}(\mathbb{T}) ).
devout snowBOT
,tex\textbf{} Let \( f: \mathbb{R}^3 \rightarrow \mathbb{R}^2 \) be the linear t...
woven radishBOT
renato
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@bitter halo Has your question been resolved?
urban jungle
**renato**
Have you determined the matrix of the linear transformation?
urban jungle
So we have a system of equations, essentially
Let the output of the transformation be (y1, y2)
So we've got y1 = x1 - x2, y2 = x2 + x3
bitter halo
yeah
urban jungle
what's the coefficient matrix of the system?
are you familiar with matrix forms for systems of equations
where a system is represented as Ax=b
urban jungle
correct
this is also the matrix of the linear transformation
linear transformations and systems of equations are essentially the same thing
with that matrix, what would be the image of S under the linear transformation?
bitter halo
dunno
urban jungle
do you know how to do linear transformations/
?
(it's just computing Ax, where A is the transformation matrix and x is the vector you're transforming)
bitter halo
,w [[1,-1,0],[0,1,1]] * {{1},{2},{1}}
woven radishBOT
bitter halo
@urban jungle
urban jungle
this is correct
for the next one it wants you to go backwards and find the solution set of Ax=v
bitter halo
sometimes
urban jungle
well this is essentially taking a matrix equation and treating it as a system of equations (they're the same thing fundamentally)
you know A, you know v, now find x (which should be a solution set because you only have 2 equations and 3 variables)
woven radishBOT
bitter halo
this is the coefficient matrix augmented with the vector given v to find x
@urban jungle
do you understand what is going on? Y/N
urban jungle
yes
so what you have here is a solution set with one free variable
it helps to write this out as a system of equations
because you can then find x1 and x2 in terms of x3
bitter halo
,, \begin{cases}
x_1 + x_3 = 5 \
x_2 + x_3 = 3
\end{cases}
woven radishBOT
renato
urban jungle
isolate x1 and x2
bitter halo
,, \begin{cases}
x_1 = 5 - x_3 \
x_2 = 3 -x_3
\end{cases}
woven radishBOT
renato
urban jungle
alright
now you can make a line of the form (x1, x2, x3) = mt + b
where m and b are vectors in R^3
do you see how you could do that?
bitter halo
i need to parametrize with the free variable
urban jungle
yes
bitter halo
,tex
[
\text{Given: }
\begin{cases}
x_1 = 5 - x_3, \
x_2 = 3 - x_3.
\end{cases}
]
Let ( x_3 = t ). Then,
\begin{flalign*}
x_1 &= 5 - t, && \
x_2 &= 3 - t, && \
x_3 &= t. &&
\end{flalign*}
[
\text{The parametric line in } \mathbb{R}^3 \text{ is given by:}
\begin{pmatrix}
x_1 \
x_2 \
x_3
\end{pmatrix}
\begin{pmatrix}
5 \
3 \
0
\end{pmatrix}
+
t
\begin{pmatrix}
-1 \
-1 \
1
\end{pmatrix}
]
@urban jungle
woven radishBOT
renato
bitter halo
@urban jungle
bitter halo
what?
urban jungle
that's f^-1(v)
they use f^-1 a bit loosely because it's not actually a proper function (f^-1 only exists for bijective functions, or in linalg terms, linear transformations with nonsingular square matrices)
really it should be the preimage of v under f
bitter halo
what exactly. ?
urban jungle
f^-1(v) in this case refers to the preimage of v under f
since there are many vectors that give v when multiplied by the matrix
"preimage" is just all the inputs that give you a certain output
urban jungle
it's all the vectors given by that parametrization for t in R
every one of those vectors gives you v when f is applied to it
bitter halo
so the entire line
urban jungle
yes
bitter halo
or something
,tex\textbf{} Let ( f: \mathbb{R}^3 \rightarrow \mathbb{R}^2 ) be the linear transformation ( f(x_1, x_2, x_3) = (x_1 - x_2, x_2 + x_3) ) and let ( \bm{v} = (2,3) ); ( \mathbb{S} = \langle (1,2,1) \rangle ); ( \mathbb{T} = { \bm{x} \in \mathbb{R}^2 \mid 3x_1 - 2x_2 = 0 } ).
Find ( f(\mathbb{S}) ), ( f^{-1}(\bm{v}) ) and ( f^{-1}(\mathbb{T}) ).
woven radishBOT
renato
urban jungle
formally, it's the set of all vectors on that line
bitter halo
how do i find c?
urban jungle
thanks for the nerd emoji 😔
i'm learning analysis rn, the set theory is compulsive
anyway
for the third one they're being tricky with notation
the x_n's in that set definition should not be confused with the x_n's in the function definition
so it's good to change them to y_n's to be consistent
so they've given you an equation, 3y1 - 2y2 = 0
but what are y1 and y2, in terms of how the function determines them?
(where (y1,y2) is the output of the function, in R^2
woven radishBOT
renato
urban jungle
now make the substitution into the equation for the set (3y1 - 2y2 = 0)
and you've got an equation that gives you a solution set
bitter halo
,, 3y_1 -2y_2 = 0 \ 3(x_1-x_2) -2(x_2 + x_3)=0
urban jungle
now what would be the solution set for this equation?
woven radishBOT
renato
bitter halo
,, 3y_1 -2y_2 = 0 \ 3(x_1-x_2) -2(x_2 + x_3)=0 \ 3x_1 - 3x_2 -2x_2 -2x_3 = 0
woven radishBOT
renato
bitter halo
,, 3y_1 -2y_2 = 0 \ 3(x_1-x_2) -2(x_2 + x_3)=0 \ 3x_1 - 3x_2 -2x_2 -2x_3 = 0 \ 3x_1 -5x_2 -2x_3 = 0
woven radishBOT
renato
bitter halo
,tex\textbf{} Let ( f: \mathbb{R}^3 \rightarrow \mathbb{R}^2 ) be the linear transformation ( f(x_1, x_2, x_3) = (x_1 - x_2, x_2 + x_3) ) and let ( \bm{v} = (2,3) ); ( \mathbb{S} = \langle (1,2,1) \rangle ); ( \mathbb{T} = { \bm{x} \in \mathbb{R}^2 \mid 3x_1 - 2x_2 = 0 } ).
Find ( f(\mathbb{S}) ), ( f^{-1}(\bm{v}) ) and ( f^{-1}(\mathbb{T}) ).
woven radishBOT
renato
urban jungle
now you do have an equation for a plane so that might be satisfactory for the question, but a parametrization would be better
bitter halo
wdym?
urban jungle
3x1 - 5x2 - 2x3 = 0 is an equation for a plane
but you can also make a parametrization with two parameters
(in fact if you've never done a parametrization of a plane in R^3 before i'd recommend doing it, it helps intuition for planes to see how ax + by + cz = d relates to linear combinations of vectors)
do you see how you can parametrize the solutions to that equation?
bitter halo
,tex
[
\text{Given the plane equation: } 3x_1 - 5x_2 - 2x_3 = 0,
]
Let ( x_2 = t ) and ( x_3 = s ). We solve for ( x_1 ) as follows:
\begin{flalign*}
3x_1 - 5t - 2s &= 0 && \
3x_1 &= 5t + 2s && \
x_1 &= \frac{5}{3}t + \frac{2}{3}s &&
\end{flalign*}
With ( x_2 = t ) and ( x_3 = s ), we have:
\begin{flalign*}
x_1 &= \frac{5}{3}t + \frac{2}{3}s, && \
x_2 &= t, && \
x_3 &= s. &&
\end{flalign*}
[
\begin{pmatrix}
x_1 \
x_2 \
x_3
\end{pmatrix}
\begin{pmatrix}
\frac{5}{3}t + \frac{2}{3}s \
t \
s
\end{pmatrix}
]
woven radishBOT
renato
bitter halo
we can further parametrize by separating t and s
urban jungle
true, you could get it in the form at + bs which is probably preferred notation
but aside from reformatting, that should cover all three parts of the question
bitter halo
not that hard after you explained the tidbits
,tex
[
\text{Given the plane equation: } 3x_1 - 5x_2 - 2x_3 = 0,
]
Let ( x_2 = t ) and ( x_3 = s ). We solve for ( x_1 ) as follows:
\begin{flalign*}
3x_1 - 5t - 2s &= 0 && \
3x_1 &= 5t + 2s && \
x_1 &= \frac{5}{3}t + \frac{2}{3}s &&
\end{flalign*}
With ( x_2 = t ) and ( x_3 = s ), we have:
\begin{flalign*}
x_1 &= \frac{5}{3}t + \frac{2}{3}s, && \
x_2 &= t, && \
x_3 &= s. &&
\end{flalign*}
Thus, the parametric representation of the plane in ( \mathbb{R}^3 ) can be expressed in two ways:
As a single vector:
[
\begin{pmatrix}
x_1 \
x_2 \
x_3
\end{pmatrix}
\begin{pmatrix}
\frac{5}{3}t + \frac{2}{3}s \
t \
s
\end{pmatrix}
]
And as a linear combination:
[
\begin{pmatrix}
x_1 \
x_2 \
x_3
\end{pmatrix}
t
\begin{pmatrix}
\frac{5}{3} \
1 \
0
\end{pmatrix}
+
s
\begin{pmatrix}
\frac{2}{3} \
0 \
1
\end{pmatrix}
]
woven radishBOT
renato
urban jungle
yeah; with linalg once you realize everything is the same thing in different forms (systems of equations, linear transformations, matrices etc.) it's pretty easy
bitter halo
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7. None of the above
I suppose it's 5?
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full rover
i didn't understand the underlined thing, is it a rule or anything to get A inverse?
silk sky
If you do the product of a matrix A with any other B, and this product gives you I then B is the inverse of A
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eternal flint
I'm just a little confused on what to do first, would appreciate some help with this problem!
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I'm just a little confused on what to do first, would appreciate some help with ...
weak cove
Take it one step at a time
if that blue is the graph of f(x)
what is the graph of f(x)-2
could you graph that
eternal flint
So I would start off with the horizontal shift first, right?
shifting the graph to the right by 2?
eternal flint
oh, so it doesn't matter what order of transformations used? i thought they did matter
or is that only when it comes to certain transformations
if it's asking me to reflect the graph over the x-axis (-f) do I not do that one first?
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inner ibex
Hello i had a test on fri
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Hello i had a test on fri
inner ibex
And there was a question like
Find a counterexample for the statement " the product of 2 irrational numbers, m and n is irrational where m is not equal to n"
And i did sqrt(2) x sqrt(8)
=sqrt(16)
=4
is this right
inner ibex
Where the product is rational you mean?
No it was counterexamples
So i disproved the statement
Ok thank u
uncut crow
but what was the statement
inner ibex
Find a counterexample for the statement " the product of 2 irrational numbers, m...
everything after the word for
lusty sapphire
then yea, your example works fine
lusty sapphire
Also, for any irrational number r, 1/r is also irrational and their product is rational
cobalt mauve
Probability of player A winning above player B is 0,7. What is the probability that in the next ten rounds the player B wins atleast once?
I understand that it should be solved as
$$
P(B)=1-P(B^c)
$$
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Probability of player A winning above player B is 0,7. What is the probability t...
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konxmok
cobalt mauve
but in the solution its $$1-(0,7)^{10}$$ and my question is what formula is used so that the 10 is in the power, is that some theorem? Why is it not f.e. multiplied by 10?
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konxmok
cobalt mauve
I know that if you multiply it by 10 you would be abobe probability of 10 but why specifically the exponent?
tall knoll
P(B)=1-P(B^c)
what is your event B in this situation?
cobalt mauve
that player B wins at least once
tall knoll
and so B^c would be?
cobalt mauve
the probability of player A winning
tall knoll
winning what?
cobalt mauve
winning what?
winning that round
tall knoll
no
if the event B is "player B wins at least once in 10 rounds," then B^c is not "A wins that round"
you'll have to be more specific
cobalt mauve
this is exactly translated from the given problem so i dont know how to specify it more
The probability of Player A winning against Player B is 0.7. What is the probability that during ten consecutive matches (1). at least once B has won
this is translation using deepl, hopefully its better
tall knoll
i'm not concerned with the problem, I'm concerned with your response
sure, we can use matches instead of rounds
what is the complement of "player B wins at least one of the next ten matches"?
cobalt mauve
player A looses atleast one of the next ten matches?
cobalt mauve
im not really sure, we had just one lecture so im still little bit lost in it
tall knoll
what is your definition of "complement"?
cobalt mauve
what is your definition of "complement"?
we had only the set defition at the lecture and before hand we did some problems at the labs, so its possible that i undesrtand it wrong, do you mind defining it using word since we havent done that?
during the lecutre we have defined it as a property of sigma algebra and that this identity holds P(A^c)=1-P(A) for all A in Omega, where omega is sigma algebra
tall knoll
well we can work with that
P(A^c)=1-P(A)
so P(A) + P(A^c) = P(S) = 1 (S being our sample space)
cobalt mauve
>P(A^c)=1-P(A)
so P(A) + P(A^c) = P(S) = 1 (S being our sample space)
okey
tall knoll
so A U A^c = ?
cobalt mauve
so A U A^c = ?
P(A)+P(A^c)-P(A ∩ A^c)
tall knoll
A U A^c is a set, this has nothing to do with probabilities
cobalt mauve
we havent had set theory so idk what should i write
tall knoll
sigma algebra is a set, how did they define it without using set theory?
cobalt mauve
sigma algebra is a set, how did they define it without using set theory?
they defined it using set theory, we just havent had set theory ever before, i study engineering so its little bit harder to understand for us
tall knoll
the point I'm trying to make is that A U A^c = S
ie the elements that make up A, and the elements that make up A^c, together make up S
cobalt mauve
oh you meant it like that, okey yes i know that
tall knoll
so if event B is "player B wins at least one game," what is B^c?
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wintry ferry
So I'm doing the differential equation of y-xy' = 3 - 2x^2y' after which I get dx/(-2x+x) = dy/y-3 then ln(x/(2x-1)) + C = ln(y-3) my answer of y = ((Cx)/(2x-1)) + 3
However, when I do dx/(2x-x) = dy/3-y which is
y-xy' = 3 - 2x^2y'
turn into 3-y = 2x^2y'-xy'
I'm getting a different answer because the partial fraction would flip and cause the ln fraction to flip as well making a totally different answer of 3-(2Cx-1)/x=y.
I was wondering what I'm doing wrong
devout snowBOT
So I'm doing the differential equation of y-xy' = 3 - 2x^2y' after which I get d...
wintry ferry
The path to my first answer is
y-xy' = 3 - 2x^2y'
y-3 = y'(-2x+x)
dx/(-2x+x) = dy/y-3
1/(-2x+x) = -1/(2x-x)
Left side using partial fractions is
a/(x)+ b/(2x-1)
A = 1 B = -2
then it's ln(x) - ln(2x-1) which is just ln(x/(2x-1)) + C
Right side is just ln(y-3)
After integration it's
ln(x/(2x-1)) + C = ln(y-3)
Taking e of both would get you
x/(2x-1) * C= y-3 NOTE: e^C = C
((Cx)/(2x-1)) + 3 = y <-- My answer
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rose bobcat
wondering what i did wrong on these questions, prepping for an exam tmr
restive river
d one is wrong
think of -x as some negative number
let us assume it for now to be -2
rose bobcat
so |-x|+14?
void obsidian
both would be correct
rose bobcat
what about f(x+7)?
void obsidian
that one's correct as well
rose bobcat
is it just x+7+14 = x+21 instead?
void obsidian
i don't know why it marked them as wrong but all of your answers are correct
rose bobcat
weird
soft umbra
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stuck cypress
what is the answer to lim (x+1)/(x^2secx) as x approaches infinity
soft umbra
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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
2
late orbit
it is basically $\frac{x+1}{x^2}$ *cos(x). Now since the numerator has higher power than denominator, $\frac{x+1}{x^2}$ goes to 0 and cos(x) goes to 1.
woven radishBOT
Solomaniac
late orbit
Thus, whole thing goes to 0 as $x \rightarrow \inf$
woven radishBOT
Solomaniac
late orbit
@stuck cypress
restive river
cosx goes to 1 ?
late orbit
Oh no sorry.
restive river
🤔
soft umbra
stuck cypress
ok, thank you so much understood now
late orbit
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twilit granite
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twilit granite
why do the answers give -7pi/9 as a third answer?
twilit granite
cheers
twilit granite
<@200793484053970944>
sorry could you explain where the 5pi/6-2pi comes from?
granite arch
sorry could you explain where the 5pi/6-2pi comes from?
sin(x+2kpi)=sinx
think of it as a clockwise/anticlockwise revolution in the unit circle
twilit granite
think of it as a clockwise/anticlockwise revolution in the unit circle
im starting to understand, but if the domain is between [-pi, pi] how is -7pi/6 possible if it's outside the domain?
granite arch
im starting to understand, but if the domain is between [-pi, pi] how is -7pi/6 ...
the domain got modified
twilit granite
oh so simply replacing x in 3x/2 with pi?
granite arch
if -pi ≤ x ≤ pi, what happens when x is 3x/2
twilit granite
then -3pi/2 ≤ 3x/2 ≤ 3pi/2
correct?
twilit granite
if -pi ≤ x ≤ pi, what happens when x is 3x/2
im still having trouble understanding why the domain must be modified. the answer being outside the initial domain throws me off.
granite arch
then -3pi/2 ≤ 3x/2 ≤ 3pi/2
correct?
yes
well u could do a substitution if that helps
let u = 3x/2
sin u = 1/2
twilit granite
well u could do a substitution if that helps
now i know how to do it, but is there an explanation that can help me understand the reason for changing the domain?
sorry if its a bit much, but it seems like the initial domain is contradictory to the answer
granite arch
now i know how to do it, but is there an explanation that can help me understand...
its just that x is changed
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sonic ravine
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uneven notch
What have you tried so far?
sonic ravine
at first i thought of sin^2x + cos^2x = 1
uneven notch
yes
sonic ravine
hm
make a the subject of smth
then replace a of the first equation with smth from the second equation
sonic ravine
oh
final token
add the equations
final token
add both sides of both equations and you get a quadratic
uneven notch
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simple pewter
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restive river
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faint hearth
Try to think of how many times a fixed number will appear in the units place, how many times it will appear in the 10s place and how many times it will appear in the hundreds place
restive river
for 3 digits?
faint hearth
Yes
restive river
in the units place, we can have 3,5,9,6,4
faint hearth
No, for example, how many times will 3 appear in the units place if you listed out all the numbers
late orbit
In any place, you can have 3,5,9,6,4.
restive river
No, for example, how many times will 3 appear in the units place if you listed o...
oh that i dont know
thats what i am trying to figure out
how do i figure that out
do i need to list all the numbers?
late orbit
The sum of all numbers would just be 100(3+5+9+6+4)+10(3+5+9+6+4)+1(3+5+9+6+4)
late orbit
do i need to list all the numbers?
Nope. Just some common onservation.
faint hearth
The sum of all numbers would just be 100(3+5+9+6+4)+10(3+5+9+6+4)+1(3+5+9+6+4)
Times P(4,2)
Also, don't give out answers like this
restive river
why P(4,2)
faint hearth
A 3 digit number will look like 100a+10b+c right?
late orbit
Also, don't give out answers like this
No buddy. This is a little hard to think.
restive river
A 3 digit number will look like 100a+10b+c right?
yes
faint hearth
So if you added them all up, you would get 100(....)+10(....)+1(....)
restive river
yes
faint hearth
Try to figure out what those .... are
restive river
well it will be 3,4,5,6,9
in there
that i understand
but what i do not understand is the P(4,2)
i was faced with a problem like this in my textbook
and i did not understand why they use permutation
faint hearth
No buddy. This is a little hard to think.
No to what, your answer was wrong
The whole thing will be multiplied by P(4,2)
faint hearth
but what i do not understand is the P(4,2)
Because for the units place for example, a certain digit will occur P(4,2) times
restive river
yes
late orbit
You said don't give out answers directly, right ? If it was an easy question I would've given a simple hint. But this a little hard.
late orbit
The whole thing will be multiplied by P(4,2)
I was just typing that thing. Until you pointed it out.
restive river
Because for the units place for example, a certain digit will occur P(4,2) times
but how do we come up with P(4,2) times
faint hearth
Well, fix the units digit
There are 2 places left to fill
And 4 digits to fill them up with
restive river
ohhh
faint hearth
Same goes for the 10s place
restive river
ahhhhh
faint hearth
And same for the hundreds place
faint hearth
This is hard, like solomaniac pointed out
restive river
yeah it is but now that i know, i will keep practicing these types of problems
thank you very much!!
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maiden zinc
Pn a polynom of degree n
Q a polynom of degree p <= n
Why if PnQ = 0 we have p = n (they have the same degree)
For me it is false, degree of Q must be 0
magic thicket
A product of polynomials can only be 0 if one of the polynomials is 0
(on infinite fields)
(btw, the convention is that the zero polynomial has degree -inf, not 0, that way all formulas regarding the degree and polynomial operations hold also for the 0 polynomial)
maiden zinc
Pn a polynom of degree n
Q a polynom of degree p <= n
Why if PnQ = 0 we have p ...
ok thx
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flat lark
I need help with this ap statistics problem, not really sure where to start after doing the conditions
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I need help with this ap statistics problem, not really sure where to start afte...
graceful cosmos
What does "doing the conditions" mean
There's a very obvious mistake right at the beginning, can you spot that one?
flat lark
like figuring out random, 10% and normality
do i need a null hypothesis
is
Ha supposed to be p-hat > .75
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tender dagger
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tender dagger
a) 2 theta - 2sin theta
is there any way to get a value for theta?
for b) i got 5 x 4 - 2(2theta - 2 sin theta) = 13.4 and idk how to find theta
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winged timber
"construire Le point D image de A par la translation de vecteur BC
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limpid hemlock
could someone explain why this is true
devout snowBOT
could someone explain why this is true
limpid hemlock
Im kind of lost wht does B stand for in this chart
smoky nimbus
Just a variable to represent a number
smoky nimbus
It gives you points
smoky nimbus
So the transformation is -f(3x)
devout snowBOT
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late orbit
Yep.
smoky sentinel
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next moth
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next moth
does anyone know how to do this
i think im meant to do the partial derivatives of u and v and find the determinant but them im not sure if im supposed to take the inverse or something
restive river
indeed you must first find the partial derivatives, then put them in the Jacobian matrix (or its transpose, doesn't matter I think) and then calculate its determinant
next moth
right ive done that so its either d or e ig. im still not sure on which because theres some stuff in the definitions given about taking the inverse
restive river
Can you share your definition?
next moth
yeah one sec
theres that
and theres also this other one which follows from it
im pretty sure youre meant to use the second formula for this question but im kind of confused on the notation
restive river
the second formula gives you the Jacobian in term of the derivative of u and v according to x and y
the first one is the derivative of x and y as functions of u and v
So yes you should use the second one since you have u and v in terms of x and y
next moth
yeah that makes sense i was just being stupid
thank you for explaining it to me
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glass trellis
The set B = {1-t^2, t+t^2, 1+2t+t^2} is a basis for P2. The coordinate vector of p(t) = 1 + 4t + 7t^2 relative to B is what
devout snowBOT
The set B = {1-t^2, t+t^2, 1+2t+t^2} is a basis for P2. The coordinate vector of...
glass trellis
i can't help but feel like this is totally wrong
i tried $\begin{bmatrix}a1 + 1 & 0 & a1 & 0\a2 & -4 & a2 & 0\a3 & 2a3 & a3-7 & 0\end{bmatrix}$
woven radishBOT
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glass trellis
but where would i go from here?
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glass trellis
<@&286206848099549185>
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faint zinc
@glass trellis this is equivalent to asking how to express the vector [1 4 7] using the basis set [1 0 -1], [0 1 1], [1 2 1]
Do you know how to do this?
Do you see why it is equivalent?
glass trellis
<@385611499881889794> this is equivalent to asking how to express the vector [1 ...
does it not matter that the values are related? t and t²?
faint zinc
does it not matter that the values are related? t and t²?
It actually does not
Because the equality has to hold for all values of t
so you can think of the "vector" corresponding to t as being orthogonal to the one corresponding to t^2
glass trellis
so I can basically just replace t⁰, t¹, and t² with x1, x2, and x3?
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kindred kayak
can someone help me with this? i dont know how to solve for matrix with a variable
devout snowBOT
can someone help me with this? i dont know how to solve for matrix with a variab...
urban harbor
how can you use definition of linear (in)dependence here?
kindred kayak
how can you use definition of linear (in)dependence here?
linear independence is when each row has a pivot right
restive river
Try to set it up as a(1,-1,5) +b(6,-5,30) = (6,3,h). Try to solve for a and b using the first two coordinates. And once you have that, plug that into LHS to get what h should be from this equation
urban harbor
there's various ways to think about it, but at its core, linear independence is when a linear combination of the vectors equaling the 0 vector implies the only coefficients that work are the trivial one where all the coefficients are 0:
so you can think of it as a(1, -1, 5) + b(6, -5, 30) +c(6, 3, h) = 0 implies a =b = c = 0 for linearly independent vectors
that's pretty much directly the definition of linear independence
kindred kayak
Try to set it up as a(1,-1,5) +b(6,-5,30) = (6,3,h). Try to solve for a and b us...
1+6 = 6?
urban harbor
Try to set it up as a(1,-1,5) +b(6,-5,30) = (6,3,h). Try to solve for a and b us...
actually this isn't quite right subtly, in a set of linearly dependent vectors you will be guaranteed to have one of them be a LC of the others, but you can't be sure which one. Example: (1, 1), (-1, -1), (2,0)
you won't have a nontrivial combination where a(1, 1) + b(-1, -1) = (2,0)
restive river
that only applies if the two vectors you're considering aren't linearly independent. In this case that's not true
so it's fine
urban harbor
ya fair enough
kindred kayak
a = 6-6b
restive river
but that's a good point to make in general fs
kindred kayak
and then plug in a in the second equation
restive river
No, you have two variables a and b
kindred kayak
i thought i was substituting
restive river
then plug that into the third equation
urban harbor
a + 6b = 6
-a -5b = 3
realy easy to just add those equations together
kindred kayak
b = 9
kindred kayak
-48?
restive river
happy to help!
hidden hinge
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hidden hinge
Am I supposed to solve for x, y, and z?
Show that it's consistent?
Or do I show that the three vectors are part of the same plane
Where the three vectors are like <1,2,3> <1,2,4> and <2,4,7>?
Or are the three vectors <1,1,2> <2,2,4> etc?
And what does the last column have to do with it?
outer kestrel
Am I supposed to solve for x, y, and z?
i think maybe its that
or else itd jusy be trivial
magic pine
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red shale
can someone check this rq
pseudo rain
How would i do this mathematically.
devout snowBOT
How would i do this mathematically.
stone stump
to get from A to B you need to make 10 moves, and 6 of them need to be right and 4 need to be up
so you could for example write one such option as RRURUURRUR
pseudo rain
yes
stone stump
how many ways are there to write such a string
restive river
the number of rows (up moves) be m and the number of columns(the number of right moves) be n
pseudo rain
where did u get the -2 and 01 from
restive river
in this case, it's 10!/4!6!
pseudo rain
so its just the 4 posisble rows and the 6 possible columns to move to
the formula u used is there a gneeral form for it
like in what type of questions do you use the specific formula
for part b i used the same formula however i only got 10 possible ways u can get to point D
restive river
yeah
for part b
it's more subtle
you first move to D
and then in that many ways, move to B
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restive river
yeah
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hard knot
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hard knot
Having a problem
i made then equal to eachother
But i just ended up with 3sin2x = 2cos2x
Im guessing im supposed to just play around with it
Just to get it into the form of b they want
green kelp
Yeah ur not supposed to actually solve it
late orbit
Just write cos^2x = (cos(2x)-1)/2
green kelp
Try c2-c1=0
late orbit
You''ll get what you are looking for
late orbit
Try c2-c1=0
He's actually done that, unkowingly
green kelp
It's basically same as c1=c2
hard knot
oh yeah
green kelp
We just subtract c1 both sides
green kelp
So we have 4cos2x + 3sin2x - 6cos²x=0
hard knot
mhm
hard knot
ok ill do that