divide by 2?
#help-28
fleet harbor
fleet harbor
oh right
sharp lagoon
But a problem with row 4 is that it has two empty boxes.
So, we'll have something to add, but we won't know how much to put in box 1 and how much in box 2.
Pick a row or column that has only one empty box.
sharp lagoon
OK, sounds good.
fleet harbor
okay so then i do row 4 again but with 6-3x
fleet harbor
yeah that is the last box
sharp lagoon
The filled in boxes add up to 6 - 3x.
fleet harbor
oh
sharp lagoon
But we want the total with the last box to be 2x - 2 like before.
fleet harbor
so -4+5x?
sharp lagoon
Almost. You got the 5x right.
fleet harbor
-8?
sharp lagoon
Good, so 5x - 8.
sharp lagoon
OK.
fleet harbor
i did?
sharp lagoon
Yep, you did -5x + 6x = 1x.
fleet harbor
no cuz 5x-5x=0
sharp lagoon
Right, so then you have 0 + 6x.
fleet harbor
-4x
sharp lagoon
OK, and what do you add to -9 to get -2?
sharp lagoon
Good.
fleet harbor
so now we do row 4?
sharp lagoon
Oh, that was for row 4.
sharp lagoon
We did box 1 row 4 when we solved column 1.
fleet harbor
yeah
fleet harbor
so we need box 3 row 4
sharp lagoon
Right, we got 5x - 8 for box 4 in column 1.
sharp lagoon
Good, so 5x - 8.
From here ^
fleet harbor
so 5x-8+6-5x+6x-7
sharp lagoon
5x-8+6-5x+6x-7
Right, we did that here ^
sharp lagoon
Do you have boxes 2, 3, and 4 in row 1 unfilled in?
fleet harbor
yeah
sharp lagoon
Do you have all the other boxes filled in?
fleet harbor
yup
sharp lagoon
OK, good 🙂
fleet harbor
alright ill do that
sharp lagoon
alright ill do that
Sorry, I think I made a mistake.
We got that the sum for each row and column needs to be 2x + 2. Somehow, that got changed to 2x - 2.
It was still 2x + 2 here ^
sharp lagoon
Now we need it to add to 2x - 2 again.
But then it incorrectly became 2x - 2 here ^
Sorry about that.
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woven ermine
How to do this
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How to do this
woven ermine
The first step shows the equation without the sigma??
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woven ermine
<@&286206848099549185>
verbal leaf
Does thou needest help?
“Hū mæg ic þē helpan, good friend? If thou art in need of aid, speak, and I shall do mine utmost to ease thy burden. Whether thou seekest counsel, guidance, or a steady hand, let it be known, and I shall stand ready to serve.”
verbal leaf
<@&286206848099549185>
Hey buddy
grand hatch
How to do this
grand hatch
$\frac{\tan(\frac{x}{2^{r+1}}) + \tan^{3}(\frac{x}{2^{r+1}})}{1-\tan^{2}(\frac{x}{2^{r+1}})} = \frac{1}{2}\tan\left(\frac{x}{2^{r}}\right)\sec^{2}\left(\frac{x}{2^{r}}\right)$
woven ermine
Is this a result?
glossy valveBOT
Asteroid
grand hatch
its simple algebra
just factor out a tan and divide and multiply by 2. theres a typo in the second term
grand hatch
Aight i got it thanks
theres a small typo in the second term of what ive sent. the denominator is 2^{r+1}
grand hatch
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woven ermine
theres a small typo in the second term of what ive sent. the denominator is 2^{r...
👍
woven ermine
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frigid crag
I have a rectangle ABCD where A(0,0), AB=2BC, the point (5,10) is somewhere on BC and the point (7,-1) is somewhere on DC. I need to find the equations and lengths for AB and CD. Note that we don't know if any of the sides are perpendicular to any of the axes. No matter what I try I get 4 simultaneous equations, which I can't solve.
frigid crag
Should I try a simultaneous equation for the Pythagorean theorem between AB, B to (5,10) and (5,10) to A, and the equation for the product of their slopes?
Should I try a simultaneous equation for the Pythagorean theorem between AB, B to (5,10) and (5,10) to A, and the equation for the product of their slopes?
Then I could find the distances and slopes
Ok hold on maybe I solved it myself
sharp lagoon
I have a rectangle ABCD where A(0,0), AB=2BC, the point (5,10) is somewhere on B...
Well, (5, 10) is somewhere on BC.
Notice that BC is a horizontal line, which means the y value will be the same throughout.
What is the y value on BC?
frigid crag
That's just the way I drew it, it isn't perpendicular to any axis
sharp lagoon
Oh, missed that part, sorry.
late blade
waitwaitwait why are there differential equations in this problem
frigid crag
Not familiar with that term, I wasn't taught in English xd
late blade
what does y' mean
frigid crag
X' and Y' are just the names for the variables, you could use x and y
frigid crag
Didn't think about that
somber trench
i think u just find distance between (0,0) and (5,10)
and then x^2 + 2x^2 = (the distance)^2
idk if thats what ur doing rn
frigid crag
Exactly what I'm doing
somber trench
whats the distance formula
frigid crag
Wait no thats incorrect
The distance between 0,0 and 5,10 isn't the distance between A and C
frigid crag
Aw man
frigid crag
Thanks
I could say the distance of AB is twice the distance of BC but then I need two more equations
brisk obsidian
I could say the distance of AB is twice the distance of BC but then I need two m...
A ratio between 3/2 and 5/3 are the only ratios that seem possible unless I am misreading the problem.
frigid crag
The question was already solved by someone else, and it's from a university course so I think it should be solvable, I just have no idea how
frigid crag
https://www.geogebra.org/classic/cpjpzegn
Also I'm not sure what you linked, these are all magic numbers to me
brisk obsidian
The decimals are the lengths of the line segments AB and BC.
Alpha and Beta are the constraint angles for the rotation of the rectangle that keep (5,10) on the line segment BC and (7,-1) on line segment CD.
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@frigid crag Has your question been resolved?
tropic yarrow
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how do i graph this?
proven dirge
I would suggest factorising the numerator so you can simplify f(x)
hot herald
use the info you gained in part a)
hot herald
if you were to ignore domain restrictions
f(x) = x - 3
which when graphed would be a straight line, would you be able to graph that?
hot herald
when graphing, if in doubt make a table of values, but its a bit overkill for a straight line
finding two points is sufficient, so no real need to make a table just for that
choose 2 (reasonable) values for x, sub them into the equation to get the respective y values
hot herald
yes
hot herald
make your xy plane if you haven't already
ideally you'd want it big enough that so that the point (-5,-8) can be represented on there
what you've just done indicates that the points
(1,-2) and (2,-1)
are on your line
plot those two points and draw a line through them and extend that line in both directions
hot herald
did you not have a ruler?
hot herald
its generally a good idea to determine the intercepts
so it should be intersecting the x-axis at 3, not around 3
now, in part a), you also determine that there is a discontinuity at (-5,-8)
which you should represent with an open circle
hot herald
try to keep everything to scale
hot herald
the -5 is too close, compared to your 3
the point would be better represented more to the left
hot herald
weird how
hot herald
you're mixing up x and y coordinates
twin wolf
dr du
strong merlin
strong merlin
is this better
hot herald
yep
twin wolf
r u year 11
strong merlin
yeah
twin wolf
oh nice r u planning on doing 4u
hot herald
if in doubt and/or you have no idea what the function looks like,
make a table of values
hot herald
pick values for x
to get their respective y values to get points
and plot them
just like you did just now
hot herald
no
strong merlin
at 1
hot herald
its not asking you to draw a vertical line
its asking you to consider what's happening around x=1
what's happening to the left of x=1,
what's happening at x=1
what's happening to the right of x=1
strong merlin
um
glossy valveBOT
ℝαμOmeganato5
strong merlin
oh
to the right of x
x= 1
and to the left of x
x = 2
i think
is this right
hot herald
not x=
strong merlin
is f(x) my y value
hot herald
yes
strong merlin
whats the eqaution of the line for the graph
hot herald
its given to you
strong merlin
x=1 ?
strong merlin
huh
hot herald
that's your function
strong merlin
yeah
hot herald
and you've just used that to see what's happening around x=1
use that same thing to determine what you get at other x values
hot herald
yes
strong merlin
how do i know where the discontuity is
hot herald
from the conditions
which separates behaviour around x=1
is also what they requested that you analyse
hot herald
which part
hot herald
for x**<1**
for x**>= 1**
do you see that the function behaves differently to the left and right of x=1?
hot herald
so there is a potential discontinuity there
which the question asked you to anaylse
and you found that the left limit is 2
and the right limit is 1
hot herald
graph is incorrect
strong merlin
bruh
hot herald
recall what you calculated for the left limit
strong merlin
1
hot herald
no
strong merlin
2
hot herald
yes
hot herald
no
hot herald
so to the left of x=1, the function value should be approaching the value you calculated 2
and since that piece has <, indicating non-inclusion
that would be represented with an open circle there (at (1,2))
hot herald
both
strong merlin
😭
hot herald
because your graph didn't represent what you calculated
the limits you calculated were 1 and 2
so its discontinuous and there will be a break
you seemed to got them mixed up and/or try to forcibly connect the pieces
strong merlin
i honestly
strong merlin
the limits you calculated were 1 and 2
so its discontinuous and there will be a ...
dont know what this means
hot herald
can you identify left and right side limits looking at a graph?
hot herald
you follow the curve on the graph from each side towards the specified x value
and observe what's happening to the y value
e.g. from what you graphed in the previous question
as you follow the graph towards x=-5, the y values approaches -8 in both directions
from your calculations you determined
$$\lim_{x\to 1^-} f(x) = 2$$
$$\lim_{x\to 1^+} f(x) = 1$$
glossy valveBOT
ℝαμOmeganato5
hot herald
draw a circle at those points
and fill them in or leave it depending on whether equality is indicated in the piecewise function definition
hot herald
why (2,1)
strong merlin
idk
hot herald
(x value, y value)
you'd draw circles at
(1,2) for the left piece
and
(1,1) for the right piece
strong merlin
oh
hot herald
and fill them in or leave it depending on whether equality is indicated in the piecewise function definition
for the left piece you only have <, so leave that as an open circle
for the right piece you have >=, so fill that in
stuck fiber
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stuck fiber
Part (b)
I got the coordinate of A by getting in y(0), that gave me +-2
As you can see from the graph A is located where x is positive so its +2
I then subbed in y(2) to get -3
So A is located at (2, -3)
slate violet
I got the coordinate of A by getting in y(0), that gave me +-2
,calc (-2)^2 - 2(-2) - 3
glossy valveBOT
Result:
5
,calc (2)^2 - 2(2) - 3
glossy valveBOT
Result:
-3
this reasoning is incorrect
stuck fiber
this reasoning is incorrect
How come sry?
slate violet
you want to find the value of t that corresponds to point A
so y(t) = 0 means t = -2, 2 and that's good so far
then you want x(t) to be positive from the graph
slate violet
How come sry?
you don't know which direction the curve traces as t increases
stuck fiber
Ahhh true
slate violet
Result:
```
5
```
yeah, so subbing in t = -2 into x(t) and y(t), what must be the coordinates of A?
stuck fiber
Wait but how did you figure out t = -2
slate violet
then you want x(t) to be positive from the graph
.
stuck fiber
then you want x(t) to be positive from the graph
It was right?
I got the coordinates (2, -3)
slate violet
It was right?
no, t = -2
slate violet
I got the coordinates (2, -3)
and that doesn't make sense cause obviously the y-coordinate of A is 0
stuck fiber
Oh ya 😅
Ok 1 moment
Ok yes the coordinates of A is (5, 0)
Now I was stuck on what to do next
slate violet
Ok yes the coordinates of A is (5, 0)
right
stuck fiber
Cause I know there is some formula or equation or something that gives the exact tangent vector at a point
slate violet
so do you know how to find dy/dx if you have dx/dt and dy/dt ?
stuck fiber
I forget what it is though
slate violet
I forget what it is though
think of dx/dt and dy/dt as fractions
so you have $\frac{dy/dt}{dx/dt} = \frac{dy}{dx}$
glossy valveBOT
south
slate violet
yeah and remember we had that t = -2
slate violet
x'(t) = 2t - 2
actually for the vector representation, there's no need to divide
slate violet
x'(t) = 2t - 2
you can just sub in t = -2 into these
so the direction vector would be (-6, 4)
and yes that has slope -2/3 if you want to check
stuck fiber
Oh
stuck fiber
actually for the vector representation, there's no need to divide
So is it the vector for the tangent? Or is it the vector perpendicular to the tangent?
slate violet
So is it the vector for the tangent? Or is it the vector perpendicular to the ta...
yep, it's the tangent vector
stuck fiber
yep, it's the tangent vector
Ok so then the tangent we want is (4, 6), right?
slate violet
Ok so then the tangent we want is (4, 6), right?
nope
slate violet
it's just (-6, 4)
stuck fiber
yep, it's the tangent vector
But you said its the tangent vector no?
We want the vector perpendicular to the tangent
For the direction vector in our line equations
slate violet
But you said its the tangent vector no?
yes, that vector is already tangent to the curve at A
slate violet
For the direction vector in our line equations
you're confusing it with 3D or something, where the normal vector is perpendicular to any vector on the plane
(dx/dt, dy/dt) has slope (dy/dt) / (dx/dt) = dy/dx already
stuck fiber
Wait I will draw a diagram 1 sec
stuck fiber
yep, it's the tangent vector
The tangent vector is the blue one, no?
Its tangent to the curve
We want the red one right?
slate violet
We want the red one right?
no, you want the blue one
the tangent line doesn't change definition just cause you have a parametric function
stuck fiber
Ohhhhhhhhh
I misread the question
I thought we wanted the red vector for some reason 🤡
Okok
So (-6, 4)
That means the vector line equation is (-6, 4)t + (5 ,0), right?
Yep I just checked, thats the answer
Thank you so much for your help!!
❤️
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faint iron
When writing the negation of a statement, I've observed that sometimes the quantifier is negated, while other times it is not. In what situations would the quantifier be negated as opposed to left unchanged? eg, ∀ into ∃
spiral vigil
~[∀x : f(x)]
is equal to
∃x : ~[f(x)]
faint iron
does the colon mean implies
onyx glen
so, it's a decoration that does not have any meaning on its own
onyx glen
When writing the negation of a statement, I've observed that sometimes the quant...
when did you observe a negation in which a quantifier did not swap?
faint iron
in some cases when the quantifier simply provides a condition
so for example
well actually I don't have an example in hand, but sometimes the statement begins with 'for all x, p=>q,' where the 'for all x' isn't negated
onyx glen
you sure?
faint iron
if I were to say, for all x integers, x^2+1 is not divisible by 3, would the first part of the statement be negated?
or would it be, for all integers x, x^2+1 is divible by 3
onyx glen
the negation of "for all integers x, x^2+1 is not divisible by 3"
would be "there exists an integer x such that x^2+1 is divisible by 3"
idk where you are getting a retained forall from
can we be sure you did not misunderstand or misremember
faint iron
yes but perhaps it was something more trivial than this
i'll try find an example
in the meantime, could you take a look at this one
Ok getting back to the quantifiers,the question was saying something like "if n is a natural number, p=>q
onyx glen
in the meantime, could you take a look at this one
what's the goal here? prove, or just translate?
onyx glen
ok, any progress thus far?
faint iron
I've expanding it into its factorial notation, which gives me p(p-1)...(p-r+1) / r! , which has a factor of p clearly but i gotta prove the rest of the expression is an integer
and whats confusing me is that I don't see how the condition of being prime plays into anything
onyx glen
I've expanding it into its factorial notation, which gives me p(p-1)...(p-r+1) /...
better to do it in a different way
pCr * r! * (p-r)! = p!
now, what do you know about prime numbers and divisibility?
faint iron
p is divisble by p and 1
onyx glen
this is missing a very key detail even for a definition of "prime number"
a crucial detail
faint iron
ah, is the idea that the LHS is divisble by p, yet the (p-r)! is not divisble by p clearly, hence pCr is?
onyx glen
ah, is the idea that the LHS is divisble by p, yet the (p-r)! is not divisble by...
yes but too early for that
faint iron
what is this crucial detail
onyx glen
the fact you misquoted the definition of a prime number is something that needs to be addressed
like
let's put it this way
everybody is divisible by themselves and 1. even numbers like 8465
what makes primes special?
faint iron
ig i've omitted an only
onyx glen
ig i've omitted an **only**
yes you have. and it's quite a grave omission!
faint iron
yep
althought
what would exactly be the wording to use to prove (p-r)! is indivisble by p despite it being quite obvious
onyx glen
right anyway the other thing you should know is
for a prime p and integers x and y, if xy is divisible by p, then at least one of x and y must itself be divisible by p
faint iron
yep that makes sense
onyx glen
(this does NOT work if we relax the assumption that p is prime)
faint iron
yeah because
(x-r)! can be divisble by x since the factors within (x-r)! can multiply w each other
onyx glen
what would exactly be the wording to use to prove (p-r)! is indivisble by p desp...
the property i just stated is within spitting distance of "the product of two non-multiples of p is a non-multiple of p"
faint iron
two or more right
onyx glen
sure
you can get the "two or more" by induction if you so wish
and r! * (p-r)! is a mighty long product but all of the factors going into it are less than p and thus aren't divisible by it
dusk drum
if we have a bond bearing n coupons per year and we are given an interest rate of x per annum, do we just generallyassume that the interest rate is compounded n times per year?
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wide wind
What is the difference/similarity of "Relation" and "Incident system"? Say $a \in A$ and $b \in B$ $$a ; \mathscr{R}; b \quad \text{and} \quad a ; I ; b $$
glossy valveBOT
Good
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fast peak
an incidence system is a specific type of relation
or rather you just think about it differently
say, points on lines
vertices on edges
things like that
cinder wolf
.opem
.opem
.open
can somebody please solve this for me and give me a step by step explanation?
lime ether
help you
lime ether
not solve it for you and give you a step by step solution
lime ether
wouldn’t be here if i couldn’t sir
cinder wolf
alright so where do we start
lime ether
vertical asymptote
what conditions should be satisfied if a rational function has a vertical asymptote
and do you know what that is
cinder wolf
ok so is my vertical asymptote 5?
spring vessel
I’m trying to solve a Ax=b using low precision LU factors. A is a highly il conditioned random matrix.
At first, I was randomly generating A and b randomly on MATLAB, and I was trying to solve for x.
However, my supervisor suggested that I instead randomly generate x, compute b = Ax, and then feed the A and b to my solver. He mentioned something about this being much better in terms of complexity or cost. Does anyone possibly have any idea of what he was referring to?
cinder wolf
@lime ether
lime ether
<@834774466290253834>
don’t ping me brother it’s been like 10 seconds
cinder wolf
my bad
i just pinged cus it got buried
wont againt though
i also think my horizontal asymptote is -5/4
lime ether
$f(x) = \frac{5(x-2)(x+2)}{-4(x-5)(x+2)}$
glossy valveBOT
knief
lime ether
ok so is my vertical asymptote 5?
it’s an equation not just a number
cinder wolf
yes but for graphing purposes
cus im having difficulty with the graphing
I have my numbers i believe
cinder wolf
Vert: x=5
Horizontal: -5/4
x: 2,0
y: 0,-1/2
hole at x=-2,0?
cinder wolf
so my issue is how do i put this on the graph now
cus i get two chances and put it on the graph the first time and got it wrong
cinder wolf
so i have one more chance before it moves me onto another question
cinder wolf
yeah i had hole is x=2 at first
lime ether
no you had x = -2
cinder wolf
sorry yeah
cinder wolf
so its a point
lime ether
but x = -2,0 is interpreted as x = -2 and x = 0
cinder wolf
ok gotcha
so how do i put this on the graph
Vert: x=5
Horizontal: y= -5/4
x: (2,0)
y: (0,-1/2)
hole at x=-2
so this is my final list?
lime ether
horizontal is an equation
not a number
and use parentheses for points
(2, 0)
(0, -1/2)
better
what happens when you click the feature
like for vertical asymptote does it give you an option to plot a dashed vertical line
lime ether
how do i plot y= -5/4 though?
what kind of line is it
cinder wolf
its a horizontal dashed line
lime ether
it’s the set of all points whose y coordinate is -5/4
lime ether
its a horizontal dashed line
smack it on at y = -5/4
cinder wolf
is there a way to send a example in discord
lime ether
idk what you mean
cinder wolf
yeah but where is that on the graph
lime ether
5/4 units below the x axis
cinder wolf
i get that but idk how to put that there
like reading it is fine
but where on the axis
lime ether
what integers is it between
cinder wolf
its like inbetween 1 and 2 right?
cinder wolf
or sorry between negative 1 and 1.5
lime ether
1.5 isn’t an integer but yes it’s between -1 and -1.5
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unborn acorn
hi again
unborn acorn
show that any rational function can be expanded into a power series in any disk where it is defined
gritty rose
unborn acorn
f(z) = P(z)/Q(z) ; P and Q polynomials and i dont know
gritty rose
so just the definition of rational function
solid inlet
What’s the polynomials ?
solid inlet
And ?
unborn acorn
Something like (x+a)(x+b)?
?
solid inlet
Try to take a picture
raven river
Woops, my bad. It's this definition:
unborn acorn
what is this?
raven river
An example of polynomial.
unborn acorn
it's not even my question
solid inlet
Woops, my bad. It's this definition:
That’s so simple
But like give me the equation
I’ll solve it
raven river
show that any rational function can be expanded into a power series in any disk ...
Is this the main question?
unborn acorn
yes
idle kelp
You should think about how transforming Q(z) into a geometric series helps you get the power series for P/Q and what is required to make this possible.
idle kelp
By thinking about it for longer than 15 seconds before asking for the next step
unborn acorn
well i really dont know, Q(z) = a0 + a1z + a2z² + ...
Q(z) = sum of
a_n z^n
how do i get a geometric series from this?
@idle kelp
idle kelp
You should probably try this with a concrete example, like write x/(1+2x+x^2) or something easy as a power series then look at what you needed to do to write down that power series and use it to turn it into a proof.
gritty rose
you don't need exponents. use fundamental theorem of algebra
oh i guess you're trying to account for multiplicities. ignore me
but you shouldn't need R(z) either
unborn acorn
so it's just Q(z) = (z-z0)^m0 (z - z1)^m1 ....
uhm??
P(z)/Q(z) = P(z)/(z-z0)^m0 (z - z1)^m1 ....
idk
idle kelp
I don't know where you are going with this and you should really just try writing the concrete example I gave you as a power series to understand the mechanics of it
The idea is that if you do the exercise I suggested, you would see that you can decompose Q(z) into two functions M(z)N(z) and 1/N(z) can be written as a geometric series and P/M is a polynomial. Then a polynomial times a geometric is a power series.
Once you understand that, you need to go backwards and think about what can go wrong with Q and show that these are repairable or not relevant
unborn acorn
ok
idle kelp
but above all, trying an actual example to see what is mechanically happening is important
You are never going to write a proof by just writing down an arbitrary factored form of a polynomial, staring at a long list of zi's and mi's with no sense of what to do and then declaring "idk"
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@unborn acorn Has your question been resolved?
spare pecan
Could you help me understand this problem? I kept getting it wrong
devout valley
Can you explain how you went about solving it?
brave holly
those two solutions you gave are redundant btw
spare pecan
im not asking for the answer but I just need guidance on how to solve these equations in general. like a step by step
devout valley
That's why I asked for how you went about solving it(!)
tulip oriole
isolate cos(theta)
???
profit
brave holly
and know the trigonometry table of common angle-value pairs
devout valley
Pretty much, that's better 
spare pecan
another one?
devout valley
Do the same thing for the second to last line
(you also preferably should add in some 
for both of these, to indicate that you're multiplying all terms by 3, as a minor comment)
spare pecan
ohh okay
devout valley
Well, if I were being extra strict, I would not be happy with it, as it's only the context (and the last line she's done) that give away that the intention is to multiply everything by 3, so I would count it as necessary
However some people may be less strict than that, so there's that 
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eternal yoke
stuck on this problem
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stuck on this problem
eternal yoke
what I tried. F=300N is given
wide wind
You should probably give the whole question.
vocal sonnet
what is the situation that is given for the overall group of questions
eternal yoke
find moment about point O
I'm not too sure but I think the horizontal component produces no rotation
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umbral dome
you can either split into components and use the moment arm for each component or use the sin formula directly
frigid phoenix
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frigid phoenix
initially I've picked Nx = p(x^2+y^2) and My = 0
is that valid as well?
i could have picked also the other way around of what they did, right?
also, as another note, shouldn't i M and N have also a +c for completness?
umbral dome
there are many different choices of M and N, they are just trying to choose one that is simple
yours would work as well
umbral dome
also, as another note, shouldn't i M and N have also a +c for completness?
this also goes back to the fact that we are just using one choice of M and N, not attempting to find a "most general formula"
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fickle shore
Hello
clear pumice
r is just the ratio that the function increases per x
well r - 1 is the ratio
in that case
in an exponential relationship you know that for each (integer) x, the next number will be 1 + r times the previous
which applies between any value
you just have to find the factor it multiplies to get the next value
fickle shore
I don't understand, how to find r in this question?
clear pumice
take 2 consecutive values such as day 1 and 2
fickle shore
Okay
clear pumice
divide day 2 by day 1 to get the factor
fickle shore
Ahh
clear pumice
you'll get the same thing if you divide day 3 by day 2
that result is r - 1
(if using the formula a(1 + r)^x)
fickle shore
So now it's, 250000(1+2)^x? And we put x value for ever option
clear pumice
not 1 + 2
fickle shore
And match it with ans given?
fickle shore
What why is that
clear pumice
since you're doubling the value every x
2^x
not 3^x
that would be tripling
1, 2, 4, 8, ...
vs 1, 3, 9, 27, ...
fickle shore
So r is 2 right?
fickle shore
i think it's better if you just use ar^x
Why in this case?
clear pumice
i think it's better if you just use ar^x
in this equation
clear pumice
Why in this case?
it makes it a bit less confusing
since in the other case r = 1
(1 + 1)^x = 2^x
clear pumice
how did you get that
fickle shore
Thanks
clear pumice
i mean that's the way to brute force it i guess
do u know how to do it with log though
(the proper way)
clear pumice
dang
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limpid pivot
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limpid pivot
professor was mad that we had to provide us a “take-home exam” so he gives us this, which we have never done officially
I need step by step help
I know that, for instance, between x= -7 and -4 the slope will be 2x
and then - 2x between -4 and -1
yeah?
please. tell me more.
it is the tangent line of a specific point
rate of change
so f’(-4) = 0
what about the point near -7,0 ?
i don’t appreciate how it’s not exactly -7
alright so we have our horizontal tangent lines, 7 of them
now what
can I use the help command?
<@&286206848099549185>
thanks thus far
have a sound sleep
that is very helpful
uncut badge
lol
uncut badge
i didnt say allat
i thought the prof part was funny
what part do you need help with
limpid pivot
ohhhhzzz
uncut badge
that mq didnt
limpid pivot
hahah was mq right about no horizontal tangent line at x=0
uncut badge
oh okie
pseudo cape
[because a limit, by def the derivative is a limit, only exists when the limits from both sides exist (I mean takes on a finite value with this) and match]
pseudo cape
That's not the limit that's the definition of the derivative
And geometrically, that's just the slope of the tangent at a
I don't agree with the -1
Make that a -1/2
limpid pivot
the limit is zero right? why wouldn’t they both be the same if it’s continuous?
pseudo cape
This is the function
And this is the derivative
The function is continuous and yet it's derivative is not
[This is just an example, not relating to your exercise]
limpid pivot
woah yes. i’ve seen this before
pseudo cape
Generally, whenever there is a sharp edge, the derivative will be discontinuous (jump discontinuity)
limpid pivot
how would you algebraically solve for the limits from both sides of the function from my question?
pseudo cape
You wouldn't, because the piecewise function isn't given
You could only deduce the algebraic representation graphically, but that could be inaccurate and just not worth it
Just do it as suggested before, graphically using the tangent
limpid pivot
okay, what if the function from (-1, 1) is actually curved and not a straight line as we assumed?
pseudo cape
From (-1, 0) you mean
limpid pivot
oh yes
pseudo cape
Well, it doesn't matter that much; if it's curved then the derivative would not be a horizontal line but going into a very small dent; and if we can't notice the curve then you won't notice the dent either
Put your ruler down accordingly; if it fits the graph on (-1, 0) then you may assume it's actually line
limpid pivot
too pixelated to be sure of it
but
since he has brought this up once
it’s almost definitely straight
meaning
horizontal lines at y=-1/2 and y = 4?
pseudo cape
on the interval (-1, 0) it will be the horizontal line y = 4, yes
On (0, 4), it will be the horizontal line y = -1/2, yes
limpid pivot
on the interval (-1, 0) it will be the horizontal line y = 4, yes
is there any way to prove this using mathematical/linguistic notation? i’m interested in how one might represent this in words and stuff
pseudo cape
This whole limit evaluation directly translates to adjusting your ruler to match the curve at some point
limpid pivot
tysm.
still a lot to chew for a newbie like me
super cool to see that
another question
in dy/dx = h’(x)
on the interval we’ve been discussing (-1, 0)U(0,1)
does this derivative ever actually touch x=0? because above the limit implies it never actually touches 0?
pseudo cape
(-1, 0) u (0, 4) even. No it doesn't touch the x-axis, it has a jump. For (-1, 0) it's the horizontal line y = 4, then it jumps down to the horizontal line y = -1/2
Inbetween the jump you should draw a little open circle (or whatever your class uses) to indicate that we can't determine the value at the position x = 0
limpid pivot
that’s what my follow up question was going to be
awesome
I think I can crack this now
i’ll be back with a sketch
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pseudo cape
With what part?
limpid pivot
pseudo cape
Can you send it so that the left part is in the picture too
pseudo cape
Are the dark points indicating that the derivative has a zero there?
pseudo cape
limpid pivot
the last one
pseudo cape
That's not true then
The derivative is zero where there would be a horizontal tangent
limpid pivot
the one near 7 isn’t a zero
pseudo cape
Yes
limpid pivot
so then, picking up from the horizontal at -4 coming from the left, would I draw an -x^2 shape
connected to that line?
pseudo cape
If you are very strict the derivative of the line left to -7 does not exactly have slope -4, but something more like -3.9
Because it hits the x-ais not exactly at -7
pseudo cape
so then, picking up from the horizontal at -4 coming from the left, would I draw...
No, it'll look like a line, because if you take
Then the derivative of that will be
But not a horizontal line this time
pseudo cape
Because the slope increases
limpid pivot
pseudo cape
It's the derivative of a cubic function, not the derivative of a quadratic
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limpid pivot
.reopen
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limpid pivot
okay okay
so from y= -3.9
we’re gonna have a slope of 2x
then that hits 0
and this line will start from where y = 3.9 ended
wait no
it will start from somewhere outside of the graph, below quad iv
3**
pseudo cape
Be careful with the (-7, -4) interval
At -7, is the derivative from the right positive or negative?
limpid pivot
nega
limpid pivot
posi
pseudo cape
yes
The value will be positive and the slope of that line
It seems from the picture it's around 2
Again maybe a bit bigger because the zero isn't at exactly -7 but I don't think your prof cares
pseudo cape
With some shift to the left
Also it must have a negative factor
Because it's opened downwards
And shifted upwards too
Anyways, I will go to sleep, it's pretty late
limpid pivot
is this question evidence of a professor who likes to give hurt?
pseudo cape
ah, and this is on-going?
limpid pivot
yeah 3/8
pseudo cape
Well, I will head to sleep now as I said, good luck
limpid pivot
sounds good, thanks for all the help
also
can I ask
what it is that enables all of you helpers to keep doing this?
just the love of knowledge?
pseudo cape
Your professor might not care about the -3.9 by the way and you could make it -4, I don't know how strict he is on this
pseudo cape
what it is that enables all of you helpers to keep doing this?
I just like helping when I have time
I love maths
pseudo cape
It's all a process, I'm still in high school (well, and going to uni in parallel)
There will always be some area you have to get fluent in
limpid pivot
that is super awesome and respectable
pseudo cape
It's not like you can master everything and you're done
lyric jewel
hello would u be able to help my friend in https://discord.com/channels/268882317391429632/1347751082675277825
limpid pivot
sounds like hell
a tropical one
I made a U turn when I turned 20 years old
I realized men were made to be more than a blue collar slave
pseudo cape
Are you majoring in maths?
limpid pivot
chemistry, ex psychology; ex ex journalism
I just hope i’m ready
wow, bilingual and math stud?
pseudo cape
I'm from Germany
pseudo cape
Yeah
limpid pivot
I forget that english is a global language
pseudo cape
Alright, I will head to sleep now
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regal scroll
guys if i differentiate x*d/dt(sec^2(theta))dtheta/dt what do i get?
regal scroll
it is confusing me
narrow ermine
Do you have a picture or any better way of formatting it?
modest obsidian
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hot herald
is that the full thing?