@true spire ?
#help-10
royal basin
obtuse pebbleBOT
@true spire Has your question been resolved?
tawdry vigil
This is a problem from the SAT. We have to find the points B(-1, 10) and C(8, 10) which I got correct. But when I substituted B(-1, 10) into the equation of y=a(x-3)^2 I got B, which is the incorrect answer. According to the answer we had to use C(8, 10) to arrive at A, which was the correct answer. Why can't we use either value here like we can in any other question?
obtuse pebbleBOT
This is a problem from the SAT. We have to find the points B(-1, 10) and C(8, 10...
@tawdry vigil Has your question been resolved?
obtuse pebbleBOT
@tawdry vigil Has your question been resolved?
tawdry vigil
<@&286206848099549185>
timid silo
Isnt there a mistake in the question. If the vertex of the parabola is at (3,0) and points B and C have the same y-coordinate then the average of their x-coordinates should be 3 because of symmetry. So point B should be (-2,10) which would lead to the answer A instead of B
So the drawing is incorrect i think
clever pier
coarse laurel
can anyone help me
fierce lagoon
.close
obtuse pebbleBOT
fierce lagoon
You have a channel already
coarse laurel
ya i know but i cant find it
timber fox
coarse laurel
thanks sorry about that
obtuse pebbleBOT
Available help channel!
crude estuary
obtuse pebbleBOT
crude estuary
Im thinking
A)7
B)6
C)4
hard igloo
A is seven
crude estuary
b and c?
hard igloo
I dont know yet can you explain how did you derived them
crude estuary
so its an exponential function
t/6 the exponent
every 6/6 it doubles
im assuming
hard igloo
But at t=0 the value is 7 and t=6 the value is 10
crude estuary
so how often would it double then
hard igloo
No clue
obtuse pebbleBOT
@crude estuary Has your question been resolved?
obtuse pebbleBOT
@crude estuary Has your question been resolved?
crude estuary
tardy epoch
equation for h is cut off
tardy epoch
,calc 3 * (2)^(0/6) + 4
warm shaleBOT
Result:
7
tardy epoch
is t >0 ?
tardy epoch
screenshot the whole page
crude estuary
crude estuary
cant
tardy epoch
as t increases, h(t) increases to + infinity
crude estuary
what about c
tardy epoch
as t increases, h(t) increases to + infinity
this is also used to contradict c)
if t > 0
if t < 0, then it's the left horizontal asymptote
crude estuary
so 4
obtuse pebbleBOT
@crude estuary Has your question been resolved?
worthy cargo
.close
obtuse pebbleBOT
Available help channel!
fallow rampart
I need help with why this simplifies the way that it does. I don't understand the -65
obtuse pebbleBOT
I need help with why this simplifies the way that it does. I don't understand th...
shadow tangle
They first distributed the 4 into the numerator, then they turned the -9 into fractional form and found a common denominator. Finally they combined the two fractions.
fallow rampart
-9/1?
fallow rampart
1
shadow tangle
Not quite
The denominators are 9 and 1
Could you possibly multiply one of the denominators to make them equal to eachother?
fast cliff
well my channels was just ignored
fallow rampart
I'm sure there are limited people with limited knowldge. Did you ping a tutor?
fast cliff
yes
fallow rampart
shadow tangle
Great job
fallow rampart
ty
shadow tangle
Np
whole nimbus
hi
hexed gull
I need help with finding the length thats right of the letter B
as a tip: the two inner triangles have the same side relations as the outer triangle
since their angles are the same
once that becomes clear to you solving for any length in this diagram becomes simple
@whole nimbus
whole nimbus
as a tip: the two inner triangles have the same side relations as the outer tria...
wdym by side relations
the ratio between adjacent sides?
hexed gull
note that this red triangle has the same angle as the outer triangle
which means it's just a smaller version of the outer triangle
both of them share a right angle and the angle on the very left, which means the third angle is also equal.
the outer triangle has side lengths 1.7m 5.6m 5.85...m right?
whole nimbus
yes yes
hexed gull
which means the red triangle has the same lengths times a scaling factor
since its the same triangle just smaller
and you already know the hypothenuse (longest side) of the red triangle has the length 5.6
which means you can calculate the scaling factor
whole nimbus
5.6 and 1.7?
hexed gull
wdym
whole nimbus
dont i need to get the scale factor between 5.6 and 1.7?
hexed gull
between 5.6 and 5.85...
since the longest side of the red triangle has length 5.6
and the longest side of the outer triangle has length 5.85...
just to be clear this is the outer triangle
is that part clear?
whole nimbus
oh...
whole nimbus
is that part clear?
yes yes
hexed gull
then you get the scaling factor by dividing the length of the two longest sides: 5.6 / 5.8... = 0.9568805766427723...
whole nimbus
where do we apply that scale factor?
hexed gull
so now you know the red triangle is about 95.68% the size of the blue triangle
whole nimbus
mhm
hexed gull
and you know the blue triangle has lengths 1.7m 5.6m 5.85...m
whole nimbus
yes
hexed gull
so just multiply these lengths with the scaling factor
and you have all side lengths of the red triangle
including the length you're looking for
whole nimbus
so do I apply that scale factor to 5.85
because if I have that I can use pythagorean theorem
hexed gull
you're already done if you just multiply the values:
0.9568805766427723*1.7 = 1.6266969802927127
0.9568805766427723*5.6 = 5.358531229199524
0.9568805766427723*5.8... = 5.6
that way you know the red triangle has the smallest side length 1.62... (the length you're looking for)
second longest side length 5.35...
and the longest side length 5.6
which was already given
whole nimbus
second longest side length 5.35...
Im very sorry but which part of the triangle is this length?
hexed gull
oops I wrote the (the length you're looking for) in the wrong line
you're looking for 1.6266...
hexed gull
shouldnt it be 5.8 x scale factor?
because 5.6 is the second longest length of the blue triangle
whole nimbus
I can apply it to 1.7 and it still works?
hexed gull
I can apply it to 1.7 and it still works?
it's the same triangle, just smaller yes
longest_length_of_blue * scale_factor = longest_length_of_red
2.longest_length_of_blue * scale_factor = 2.longest_length_of_red
shortest_length_of_blue * scale_factor = shortest_length_of_red
maybe this makes it easier
whole nimbus
but this length is the longest length in blue triangle?
hexed gull
5.6 * scale_factor = 5.35...
hexed gull
but this length is the longest length in blue triangle?
no the longest length in the blue triangle is 5.85...
whole nimbus
you're already done if you just multiply the values:
`0.9568805766427723*1.7 = 1...
exactly, you applied the scale factor to 5.8... which is the longest in blue
whole nimbus
yes
hexed gull
yes
was confused since it points at the second_longest_length of the red triangle
but in regards to the blue triangle yes the full side is the longest length of blue
hexed gull
yes
hexed gull
yep
hexed gull
np :)
btw if you want to know the quickest way to solve this
you can calculate the triangle's area
via 1.7m * 5.6m * 0.5
in general Base * Height * 0.5
and then divide that area by the longest side (5.85...)
and you already get the result
whole nimbus
how does this work?
obtuse pebbleBOT
@whole nimbus Has your question been resolved?
obtuse pebbleBOT
@whole nimbus Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
untold mist
i only need help on part c. i know that the derivative is underfined at x= 2, and x=4 but i dont know the third one
obtuse pebbleBOT
i only need help on part c. i know that the derivative is underfined at x= 2, an...
@untold mist Has your question been resolved?
slate kayak
obtuse pebbleBOT
https://cdn.discordapp.com/attachments/808388205867040778/1129250872795332639/im...
slate kayak
Anyone can help? I counted and got 15
high lily
show work
slate kayak
show work
I counted all multiples of 4
but okay let me write it out
4 -> 1
8 -> 1
12 -> 1
16 -> 2
20 -> 1
24 -> 1
28 -> 1
32 -> 2
36 -> 1
40 -> 1
44 -> 1
48 -> 2
adding everything you have 12+3 = 15
@high lily
high lily
note that two factors of 2 will result in another factor of 4
slate kayak
note that two factors of 2 will result in another factor of 4
yeah
high lily
you didn't account for those
slate kayak
i tried rewriting it as (2^(2x))
slate kayak
but the result you get from multiplying 2 and 6
i understand you, but could you tell me a number i missed
because i can't seem to see any number i missed
high lily
you don't seem to be understanding me
slate kayak
if i'm understanding you right
you are saying that i need to account for every 2 occurences of 2?
high lily
yes
slate kayak
6 = 2*3 so multiplying by 2 already gives you 2 of 2's
so i have to find numbers that have only 1 2 and then multiply with 2
high lily
2 , 4 , 6 , 8 , 10 , 12 , 14 , 16
you've only considered the bolded numbers
but not whether the other values contribute to more factors of 4
slate kayak
2 , **4** , 6 , **8** , 10 , **12** , 14 , **16**
you've only considered the bol...
14 has 1 factor of 2
so multiplying by 2 gives 2 of 2's
but 28 is already included
same with 10
10 has 1 factor of 2 -> 5*2
slate kayak
sorry, i lost you
theoretically, we want to count the number of 2 2's in range [1,50]
right?
high lily
yes
slate kayak
why would i count 20 (for example) twice then
slate kayak
10 = 5 * 2
20 = 5 * 2*2
high lily
like
10 * 10 * 4 / 4^n
are you implying that the max n where that's an integer is 1?
slate kayak
like
10 * 10 * 4 / 4^n
are you implying that the max n where that's an integer i...
it would be 2, right?
high lily
yes
(4 * 8 * 12 * ... * 48)/4^15 is an integer
but you've ignored
2 * 6 * 10 * 14 * ... * 50/4^n
slate kayak
oh i guess then the question is more along the lines of how many pairs of two 2's are there rather than considering each number?
high lily
yes
slate kayak
(4 * 8 * 12 * ... * 48)/4^15 is an integer
but you've ignored
2 * 6 * 10 * 14 * ...
there's 38 numbers that aren't multiples of 4
from each you can pick 1
so 17 pairs of two 2's
so 17 + 15 ?
high lily
there's 38 numbers that aren't multiples of 4
some of those are odd
and aren't multiples of 2
slate kayak
ohh
okay let me see
wait
okay 12 groups of all numbers before multiples of 4
in each group there's one multiple of 2
so 12 there as well
?
high lily
the way i'd count the total factors of 2 in the prime factorisation is
count multiples of 2
count multiples of 4
count multiples of 8
count multiples of 16
count multiples of 32
(from 1 to 50)
slate kayak
the way i'd count the total factors of 2 in the prime factorisation is
count mul...
you mean the number of 2's?
isn't it total multiples of 2
or is that how it's called
high lily
i mean what i said
slate kayak
yes but i don't understand
this
the way i'd count the total factors of 2 in the prime factorisation
high lily
oh that,yeh , total number of 2s in the prime factorsation
slate kayak
the way i'd count the total factors of 2 in the prime factorisation is
count mul...
wouldn't this be double counting
high lily
no
slate kayak
or is the onus on me to not double count
for example:
count multiples of 2
count multiples of 4
count multiples of 8
count multiples of 16 -> 16, 32
count multiples of 32 -> 32
do i only count 32 once?
i'll get more 32's when i go up i guess
high lily
note that 32 will appear in all 5 lists
slate kayak
yes
high lily
32 = 2^5 and contributes five 2s in the prime factorisation
overall you count the right amount
slate kayak
overall you count the right amount
owh i see
had the range been extended to 64
wait
oh then i'd have to add another row
high lily
count up to multiples of 2^k
high lily
yeh
high lily
yeh
slate kayak
oh nicee
high lily
,w 50!/4^23
slate kayak
what is this method called? considering it seems like it can be generalized
warm shaleBOT
high lily
,w 50!/4^24
warm shaleBOT
slate kayak
,w 50! mod 4^(24)
warm shaleBOT
slate kayak
,w 50! mod 4^(23)
warm shaleBOT
high lily
it probably has a name, but i don't know it
high lily
yeh
high lily
essentially what we just went through
slate kayak
you're right idk where the logic of that comes from though
it doesn't seem obvious at first glance tbh
but i guess yeah it seems
very reminscent of what we did
high lily
multiples of 2 (each multiple of 2 is counted once so far)
multiples of 4 (multiples of 4 have at least an additional factor of 2)
multiples of 8 (have at least one more on top of that)
etc
slate kayak
multiples of 2 (each multiple of 2 is counted once so far)
multiples of 4 (multi...
i still don't get it tbh, maybe i'm missing something
i get how to solve the question and what to do
not sure how to make this logic myself though
2 -> 2,4,6,8,10,...
4 -> 4,8,12,....
8 -> 8,...
.
.
.
it conveniently counts the number of factors though π
obtuse pebbleBOT
@slate kayak Has your question been resolved?
slate kayak
ah i see
i think i vaguely get it
thansk for trying to help me w the intuition
and thanks for showing me the procedures which i understand
summer owl
Just making sure if this is correct, there should only be one restriction for this question right?
obtuse pebbleBOT
Just making sure if this is correct, there should only be one restriction for th...
leaden ginkgo
what did the question ask you to do
summer owl
simplify the rational expression and state all the restrictions
leaden ginkgo
the restriction would be -2 too, no?
summer owl
i think restrictions are the values that are located in the reduced expression and the original expression
i might have just answered my own question lol
lethal sand
i think this question is rather for your teacher/marker. It depends on whether or not they think x!=-2 is clearly implied in the expression
summer owl
iβll make sure to ask them about it tomorrow π
donβt know what happened here, these are from the answer sheet that my teacher gave us btw
leaden ginkgo
which step are you confused about
summer owl
3x + 1
leaden ginkgo
what's $\frac {4x}{4x}$
warm shaleBOT
__saad
wild shoal
This was my working out for part a, I don't have answers would anyone be able to clarify if this is correct?
obtuse pebbleBOT
This was my working out for part a, I don't have answers would anyone be able to...
@wild shoal Has your question been resolved?
unreal siren
youre working for a is correct
wild shoal
ok cool, for b i don't really know how to approach it
ill send my working out so far
im pretty sure its wrong
unreal siren
ok
wild shoal
unreal siren
,rccw
warm shaleBOT
unreal siren
for x try writing in terms of sin 38
wild shoal
so like sin38=sin(100)x/6?
willow tulip
Just a random question I had in my head. Say you had the equation, mx + b where m and b are integer values. If you were to have say 5x + 3, then you go from x = 0 to x = [infinity] in integer values where 5x + 3 at any integer value of x from 0 to infinity. In general, mx + b with plugged in values for m and b, find all values of mx + b when x is an integer from 0 to infinity. Lining up all the values, my question is, is there any value of mx + b where all total values will be a prime number?
obtuse pebbleBOT
Just a random question I had in my head. Say you had the equation, mx + b where ...
plucky rivet
Well for that you just need x as multiple of b
unreal siren
are you asking for what x is y prime?
plucky rivet
They are asking a line where all values would be prime i think
polar fossil
Just a random question I had in my head. Say you had the equation, mx + b where ...
"all total values" meaning mx + b is always prime for x = 1, 2, 3, 4, ...?
there's a way to prove that this isn't true π
plucky rivet
For every x in IR
polar fossil
For every x in IR
no, |N
willow tulip
yes
polar fossil
natural numbers
unreal siren
oh
willow tulip
ok let me explain my question more better
unreal siren
well x^2 + x + 1 is always prime till x=41 i think
but i dont think it can exist for linear expressions
finite mirage
Just a random question I had in my head. Say you had the equation, mx + b where ...
I think never
plucky rivet
Well for that you just need x as multiple of b
Since b is an integer, you can sub x=bk where k is a natural number to prove this is false
willow tulip
mx + b = prime where m and b are constants and x is a variable that is limited from 0 to infinity and can only be an integer.
Is this true
polar fossil
nope there will always be a composite number somewhere down the line
but! there will also always be another prime number
i think
grizzled shore
hayley's conjecture
finite mirage
Because b = integer, there is a time when x=b, then equation is easily divided by x or b
obtuse pebbleBOT
@willow tulip Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
ripe terrace
how to write this in set builder notation ?
obtuse pebbleBOT
how to write this in set builder notation ?
rich plume
what have you tried?
ripe terrace
I guess there is a pattern 3 , 3+3 , 3+3+5 , ... but can't figure out a formula for it
rich plume
no
rich plume
It is something related with squares
hard igloo
+3, +5, +7, +9, +11
ripe terrace
{n : n = kΒ²+2 , kββ€βΊ} ?
rich plume
kβN
ivory coral
Does a β’ b + 1 : b = a + 1 ?
obtuse pebbleBOT
Does a β’ b + 1 : b = a + 1 ?
ivory coral
robust sleet
Does a β’ b + 1 : b = a + 1 ?
No
robust sleet
π¦
ivory coral
But a β’ b : b = a
robust sleet
Yes
robust sleet
No
robust sleet
I have no idea what you did
ivory coral
(a)(b) + 1 : (b) = a + 1?
tranquil sonnet
no
robust sleet
no
ivory coral
(a β’ b + 1) : (b) = a + 1?
robust sleet
$\frac{a+b}{c+d}$
warm shaleBOT
Fucktalogist
robust sleet
Expand this
robust sleet
(a β’ b + 1) : (b) = a + 1?
No stop
robust sleet
You can expand jt
ivory coral
How
robust sleet
Now try (a+b)/c
robust sleet
I would recommand you watch videos
ivory coral
a/c + b/c
robust sleet
Yes
ivory coral
Γ
proper kelp
not quite sure how to do c)
obtuse pebbleBOT
not quite sure how to do c)
whole dock
Induction?
proper kelp
yep
whole dock
What have you tried
proper kelp
my writing is very messy here but ill send anyways
actually ill write it again
And then I tried to write the last expression in terms of m but I couldnβt
elfin burrow
how did k+1 become k+2
proper kelp
Yeah oops but regardless canβt do
elfin burrow
proceed further, use distributivity
proper kelp
Should I fully expand
elfin burrow
yes
whole dock
$\br{9^{k+1} - 1} -8\br{k+1} = 64m$
warm shaleBOT
coldtee
proper kelp
$\br{9^{k} - 1} -8\br{k} = 64m$
warm shaleBOT
yooyooyoyooyooyooyoo
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
elfin burrow
And then I get 81(9^k)-8k-17
compute the difference between this and the expression in your induction hypothesis and show that the difference is a multiple of 64
that's the easiest way i can see
brisk arrow
obtuse pebbleBOT
brisk arrow
I can accept the fact that if a point, which we assume is (a,f(a)), is the inflection point of a function. Then fβ(a)=0
It is true
But what does fβββ(a) not equals to 0 implies? I canβt understand why the necessary conditions for the inflection point would be fββ(a)=0 and fβββ(a) not equals to 0
glossy basalt
i dont know the formal proof, but you may consider y=xΒ³ and y=xβ΄
it's because there are uncertainties with f'''(x)=0
brisk arrow
i dont know the formal proof, but you may consider y=xΒ³ and y=xβ΄
I know, letβs name your example as f(x)=xΒ³ and g(x)=xβ΄ for convenience
We know that gββ(0)=0, however it isnβt a inflection point.
Therefore we take further action to confirm it, which is gβββ(0) should not be equals to 0
And yet, gβββ(0) is 0 and that implies it is not a inflection point
brisk arrow
i dont know the formal proof, but you may consider y=xΒ³ and y=xβ΄
Is there any explanation for the methodology
brisk matrix
where did this come from
glossy basalt
with 3rd derivative=0 and 2nd derivative=0, we cannot conclude if it is a an inflection point or not (?)
brisk arrow
where did this come from
From my math teacher
brisk matrix
if the third derivative is not 0, and the second derivative is 0, the second derivative must be crossing the x axis at that value
brisk arrow
with 3rd derivative=0 and 2nd derivative=0, we cannot conclude if it is a an inf...
But with 3rd derivative does not equals to 0 while 2nd derivative=0, we can determine that it is a inflection point
brisk matrix
if the third derivative is not 0, and the second derivative is 0, the second der...
so there is a change in the sign of the second derivative
brisk arrow
if the third derivative is not 0, and the second derivative is 0, the second der...
Why? I canβt imagine
brisk matrix
,w plot x^2
warm shaleBOT
brisk matrix
this is what a derivative of 0 can look like
we may touch and go back to being positive
,w plot x
warm shaleBOT
brisk matrix
this is what a nonzero derivative looks like
so if g(a) = 0, and gβ(a) is not 0
then we cross the x axis at x = a
now replace g with fββ
fββ(a) = 0
fβββ(a) not 0
then fββ(x) crosses the x axis
meaning it changes sign
the definition of a point of inflection
obtuse pebbleBOT
@brisk arrow Has your question been resolved?
glossy basalt
Why? I canβt imagine
as maximo said
f''(x)=0 at x=a
means f''(x) touches x-axis at a and
f'''(x)β 0 at x=a, let's say f'''(x)=c for some real c.
then
for c>0, f''(x) will be increasing at a,
for c<0, f''(x) will be decreasing at a,
that is, for both cases
f''(x) is changing.
that means it will goes from + to - or - to + at a.
furthermore
f'(x) will increase (decrease) until it reaches a, and then start decreasing (increasing) after a.
that is f'(x) is a maximal (minimal) at a.
for explaination for f(x), you can continue by yourself and finally get to a point that (a,f(a)) is an inflection point.
obtuse pebbleBOT
@brisk arrow Has your question been resolved?
glossy basalt
sigh.... what do you need? @brisk arrow
brisk arrow
Itβs seems very abstract to me. Namely I canβt understand
glossy basalt
i see
let's head back to step one then
let's take f(x)=xΒ³
f'(x)=3xΒ²
f''(x)=6x
f'''(x)=6
all good till here?
brisk arrow
all good till here?
Yes
glossy basalt
Wait what is a? is it (a, f(a)) the assumed inflection point
yea, or do you want i just say 0?
if that's too ambiguous
brisk arrow
No, Letβs go with the a
glossy basalt
okay
glossy basalt
we know that f(x) has a local minimum at a
this is because we use the way that f'(x)=0 and +/- checking right?
ohhhh
sorry
I thought i was doing xβ΄ lol
I'll type again
brisk arrow
Ohh
glossy basalt
we know that f(x) has a local minimum at a
we know that f'(x) and f''(x)=0 at a
therefore we dont know if f(x) is a local Extrema or not
(this is correct now)
any question up till now?
brisk arrow
No I have no question
glossy basalt
good
next
f'(x) represent the rate of change of f(x)
and
f''(x) represent the rate of change of f'(x)
sorry for the typos
brisk arrow
Okay youβre right
glossy basalt
so, from f'(a)=0, we know that f(x) has no rate of change at a
brisk arrow
Yes
glossy basalt
and from f''(a)=0, we know that f'(x) has no rate of change at a
brisk arrow
Yes
glossy basalt
now finally
f'''(a)=6, which means f''(x) is changing at a
brisk arrow
Yes
brisk arrow
For sure
glossy basalt
f''(a)=0 , and it is increasing at a
therefore
f''(x) before a is negative
and
f''(x) after a is postive
brisk arrow
now finally
f'''(a)=6, which means f''(x) is changing at a
Iβm not sure whatβs the effect that fββ(x) is changing at a
Okay
glossy basalt
Iβm not sure whatβs the effect that fββ(x) is changing at a
that is, since f'''(x)=6, f''(x) keep increasing (since it's postive).
brisk arrow
Yes, youβre right
glossy basalt
good, next step inwards
brisk arrow
For sure
glossy basalt
f''(a)=0, and it is negative before a, and postive after a
f'(x) before a is decreasing
f'(x) reaches 0 at a
f'(x) after a is increasing
all good till here?
brisk arrow
f''(a)=0, and it is negative before a, and postive after a
f'(x) before a is dec...
How do you get that it is negative before a and positive after a
glossy basalt
f''(a)=0 , and it is increasing at a
therefore
f''(x) before a is negative
and
...
here
glossy basalt
f'(x)=3xΒ²
f''(x)=6x
f'''(x)=6
and when you have doubts, just read these graphs near a
brisk arrow
f''(a)=0, and it is negative before a, and postive after a
f'(x) before a is dec...
Yes itβs right
brisk arrow
f''(a)=0, and it is negative before a, and postive after a
f'(x) before a is dec...
I think itβs enough to proves that a is the inflection point
Like if you draw the graph out
brisk arrow
that is, since f'''(x)=6, f''(x) keep increasing (since it's postive).
But is it related to the 3rd derivative?
glossy basalt
But is it related to the 3rd derivative?
yea, because we get the increasing/decreasing of f'(x) from f'''(x)
brisk arrow
f''(a)=0, and it is negative before a, and postive after a
f'(x) before a is dec...
It more like we know the trend of the function then we could know if itβs inflection point upon that
glossy basalt
It more like we know the trend of the function then we could know if itβs inflec...
very true
brisk arrow
very true
3rd derivative basically means the rate of change of fββ(x) isnβt zero, however it never says that the sign fββ(x) would change
Therefore how could we reach to the result that it is the inflection point
Only when fβ(x) change sign in a then we could tell it is a inflection point
glossy basalt
well,
3rd derivative basically means the rate of change of fββ(x) isnβt zero
and f''(x) is 0 at a
therefore we know theres a sign change
brisk arrow
I see and if the sign of fββ(x) remains the same then βaβ is not a inflection point
In contrast if fββ(x) changes sign in a then it is
glossy basalt
you can say that
brisk arrow
But how do we know if it change its sign or not?
By the 3rd derivative
No, I mean we cannot determine if it does change sign or not by 3rd derivative
glossy basalt
No, I mean we cannot determine if it does change sign or not by 3rd derivative
that's why we also need the 2nd derivative to be 0 at a
brisk arrow
that's why we also need the 2nd derivative to be 0 at a
What is the general meaning of 3nd derivative, Iβm confused by that
glossy basalt
to be honest, i am not sure about that. you might have to google a bit to see, but i remember i tried that before and got no conclusion myself
brisk arrow
that's why we also need the 2nd derivative to be 0 at a
Even with that we cannot say it is a inflection point
glossy basalt
Even with that we cannot say it is a inflection point
why?
brisk arrow
the 3rd derivative isnβt zero
Could lead to the red or blue on the graph of fβ(x)
However if it leads to the red, then itβs not a inflection point
Therefore we couldnβt determine whether it is inflection point or not by that
F(x) is just a function which qualifies the conditions of 2rd is 0 while 3rd isnβt
glossy basalt
lemme re-read my typings
okay, so i was looking at another example
g(x)=xΒ²(x-1)Β²
this example reminds me of the g''(x) related to the concavity of the function
the way point of inflexion behaves is where the graph is changing concavity
so, given g''(a)=0 and g''(a) is changing signs around a, it will be a saddle point
brisk arrow
so, given g''(a)=0 and g''(a) is changing signs around a, it will be a saddle po...
Does it explains that 2rd is 0 and 3rd isnβt the conditions for inflection point
glossy basalt
yep, but in the above, we didn't consider when 3rd tends to Β±infinity
all i can say from the previous conclusions is that 2nd is 0 and 3rd is a real number at a, then (a,f(a)) is an inflection point
obtuse pebbleBOT
@brisk arrow Has your question been resolved?
fierce geyser
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obtuse pebbleBOT
Available help channel!
rain loom
obtuse pebbleBOT
polar fossil
hmm ok
do you know what sin, cos, and tan mean?
rain loom
hmm ok
do you know what sin, cos, and tan mean?
I only know what TOA-CAH-SOH is.
grizzled shore
what is TOA-CAH-SOH?
polar fossil
okay, so what does tan ACB = 4/3 tell you?
rain loom
5?
polar fossil
yea
3-4-5 is one of those that will come up a lot
that should be enough to get you the first two answers
polar fossil
the third one is more difficult
cosine of that angle would be adjacent over hypotenuse, right
but it doesn't look like we have a right triangle here, since the angle is obtuse (bigger than 90Β°)
rain loom
yes
polar fossil
but we actually do have a right triangle here, it's the same one we've been using
rain loom
mhm.
polar fossil
it's just that from angle ACD's perspective it's horizontally reversed sorta
so any horizontal distances we measure need to be negative
rain loom
Yup
polar fossil
so with that in mind, what are your thoughts on adjacent and hypotenuse?
polar fossil
that's understandable! it's a strange concept
let's try this, what is happening to the cosine of this angle?
as this triangle gets longer and skinnier
rain loom
alright.
polar fossil
and the angle gets closer to 90β’, what's happening to the cosine?
rain loom
it's like
polar fossil
you can just kinda make up side lengths if you need them
polar fossil
it does indeed reduce, and quite dramatically!
rain loom
oh!
polar fossil
what happens when the bottom side is basically not even there?
rain loom
0?
polar fossil
yeah! so we say that cos(90β’) = 0
rain loom
ah!
polar fossil
because it's not really a triangle anymore so our normal C = A/H definition doesn't really apply
but that's what it seems to want to be
if it's continuous that's what it should be
rain loom
yup
polar fossil
so what do you think about this angle?
rain loom
oh negative.
polar fossil
yep negative and only barely so
rain loom
yes!
polar fossil
it's still 'adjacent' / 'hypotenuse' it's just that we're using different ones
rain loom
Yup
polar fossil
so do you think you can figure out cos(<ACD) now?
rain loom
is it negative cos angle acb?
polar fossil
yes exactly!
polar fossil
if you've seen a sine wave before that's what's going on here
that's why it drops below 0 half the time
rain loom
Oh I see!
Thank you very much! I am sorry for any hassle caused, during time where you had to explain this concept to me.
.close
polar fossil
Thank you very much! I am sorry for any hassle caused, during time where you had...
noooo i appreciate the opportunity, i've been thinking of how best to explain this
obtuse pebbleBOT
noooo i appreciate the opportunity, i've been thinking of how best to explain th...
polar fossil
.close
mint nebula
obtuse pebbleBOT
mint nebula
Prove that this is irrational
@polar fossil any idea? π
So I tried to prove that 2^n+1.... is not a perfect square
Expanded the powers etc etc
And now I'm stuck with uh
10^n*2+1
No, the whole thing
mint nebula
how did you get that?
math
elfin burrow
pretty sure they mean 2(10^n) + 1
empty cypress
$\sqrt{2\cdot 10^n + 1}$
warm shaleBOT
mint nebula
elfin burrow
just have to show that the inside is never a perfect square
empty cypress
just have to show that the inside is never a perfect square
It could be the square of a rational number
mint nebula
nawww
elfin burrow
It could be the square of a rational number
yes, but always the square of an integer
the square root of an integer is either irrational or another integer (this is a result)
elfin burrow
hence it suffices to prove that the inside is not a pefect square
mint nebula
Isnt a perfect square
mint nebula
hence it suffices to prove that the inside is not a pefect square
Thats what ive been saying
Yeah so I've tried factorizing it
Not worth
mint nebula
impress me
elfin burrow
Write $2 \cdot 10^n + 1 = k^2$ for some integer $k$. Then $$2 \cdot 10^n = (k+1)(k-1)$$
warm shaleBOT
tushar
rigid lintel
It reminds me of aΒ²-bΒ²
thats what it is with b = 1
rigid lintel
show that no such k can exist
mint nebula
show that no such k can exist
No idea how.
elfin burrow
No idea how.
LHS and thus the RHS is 2 mod 3, so figure out k mod 3
k can be either 0, 1, or 2 mod 3. check which yields 2 mod 3 in the product
obtuse pebbleBOT
@mint nebula Has your question been resolved?
obtuse pebbleBOT
@mint nebula Has your question been resolved?
tranquil sonnet
k is always 0 mod 3, so after expanding with 3a=k we have 2*10^n=9a^2-1
LHS is 2 mod 9, RHS is 8 mod 9, so no such integer solutions exist
@mint nebula
obtuse pebbleBOT
@mint nebula Has your question been resolved?
tranquil sonnet
What didnβt you understand?
mint nebula
like
i need to see that on paper
or written by the bot
if that's possibl
possible
tranquil sonnet
Oh
Ok
$2 \cdot 10^n+1=k^2$
$2 \cdot 10^n+1 \equiv 2 \cdot 1^n+1 \equiv 3 \equiv 0 \pmod{3}$
This means that $k$ is divisible by $3$, so let $k=3a$ where $a$ is an integer. Then,
$2 \cdot 10^n+1=9a^2$
Taking modulo $9$, we get
$2 \cdot 1^n+1 \equiv 0 \pmod{9}$
which is not possible, thus there are no integer solutions.
warm shaleBOT
kappa_07
glossy juniper
So that doesn't mean they're equal $10^{2} \neq 1^2$, but that when you take them mod 3, they give the same result.
The divisibility trick for 3 is to find if the sum of all the digits in a number is divisible by 3 (ex. 453 is bc 4+5+3 = 12 and 12 is divisible by 3) if any power of 10 has 1 for a sum of its digits, then $10^n\equiv1^n (mod 3)$, aka any power of ten will be 1 more than a multiple of three.
Same is true when the sum of all digits is 2 ($2\cdot10^n$) and adding 1 to that will make the sum of the digits 3, so it will always be divisible by 3 ($2\cdot10^n + 1\equiv3\equiv0 (mod 3)$
Sry that was probably too much
warm shaleBOT
mking625
mint nebula
So that doesn't mean they're equal $10^{2} \neq 1^2$, but that when you take the...
i don't think I can use that sadly.
glossy juniper
Ok, sorry that didn't help then π
tardy epoch
i don't think I can use that sadly.
what part can't you use? mod 3?
fringe rivet
- Describe the transformations to the graph of f(x)=3^x to produce each of the following functions.
obtuse pebbleBOT
1. Describe the transformations to the graph of f(x)=3^x to produce each of the ...
fringe rivet
a) $g(x)=2f(x-3)+1$
b) $g(x)=-\frac{1}{2}f\left (\frac{1}{2}x \right )$
c) $g(x)=4\cdot 3^{-\left (x+1 \right )}$
d)$ g(x)=3^{5\left (x-3 \right )}-2$
warm shaleBOT
deviousglxy
obtuse pebbleBOT
1. Describe the transformations to the graph of f(x)=3^x to produce each of the ...
What step are you on?
1. I don't know where to begin
2. I have begun but got stuck midway
3. I got an answer but I'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
nocturne minnow
It's asking what transformations were done
fringe rivet
how did 3^x turn into (x-3)
nocturne minnow
fringe rivet
It's asking what transformations were done
like it moved to the right 3 times and up once also it vertically streched by a factor of 2
nocturne minnow
Yes
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
obtuse pebbleBOT
timid silo
khan academy
pseudo swift
le truc est globalement ordonnΓ©
@timid silo
si t'as vraiment envie de commencer avec les 4 opΓ©rations
dull rune
Personally, my go to precalc book was allendoerfer
It requires a dedicated sit down, but is excellent to learn everything before calculus really well
obtuse pebbleBOT
@timid silo Has your question been resolved?
echo gazelle
obtuse pebbleBOT
echo gazelle
i did chain rule derivative of cosh (which is sinh) of e^5t* derivative of e^(5t) which is 5e^(5t)
fathom flicker
make better use of parenthesis
empty cypress
Yeah the answer is right, your formatting isn't
echo gazelle
i tried before it changed it to that
okay ill try again
format it like this?
ok ya
thanks
.close
echo gazelle
obtuse pebbleBOT
echo gazelle
ik
echo gazelle
im confused how though bc its all to the -1/2
does it make it c-ax / -b^(-1/2)
wait no
echo gazelle
i tried to i isolated y and made it y' and made the right side ' d
echo gazelle
okay
polar fossil
remembering the chain rule
echo gazelle
wait since a b and c are constants is the derivative of c^2 on the right just 0
polar fossil
yep
polar fossil
is it 2ax-2by=0
chain rule!!
polar fossil
or -2by*y'
yeah that (but probably write it as dy/dx not y' )
anyway now solve for that
boreal edge
I really need helps
Its something simple
but I just dont get it
echo gazelle
I really need helps
Its something simple
but I just dont get it
here take over this channel
.close
obtuse pebbleBOT
echo gazelle
o idk i think from typing but nvm
type in there
and post ur problem
go to #help-7ο½zen1thxyz
obtuse pebbleBOT
Available help channel!
timid silo
I have a question, if a polynomial is divisible by some factor, it leaves no remainder correct?
obtuse pebbleBOT
I have a question, if a polynomial is divisible by some factor, it leaves no rem...
violet sentinel
correct
timid silo
Okay thanks π
obtuse pebbleBOT
@timid silo Has your question been resolved?
empty cypress
.close
shut plover
is this correct
obtuse pebbleBOT
is this correct
obtuse pebbleBOT
@shut plover Has your question been resolved?
shut plover
thanks
gusty night
need help with this
obtuse pebbleBOT
need help with this
obtuse pebbleBOT
<@&286206848099549185>
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.
timid silo
For part a
Imagine the square as two triangles
The base of the triangles would be the diameter of the circle
So you find the area of the triangles which is also the area of the square
Then square root the number and thats your side length
And since you did part a pretty sure you can also do part b
winged mirage
obtuse pebbleBOT
winged mirage
all the bottom numbers are squares
ohhh
how does that help me
how do i use that to find out my answer
slim garnet
simplifies the numbers
high lily
consider 99 = 11 * 9
80 = 10 * 8
etc
and you'll have mass cancellation of common factors
winged mirage
.close
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
outer moth
Hallo there, i wanted to know how would you do part e since the first piecewise is a modulus of x.
You have to check if its differentiable or not
obtuse pebbleBOT
Hallo there, i wanted to know how would you do part e since the first piecewise ...
prime reef
so what is f(x) for -1<=x<=0
outer moth
so what is f(x) for -1<=x<=0
Oh wait, im dum the modulus would jus make it to root x squared
Its fine now, mb. Thanks anyways
.close
obtuse pebbleBOT
kind hawk
eg the function $\gamma:[0,1]\to \bC, x\mapsto \exp(2\pi i x)$ parametrizes the unit circle
warm shaleBOT
denascite
kind hawk
which notation
function notation?
f:A->B ?
gamma takes a value in [a,b] and outputs a value in C
yes
brisk matrix
$\gamma$
warm shaleBOT
maximofs
kind hawk
z is a parameter. as you increase it you run along the curve
kind hawk
**denascite**
.
think z instead of x
yes
yes
the unit circle has center at 0
you are only walking around the circle, not the disk
warm shaleBOT
maximofs
kind hawk
its sad that desmos doesnt have native C support
brisk matrix
yeah especially given they basically have all the components already
i think it helps to refer to the graphs
(x, f(x))
that you have been using for whoknows how long
they can help demistify this parametrization thing
kind hawk
because desmos doesnt know complex numbers so you have to do it in R^2
brisk matrix
e^i(2pi * t) is basically the same thing as you saw in that desmos thing
brisk matrix
<https://www.desmos.com/calculator/snxbqf0tio>
try it with this one
in the other one you need to put in pi for t
brisk matrix
we are mapping z -> e^(2pi * i * z)
so the image is
cos(2pi * z) + i sin(2pi * z)
which represented in R^2 is just the vector (cos(2pi * z), sin(2pi * z))
kind hawk
because desmos doesnt understand C and we cant use i
so we have to translate from C to R^2
kind hawk
yes
obtuse pebbleBOT
@shy solar Has your question been resolved?
kind hawk
well the complex number -1 is on the x-axis (aka the real axis)
the point (0, -1) corresponds to -i
(a,b) corresponds to a+bi
obtuse pebbleBOT
obtuse pebbleBOT
β
kind hawk
actually, I think they mean that gamma=z(t) so t is the input and z the output
drifting badger
