I would suggest fractions
#help-13
hot python
decimals work here though
my program doesnt mind em
is my work looking good?
i have to rank them from least to greatest so they have to be correct or the ranking gets messed up
<@&286206848099549185>
left edge
i have to rank them from least to greatest so they have to be correct or the ran...
I would say most are correct based on a slight glance but I can't help you right now
cedar kilnBOT
@hot python Has your question been resolved?
cedar kilnBOT
@hot python Has your question been resolved?
jaunty mural
Please do not call it a 3d pentagon
i see nothing relating to a pentagon on there, only a hexagonal prism.
hot python
<@&286206848099549185>
dire geode
is my work looking good?
show how you got your numbers
hot python
i just used the formula but again i dont have my work
i can check it with the google calculator but i feel like im doing something wrong
dire geode
i can check it with the google calculator but i feel like im doing something wro...
why can't you just do it here
,calc 2 + 2
wraith daggerBOT
Result:
4
use that as a template
hot python
wait how
dire geode
and then type your formula
hot python
,calc
wraith daggerBOT
dire geode
well you got these numbers doing some calculation right?
type those calculations after ,calc
dire geode
type the formulas here
hot python
.close
wispy harbor
Does anyone offer 1v1 tutoring, through voice chat? I'm really struggling. I feel like I'm drowning in math. I'm currently in trigonometry
cedar kilnBOT
Does anyone offer 1v1 tutoring, through voice chat? I'm really struggling. I fee...
@wispy harbor Has your question been resolved?
cedar kilnBOT
@wispy harbor Has your question been resolved?
round bone
please tell me im going insane and there is something wrong with my work
nimble mulch
I think there's an error
long sleet
Yeah its 5.4, just a typo from her
nimble mulch
they also have 2.5 as the mean of the roots
round bone
stupid college board
nimble mulch
so -0.5 and 4.4 is wrong
long sleet
Well 5.5 technically cuz you round up
round bone
ok thank lord i thought i was going insane
nimble mulch
np
cedar kilnBOT
Available help channel!
crimson sedge
I’m confused with this
cedar kilnBOT
I’m confused with this
crimson sedge
How do I solve for infinity? Do I need to graph this and if so, how? The e with the exponent got me confused
do you know what e^-x means?
No
do you know what e is?
Like the natural log thing
yea just a number
its special but you will learn later probably
do you know what a negative exponent does?
Umm I don’t think I remember
blazing dune
its special but you will learn later probably
I don't think this question needs him to know about e to be solvable
Just deal with it as a positive constant and you will be able to solve it
crimson sedge
hih thats what i said
blazing dune
If you understand the limits of $C^{-X}$ as X approaches infinity
crimson sedge
Umm I don’t think I remember
just makes it 1/e^x
wraith daggerBOT
Sherif Player
crimson sedge
he doesnt
crimson sedge
just makes it 1/e^x
Okay
so as e^x gets bigger
1/e^x gets smaller
e^-x as x goes to infinity is zero
blazing dune
So when it approaches infinity it approaches to 0
crimson sedge
you can look at the graph of the function to understand it better
blazing dune
You should be able to solve it from here
crimson sedge
Oh so what you did was you made the e^-x to 1/e^x? Then you substituted infinity to x and now it becomes 1/e^infinity? And as infinity gets greater, then the value gets smaller till it equals 0? Then the answer is literally just 2/3??
yes!
e^-x = 1/e^x
blazing dune
Yes
crimson sedge
its the same
O okay so e is just a random number?
nono
e is a specific number
What is e exactly?
2.4 something
Oh okay
you should google it
its like pi a special number we need for calculus and stuff
Okay
blazing dune
O okay so e is just a random number?
It is just a special character you will study when you get to know about logarithms
crimson sedge
I’m in AB Calc right now and my teacher expected us to know it but i don’t remember my last teacher teaching it
blazing dune
2.4 something
2.71828
crimson sedge
yup sorry
crimson sedge
I’m in AB Calc right now and my teacher expected us to know it but i don’t remem...
Yea its hard to learn only from school and teachers
blazing dune
I’m in AB Calc right now and my teacher expected us to know it but i don’t remem...
Just study logarithmic functions and you will know about it
crimson sedge
Definitely everyone in my class is struggling lol
i dont think logarithm will explain where e comes from lol
Well thanks guys so much
no problem! close the channel if u dont need any more help!
Okay!!!
blazing dune
i dont think logarithm will explain where e comes from lol
Yeah but it has a little connection with it so I think it should be fine
crimson sedge
Yeah but it has a little connection with it so I think it should be fine
very little lmao
.close
peak bough
What's the difference between arccos and secant?
cedar kilnBOT
What's the difference between arccos and secant?
crimson sedge
What's the difference between arccos and secant?
arccos is the inverse function of cos, secant is 1/cos
peak bough
isn't that the same?
crimson sedge
nop
peak bough
how?
crimson sedge
sqrt(x) is the inverse of x² not x^-2
you might be confused because of the notation, sometimes arccos is written as cos^-1
peak bough
sqrt(x) is the inverse of x² not x^-2
sorry im confused with this
peak bough
can u give more examples, please?
crimson sedge
you might be confunding the ^-1 notation
when you think of exponents, 5^2 = 5 * 5, or 5^-3 = 1/(5* 5*5)
however, there is some guy who decided to name the inverse of cos(x) cos^-1(x), even though it doesn't mean 1/cosx
to denote 1/cosx you need to write ( cosx ) ^-1, not cos^-1x
cedar kilnBOT
@peak bough Has your question been resolved?
dull magnet
Hi
dull magnet
Pls solve (The recommended mass of a soccer ball is 0.43 kilograms. The actual mass is allowed to vary by up to 20 grams. A. Write and solve an absolute value equation/equation to find minimum and maximum acceptable soccer ball masses) equations
obsidian coral
Pls solve (The recommended mass of a soccer ball is 0.43 kilograms. The actual m...
We are not here to do your work for you
dull magnet
Ok but I do not understand
cedar kilnBOT
Available help channel!
modest dove
cedar kilnBOT
modest dove
I am familiar with a few trig rules, just not too confident on which ones to use and how to apply them.
wanton grove
I am familiar with a few trig rules, just not too confident on which ones to use...
cos(240deg) = cos(180deg + 60deg)
the denominators should be easy enough to simplify
modest dove
cos(240deg) = cos(180deg + 60deg)
Bit confused on why to do this?
Are we trying to get exact values?
modest dove
Right so, cos 60/1-cos(90-theta) + cos 180 * cos 240/1-sin(360-theta)
wanton grove
Right so, cos 60/1-cos(90-theta) + cos 180 * cos 240/1-sin(360-theta)
thats not what I meant
cos(240deg) = cos(180deg + 60deg) = -cos(60deg)
modest dove
Okay yeah that makes sense now, that is what I have done.
wanton grove
right so just replace cos(60deg) with 1/2, then we can deal with the denominators
do you know an identity like cos(90deg - x) = sin(x)?
modest dove
wanton grove
,rccw
modest dove
I am working with a friend, would this be in the right direction?
wraith daggerBOT
wanton grove
the concept is correct
cant read a few things clearly so I can verify the final answer when you're done
full tulip
Yet another "1=0"-esque proof. I want to know what it means to "un-sine" an equation. As in, how do you go from the second line to the third line and why is this wrong? What are the restrictions? Thanks in advance.
versed fulcrum
What do you mean by "un-sine"
spice kraken
sin is not injective
so you can't cancel sin
if f(x) = f(y) implies x = y then f is called injective
cedar kilnBOT
@full tulip Has your question been resolved?
full tulip
What do you mean by "un-sine"
I'm not entirely sure what to call it, but it's the last operation in this pic. My education system doesn't really get into this sorta stuff, and i'm still not too experienced.
spice kraken
,w graph sinx
wraith daggerBOT
spice kraken
as you can see there are multiple x values such that sinx = 0.5
you either write down all solutions
or limit the domain such that sinx is injective on that domain
crimson sedge
I need help with my summer dossier and I can’t get to the answer of exercises 1) (b) and 2) (a). I have a picture of 1) (a) as a reference of how to work on it. (Answer sheet included).
crimson sedge
anyone wanna help me with algrbra
can anyone check these for me, and also i need help understanding the question 1
lunar lynx
anyone wanna help me with algrbra
This channel is taken. Open another channel. Refer #❓how-to-get-help
cedar kilnBOT
@crimson sedge Has your question been resolved?
cedar kilnBOT
@crimson sedge Has your question been resolved?
crimson sedge
<@&286206848099549185>
cedar kilnBOT
@crimson sedge Has your question been resolved?
midnight wadi
???
crimson sedge
Referring to this question... I forgot to reply to it.
midnight wadi
yea but this is already taken by other question
crimson sedge
Oh? I didn't see it. I apologise.
midnight wadi
I need help with my summer dossier and I can’t get to the answer of exercises 1)...
so or 1(b) i got: f(x) -> f(x+2) -> 1/2f(x+2)
so g(x) = 1/2 (2(x+2)^2-4)
expanded= x^2 + 4x + 2
therres a possibility that the answer sheet you sent is wrong
cedar kilnBOT
cedar kilnBOT
Available help channel!
bitter kayak
Can someone explain this answer to me? I don't understand why they are saying that the velocity vector is perpendicular to the radius vector at the critical time of intersection.
https://math.stackexchange.com/questions/4755129/minimization-problem-of-parametric-equations-in-ballistics-minimum-projectile-s/4757765#4757765
cedar kilnBOT
Can someone explain this answer to me? I don't understand why they are saying th...
@bitter kayak Has your question been resolved?
cedar kilnBOT
@bitter kayak Has your question been resolved?
cedar kilnBOT
@bitter kayak Has your question been resolved?
ebon drift
When the liquid is added to a person’s blood sample, it is expected to turn blue if the person is allergic to nuts and to turn red if the person is not allergic to nuts.
The company claims that the probability that the test result is correct is 98% for people who are allergic to nuts and 95% for people who are not allergic to nuts.
It is known that 6 in every 1000 adults are allergic to nuts.
ebon drift
Find the probability that the tested adult is allergic to nuts given that the liquid turned blue.
how would I do that
would I just find the average of 98% and 95%
or would I subtract 7% from 100%
cedar kilnBOT
@ebon drift Has your question been resolved?
ebon drift
.close
wraith daggerBOT
Dyssrupt
junior dome
comparing with distance formula
you can see it represents a locus of points whose distance from origin (centre of circle) is constant = 1
lyric swallow
obsidian coral
Another way to do it, is take $x^2+y^2=1$ , solve for y, and you get $y = \pm \sqrt{1-x^2}$, then you can plot $y = +\sqrt{1-x^2}$ and $y = -\sqrt{1-x^2}$ indivdually
wraith daggerBOT
dldh06
wintry wasp
I feel really silly asking this but could anyone please refer me to a website where i can read up on the solvability of a coefficient matrix purely based on its row echelon form. Its a stupid little thing that i never got the hang off in my LA class.
wintry wasp
My book has a (very) small section written about it buts its horribly explained so i skipped it for like the whole LA course so here i am grasping spectral theorem but not when a mxn coefficient matrix is solveable lol
crimson sedge
somewhere in this playlist
but also , try to find gilbert strangs introduction to linear algebra, that book is great
wintry wasp
https://youtu.be/yGJjYWEcQs8?si=wlCIg78z5m6rdG_H
Thank you but that doesnt help since my mind cant translate the 2x2 case to general mxn matrices 4 some reason
wintry wasp
but also , try to find gilbert strangs introduction to linear algebra, that book...
okey i'll check it out
cedar kilnBOT
@wintry wasp Has your question been resolved?
wintry wasp
.close
wise igloo
When solving these inequalities, how does the number line give you the answer in interval notation? because it looks like in number 1 the answer is the red and blue parts of the number line but in number 2, the answer is only where the two red and blue parts intersect
bold lagoon
This is affected by the direction of the inequality sign. In 2, The answer is the common part of these lines, because |x| is the distance from 0 to x on the number line. If you go any further, the distance would increase and the value of |x| would be too big to satisfy the inequality.
@wise igloo Do I have something to clarify?
wise igloo
it makes sense when i think about what the answer would be but is there a way to definitively know when to use the intersection of the lines vs the whole thing
or do you have to like think about what the answer would be to see if it makes sense
bold lagoon
it makes sense when i think about what the answer would be but is there a way to...
If it's easier for you, you can think about the answers, but the other, less fun way is just to memorise (give me a sec)
$\left|x\right|<a:\rightarrow :x\in \left(-a,:a\right)$
wraith daggerBOT
themathboi #2137
bold lagoon
$\left|x\right|>a:\rightarrow :x\in \left(-\infty ,:-a\right)\cup \left(a:,:+\infty \right)$
wraith daggerBOT
themathboi #2137
bold lagoon
$\left|x\right|\ge a:\rightarrow :x\in \left(-\infty ,:-a\right]\cup \left[a:,:+\infty \right)$
wraith daggerBOT
themathboi #2137
bold lagoon
$\left|x\right|\le a:\rightarrow x\in \left[-a,:a\right]$
wraith daggerBOT
themathboi #2137
bold lagoon
it makes sense when i think about what the answer would be but is there a way to...
That's the whole boring theory, but I find it more interesting to think about the distance from 0 being larger or smaller.
@wise igloo Are you there?
wise igloo
yeah
makes sense. i’ll probably just stick with thinking about if the answer makes sense
thanks
bold lagoon
np
wise igloo
.close
blazing aurora
I have gotten stuck on this problem. It is a problem that relates to compound interest. What I attempted to do was create two separate equations that equal the same value (final amount A) and then set them equal to each other. However, I do not have enough information to solve this, as there are two variables -- is there a third equation that I could make so that I could have a system?
blazing aurora
To calculate compound interest I used this formula
ok obviously i can divide the z out
but when put into a calculator i get this
this cannot be correct
when solving for pedro's amount of time, i get this. as this is negative i believe i must have made some mistake
cedar kilnBOT
@blazing aurora Has your question been resolved?
cedar kilnBOT
@blazing aurora Has your question been resolved?
blazing aurora
.close
lilac mural
hello
vagrant elbow
Just post your question
cedar kilnBOT
@lilac mural Has your question been resolved?
cedar kilnBOT
I want to ask for clarification about thisss
high coyote
cinder venture
So wrong
high coyote
Take a look again into them and try again
deep oriole
your values for A,B,C arent all labelled correctly
shrewd blaze
Thenn
deep oriole
then what?
shrewd blaze
What did i do wrong
deep oriole
i just told you
deep oriole
also your formula for the sum for number 2 is wrong, even accounting for incorrect labelling
there is no B
B=0 was correct initially
shrewd blaze
A=2 B= -5 C=1 So Sum=B/A substitute -5\2, Product = C/A substitute 1/2? Whats wrong theree
shrewd blaze
c isnt 1, sum isnt b/a
Ops
For got something
Wait let do it again
Imma just finish this in school, Thxxxx for your help heheheh
.close
sharp kestrel
cedar kilnBOT
sharp kestrel
What is the domain is it x greater than or equal to square root of 2 or is it all real numbers
sharp kestrel
So what is it in interval notation
naive zinc
Idk 💀
naive zinc
I literally have no clue
I just searched up what an interval was and i guessed
Sorry
sharp kestrel
.close
mild nexus
I have a probability question
cedar kilnBOT
I have a probability question
mild nexus
Say I have coins whose probability of H is uniform 0,1 what is the probability of getttinf 100 heads if I draw one at random
I got 1/101 but seems odd
<@&286206848099549185>
cedar kilnBOT
@mild nexus Has your question been resolved?
brisk cape
Answer is 58, what did I do wrong?
cedar kilnBOT
Answer is 58, what did I do wrong?
stuck walrus
(please get a better eraser)
brisk cape
yeah
stuck walrus
it is negative 39 subtract 10
brisk cape
so 49
brisk cape
forgot -
stuck walrus
and also idk how the left side became 37
brisk cape
47-10
stuck walrus
you cancelled out the 10 already
brisk cape
.close
mental crag
Could someone help me with this question using Excel? I'm unsure whether the 103936.47 is the future value or the present value
mental crag
This is what I have so far but I feel like I'm doing it wrong since I'm getting a negative number...
<@&286206848099549185>
<@&286206848099549185>
<@&286206848099549185>
cedar kilnBOT
@mental crag Has your question been resolved?
mental crag
Anyoneee? <@&286206848099549185>
mental crag
It's ok, another person can help me
<@&286206848099549185>
<@&286206848099549185>
Hi, is there any helper that could help me😭
blazing dune
I would try so can you write the question here since I have problems reading it from that Excel
cedar kilnBOT
@mental crag Has your question been resolved?
mental crag
I would try so can you write the question here since I have problems reading it ...
Let me type it up for you
On January 1, 2023 Suresh Ltd. issues bonds and receives proceeds of $103936.47 to yield 7.00%. The bonds pay interest annually at 8.5% and mature in 3 years.
Required
a. Show the journal entry to record the issuance of the bonds.
b. What is the face value of the bonds?
c. Are the bonds issued at par or with a discount or premium? Will the carrying value of the liability for these bonds increase over time, or decrease? Explain briefly.
d. Show the journal entries to record the first two interest payments including premium/discount amortization on these bonds.
blazing dune
On January 1, 2023 Suresh Ltd. issues bonds and receives proceeds of $103936.47...
I appreciate the effort done to write all of this
But I don't understand half of the vocab used here sooooo.......
cedar kilnBOT
@mental crag Has your question been resolved?
indigo notch
\
indigo notch
So I understand that we want to get it into the reverse chain rule form then integrate
but I dont understand how we get there
Like I dont understand how wolfram got to the bottom left of the equation
celest ledge
Called separation of variables, must be in your textbook. Just find it and read it
indigo notch
God Bless your soul
Bruh
But how did he get to the separtion of variables
I understand what it is but how did he get to the equation to be separable
cedar kilnBOT
@indigo notch Has your question been resolved?
viral pulsar
cedar kilnBOT
viral pulsar
Hello! For this problem I got 42 but it’s wrong so I’m not sure on what to do
I did 7 nPr 2
tropic oxide
why tho
viral pulsar
Cus there’s 7 choices
tropic oxide
no there aren't...
3 bagel choices * 4 drink choices
it's just that
you overthought it
viral pulsar
So the answer is 7?
tropic oxide
yes..
cedar kilnBOT
cedar kilnBOT
Available help channel!
still coral
cedar kilnBOT
still coral
Does anyone know why there is a negative sign on both sides
it goes from that to this
prisma gull
Here from step 1 to step 2, it looks incorrect
haven’t distrusted the negative it seems
fallen moat
iirc, you were asking about the same problem just now
and iirc, it's about the integral
still coral
thats what im saying but somehow he makes it work still
theres no integrals in this part
fallen moat
i know
still coral
fallen moat
can you post the part with integral? i think there are some problem with the ln(Q/C+V)
prisma gull
shouldn’t the numerator be -t-k1?
fallen moat
yea
still coral
shouldn’t the numerator be -t-k1?
hmm maybe ill look more into it
still coral
I think he brought out the negative
and then put it on the other side while he was dividing
still coral
im confused on where the 2nd negative comes from
whatever he did somehow is correct lol
idk about the writing
fallen moat
because he "added" a negative "magically" after he did wrong.
I'll draw on your pic real quick
brb
still coral
I gotchu ty
fallen moat
still coral
the thing about integral is you can just take negatives out of them so it can be -c (int) 1/u
am I wrong in that
slate lintel
the thing about integral is you can just take negatives out of them so it can be...
that is correct, you can do that
still coral
the thing about integral is you can just take negatives out of them so it can be...
how he got the other negative
still coral
the thing about integral is you can just take negatives out of them so it can be...
since this rule exists
fallen moat
how he got the other negative
which negative lol, circle it on my pic
there are many negatives in the pic
still coral
oh I understand it now
lol
I real ized you cant just pull out a negative
you would have to do -(q/c - v)
ty yall
.close
cedar kilnBOT
fallen moat
you would have to do -(q/c - v)
and also, you can just pull the negative out of the ln
anyways cheers!
cedar kilnBOT
Available help channel!
bold lagoon
Hello.
Given the value of Δ and all roots of a quadratic equation (in its basic form), what will be the method of finding values for a, b, c, for which the equation is true?
( I'm talking about an equation that will give out values for a, b, c after the input of the known values)
wanton grove
Hello.
**Given the value of Δ and all roots of a quadratic equation (in its basi...
so you have the roots x1, x2 and you want an equation ax^2 + bx + c = 0?
mental trail
If you know the roots of a quadratic, you can write the quadratic equation as a(x-x1)(x-x2) = 0
Then all you have to do is expand
wanton grove
this^
bold lagoon
If you know the roots of a quadratic, you can write the quadratic equation as a(...
Fair enough, but my question is "where did you get that from?"
fallen moat
Fair enough, but my question is "where did you get that from?"
from the fact that if ab=0, then a=0 or b=0
mental trail
Fair enough, but my question is "where did you get that from?"
ax² + bx + c = a(x² + bx/a + c/a)
= a(x² + 2x(b/(2a)) + b²/(4a²) - b²/(4a²) + c/a)
= a[(x+b/(2a))² - Δ/(4a²)]
= a(x-x1)(x-x2)
You might have seen applications of this while "completing the square"
This is just completing the square in the general case
lyric widget
ax² + bx + c = a(x² + bx/a + c/a)
= a(x² + 2x(b/(2a)) + b²/(4a²) - b²/(4a²) + c/...
How do you justify the final equality?
mental trail
How do you justify the final equality?
It's of the form A²-B²
bold lagoon
ax² + bx + c = a(x² + bx/a + c/a)
= a(x² + 2x(b/(2a)) + b²/(4a²) - b²/(4a²) + c/...
This may be a not-so-smart question, but doesn't b²/(4a²) - b²/(4a²) just cancel out? (2nd line)
lyric widget
You deliberately add b²/(4a²) - b²/(4a²) into the expression in order to complete the square
because
x² + 2x(b/(2a)) + b²/(4a²)
simplifies nicely to
(x+b/(2a))²
and then the rest
- b²/(4a²) + c/a
is just additional garbage that you don't really care about, but fortunately (at least for deriving the quadratic formula) it's garbage that doesn't include x
bold lagoon
Thanks, just one more thing stands out to me
How did you get Δ in the 3rd line? I don't see any expressions that could be simplified to b²-4ac
mental trail
Put the two final terms under the same denominator of 4a²
lyric widget
b^2 / 4a^2 + c / a
= b^2 / 4a^2 + 4ac / 4a^2
= (b^2 + 4ac) / 4a^2
mental trail
b^2 / 4a^2 + c / a
= b^2 / 4a^2 + 4ac / 4a^2
= (b^2 + 4ac) / 4a^2
Don't forget the - in front of the first fraction
-b^2 / 4a^2 + c / a
= -b^2 / 4a^2 + 4ac / 4a^2
= -(b^2 - 4ac) / 4a^2
zealous portal
I just need some verification? This is not a cyclic group, correct?
cedar kilnBOT
@zealous portal Has your question been resolved?
cedar kilnBOT
cedar kilnBOT
Available help channel!
next spruce
can someone explain
"the minimum value of () is 0, so the minimum value of f(x) = () is 1" how did they get the minimum value
tropic oxide
sqrt(x-4) is always nonnegative by defn
so it is >= 0
and there is a point at which it is 0
tulip stream
ahh so since the sqrt is 0 then the other one is 1 because +1
square root equals 0 when x = 4 yes
minimum value a square root function is 0
$$\sqrt {x-4} \geq 0$$
wraith daggerBOT
Cyrenux
tropic oxide
ahh so since the sqrt is 0 then the other one is 1 because +1
exactly
tulip stream
your function is $\sqrt{x-4} +1 $
\
so add 1 to both sides
$$\sqrt {x-4} +1 \geq 1 $$
wraith daggerBOT
Cyrenux
next spruce
ohhh
tulip stream
be careful though
domain of your function includes 4
if it hadnt included 4, then range wouldnt have included 1 of the range [1, infinity)
tulip stream
no, one term is in square root, other is outside the square root
tulip stream
yeah
next spruce
if it hadnt included 4, then range wouldnt have included 1 of the range [1, infi...
still processing this
crimson sedge
still processing this
do you know what bracket and parenthesis difference
next spruce
yeah
tulip stream
**Cyrenux**
well i didnt explain here but notice since domain is $[4, \infty)$
\that means:
$$ 4 \leq x < \infty$$
wraith daggerBOT
Cyrenux
tulip stream
extract 4 on all sides
vapid edge
range of f(x) = sqrt(x - 4) + 1 shifts by 1
tulip stream
$$ 0 \leq x-4 < \infty $$
wraith daggerBOT
Cyrenux
wraith daggerBOT
Cyrenux
wraith daggerBOT
Cyrenux
crimson sedge
in still confused on how greater than and less than works when there is infinity
tulip stream
this wont work on every question so dont attempt this lol
just use the fact that
inside of square root must be always positive
for that number to be defined in REAL numbers
tulip stream
you can kinda see it from the operations i did here too
next spruce
yupp
tulip stream
**Cyrenux**
this inequality here wouldnt have the equal symbol
dont do it the way i did though
there will be questions where you wont be able to do what i did
tulip stream
just remember that inside of a square root is greater or equal to 0
or 4th root, 6th rooth , 8th rooth ... so on
if root is even
it must be positive or zero
next spruce
nice nice
vapid edge
in still confused on how greater than and less than works when there is infinity
you can still compare using limits tho
tulip stream
in other words, non-negative value
tulip stream
yes
do you know how relations work?
you take the set A and match its elements with another set which would be set B
next spruce
yup, I just watched a vid on it to understand my hw better
crimson sedge
you can still compare using limits tho
i haven’t learned limits
i wanna master function before i proceed to limit
function is already hard for me man
i wanna know how to shortcut in foil method
next spruce
do you know why functions are special relations?
Idk that..
crimson sedge
i wanna know how to shortcut in foil method
nvm i just learned it
next spruce
ohh that
next spruce
that's the relation and function thing
next spruce
if its a function
no, it's just a relation
next spruce
ay i read wrong
next spruce
tulip stream
domain can only match one element of range
so (a,2) only (This would be a function)
if its (a,2) (a,3) its NOT a function
but assume if c didnt match any element in B
c is element of the domain
would that make it a function?
next spruce
would that make it a function?
it would not..
tulip stream
yes
next spruce
NICEE
tulip stream
now when you are defining inverse functions
you go from B to A instead
B becomes the domain
A becomes the range
vapid edge
i haven’t learned limits
sure, you can still check by finding their ratios, f(x) and g(x)
tulip stream
an example
we can define function f such that element of B, 3 doesnt get matched with anything of the domain
this still makes it a function
but when you are going from B to A it wont be a function since 3 WONT match with anything
next spruce
can you make an illustration of this please?^
next spruce
oooo
tulip stream
its better to ask in another channel since this is claimed
crimson sedge
no
im not asking for an answer
i just said i what i feel
oops also i forgot that u guys are discussing something
sorry
next spruce
bonk
tulip stream
ok fixed
so top is a function
but its inverse isnt
domain of functions can only match one element
think domain as a child
and range as mother
every child has only 1 mother
now for inverse to exist
mother becomes the child and child becomes the mother
so in other words every mother must have exactly one child
next spruce
damn
tulip stream
every child has only 1 mother
(didnt mention but every child also guaranteedly has 1 mother)
i think child-mother explanation is best way to memorize it
reminder that in real life, a mother can have more than one child
next spruce
i think child-mother explanation is best way to memorize it
real
tulip stream
so on function f: b and c are children of 2
3 is a mother who just doesnt have a child
next spruce
okay i understand
tulip stream
if every mother corresponds to only one child we call it 1-1 (one-to-one / injective)
next spruce
ohh I remember my teacher mentioning this^^
tulip stream
algebracially one-to-one(injective) functions are expressed as:
for all elements a and b in the domain:
if f(a) = f(b) satisfies then a = b
tulip stream
if every mother corresponds to only one child we call it 1-1 (one-to-one / inje...
(also i had typed bijective here earlier, its wrong, bijective means its injective and surjective at the same time, i confused them)
geometric approach will help too
tulip stream
yeah its sent to left hand side
we want to isolate y
normally functions are typed as y=f(x)
to find inverse of them we try to find $$ x=f^{-1}(x) $$
but this guy just swapped x and y straight up
tulip stream
well its kinda up to you
you can just isolate x and type $$f^{-1} (x)$$
\ where you see it
wraith daggerBOT
Cyrenux
tulip stream
or just swap x and y and isolate y
tulip stream
to find inverse of them we try to find $$ x=f^{-1}(x) $$
i typed wrongly here , i will fxi that ( now fixed)
wraith daggerBOT
Cyrenux
next spruce
ooooo
tulip stream
here is a geometric appraoch to bijective (one-to-one ) functions
you can draw horizontal line
and if you they, its not bijective
if a function is bijective
their input is also equal
next spruce
can you explain this?
tulip stream
we add 4 to both sides
next spruce
thanks for taking the time teaching me too OEIFJOSEIJFSEF
tulip stream
np, thanks for baring to listen to my explanation
next spruce
we add 4 to both sides
ahhh
okay completed the example
now I'll actually try solving the real problem
tulip stream
here is an example of how to check bijectivity on graphs
i drew the line y=2
and saw that it intersects two points of the graph
red curve is the graph
it intersects one point in first quadrant
and another point on second quadrant
next spruce
i drew the line y=2
it can be any postive right?
tulip stream
any line that intersects the graph
not only positive
we could have drawn below x axis too
i didnt specify what line i drew
but its like y= -3 or something
whatever you want to be
notice y= -5 doesnt cut the graph though
least value y can get is -4
wraith daggerBOT
Cyrenux
tulip stream
if $$ x<0 $$
\ then
$$ \sqrt{x^2} \neq x$$
$$ \sqrt{x^2} = -x$$
wraith daggerBOT
Cyrenux
tulip stream
hint: look at the domain of the inverse
tulip stream
F
next spruce
hint: look at the domain of the inverse
i give up, I understand the mistake but don't know how to fix it
at least the domain is correct hehe
next spruce
yeah
tulip stream
on inverse it would describe interval of y
next spruce
yuh
wraith daggerBOT
Cyrenux
tulip stream
when you square you get $$ (y-5)^2 = |2-x| $$
wraith daggerBOT
Cyrenux
tulip stream
we are free to use interval of x and y
doesnt matter
hmm
what was interval of x?
next spruce
ohh
tulip stream
interval of x is the domain of f
next spruce
so 5?
next spruce
aww
tulip stream
domain of f is $$ (-\infty , 2]$$
wraith daggerBOT
Cyrenux
next spruce
ay it's that one
tulip stream
now, figure how |2-x| comes out of the absolute value
by using our interval
the way i would do it is (i dont recommend this way):
since $$(-\infty,2] $$ then
$$ -\infty < x \leq 2$$
wraith daggerBOT
Cyrenux
tulip stream
now try to make it look like 2-x
to do that we'll do following operations:
i) multiply all sides by -1
ii) add 2 to all sides
by operation i) we get :
$$ -2 \leq -x < \infty$$
\
and by operation ii) we get:
$$ 0 \leq 2-x < \infty$$
wraith daggerBOT
Cyrenux
next spruce
ahh the thing you did earlier
tulip stream
so its positive when its extracted as |2-x| = 2-x
this is what you did though, hmm i think im high then
you unintentionally did it correctly without noticing
next spruce
slay?
next spruce
I hate functions SISEFOJSENSEFSLEF
tulip stream
you dont have to do it like the way im doing though
its bad
the way i recommend is by typing values in it
recommended way :
range of x is $$(-\infty ,2]$$
\ what we want:$ |2-x| = ? $
\ i will take a number in that interval, 1 looks nice enough
\$ |2-1| > 0 $
\ so $|2-x| = 2-x$
wraith daggerBOT
Cyrenux
next spruce
so how do i continue the solution..
tulip stream
since absolute value spits out positive value
tulip stream
,rotate
nvm didnt work lol
i was trying to repost it
multiply both sides by -1 and its over
or divide by -1 as you said
same operation
then its over
next spruce
i'll multiply since I don't know how to continue when I divide
next spruce
i'll multiply since I don't know how to continue when I divide
that or I just divided it wrong
tulip stream
i'll multiply since I don't know how to continue when I divide
same thing as division when you multiply
i think you are confusing the fact that $$ \frac{a}{-b} = \frac{-a}{b} = - \frac{a}{b}$$
wraith daggerBOT
Cyrenux
tulip stream
also i didnt type it but suppose b is not 0 there lol
tulip stream
**Cyrenux**
you now know this, notice however:
\ for any number a,b,c and where $b \neq 0$
\ we have $$ -\frac{a+c}{b} = \frac{-a-c}{b} \neq \frac{-a+c}{b}$$
wraith daggerBOT
Cyrenux
next spruce
cmds are hard here
tulip stream
nah im just bad
next spruce
oop
tulip stream
if you want to move -1 on the denominator from denominator to numerator, you multiply denominator and numerator by -1 also
next spruce
weeeeeeeeee
next spruce
if you want to move -1 on the denominator from denominator to numerator, you mul...
oh
tulip stream
if you want to move -1 on the denominator from denominator to numerator, you mul...
$$ \frac{a+c}{-b} = \frac{-(a+c)}{-(-b)} = \frac{-a-c}{b}$$
wraith daggerBOT
Cyrenux
tulip stream
also we have $$ -(a+c) = -a -c$$
\ notice however $$ -(ac) = -ac = a*-c$$
wraith daggerBOT
Cyrenux
wraith daggerBOT
Cyrenux
next spruce
**Cyrenux**
ahhh so -1 ((y-5)^2 -2) then distribute -1
wraith daggerBOT
Cyrenux
next spruce
did i do it right
tulip stream
shouldnt affect inside bracket
tulip stream
yes
next spruce
YAYYYYY
tulip stream
also did you know this ? $$ (a-b)^2 = (b-a)^2$$
\just like how $$|a-b| = |b-a|$$
wraith daggerBOT
Cyrenux
tulip stream
try to expand both sides
next spruce
(a-b)(a-b) = (b-a)(b-a); Foil method?
tulip stream
you can do that
or factor the right handside like this:
$$ (a-b)^2 = ( -(a-b) )^2 = ( (-1) (a-b) )^2$$
$$ (ab)^c = a^c * b^c $$
$$ ( (-1) (a-b) )^2 = (-1)^2 (a-b)^2 = (a-b)^2$$
wraith daggerBOT
Cyrenux
tulip stream
**Cyrenux**
do you understand this
next spruce
**Cyrenux**
with the given/sol no
tulip stream
which line
next spruce
tulip stream
do you understand the 2nd line at least
next spruce
yeah
tulip stream
its power of two multiplications
next spruce
oh your explaining the distributive
tulip stream
(b-a) = -(a-b) but didnt type
tulip stream
**Cyrenux**
im expanding the line 1 from this
i just didnt type right hand side and straight up typed -(a-b) instead of (b-a)
next spruce
ohh
tulip stream
$$ (b-a)^2 = ( -(a-b) )^2 = ( (-1) (a-b) )^2$$
then use the fact that:
$$ (ab)^c = a^c * b^c $$
$$ ( (-1) (a-b) )^2 = (-1)^2 (a-b)^2 = (a-b)^2$$
if was like this you would have understood
wraith daggerBOT
Cyrenux
tulip stream
then use commutativity of equality (actually symmetry property)
if RHS = LHS then LHS = RHS
(Left Hand Side , Right Hand Side)
next spruce
oh what's that?
tulip stream
when we are talking about injectivity (one-to-one ness) of a function
\newline
we observe for ALL real numbers a,b satisfying $$f(a)=f(b)$$
wraith daggerBOT
Cyrenux
tulip stream
we know that its a=b for every a,b satisfying f(a)=f(b) if injective
notice how im saying ALL a,b
because you can find infinitely many a,b satisfying a=b for f(a)=f(b)
BUT despite infinitely many satisfying $a=b$ ,
\ if there is one exception such that $$ a \neq b$$ for $ f(a) = f(b)$
\ then that function is NOT injective
next spruce
ooooo
wraith daggerBOT
Cyrenux
tulip stream
yeah keep this in mind
"every" and "all" make a huge difference
you can copy this definition to make definition of \textbf{non injective} function :
\
for some a,b satisfying $f(a) = f(b)$ we have $a \neq b$
wraith daggerBOT
Cyrenux
primal mulch
primal mulch
<@&286206848099549185>
cedar kilnBOT
@primal mulch Has your question been resolved?
primal mulch
<@&286206848099549185>
primal mulch
<@&286206848099549185>
sand cradle
This has been asked on AoPS: https://artofproblemsolving.com/community/c6h116470p661463 and https://artofproblemsolving.com/community/c6h1432694p8091262
primal mulch
This has been asked on AoPS: <https://artofproblemsolving.com/community/c6h11647...
this one is the correct way is solving it?
sand cradle
this one is the correct way is solving it?
Well, the person who wrote this claims that the shorter solution above is also correct
sand cradle
This has been asked on AoPS: <https://artofproblemsolving.com/community/c6h11647...
... and Math StackExchange: https://math.stackexchange.com/questions/1401440/prove-the-root-is-less-than-2n?noredirect=1
quartz yew
Hello, I tried to answer this but I am confused about the whole process of solving so Im wondering if I did anything wrong
quartz yew
These are the questions
And these are what I did
Im pretty confident about my answer in no. 1 tho as it seemed simple enough
cedar kilnBOT
@quartz yew Has your question been resolved?
nimble stream
whats the original question
quartz yew
This one
dapper patio
Wht do we need to find in This
quartz yew
We need to use the epsilon-delta definition to show proof of the limit
dapper patio
Ayo u in University?
dapper patio
I see sorry cz am still in last year of high school
12th grade
Idk epsilon delta thing
😔
dapper patio
Idk am new
quartz yew
Oh Haha I see
dapper patio
<@&286206848099549185>
quartz yew
Oh thanks man
dapper patio
Ye np
Hey umm if no one replies do u know abt the brainy app?
It might help with your problem
quartz yew
Oh yeah I'll try that too ty
cedar kilnBOT
@quartz yew Has your question been resolved?
quartz yew
<@&286206848099549185>
dim tiger
This answer is correct
You guess a value of δ in terms of ε and then confirm that by substituting the value of δ and check if you reach |f(x)-L|<ε
quartz yew
Nice how about the rest?
dim tiger
I didn't check them gimme a sec
quartz yew
Ok cool
dim tiger
Incorrect
quartz yew
😦 where did I go wrong?
dim tiger
|6-x/3 -7|=|-1-x/3|=|1+x/3|=|(x+3)|/3
quartz yew
Oh right from the start ahaha
dim tiger
From here you can see that |x+3|/3<ε which motivates δ/3ε or δ=3ε then you verify as you did in part 1
quartz yew
Isee Isee I'll work on it tomorrow cause it's very late at night for me here
How about my other answers?
dim tiger
I didn't check them yet
brb when I am back I'll check them and write the comments here
Remember the number of the channel in case it closes because of timeout
To check the notes when you wake up tomorrow
Write on a piece of paper or something that this channel is help 13
quartz yew
Nice will do just that, have a good night/day!
dim tiger
You too
cedar kilnBOT
@quartz yew Has your question been resolved?
dim tiger
here it should be $\delta = min{1,\frac{\epsilon}{5}}$
wraith daggerBOT
calculus is fun
dim tiger
you need this because you will get the following in the process of your work
if $|x-2|< \delta$ then $|x^2-4|< \epsilon$
wraith daggerBOT
calculus is fun
dim tiger
so $|x-2||x+2|< \epsilon$ no if you can find a constant upper bound of $|x+2|$ you will be good to go
wraith daggerBOT
calculus is fun
