help-27

[winter patrol]

and it's flawed

[ancient basin]

I appreciate this but on the answer sheet the answer is option D

[modern nimbus]

okay 👍

[ancient basin]

I think since g(x) is the denominator

It shouldnt be 0

The entirety

[winter patrol]

#help-27 message

[ancient basin]

Not just the denominator of g(x) itself

[ancient basin]

Yeah but how do i define them

Or how to get the domain

For g(x)

Thats what i didnt know

[winter patrol]

are there any situations where you'd be unable to evaluate g(x)

[ancient basin]

Is this right?

[winter patrol]

yes

[ancient basin]

Thanks

[devout snow]

@ancient basin Has your question been resolved?

[devout snow]

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[deep vortex]

\begin{problem} The sequence with [a_1 = 1; \qquad a_{n + 1} = \sqrt{1 + a_n}] is given. Show using induction that it is bounded above by $2$ and strictly monotonic increasing. \[5pt] (You can take the monotonicity of the square-root-function for granted) \end{problem}

[deep vortex]

So, I showed that it is bounded above by 2

How do we show it's strictly monotonic increasing by induction?

[feral agate]

Do you even need induction for this?

Ah yeah you do nvm

Assume that $a_k>a_{k-1}$, then prove that $a_{k+1}>a_k$

[woven radish]

kheerii

[feral agate]

Notice that $a_{k-1}=a_k^2-1$

[woven radish]

kheerii

[deep vortex]

What about a second induction? We first show it's bounded above by 2

Then, in a seperate one,

[feral agate]

What?

[deep vortex]

$a_{n + 1} = \sqrt{a_n + 1} > a_n \iff a_n + 1 > a_n^2 \iff 0 > a_n^2 - a_n - 1$

[feral agate]

Yeah, that works

[woven radish]

Kepe

[deep vortex]

How do we show this though

[feral agate]

What does the last condition give us

Wait, the limit of this sequence is not meant to be 2 right?

That’s just an arbitrary number they chose for you to prove upper boundedness

[deep vortex]

It is supposed to be bounded above by 2

[feral agate]

Yeah

But that doesn’t mean the limit is two

[deep vortex]

Yeah

[feral agate]

Anyway, this tells you $$0<a_n<\frac{\sqrt{5}+1}{2}$$

[woven radish]

kheerii

[deep vortex]

Why?

[feral agate]

Just solve the inequality

[deep vortex]

Where does the 0 arise from

[feral agate]

a_n must be positive

Since it’s defined as a square root

[feral agate]

This is true due to your boundedness condition, so we are done

Unless I messed up something

[deep vortex]

Yeah, sqrt(5) < 3 and (3 + 1)/2 = 2

Thank you!

[feral agate]

can I just ask, how did you prove boundedness?

[deep vortex]

Though we need two seperate inductions, right?

First one, boundedness

And only after we are done with that can we do this

[feral agate]

yeah the logic I used is based on your former proof of boundedness

which would also require an induction

[deep vortex]

Induction step: Assume $a_n \leq 2$. Then $a_{n + 1} = \sqrt{a_n + 1} = \sqrt{3} < \sqrt 4 = 2$

[feral agate]

so yeah two inductions should be good

[woven radish]

Kepe

[feral agate]

yep, looks sound

[deep vortex]

Alright, thanks

.close

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[feral agate]

just make sure you mention $a_n>0$ in your proof

[woven radish]

kheerii

[devout snow]

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[copper birch]

you take the inverse negative of the given slope and put that into point-slope form and solve, right? i'm getting (1.615, 3.077) -> 4.692 but it's telling me i'm wrong 🤔
Image

[trail eagle]

This looks right. Maybe try and change the number of decimals you input

[copper birch]

still not working but thanks!

[devout snow]

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[barren prawn]

Physics related

[barren prawn]

A small body slides from the top to the bottom of a wedge-
shaped stationary slope with a base of 1 m. At what slope
will the sliding time be the smallest if the coefficient of friction
is equal to 0.25? By how much does the result change if the
slope also moves while the body slides, namely with a constant
acceleration of 1 m/s2 in the horizontal direction, opposite to
the body’s sliding direction?

[woven radish]

OHHELLNAH
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[barren prawn]

but how do i solve the second part?

I need to apply that the whole wedge is accelerating backwarts with 1m/s^2

[devout snow]

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[topaz mica]

Is this valid algebra

[topaz mica]

Or can I not do that

[solid trail]

could you tell me the thought process?

[topaz mica]

Just not sure if I’m only able to that if it’s a term by itself rather than being added to another term

I’m PRETTY sure it’s ok but not completely

[solid trail]

If we had (1 + 2)/(99 + 100), we can't turn it into (99 + 2)/(1 + 100) can we

[topaz mica]

The book leaves the answer at the first part too

Oh

No we can’t

Shoot so i can’t do that

[solid trail]

In general, the only "communication" between numerators and denominators is cancellation

Like (2x)/(3x + x) = (2x)/(x (3 + 1)) = 2/(3 + 1)

That is valid

[topaz mica]

Rightttt

Ok thank you!! Glad I asked

[solid trail]

Or going the other way around multiplying top and bottom by the same thing

[solid trail]

Np feel free to ask here anytime

[topaz mica]

.close

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[restive river]

Can someone please explain and give me some intuition for Vieta formula?
Im familiar with calc 2 so you can use some stuff from there if it helps with the intuition

[pastel pasture]

There's no need for calculus

[woven radish]

Bob l'éponge

Bob l'éponge

[restive river]

whats beta and alpha?

[pastel pasture]

roots

[restive river]

what do you mean by roots?

[pastel pasture]

solutions

values for which the quadratic is 0

[restive river]

makes sense
what about the intuition?

[pastel pasture]

intuition?

honestly it's probably something to do with combinatorics

nothing too obvious about it (to me) until you expand it and realize "oh well, that makes sense"

[restive river]

I dont quite understands why we use the roots and what role they play

in your first message

this part to be specific

[dark dawn]

it reasons with this I suppose

you have an nth degree polynomial with n roots, it can be factored as such (x-a)(x-b)...

I guess?

for intuiton you could look at the proofs

[restive river]

how do you get from the roots to the factored?

[dark dawn]

im not sure what ya mean

[restive river]

how you get from the left side to the right

[dark dawn]

have you tried expanding the right hand side of that expression

like bob leponge mentioned before

[restive river]

Ill try

[dark dawn]

maybe what you mean to ask is why we can write a relationship like that? , after that has been setup the relation between coefficients and roots kinda just plops out

the reasoning is found from https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra and https://en.wikipedia.org/wiki/Vieta's_formulas in the proof section

check the base case for n=2 and I think you'll be happy

[devout snow]

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[devout plover]

confuse

Ik 4,5,6 isn't one of the pythagorean triples

but 4, 7.5, 8.5 aren'teither

[dense lynx]

multiply these numbers by a constant so that they're all integers

[devout plover]

?

can I multiply them by 2?

[dense lynx]

yes

[devout plover]

but so what

[dense lynx]

what do you get?

[devout plover]

8, 15, 17

[dense lynx]

and?

[devout plover]

and what

[dense lynx]

that's a pythagorean triple

[devout plover]

what about if I multiplied it by 3

[dense lynx]

the idea here is triangle similarity

[devout plover]

I would get 12, 22.5, 25.5

that isn'ta triple

[dense lynx]

why not?

[devout plover]

idk

because

[dense lynx]

12^2 + 22.5^2 = 25.5^2

[devout plover]

my teacher said tht

3,4,5

[dense lynx]

the numbers are the valid sides of a right triangle

[devout plover]

8, 15, 17

7,24,25

5,12,13

are the pythagorean triples

no others

[dense lynx]

are the pythagorean triples you will be using in your class* is most likely what they meant

[devout plover]

so how many are there

[dense lynx]

as you just saw, infinitely many

[devout plover]

why can't 4,5,6 be a pythagorean triple

if

3,4,5 is

[dense lynx]

well

what do you multiply 3, 4, 5 by to get 4, 5, 6?

[devout plover]

what?

[dense lynx]

scaling the sides of a shape by the same factor keeps the structure of the shape the same

if you multiply each of the sides of a right triangle by some number, you will have a right triangle as a result, no matter what numbers you use

[wooden veldt]

worth checking if you know what a pythag triple is, rather than just some examples of them

[devout plover]

so what do I multiply 8,15,17 to get 7, 24,25

what does that even prove

[dense lynx]

this is a good point

[devout plover]

https://www.youtube.com/watch?v=F9YtVqyps1s

[dense lynx]

@devout plover can you explain what you think a pythagorean triple is?

[devout plover]

I wathced this video

[dense lynx]

right

8-15-17 and 7-24-25 are both pythagorean triples, but they aren't multiples of each other

[devout plover]

what the heck is a scalar mltiple

[dense lynx]

so as you can see, they don't have to be

just remove the word "scalar" from that explanation then

[devout plover]

I still don't get it

[dense lynx]

which part?

[devout plover]

I don't understand anything that you're saying

[dense lynx]

okay, let's start here then

[devout plover]

the guy in the video didn't tell us the definition

he just said the 4 most common pythagorean triples

and that we can memorize them to use in the ACT

[dense lynx]

did you see the part where he wrote down a^2 + b^2 = c^2

[uncut crow]

really helpful

[devout plover]

that's the pythagorean theorm

that isn't the definition of a pythagorean triple

[wooden veldt]

(the video does infact say there are infinitely many, and that those are the most common)

[dense lynx]

in your class, your teacher is taking "pythagorean triple" to mean 3 numbers a, b, c that satisfy the equation a^2 + b^2 = c^2

(normally, a pythagorean triple is limited to positive integers only, but judging by the question you posted above, i guess your teacher isn't using this definition)

[devout plover]

my teacher has made tons of mistakes in the online class I'm taking

so idk

I can't really even trust him anymore

[dense lynx]

anyway do you understand this definition

[devout plover]

so

3^2 + 4^2 = 5^2

for example

of a pythagoraen triple

[dense lynx]

that's the basic idea behind it

[devout plover]

why not 4,5,6

[dense lynx]

try evaluating 4^2 + 5^2, and compare that with the value of 6^2

[devout plover]

those are 3 numbers

so there are like

what about

5,6,7

or

100,200,300

[dense lynx]

that's not even a triangle

[devout plover]

it could be a really big triangle

[dense lynx]

no

remember that in a triangle the sum of two sides has to be greater than the third side

[devout plover]

why not

[dense lynx]

100 + 200 is not greater than 300

therefore, you cannot have a triangle with side lengths 100, 200, 300

[devout plover]

ok

101,202,300

is that a triangle

[dense lynx]

yes

[devout plover]

ok

what about

69,52,100

tht is a triangle

but how do I uickly know if it is a triple or not

in my head

[dense lynx]

in most cases when you have large numbers, you want to divide each of the numbers by a common factor

[devout plover]

ok

[dense lynx]

in this case, there is no common factor between 69, 52, and 100

and unsurprisingly, since it's just 3 random numbers, it's also not a triple

[devout plover]

how do I know that in my head

like in 5 seconds

[dense lynx]

you can do the division

like you can quickly see that 69 is divisible by 3, and 52 is divisible by 4

[devout plover]

and 100 by 5

isn't that 3,4,5

which is a triple

[dense lynx]

that's like saying 60, 60, 60, is a pythagorean triple just because the numbers are divisible by 3, 4, and 5

to maintain the shape of the triangle, you have to divide/multiply each of the side lengths by the same number

[devout plover]

what if

52,69,100

were side lengths

[dense lynx]

but they are; they're the side lengths of the triangle you drew

[devout plover]

if we knew 52

and 69

3 is by 23

hm

4 is by 13

for 52

so triple 3,4,5 wouldn't work out

because

69 is 3 by 23

and 52 is 4 by 13

[devout plover]

48+36 > than 60

so this is not a triangle?

<@&286206848099549185>

[west pewter]

It's implying that it's a Pythagorean triple

Or in other words

That you can use the Pythagorean theorem for the sides

[devout plover]

?

yeah but

the sum of 48 + 36

is greater

than

the third side

the hypotenuse

60

so

[west pewter]

Yep, as it should be

[devout plover]

this triangle is not a triangle

?

[west pewter]

It is a triangle

The triangle inequality states that the sum of any two sides is greater than the third

[devout plover]

but

when I do

36

36^2 + 48^2 = 60^2

they are eual

[west pewter]

Yep

[devout plover]

so how is it greater

[merry flax]

yu have the hint there which is 3,4,5
you can find the factor of each one for example 60 is 5 * 12, 48 is 4 * 12, sooo x is 3 * 12

[west pewter]

Squared sums and sums are different

Consider this

[devout plover]

I already know the answer

[west pewter]

3 + 4 > 5

[devout plover]

ok

[west pewter]

But 3² + 4² = 5²

[west pewter]

That this inequality holds doesn't mean that it should when we consider their powers

It may turn out to be equal as well

[devout plover]

so

suared sum

can be

eual or greater

but

sum is always greater

[west pewter]

In every triangle, it happens that the sum of any two sides is greater than the third

This is a statement that can be proven

[devout plover]

what about

the suared sum of any two sides

is eual or greater than the third?

[west pewter]

Will be greater than the third side too

If you think about it

In the case of a right angled triangle, it becomes equal

[devout plover]

okl

ok

[west pewter]

So right triangles are special, in this regard

[devout snow]

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[restive river]

tanA + tanB = 5tanC, what is tanAtanB

[restive river]

I know this

[odd island]

Hey can someone help me solve this and show me how I draw this in a coordination system

[devout snow]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

@restive river Has your question been resolved?

[devout snow]

@restive river Has your question been resolved?

[devout snow]

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[royal laurel]

Wait

[royal laurel]

Shit

Flock

0 is a multiple of any number

Flock

[royal laurel]

Is that true

[torn vessel]

is this a question

[royal laurel]

It is shrimp sushi

[graceful cosmos]

If 0 were a multiple of, let's say 2, then for some integer k:
0 = 2k

Does such an integer exist?

[torn vessel]

yes sir

[torn vessel]

3b1b mentioned

[royal laurel]

“0 is a multiple of 5”

[royal laurel]

0*2=0, so 2 is a multiple of 0

[safe jasper]

saying that something is a multiple of 0 isn't too accurate

if 2 was a multiple of 0, then there would exist an integer k for which 2 = 0 * k

no such k exist

[royal laurel]

Is it some definition or something

The “2= 0 * k”stuff

[safe jasper]

yes, it's the definition

a consequence of this is that 0 is a multiple of 0, no nonzero number can be a multiple of 0, and 0 is a multiple of every number

[royal laurel]

I see

[royal laurel]

That’s incorrect I suppose

Ahhh

Abhhhhhhhhhh

Ahhhhhhhh

It is contradictory

My mind blows

<@&286206848099549185>

Help

Why

Why he claims that 0 is a multiple of 5

[torn vessel]

no this is right

why wouldn't it be?

[torn vessel]

it agrees with this

see the "if a is NOT zero"

[royal laurel]

It doesn’t fit the definition

Ahhh

[graceful cosmos]

0 is a multiple of 5.
5 is not a multiple of 0.

[royal laurel]

Ahh

Im a human but human is not who i am

Terrific

[royal laurel]

I should memorize the definition I suppose?

[torn vessel]

square is rectangle but rectangle no square

[royal laurel]

K

By the definition if 0 is a multiple of 5

[torn vessel]

mhm

[royal laurel]

Then 0=na, where n is some integer and a is not zero

The equation would never hold!

[torn vessel]

a can be 0

what

[safe jasper]

what

to check if b is a multiple of a, show that there's an integer n for which b = na

[torn vessel]

$a\in\text{any integer}$

[woven radish]

Vѳrtєx-

[torn vessel]

that notation is NOT right but whatever

[safe jasper]

therefore to show that 0 is a multiple of 5, show that there's an integer n for which 0 = n*5

[royal laurel]

But 0=n*5 where n is integer would never hold!

[torn vessel]

n=0 💀

[royal laurel]

I didn’t know 0 is a integer

My mind

[torn vessel]

yea

(ignore how it says an integer is a negative integer thats kinda weird 🤡)

[safe jasper]

https://tenor.com/view/siege-tales-russian-badger-floor-is-made-out-of-floor-rainbow-six-gif-21093443

[royal laurel]

I see

Thank you shrimp nigiri

And Neil

Also kaynex

[torn vessel]

oh am i shrimp lmao

i am

np

[royal laurel]

.close

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[devout snow]

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[barren tundra]

help please

how do i do this porblem

[languid lichen]

do you think there are any negative zeroes based on the graph?

[barren tundra]

nope

both zeroes are positive

[languid lichen]

then the negative zero on the choices is not reasonable

so it's the answer

[barren tundra]

wait wait wait

could you explain that

i dont get it

oh shit nvm

i get it

i just read it weong

thanks bro 🙏

.close

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[next sorrel]

for a I got

I believe my answer is correct

that is textbook answer

[devout snow]

@next sorrel Has your question been resolved?

[next sorrel]

dead ahhh server

[cedar berry]

wait

shouldnt it be 30 instead of 40?

[next sorrel]

you are welcome to graph it on desmos btw

[devout snow]

@next sorrel Has your question been resolved?

[next sorrel]

bitch

stop askign me if has been resolved

if nobody talkin

[devout snow]

@next sorrel Has your question been resolved?

[devout snow]

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[candid stump]

anyone knows what this topic is called? the first column stands for a, the second for b and the third for c and the one behind the line is the result i think

[wicked turtle]

augmented matrix?

[candid stump]

th ank you

.close

[devout snow]

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[devout snow]

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[junior pecan]

This is a solution for an exam at my uni some years ago and it seems wrong to me

[junior pecan]

Second equation wouldn't yield 1 but instead 1/3

[autumn fjord]

why

-(-1/3)=1/3

so we’re still good

explicitly, the full LHS of eq 2 gives -(-1/3)+1/3+1/3 = 3(1/3) =1

if you’re on mobile the dash is actually a negative sign but discord decided to make it break up the line between the text and the rest of the equation

@junior pecan in case u wanted a ping

[junior pecan]

I see that now...I made a mistake

[autumn fjord]

happens to all of us :P

[junior pecan]

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[tawny jackal]

Grade 11

[topaz axle]

how many orders exist in total?

[tawny jackal]

order as in two people have the same entree, main and sweet
so 3 x 6 x 5 = 90?

[topaz axle]

like the menu gives you a single choice

you pick entree, main and sweet, and that's one choice

how many choices

[tawny jackal]

OH

[topaz axle]

yes

now if there's 90 choices, yo ucan find the probability that 16 people pick different choices

the opposite of that is the answer

[tawny jackal]

thats where im stuck
is it 90P16?
it cant be bcus the number is too big

WAIT
16P2 x 90?

[topaz axle]

90p16 is correct

if you divide it by 90^16, that's the probability

the numbers become smaller if you do it slightly differently

[tawny jackal]

whats the other method?

[topaz axle]

actually, that's maybe not the answer

if they mean 90 equally likely orders, you do (90/90)(89/90)(88/90)(87/90)...
this is 90p16 / 90^16, except the numbers stay small

but you could assume that each entree or each main is equally likely, and then it's a different answer

[tawny jackal]

ohhh i understand now

[topaz axle]

i don't know how to do it then have to think

maybe i'm confused and it's the same

[tawny jackal]

no i think this is the right answer

[topaz axle]

yeah okay, that's the same thing

[tawny jackal]

alr
thank you so much for helping
i appreciate it 🫶
i hope you have an amazing day!!

[topaz axle]

ty

[tawny jackal]

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[zealous umbra]

I don't get part a and b

[zealous umbra]

part a i have absolutely no clue for

for b, Im pretty sure im wrong but

[mild sorrel]

for b, first find the distance covered when its fuel is burning
now,
use v^2=u^2+2as to find the distance covered when it is under gravity
v = 0 because at the tip it will stop
find u using the v = u+at formula
a = -9.8 m/s^2

add the distances

[zealous umbra]

s = 0 x 1/2 * 20 * 3^2
s = 90

v = u + at
v = 0 + 20 * 3
v = 60

and for after the fuel runs out..

v = u + at
0 = 60 - 9.81 t
t = 6.17 s

s = ut + 1/2 at ^2
s = 60 * 6.17 - 1/2 * 9.81 * 6.17^2
s = 183.47

total s = 273.47

[zealous umbra]

okay, but could you also check where i went wrong?

[mild sorrel]

if you're calculations are correct, then absolutely correct.
for b.

,w arctan 20

[woven radish]

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[mild sorrel]

,w 1.52083*180/pi

[woven radish]

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[zealous umbra]

wait what

[zealous umbra]

NICEE

wb a?

[mild sorrel]

,w arctan -9.8

[woven radish]

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[mild sorrel]

,w 1.469107*180/pi

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[mild sorrel]

First, the velocity increases uniformly for 3 seconds. The slope will be 20/1
after 3 seconds, there is deceleration, so the slope will be negative.
negative slope = -9.8/1
20/1 will be more steep than -9.8/1 and -9.8/1 will go in the negative direction (downward until the velocity is 0).

[devout snow]

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[rugged jewel]

I have matrix A:
3 a_12 a_13
a_21 -1 a_23
a_31 a_32 -2
Given that A has 2 different eigenvalues. Determine the eigenvalues if you know that |A| = -54

The answer should be lambda_1= 3 and lambda_2 = -6.

[rugged jewel]

How do i even start with this?

I have the worked out answer with me, but i don't understand it, maybe someone can give another explanation

I know i should use a trace

[surreal night]

$\begin{pmatrix} 3&a_{12}&a_{13}\a_{21}&-1&a_{23}\a_{31}&a_{32}&-2\end{pmatrix}$

[jaunty mantle]

You know the trace

The trace of a matrix is the sum of the eigenvalues

[woven radish]

Flappie

[jaunty mantle]

You know the determinant

The determinant is the product of the eigenvalues

You know there’s only 2 distinct eigenvalues

[rugged jewel]

Yes

[jaunty mantle]

Write some equations out for these constraints

[rugged jewel]

2 distinct eigenvalues

So i could have 3

So how would i set up an equation then?

[jaunty mantle]

Just let a, b be the 2 distinct eigenvalues

[jaunty mantle]

This tells you a + b + b = trace

[jaunty mantle]

This tells you ab² = det(A)

[devout snow]

@rugged jewel Has your question been resolved?

[rugged jewel]

Wow

I see now

Thx frost

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[maiden zinc]

to integrate that do we just devide by -2t?

[sonic smelt]

See if the derivative of -e^(-t^2)/2t is e^(-t^2)

[maiden zinc]

,w integrate -e^(-t^2)/(2t)

Ei?

,w derivate e^(-t^2)

,w derivate -e^(-t^2)/(2t)

[woven radish]

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[maiden zinc]

,w integrate e^(-t^2)

[woven radish]

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[maiden zinc]

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[clever whale]

Hi guys dose anyone know how to prove that k=2

[tall stirrup]

differentiate

find the minima

[restive river]

You have to find the minima

[clever whale]

If I differentiate it I am finding acceleration no?

[restive river]

Oh shit

[clever whale]

How to find 😦

[restive river]

It is given minima of the function is 2

So basically when you derivate a function and it equals 0 , it means that function it's on its local minima or local maxima

[tall stirrup]

*differentiate

[clever whale]

But if I sub my t=2 I will get that my k is 4/10

[restive river]

Spare me in on travel 😭

[restive river]

Did you differentiate?

[clever whale]

Why need differentiate 😭😭😭I thought that is the acceleration

[restive river]

It's still a function

[tall stirrup]

how do you find the minima then

[clever whale]

That’s why I am so confused 😭

[restive river]

Think of it this way

The magnitude velocity MAY be at a minimum when acceleration is 0, right?

[clever whale]

Yes

[restive river]

There you go

[clever whale]

ohhhhhhhhhhh

Okiiii thanks guys !!

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[restive river]

Did you solve it?

[clever whale]

Yes I did

THANK UUUU

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[granite arch]

i got the general solution to be Ae^(x/2)+Be^(-x/2)-3

if x=0 y=-3, this means A+B-3=-3, meaning A+B=0

however i dont get the dy/dx=2 part

differentiating gives (1/2)(Ae^(x/2))-B(-1/2)e^(-x/2) = 2

no idea how the ans says B=-2 A=2

this is the ans

i dont understand how the answer arrived to here

[restive river]

sub x=0

[granite arch]

What?

so

the x=0 y=-3 applies to dy/dx too?

is this an english issue instead

[restive river]

yes

[granite arch]

oh my god

thanks

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[ancient basin]

How to get the quadrants

[ancient basin]

2

[agile garnet]

try making a unit circle

then find the expresion on that

and 1-sin^2x is cos^2

so u have sin/cos

which is tan

[ancient basin]

No but its in a square root

[agile garnet]

yeah

[ancient basin]

So you get cosx in absolute value

In the numerator

[agile garnet]

hmm

[ancient basin]

<@&286206848099549185>

The aboslute value is what tipped me off

[agile garnet]

same

hmm

so square the entire thing?

or consider case

cases

where sinx is +ive or -ve

hello?

[restive river]

@ancient basin

[ancient basin]

hello

[restive river]

hi!

:p

cosx=sqrt(1-sin^2(x))

do you understand how i got this ?

[ancient basin]

Yeah but it was in absolute value

Since it was cos^2x in square root

[restive river]

oh sorry
i was nto clear on my part

[ancient basin]

Np

[restive river]

writing tanx = sin / cos

i cancelled sin on both sides

[restive river]

resulting in this

get it ?

[ancient basin]

Yeah the part i dont undertsand how can you cancel sinx

Wont the tanx become secx

[restive river]

get this ?

[restive river]

,rccw

[ancient basin]

I want to know how after cancelling sinx it stays equal tanx

[restive river]

so you are left with cosx=|cosx| right ?

[restive river]

it won't

[ancient basin]

Oh yeah i understand it more now

[restive river]

now when is |cosx|=cosx ?

when cosx is positive

[ancient basin]

Yeah

[restive river]

and in which quadrat's is cosx positive ?

[ancient basin]

First and fourth

[restive river]

yep

[ancient basin]

But another thing is that in 4th quadrant

The cos stays positive

But sinx is negative

So -sinx / cosx

In 4th quadrant

Which would equal -tanx

Not tanx

[restive river]

that is wrong

[ancient basin]

I want to learn what am wrong

So i understand

So because cos is in aboslute its always postive

So as for tanx and sin

In fourth quadrant

If i multiply both sides of the equation by -1

The cosx stays positive?

[restive river]

since x belongs to 4th quad you cant write tanx = - tanx

but you can write tan x as -tan(theta) where theta = 2pi-x

[ancient basin]

Am reading but its takes a while to grasp, now in the picture how could be equal to +tanx in the equation

[restive river]

,rccw

[restive river]

the problem is

you are not changing the value inside the tan when you are adding an extra minus

what you are saying is

tanx=-tanx since x is in the fourth quad

[restive river]

now this would mean -1=1 (cancelling tanx on both sides)

which is obviously not true

get it ?

[ancient basin]

It still doesnt equal +tanx

[restive river]

now

-tanx=-sinx/cosx

multiplying on both sides

you will be left with tanx=sinx/cosx

[ancient basin]

Just the symbol of the numerator changes?

[restive river]

ig you missed the minus present to the side of sinx

[restive river]

yes

you don't change the symbol of the denominator too

[ancient basin]

Yeah ok now i understand

Thank you alot

:)

[devout snow]

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[short hare]

So

[short hare]

Weird question

[mild sorrel]

!da2a

[devout snow]

No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/

[short hare]

Ik but someone started writing in another channel i was already writing in so- i just need to claim the channel 😭

[mild sorrel]

okay np.

[short hare]

So- how far does calculus 1 go? In my third year of highschool im gonna take “second derivatives, how to derive and integrate logs, euler’s number and its limit, unspecified integration, and how to calculate the area between a curve and a straight line”

Am i finishing calculus 1 or?

Also theres more but these are the last lessons for every unit

[short hare]

About “specified and unspecified integration”

I tried to translate it

From arabic to english

And thats what i got

So its prolly not specified integration.

[wicked turtle]

looks like the university here covers chapters 1-6 of stewart in calc 1

[short hare]

Stewart?

[wicked turtle]

standard calculus book

[short hare]

You mean like- basic?

[wicked turtle]

i can paste the contents of those chapters if you want

[short hare]

I mean yeah

Maybe ill be able to figure it out

[wicked turtle]

that's it

[short hare]

2 is the start of my highschool year

[short hare]

Calculus of rainbows? Didnt know calculus was gay.

[wicked turtle]

yea i would imagine there's a lot of overlap with HS calculus

most people who took HS calculus would take a placement exam to go into either calc 2 or 3 instead

i've never read the book, i can't vouch for its quality haha

but it is used in many universities, so it probably sucks

[short hare]

What a weird conclusion

[wicked turtle]

why?

most widely used "[subject] 101" textbooks are bad, aren't they?

[short hare]

Doesnt tgat mean its good-

[short hare]

Idk

[wicked turtle]

no, it probably means the universities are getting some kind of financial incentive to use it

also it comes bundled with software for automatic HW grading so it means less work for the universities

doesn't mean it's actually good

[short hare]

5.3 is literally the end of my calculus year

[short hare]

Ooh

[short hare]

That’d make sense

[wicked turtle]

chapter 6 is grungy but standard material, you could work through it on your own

[short hare]

Yes indefinite and definite integration not specified 😭

[short hare]

We didnt learn inverse functions in algebra to learn it in calculus

[lusty sapphire]

RIP

[short hare]

That means i’ll prolly finish calculus 1 with the start of collage

[wicked turtle]

in practice in that chapter the main inverse functions you use would be exp/log and the inverse trig functions

[short hare]

I only know one inverae function

1/sqrt(1-x^2)

Sin-1(x)

And im proud

[wicked turtle]

well probably you know that x^2 and sqrt(x) are inverses of each other (for x >= 0)

[short hare]

Oh… yeah that’d make sense

Wait so

Sin(sin-1(x)) =x?

[wicked turtle]

yes if you restrict the domain of x

sin^-1 can only be applied to [-1,1]

[short hare]

Everything is making sense all of a sudden

[short hare]

I mean

Yeah

Just like her inverse function

Her

Yes

Bc

A function is a girl 😩💅

We can debate

[wicked turtle]

sure, why not

[short hare]

Tysm bungo

Bye

See you… very soon

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[wicked turtle]

haha cheers

[devout snow]

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[trim raven]

Can someone point me in the write direction with implicit differentiation? I’m in undergraduate calculus

[feral agate]

You seem to have done this correctly

What’s the problem b

?

[trim raven]

webengage says the answer is wrong, so idk

[feral agate]

Ah wait

The very first step has a mistake

cos(x+y)*(1+y’)

You missed the 1

The rest looks good

[trim raven]

so 1 + y' isn't just y'

gotcha

I guess I am good then. Thank you

[feral agate]

Of course

[devout snow]

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[restive river]

write an equation given the following x intercepts: (-2,0) and (6,0) and a point that passes through the parabola (1,-30)

[devout snow]

Please don't occupy multiple help channels.

[restive river]

I got only the x value of the vertex, which is 2

I can’t figure out the rest

[shut pagoda]

since they give you the x intercepts it would be more helpful to write the equation in factored form

ie y=a(x-r)(x-s)

[restive river]

so would the factored equation be y = 5/6(x + 2)(x - 2)?

i also have to write the equation in the standard form I think???

it’s in the form y - k = 1/4c(x - h)^2

i would also need to find c (the distance between the vertex and the directrix and the distance between the vertex and the focus)

which im confused about since I don’t have the coordinates of the focus or the directrix

[devout snow]

@restive river Has your question been resolved?

[restive river]

wait did I do something wrong

[winter patrol]

the roots aren't -2 and 2

[restive river]

I thought those were the x intercepts?

OH SSIT

WAIT

[winter patrol]

-2 is an x-int
2 is not (you mentioned thats the x-coord of the vertex)

[restive river]

is it -2 and 6

so

it would be y = 5/6(x+2)(x-6)?

[winter patrol]

how are you getting the 5/6

[restive river]

wait I messed up

am I supposed to plug in y and x

with the point that is on the graph

[winter patrol]

yes

[restive river]

ohhh

wait

but how does that correlate to the equation that I have to put in : y - k = 1/4c (x - h)^2

[winter patrol]

since you're given the x intercepts, ideally you'd want to start with the factored form

knowing the intercepts will give you
it would be
y = a(x+2)(x-6)
subbing in your point will give you the value of a

[restive river]

I got a = 2

so the factored equation would be y = 2(x + 2)(x - 6)

[winter patrol]

yes

[restive river]

then what would I do next to convert it to the other form

[winter patrol]

you've identified the x-coord of the vertex to be 2, (represented by h) so sub that in

which will give you the x-coord (represented by k)

[restive river]

the vertex is (2,-32)?

[winter patrol]

yes

[restive river]

so I have the vertex for the equation

I still need the c value

I only have y +32 = 1/4c (x -2)^2

[winter patrol]

the coefficient of (x-2)^2 will be the same as your leading coefficient a (which was 2)

[restive river]

wait is a just 2

so the equation is just y =2(x+2)(x-6)?

[winter patrol]

now you're going backwards

you already that that

[restive river]

would the 2 be in front of the (x - 2)^2

I’m confused with what you said earlier

[winter patrol]

yes

[devout snow]

@restive river Has your question been resolved?

[restive river]

should I keep this waiting or no? I have physics now

sorry-

[worn flame]

you have 3 points given, two of which are the roots

you know hence it has to be y = a(x+2)*(x-6) using the (-2,0) and (6,0) points

we'll use (1,-30) to get the value of a

plugging in x=1 -> -30 = a (1+2) * (1-6) = -15a

so a = 2

the equation is hence y = 2(x+2)(x-6)

open it up if required

[devout snow]

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[chilly ferry]

Yo, whats 4-9i/-6i

[chilly ferry]

In standard form

Well simplified

[surreal night]

!status

[devout snow]

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 was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[chilly ferry]

So

I forgot how to do it

[chilly ferry]

I guess I would say don't know where to begin

[surreal night]

Try removing the complex number in the denominator by multiplying the fraction by its conjugate

[chilly ferry]

Thank you

I got 3/2 + 2i/3

Is that correct?

[surreal night]

Yes it is

[chilly ferry]

Thank you

.close

[devout snow]

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[devout snow]

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[steel fiber]

can someone explain me this part

[steel fiber]

first of all what is modulus?

[surreal night]

modulus is the length of the vector

you can consider a complex number as a vector of real part and imag part

[steel fiber]

how?

[surreal night]

Let $z=a+bi$ then $|z|=\sqrt{a^2+b^2}$

[woven radish]

Flappie

[surreal night]

just keep this in mind

[steel fiber]

doesnt make sense when u calculate it is a^2 + 2abi -b

[steel fiber]

and like what does this has to do with here

the title is

15.7 Multiplication and Division in Polar Form

[surreal night]

which part?

[steel fiber]

what is polar form?

[steel fiber]

multiplication + divison

[surreal night]

polar form is $z=re^{i\theta}$

[woven radish]

Flappie

[surreal night]

yeah thats what the text is about, but what part in the text

[steel fiber]

every part of that

[devout snow]

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[steel fiber]

Hi

[steel fiber]

The channel closed due to timeout

[steel fiber]

.

This is my question

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[restive river]

i need help

[restive river]

i need all three answers for clarification

i alr solved for quadratic equations

but i think the answer is different for p and r

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[restive river]

<@&286206848099549185>

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[restive river]

In my case A = -1/2 while it should be positive, I cant figure out what is wrong

[sand dove]

are you sure it's not because you put 1-x on denominator instead of x-1?

[restive river]

So i should switch 1-x to x-1?

I mean in my case its (1-x)(1+x)

[sand dove]

yeah

so

stick a - in the numerator

[restive river]

but why?

[sand dove]

and denominator becomes (x-1)(x+1)

[restive river]

I saw other problems without doing that method

solved*

[sand dove]

sure but you're asking why A should be positive

if you stick with 1-x on denominator

then A being negative is fine

[restive river]

so is it wrong if its not positive in the end?

[sand dove]

?

you can keep 1-x at denominator

this won't change the end result if you do every step correctly

it's just that I know people forget that 1/(1-x) integrates to - ln|1-x|

[restive river]

Not sure if my solution is good

[sand dove]

looks good

[restive river]

thanks

[devout snow]

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[tired dagger]

The collinear points A, B and C are located on a line, so that AC/AB-BC=5. Calculate AB/AC+BC

[ripe marsh]

was the figure given or did you draw it yourself?

[tired dagger]

Draw it myself

[ripe marsh]

hmmmm

[ripe marsh]

what you are given is a/(a-b) not that (a+b)/(a-b)

and do you know about this thing called "componendo dividendo"?

[tired dagger]

No. What is it about?

[tired dagger]

I type it bad, its AC

[ripe marsh]

google it, it helps us in the expressions like the one you have

[tired dagger]

Ok

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[toxic pike]

sin2x has a trig identity, and can be rewritten as 2sinxcosx, and -tanx can be rewritten as -sinx/cosx, but im not really sure how else to solve this

my first impression was to try to make it a quadratic, but I dont see any way to do that

[surreal night]

then you can factor out sinx

[devout snow]

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[hollow walrus]

Prove sin^2(A)tanA+cos^2(A)cotA+2sinAcosA=tanA+cotA

[hollow walrus]

can I have help? I have tried it but been stumped

[kindred agate]

id start by writing everything in terms of sin and cos

[hollow walrus]

wait nvm I solved it