#help-17

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livid helm
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how do i solve for radius

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timid leaf
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timid leaf
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all good?

sly sierra
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not all good

sly sierra
# timid leaf

^ obviously 0 is a zero of this one (verify by plugging it in)

sly sierra
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all of the terms have x as a factor

vocal sleetBOT
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@timid leaf Has your question been resolved?

chrome steppe
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Bungo's right.. the y-intercept occurs when x = 0, and since all the terms have an x...

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marble carbon
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marble carbon
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Could someone solve 3, I wanna confirm my answer

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I got 11.2 as the wind speed

inner osprey
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seems suspicious

marble carbon
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Well I can show my work if you want

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Lmk if I need to clarify anything

inner osprey
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ur multiplication is fairly incorrect

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helps to take out common factors when multiplying large numbers together

inner osprey
inner osprey
marble carbon
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That's probably smart

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That changes my answer to x =4

inner osprey
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hmm

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show?

marble carbon
inner osprey
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uh...

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u did not fix ur multiplication

marble carbon
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Ain't no way I'm this bad at math and taking an honors course 💀💀

inner osprey
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do u know how to multiply numbers together when they have multiple 0s in them

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u can multiply them as usual without the zeroes, then add back on those zeroes

marble carbon
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Oh my lord I realized my mistake

inner osprey
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so like __2__00 * __5__000 would be 2 * 5 = 10... but also with the 5 zeroes from 200 and 5000

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so 1000000 (six zeroes)

marble carbon
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After realizing my stupidity, x = 112 🙏🙏

inner osprey
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show again?

marble carbon
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Realized 450 x 600 is not 27 000 😃

inner osprey
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900 * 200 isn't 18000 either

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also

inner osprey
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and when i said "i'd recommend dividing both sides of your equation by 450" earlier,

i was referring to this line

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we have a 900 factor on the left, and a 450 factor on the right

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so we can divide both sides by 450 to make our lives either

marble carbon
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.close

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trail owl
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So I got something called Factorization of a trinomial and the next is what I apparently don't understand and I just need an explanation of how it works to make my math homework and English isn't my native language so forgive the inaccuracies I might have while spelling
Factor the trinomial X^2-4x-45. .The square root is obtained to the quadratic term (x^2) and two numbers are found that added together give the coefficient of the linear term (-4); and multiplied by the independent term (45) -45=(-9)(5)-4=-9+5 The factorization of the trinomial X^4X-45 is as follows: X^4X-45=(X-9)(X+5)
this is just an example so could anyone please explain it to me? All I have is something that mightn't be correct
the trinomial I have which was displayed as an example in class was the next one X^2-12x+32=(X-4X)(X+6) I'm not entirely sure if it's truly like that or it's another thing that I missed since I did pay attention to the explanation of the teacher but I still didn't understand anything since I can't interpret Spanish as greatly as English. I would add an image but it's in Spanish

livid tapir
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Pon la imagen

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Puedo hablar español

trail owl
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Ah muchas gracias

livid tapir
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,w (x - 4x)(x + 6) simplify

trail owl
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Solo ocupo la explicacion de como se hace no necesito nada mas

livid tapir
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vez que si dice 4x so son iguales

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Pero

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Ok, lo que pasa es que si tienes una equación quadratica $ax^2 + bx + c$, a veces se puede factorizar como $a(x - r)(x - s)$, dónde $a, b, c$ son numeros reales, y también $r, s$.

twin meteorBOT
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992qqoloy

livid tapir
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Oops ars

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$ars$

twin meteorBOT
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992qqoloy

livid tapir
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En todas estas expresiones $a = 1$

twin meteorBOT
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992qqoloy

livid tapir
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Asi que $b = -(r +s)$ y $c = rs$

twin meteorBOT
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992qqoloy

livid tapir
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Asi que para factorizar, puedes intentar algunos numeros r, s cuales que el producto sea igual a $c$, asta que $-(r + s)$ también sea igual a $b$

twin meteorBOT
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992qqoloy

livid tapir
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Por ejemplo con la primera expresión quadratica

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8*4 = 32

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Y -(8 + 4) = -12

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Asi que con esa información, saves que $(x - 4)(x - 8)$ es igual a $x^2 - 12x + 32$

twin meteorBOT
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992qqoloy

livid tapir
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Pudieras haber intentado 1*(-32) o 8*(-4),etc. Tambíen

trail owl
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Tiene sentido

livid tapir
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Pero $-(1 - 32) = - 31$ y$ - (8 - 4) = -4$ asi que no trabajan porque no valen a $'b'$

twin meteorBOT
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992qqoloy

livid tapir
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Ahora intentalo con $x^2 + 16x + 63$ happy

twin meteorBOT
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992qqoloy

trail owl
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Ahi vere

livid tapir
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Por ejemplo -3*-21 = 63 pero no trabaja con b hmmCat

trail owl
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Muchas gracias por tu ayuda dios te bendiga

livid tapir
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De nada

trail owl
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.close

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keen quail
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.close

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vast shale
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vast shale
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where did he get 2x^2/2 from?

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i thought it would've been x^2/2?

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oh wait

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💀

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raven badger
vocal sleetBOT
raven badger
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I have no idea where to start. I tried to express it in the form ax^n+b^n-1 .. and subsituted the function in. it didnt get me anywhere

vocal sleetBOT
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@raven badger Has your question been resolved?

raven badger
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<@&286206848099549185>

vocal sleetBOT
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@raven badger Has your question been resolved?

maiden sphinx
# raven badger

Usually a good place to start with these kinds of problems is to start plugging in values for x and seeing what you can find.

main knot
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p(x)2+1 = p(x2+1)

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it's pretty complicated

maiden sphinx
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It’s usually less complicated than it seems

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You can immediately find that p(1) = 1 and p(2) = 2

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So you might conjecture that p(x) = x works, which it does

main knot
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You need an equation

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They are infinite

maiden sphinx
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The hard part is to prove what doesn’t work

maiden sphinx
main knot
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no it's a function it's not really the same you need some way to show every P(x) that meets the criteria

maiden sphinx
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yes that's what I'm trying to explain

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if $P(x) = x$ then $P(x^2 + 1) = x^2 + 1 = (P(x))^2 + 1$

twin meteorBOT
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Awesam

main knot
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yeah sure, I already figured that

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keep going though

main knot
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There doesn't seem to be any good info, and I can't really understand much of it anyways but it has something to do with the combined functions idk much more than that.

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@raven badger Has your question been resolved?

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vast shale
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why does LCM include all the ements

vocal sleetBOT
hardy vector
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This comes directly from the definition

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If it didn't include all factors it won't be a multiple

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marsh orbit
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30 60 90 Triangle.

ABC is Right Angle Triangle AB⊥BC

AD Is angle bisector of BAC

Given DC=2BD

Calculate angle C

mental falcon
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what's a 30 60 90 triangle?

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or why did you mention that in the first line?

inner osprey
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as it would turn out, ABC is a 30-60-90 triangle with the given information

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and so is ABD

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which is pretty interesting since that's apparently what you're supposed to find 🤔

marsh orbit
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@marsh orbit Has your question been resolved?

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hoary cypress
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how to solve this eqn for x
k1 - k2(x) = dx/dt

hoary cypress
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k1 and k2 are constants

signal pendant
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separate the variables

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dt=dx/(k1-k2x)

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You will get x as a function of t

hoary cypress
whole gale
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by integrating both sides

hoary cypress
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yea ik

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but how

signal pendant
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Substitute k1-k2x=u

whole gale
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why

hoary cypress
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ok thanks

whole gale
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you can directly integrate it

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its a linear equation

signal pendant
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I know

hoary cypress
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its power is -1

whole gale
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so integral 1/x = ln|x|

whole gale
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well carry on

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you can proceed in a way which you find convenient

hoary cypress
whole gale
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ill tell you once you solve it

hoary cypress
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i solved it

signal pendant
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They both are really the same, the integral will be a logarithmic function and you can guess ln(k1-k2x)/-k2

whole gale
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yes

signal pendant
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Its just the chain rule in reverse

whole gale
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my bad if you got confused

hoary cypress
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oh ok

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i diid think abt the antiiderivative

hoary cypress
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.close

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tawdry turret
#

Given: Quadrilateral ABCD is a square.

Quadrilateral EBGF is a square (see diagram).

I need to prove that EG || AC

tawdry turret
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So I thought of passing a diagonal in the square EBGF

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And then continuing it to D

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But, I realized, I can't force it to both be the diagonal of EBGF and ABCD

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If I could, then it would have formed a 90° angle with both EG and AC and that's it

vocal sleetBOT
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@tawdry turret Has your question been resolved?

tawdry turret
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<@&286206848099549185>

tawdry turret
#

<@&286206848099549185>

vocal sleetBOT
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@tawdry turret Has your question been resolved?

tawdry turret
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<@&286206848099549185>

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.reopen

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/reopen

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It's already open

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Why?

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.close

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tawdry turret
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.reopen

vocal sleetBOT
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tawdry turret
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<@&286206848099549185>

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@tawdry turret Has your question been resolved?

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dusty nova
vocal sleetBOT
dusty nova
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Why does the general solution just have x^2

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Wouldn't it be some constant multiplied by it?

sullen notch
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x^2 gets you the inhomogeneous term

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The other terms in the solution are for the homogeneous equation so you can multiply them by whatever (since putting them into the LHS gives 0)

regal bane
dusty nova
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Oh right so c1 and c2 would just adapt to if it were 18x^2 or any x^2 term?

sullen notch
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Not really.. c1 and c2 can be any numbers, with no constraints

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That’s because you change the RHS of the DE to 0, those last two terms are the solutions to it

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And since the equation is linear, you can use the superposition principle to add any multiple of the homogeneous solutions

dusty nova
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Thank you. Wasn't sure if the multiple added needed to have a constant to show it could be any multiple.. idk

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but that sounds fine then

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cobalt wren
#

Hi I'm just strugling with this problem and dont really know how or what format to derive the non linear ODEs in?

cobalt wren
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This is the step I am working on and confused about

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I get as far as balancing froces and confused where to go

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<@&286206848099549185>

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@cobalt wren Has your question been resolved?

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@cobalt wren Has your question been resolved?

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true cliff
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how do u

vocal sleetBOT
true cliff
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solve the definite integral of

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x^2cosx/1+e^x

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considering the upper limit is pi/2 and lower limit is -pi/2

opal obsidian
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what have you tried?

inner osprey
true cliff
twin meteorBOT
true cliff
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hmm

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if you replace x with -x

twin meteorBOT
true cliff
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wait no

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wait a min

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,w integrate x^2cosx

true cliff
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eh?

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so lost

inner osprey
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uh

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how much integration do you know

true cliff
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uh

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i can integrate like

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chain functions

inner osprey
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if u are unfamiliar with IBP then this is def not smth u should be looking at

true cliff
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trig functions

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normal functions

inner osprey
true cliff
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interesting

inner osprey
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"integration by parts" maybe you know it by another name

true cliff
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what does it say

inner osprey
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internet will help you here

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u may know u-substitution is reverse chain rule

IBP is reverse product rule

kind of

true cliff
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OHH

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OKAY

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well

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the formula looks easy

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considering ibp then

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how do u solve it

true cliff
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if you replace x with -x, it'd become (-x)^2 cos(-x)/1+e^-x

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correct?

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.close

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rose shard
#

I’m currently in the inductive step

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I was thinking on inserting these two based on our assumption (IH)

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@rose shard Has your question been resolved?

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@rose shard Has your question been resolved?

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rose shard
#

Same issue

vocal sleetBOT
rose shard
rose shard
# rose shard

I have arrived at this I don’t know how I can simplify it further For my induction proof

edgy sapphire
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adding powers of 2 gets you close to the next power of 2

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it's sort of just a thing to remember

rose shard
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What?

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Hi again😂

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I did fully understand what you meant yesterday

rose shard
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Are you talking about this though

edgy sapphire
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yea the sum of powers of 2 lets you simplify that sum

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does this help understand the backwards thing?

rose shard
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No no

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I understood it here

edgy sapphire
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okok

rose shard
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You see

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I just inserted k=i for the first element in the sun

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And the end element in the sum where k=j-1

edgy sapphire
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alright what part are we at

rose shard
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Here I say, we could write it as two times the sum of two to the power of Jay minus K

edgy sapphire
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right

rose shard
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And then we get to this point

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And the task is again to show this

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And now I’m showing it holds for K plus one as well

edgy sapphire
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hm I'm not seeing how to get the right inequality you want to show in the end, it seems like it's sometimes less

rose shard
edgy sapphire
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but usually the simplificaiton is that a sum of 2^n from anything to k is greater than 2^k and less than 2^(k+1)

rose shard
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Here is the whole thing by the way maybe I’ve done something wrong

edgy sapphire
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oh it's really jank omg

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so like the base case was j-i=1, so for induction you're allowed to assume j-i >= 2

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so you can make a tighter restriction if you turn the giant summation into its largest 2 terms

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like this

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that should all work out

rose shard
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before I do that I was just wondering can I not use geometric series to simplify the sum?

edgy sapphire
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uh that should work too, if it gives something pretty simple at the end

rose shard
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Yeah, I’m not quite sure how😂

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I’m going to watch a video on it real quick

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So ours is 2^1 + … + 2^(j-i)

edgy sapphire
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2^i + ...

rose shard
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2^i?

edgy sapphire
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like it starts at 2^i

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wait nvm go ahead

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I'm misreading

rose shard
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(2^j-i) + … + 2^(j-(j-1)) = 2^j-1 + … + 2^1

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Okay okay gotcha

edgy sapphire
rose shard
#

Does this look correct so far?

#

it’s j-i btw in the series

edgy sapphire
#

a should be 2 here since 2^1 is the first term

#

but yea then it's good and should make the induction work out in the end

rose shard
#

But what is r?

#

1

edgy sapphire
#

r is 2, the ratio

rose shard
#

So a and r us 2

#

Is

edgy sapphire
#

right

rose shard
#

Don’t think it’s right

#

When assuming i is a variable, I get this

#

Or maybe it is right and then I just have to argue that then it’s higher

#

Even though I don’t arrive at that form that you sent which would be nice

#

To end the proof like that

edgy sapphire
#

the wolframalpha result looks right

edgy sapphire
#

that's this part but I forgot to put a 2* in front

rose shard
#

So how do I connect it to the proof in the end here?

#

Because I wrote tij is greater

edgy sapphire
#

set it up >= 2^(j-i+1)-1 which was the starting goal

#

and you can move terms around to show it's true

rose shard
#

But have I technically not written it wrong then?

#

Because I have written t_i,j is greater than what we’ve just all this we’ve done

#

Where is it correct and just need to do something like this

edgy sapphire
#

yea this works because the goal is tij >= 2^... -1, and so far you have tij >= ... >= >... in a long chain

#

so if you end at that part you're good

rose shard
#

And maybe it should be equal instead of a greater than chain or what do you think?

edgy sapphire
#

I wouldn't call that right side t_ij, it's just a lower bound the problem gave

rose shard
#

Such that you know the difference that right now I am just rewriting the equation, and then the bound in the end

edgy sapphire
#

like the big picture is prove tij > lower bound, and the proof is tij > > > > > > > > lower bound

rose shard
edgy sapphire
#

the bottom row looks good to me

rose shard
#

Perfect and I should just argue that because you have the same exponent except the plus one part but in the left side you have four multiplied by it hence it outgrows?

edgy sapphire
#

eh it's not that it outgrows it has to be true for all j-i >= 2, you can split 4*2^(j-i) into 2^(j-i+1) + 2^(j-i+1) to make the comparison easier

rose shard
#

Wait damn what

#

Would that not be Nice to

#

Too

#

Then, in the exponent you could see it’s now +2 instead of only +1

edgy sapphire
#

uh I'd replace i with x when using wolframalpha

#

yea the left side is definitely a bigger exponent, but proving the >= is a little annoying

#

if you rearrange a bunch ofterms it'll have 2^... on the left and 2(j-i) on the right

rose shard
edgy sapphire
#

which is kind of a giant standard induction problem on it's own (?) but here I'd just say it works when you graph it out

rose shard
#

Yeah, I just have to say that i is a variable when using Wolfram

edgy sapphire
#

yea there's not really a great way to algebraically prove it here, but graphically youcan show it works for (j-i) > 1 so it's valid

rose shard
#

Yeah, I’m definitely not doing a whole other proof

#

I already spent so much time on this

#

😂

edgy sapphire
#

yea totally

rose shard
#

Then I’d rather go down

#

Fail the course

#

But I just don’t understand shouldn’t it be for all ING

#

I and j

#

l=j-k

#

And it has to work for all l>=0

#

Oh and when its 0 we hit the base case

#

Makes sense

edgy sapphire
#

yea it has to work for all L and L=j-i so that's sort of our single variable

rose shard
#

When l=0 j=i

#

So I should write

edgy sapphire
#

just "graphing both sides shows the left larger than the right for (j-i)>1, so we're done"

rose shard
rose shard
edgy sapphire
#

right

#

usually dominates means something else so it's a little confusing

rose shard
#

Yeah, okay is there a better term?

edgy sapphire
#

er I guess it's the same thing

#

I'm just saying larger

#

but w/e

rose shard
#

I’ll say larger as well

rose shard
edgy sapphire
#

👍 looks done

rose shard
rose shard
#

♥️♥️🤞

#

At least I’ve done what I could now I’m gonna go to sleep it’s like almost 5 (midnight)

vocal sleetBOT
#

@rose shard Has your question been resolved?

vocal sleetBOT
#
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winged burrow
#

Hi

vocal sleetBOT
winged burrow
#

,rccw

twin meteorBOT
winged burrow
#

My answer of number 3 iv

#

Didn't match with the book

timber wagon
#

Because the slope isn't 0

#

It's undefined

winged burrow
#

I got 7

#

As the gradient

timber wagon
#

Look at your own work

#

$\frac{7}{0} != 7$

winged burrow
#

What?

timber wagon
#

7/0 isn't equal to 7

winged burrow
#

Ye its 7

#

What's I wrote m=7

timber wagon
#

How are you dividing by 0?

winged burrow
#

Ohh u mean its not the same thing?

timber wagon
#

Yeah

winged burrow
#

7 and 7/0 is different?

timber wagon
#

7/0 is not defined

main knot
#

try in your calculator

timber wagon
#

,w 7/0

winged burrow
main knot
#

that's strange

timber wagon
#

What does that tell you?

timber wagon
winged burrow
#

What should I do

#

With 7/0

timber wagon
#

Look back at the points

#

What's common in both of em?

winged burrow
#

Umm

#

What do u mean by common

timber wagon
#

Can you see that they both have the same x coordinate?

winged burrow
#

Yea

#

I can see that

#

Ur saying to 🔌 the values

timber wagon
#

No

#

There's only one way that can happen

winged burrow
#

How

timber wagon
#

And that is when the line is vertical

winged burrow
#

Ohh

timber wagon
#

Another way you can see that is thru the gradient you found

winged burrow
timber wagon
#

It's undefined and gradient = tan(x) where x is the angle the line makes with the x axis

#

Tan(x) is undefined at x = 90

winged burrow
#

Yea that's true

#

So is 90 has

#

Anything to do here

timber wagon
#

Yeah it does

winged burrow
#

But tan is also

#

Undefined at 180°

#

No it isn't

timber wagon
#

Nvm

winged burrow
#

What's the value?

#

Of 180°

#

Then

timber wagon
#

Your answer I'll just be x=0

winged burrow
#

It's undefined here but whats the value normally

#

Of 180°

timber wagon
#

It's undefined

#

Mb

winged burrow
#

U said

#

It's well-defined

timber wagon
#

I made an error mb

winged burrow
#

Ohh

winged burrow
timber wagon
#

Yeah

winged burrow
#

But

timber wagon
#

If the line is perpendicular to x axis then it'll have only 1 value of x hence it should be x=something

winged burrow
#

How should I show my working

#

Tho

timber wagon
#

That something will be your x coordinate

#

Just show that your slope is undefined

winged burrow
timber wagon
#

And then say that it means it's a line parallel to y axis

winged burrow
winged burrow
timber wagon
#

It's just defined that way

winged burrow
#

And what does perpendicular mean

#

U mean I don't need to show any work?

timber wagon
#

You just need to say this stuff

winged burrow
#

Wait lemme show

#

What does perpendicular mean tho

winged burrow
#

@timber wagon

#

<@&286206848099549185>

main knot
#

the line is on x=0 and has no slope bro

winged burrow
#

That I got

#

But

#

How should I write it

main knot
#

x=0..

winged burrow
#

Like this

main knot
#

well this is jsut a weird case where when you do the method to calculate slope it's an issue

#

so that's why he's saying to do that

#

you kind of jsut have to infer that x=0 here

winged burrow
#

No like as a answer

main knot
#

the answer is x=0

#

that will satisfy your teacher

#

you can type some thing like

winged burrow
#

Can I say that as 7/0 is undefined x=0

main knot
#

honestly that's just not necessary

winged burrow
main knot
#

you can jsut say since there is no change on the x axis we can infer that the line has no slope and x=0

#

or just type x=0

#

if you want to say 7/0 is undefined so x=0 you can

#

I'd just write x=0

winged burrow
#

Ohh

winged burrow
#

To see if I got it

main knot
#

ok

winged burrow
#

But if there is no gradient on x axis there is no gradient on y axis too

winged burrow
#

Did I grasped it wrongly

#

@main knot

main knot
#

straight line

#

with no slope

winged burrow
#

Ohh

#

But the straight line is y axis

#

Not x

#

I mean both are straight but x axis is horizontal

winged burrow
# main knot

Oh wait I think I got it
There is change in y axis becuz there is -3 and 4
But there is no change in x-axis as there isn't any value and therefore as there isn't any value there is no change of gradient

main knot
#

yes exactly

#

but there is a change in y

white acorn
#

For along y axis, the gradient is not* 0

#

the gradient is undefined

winged burrow
white acorn
#

for along x axis u get 0/whatever u choose so gradient=0

winged burrow
#

Ohh

white acorn
#

it's not 0**

#

along y axis is not 0

#

along y axis is undefined

winged burrow
#

Yea

white acorn
#

since u get anything/0

winged burrow
#

Then the whole gradient is 0?

white acorn
#

So gradient for y=0 or y=c is undefined

white acorn
#

It's rise over run right

#

rise is infinity, run is 0

#

so u get undefined

main knot
#

it's not infinity

white acorn
main knot
#

it's between points -3 and 4

white acorn
#

Run is 0

#

So undefined

winged burrow
white acorn
#

u can't change undefined to 0

white acorn
#

how did u learn it

winged burrow
#

I just learn change in y axis and change in x axis

#

And just used

#

The formula

white acorn
#

yeah so change in y axis can be any constant

#

but the change in x is 0

#

So it's undefined

winged burrow
white acorn
#

IT DOES NOT BECOME 0

white acorn
winged burrow
#

How y axis is constant then but x axis is not

white acorn
#

Like 1

#

Or 2

#

Or 3

winged burrow
#

Ohh

white acorn
#

Or 18.372

#

any number as long as there's no variable

#

That's called a constant

winged burrow
#

Then why run is constant but rise is always 0

winged burrow
white acorn
#

I mean for when there's no change in x

#

Or when there's no change in y

main knot
#

idk what you guys are arguing about honestly

white acorn
#

oh god I cannot explain this

#

@main knot u can have him lmao

main knot
#

there's no slope and x=0 there's really nothing to it

white acorn
#

anything/0 is undefined

#

Same for gradient

winged burrow
winged burrow
white acorn
#

basically it's normal division

winged burrow
#

I took it like this when there's no change in y it's rise is 0 and when there is change in y.Y is constant.Same goes for x when there is change in x.X is constant and when there is no change in x.X=0

white acorn
#

straight up is undefined, horizontal is 0

winged burrow
#

Did I grasp it right

winged burrow
white acorn
white acorn
white acorn
#

what's the gradient

winged burrow
#

0

main knot
#

lol

winged burrow
#

Noo

#

It isn't

white acorn
winged burrow
#

Umm

white acorn
#

is it straight up

winged burrow
#

It says

#

Math error

#

It should be 0 then?

white acorn
#

if it's straight up what is it

main knot
#

it does not exist dude

white acorn
winged burrow
#

For me

white acorn
main knot
winged burrow
white acorn
winged burrow
#

Forever

white acorn
#

it is undefined

winged burrow
#

Ohh and its always?

white acorn
#

when y changes and x has no change

winged burrow
white acorn
winged burrow
#

Nice

white acorn
#

I'm not the teacher but I got special rights👍👍

winged burrow
white acorn
#

Do u understand

#

If I give u the points (-1,196) and (-1,814) what's the gradient

winged burrow
#

Umm

white acorn
#

okay is there a change in y

winged burrow
#

Yea

white acorn
#

Is there a change in x

winged burrow
#

There is no x

#

So nah

white acorn
#

so what is it

white acorn
winged burrow
#

Here is only y

#

Cuz we u assumed it as y

#

So no change in x

white acorn
#

yes

#

So what is it

#

What's the gradient

winged burrow
#

Constant

white acorn
#

it's undefined

white acorn
winged burrow
white acorn
#

don't overthink

#

try (1,28) and (1,-28)

#

what's the gradient

winged burrow
winged burrow
white acorn
#

cuz u get 0/change in x

white acorn
#

It's always (x,y)

#

normal coordinate system

winged burrow
#

Umm here

#

Gimme time

white acorn
winged burrow
#

The gradient is undefined

white acorn
#

YES

#

how about (0,36) and (36,36)

winged burrow
#

Cuz x has no change and y does

winged burrow
#

Here the gradient is 0

white acorn
winged burrow
#

Cuz has x has change and y doesn't

#

When we do not get any value of x but get of y we state that as undefined but when we do get a value of x but not y we can say that it is 0

#

@white acorn

#

.close

vocal sleetBOT
#
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vocal sleetBOT
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mossy latch
#

what are the shapes of each graph?

vocal sleetBOT
mossy latch
#

btw this is maths casue its statistics

hardy vector
mossy latch
#

I don’t think I’m at that level

hardy vector
#

If you want you could describe the shape

mossy latch
hardy vector
#

somewhat like a normal distribution or logistic distribution

mossy latch
#

Cause mean doesn’t equal mode

#

Or it has to be perfectly symmetrical

hardy vector
#

key word somewhat like. Take any real world data, it is almost definitely going to follow a normal distribution

#

*not going

vocal sleetBOT
#

@mossy latch Has your question been resolved?

mossy latch
#

<@&286206848099549185>

vocal sleetBOT
#

@mossy latch Has your question been resolved?

mossy latch
#

.close

vocal sleetBOT
#
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sudden moon
#

How do I test for point symmetry not about the origin, and line symmetry not relative to just the x and y axis algebraicly

sudden moon
#

I know that even function => line symmetry with the y axis

#

and odd function => point symmetry about the origin

#

but if the function is not symmetric about the origin or the y and x axis how would I test it algebraicly

vocal sleetBOT
#

@sudden moon Has your question been resolved?

vocal sleetBOT
#

@sudden moon Has your question been resolved?

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merry prawn
#

this is a solution to a problem, i dont' understand why i need to take the transpose to check if the determinant is zero

obsidian stream
#

You don't need to, it's just there because they probably expect you are more comfortable with row operations

vocal sleetBOT
#

@merry prawn Has your question been resolved?

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tawny nacelle
#

there’s no product of functions here

#

Integration by parts is not needed

#

The - in front of the integral is distributed

#

The 3e^3x gets the -, and the -6e^x becomes a +

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vast shale
vocal sleetBOT
vast shale
#

im having a big difficulty with continuity

#

if someone could just help me understand it w the help of these questions would be much appreciated

vocal sleetBOT
#

@vast shale Has your question been resolved?

vocal sleetBOT
#

@vast shale Has your question been resolved?

vocal sleetBOT
#

@vast shale Has your question been resolved?

#
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livid void
#

Im not understanding this at all

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#

@livid void Has your question been resolved?

wraith python
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livid void
vocal sleetBOT
livid void
#

I dont get how its 0 in the question on the left

tight sonnet
#

as x go more and more left y is close and closer to 0

#

so lim x_>-inf f(x) =0

livid void
#

lim?

tight sonnet
#

limit

livid void
#

also so is that always the case?

tight sonnet
#

you havent learn limits?

#

in this case that what this is

#

a limit

#

of x->-inf

tight sonnet
livid void
#

Theres no reference to limits in the chapter for "Rational Functions and their graphs"

tight sonnet
#

so forget about a term "limit"

#

that is asking when x is going left to "- infinity"

#

you are looking for the height its getting close to

#

and this height here is 0

#

do you understand?

wraith python
#

Are you studying end behaviors?

livid void
#

It was covered in a earlier chapter

wraith python
#

In the context of the question, you can substitute the y-axis with f(x).

#

The question is asking you what is the value of function relative to the y-axis as x goes to -infinity.

#

You can visually see that the curve is approaching the x-axis, or y = 0.

livid void
#

I think the other 2 lines are whats making me confused about the -inf one

#

Is it just always f(x)->0 when x->inf or -inf?

wraith python
#

No. Have you learned about asymptotes?

livid void
#

Yeah, but I had trouble understanding them. I seem to have trouble understanding arrow notation.

wraith python
#

If you look at this graph, there is a horizontal asymptote at y = 1.

#

The end behavior of f(x) approaches y = 1 as x goes to both -infinity and +infinity.

#

So it doesn't always go to y = 0.

#

Sometimes you can have a slanted asymptote. In those cases, the end behavior would go towards positive or negative infinity, depending on the slope of the slanted asymptote.

livid void
#

So I know how to find them, but there was no real reference to a question like this in the book.

wraith python
#

You do have to use your imagination a bit with regard to end behavior.

livid void
#

Thats why I posted the example because nothing showed the bottom line or how moving it horizontally affected it

wraith python
#

It would also be f(x)=0

hard crest
#

Yes

wraith python
#

You will eventually learn how to make those types of graph like so

#

At this point, you are probably only supposed to develop a basic understanding of what end behavior is.

#

Evaluating the values of end behaviors, aka limits as x -> +inf and x -> -inf, comes much later in Calculus.

livid void
#

hm, alright then. Guess I get the general idea then.

wraith python
#

Let me add on to the concept of end behaviors a bit.

#

You will find that the end behaviors of functions will usually do one of three things.

  1. Approaches some constant y = c.
  2. Goes to +infinity.
  3. Goes to -infinity.
#

And you will find that certain types of polynomial functions will exhibit one or more of these end behaviors.

livid void
#

what would be a case of y not equaling 0 y -> to something not 0 then?

hard crest
#

The limit ?

wraith python
#

f(x) = x^2 would be one example.

#

Here you can see that the end behavior of f(x) = x^2 goes to +infinity as x-> -infinity and x-> +infinity.

hard crest
livid void
#

But like a rational function that x->inf but not f(x)->0 is what I mean

hard crest
#

Like this one

#

There are a lot of rational functions

#

That don’t go to 0

vocal sleetBOT
#

@livid void Has your question been resolved?

#
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olive heron
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two numbers x and y are chosen so that -2<x<5 and -3<y<6

what is the probability that x+y > 5

olive heron
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Is there a fast way to calculate it other than brute forcing?

pallid zenith
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uniform?

olive heron
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what?

pallid zenith
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i mean

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is each number equally likely to be drawn here

olive heron
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yes

regal bane
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Are these real numbers, or do they have to be an integer?

olive heron
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integer

pallid zenith
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its a little large so i wouldnt do it by hand unless you dont fill everything in

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but you could kind of box out the places where x+y = 5

olive heron
regal bane
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There's a nice pattern, and you will catch onto it after seeing x+y = 5

pallid zenith
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here, lemme see if i can throw it together

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but, i mean

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lets say we fix y

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and let x vary

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so, lets let y be as big as it can be

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y=6

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whats the smallest x such that x+y>5 still

pallid zenith
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now let y=5

pallid zenith
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6+-1 = 5

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its not greater than 5

olive heron
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oh yeah

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so 0

pallid zenith
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yea

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if y = 6

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then x+y>5 if x=0, 1, 2, ..., 5

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now what if y = 5

olive heron
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smallest is 1

pallid zenith
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yea

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and y=4?

olive heron
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2

pallid zenith
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yea

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do you see the pattern?

olive heron
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yes

pallid zenith
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imagine we laid out all the values like, a table

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x values going across the top

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y values going down

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maybe you say...y increasing downward, x increasing rightward

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this is going to draw a barrier that kind of looks like a staircase

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and its gonna separate one side where x+y>5 and one side where it doesnt

olive heron
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y | x
6 0 1 2 3 4 5
5 1 2 3 4 5
4 2 3 4 5
3 3 4 5
2 4 5
1 5

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like that?

pallid zenith
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you might imagine iterating this as far forward in your head as you can and ask how small you can make y such that there is an x that can provide x+y>5

pallid zenith
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lemme see here

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i suck with excel but maybe i can do something

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,w Table[x + y, {y, -3, 6}, {x, -2, 5}]

pallid zenith
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wow fantastic

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lemme just draw it

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@olive heron

olive heron
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hi

pallid zenith
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if you had this on an exam or something, you wouldnt draw this whole table

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the red x's, those are values we specifically identified as not working

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for example, look at the row 'y=6'

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you said that x=-1 won't work

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and likewise y=5, x=0 ...

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so, this is more or less it

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one side is a region where x+y > 5

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the other, it doesnt

olive heron
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and x cant be 5 because its -2<x<5

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and y cant be 6

pallid zenith
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ever better happy

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just a subset of this table

olive heron
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ok

pallid zenith
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lop off the outer frame

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can you tell which region has x+y>5 here

olive heron
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8x5-5

pallid zenith
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theres more red x's

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we just havent written them

olive heron
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oh yeah

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so since the x are the ones = 5, its the squares on the bottem right

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10

olive heron
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shaded area

pallid zenith
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i just mean, identify the region is all

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sure, yea, what area is this

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i mean combinations in this region have what property

olive heron
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the amount of squares per row is the y value -1?

pallid zenith
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what? no

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the problem is asking us to do what

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identify the probability that some integers in that range will satisfy x+y > 5 right

olive heron
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yes

pallid zenith
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you drew a black region up here

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whats true in the black region

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we add x and y

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and what

olive heron
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its bigger than 5

pallid zenith
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yea

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and, everywhere else

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its not

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so youre almost done happy

olive heron
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10/(6x8)

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10/48

pallid zenith
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,calc 10/48

twin meteorBOT
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Result:

0.20833333333333
olive heron
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20% ish

pallid zenith
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ish

olive heron
pallid zenith
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you could try deriving one

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I mean, we know where the edge of the red x area is gonna be

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if you want x+y > a

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then the line will be along the combos where x+y = a + 1

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we can probably figure out where this intersects with the edges of the region ...

regal bane
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I see you guys made a triangle. The top row has 1 entry. The second row has 2 entries. The third has 3 entries...

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If only we could quickly solve 1 + 2 + 3 +...

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hmm

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HMMMMMM

olive heron
pallid zenith
olive heron
regal bane
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I guess we'll never know

pallid zenith
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this seems like a good problem to keep chewing over tbh

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you need to incorporate the ranges youre given somehow

olive heron
regal bane
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Google "triangular numbers" for a bit of extra learning imo

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You won't need them here, but there's a pattern for sure

olive heron
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authaeuhtua

pallid zenith
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1+2+....+n = T_n where you can always get what T_n is

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i think this is jumping forward

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but maybe me and kaynex are thinking about the problem differently

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to me: its a rectangular region, and you have a nice diagonal line cutting through it

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if you can identify where that happens in terms of the ranges and the given quantity the sum needs to be greater than, youre super in business

pallid zenith
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but given a<x<b and c<y<d, requiring that x+y > s

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then whats n?

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its pretty generalized so its a pain but you can totally do it happy

vocal sleetBOT
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@olive heron Has your question been resolved?

vocal sleetBOT
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@olive heron Has your question been resolved?

vocal sleetBOT
#
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#
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Remember:
Ask your math question in a clear, concise manner.
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After 15 minutes, feel free to ping <@&286206848099549185>.
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Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

gentle umbra
#

i need help with solving these type of problems