help-10

[high lily]

you're not representing displacement ,velocity ,acceleration properly

no

[twilit loom]

do you know the unit for the answer?

[sleek sinew]

Miles

[twilit loom]

the car travels 220 miles in 5s? O.o

that's better

but your final answer will be in miles, but no where near 220

so are you sure you're looking for miles travelled?

[sleek sinew]

The book says 220 without units.

[twilit loom]

-uh where'd your comment go

they're looking for feet

[high lily]

seems to be 220 feet

[sleek sinew]

We have to convert miles to ft?

[twilit loom]

miles/hr to ft/s

[cedar lichen]

a(t) = 12t? That doesn't look right to me

[twilit loom]

rectifying that now

[sleek sinew]

Why cant we do mi/s in 5 seconds?

[twilit loom]

you can if you'd prefer, just convert to feet eventually

[sleek sinew]

So is it 1 mi/300s^2 = 5280ft/300s^2

?

[twilit loom]

where did the 300 come from?

[twilit loom]

let's go from 60mi/(1hr * 5s)

how many seconds are in an hour

[sleek sinew]

3600

So i did 60/(3600 *5)

[twilit loom]

yup, that'll be acceleration in mi/s^2

[sleek sinew]

Yeah so is it 1mi/300s^2?

Which is 5280ft/300s^2?

[twilit loom]

oh I see now

yeah that's right

[sleek sinew]

So a(t) = 17.6t, where t is in seconds?

[twilit loom]

acceleration is constant, right?

[sleek sinew]

Oh right so just 17.6?

[twilit loom]

yup

try keep it in exact form though it is exact ;-;

[sleek sinew]

Alright thanks!

[obtuse pebble]

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[sick condor]

Hello,
For this problem, the transition matrix that I came up with is
|.65 .35|
|.28 .72|
though I'm not sure where to go from here to get the "distribution of customers" tomorrow, day after tomorrow, or 3 weeks from now

[warm shale]

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

KurtDee

KurtDee

[sick condor]

what would x0 be? it says 50% of the lunchtime crowd ate at each of the two restaurants today, would it just be <0.5, 0.5>?

how would one know x0=<50,50> from the information given

Oh can we just use it to represent the percentages

Ah ok. Past problems of this type actually gave us the number of people in the crowd

[spiral maple]

it'd be [.5,.5]^T

cause probability vector

[sick condor]

what is T?

[spiral maple]

transpose

yeah true, cause you'd just have a scalar difference

[sick condor]

does my transition matrix look right?

[spiral maple]

it doesnt have numbers

so no

[sick condor]

I thought I did

guess not

this is what I had

|.65 .35|
|.28 .72|

[spiral maple]

looks right

[sick condor]

I decided that
Me & ollie's is state 1
Green bean is state 2
p_ij = probability of going from state i to state j

awesome

thanks guys @spiral maple @true rain

.close

[obtuse pebble]

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[spiral maple]

comparing to this example

[sick condor]

.reopen

[obtuse pebble]

✅

[sick condor]

OK yeah that makes sense

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[quaint badge]

what was the formulae to find area of paralleogram again?

[quaint badge]

i think i got it

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[tall rampart]

A box open at the top has a rectangular base 200mmx300mm and an
altitude of 150mm. If the base and the sides are 10mm thick, find the total
surface area of the box.

[tall rampart]

It said that the base and the sides are 10 mm thick so am I supposed to add 10mm on the altitude and also add 20mm on both 200 and 300mm?

[young tartan]

Yeah sounds good

[tall rampart]

And it said the top is open so I am going to subtract the area of the rectangle at the top in the total surface area?

[young tartan]

Yeah

[upbeat island]

am i overthinking this, or do you ned to count like the interior of the box area also?

[young tartan]

Dont overthink please

This is probably not even highschool or atmost 1st year

I have suffered a lot through overthinking basic questions back in those days

[upbeat island]

sounds good haha

[tall rampart]

@young tartan sure, thanks

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[cyan totem]

the answer for this is 9.28 degrees, but how do you work it out?

[unborn valley]

can you use trigonometry?

[short spruce]

i'd assume you have to

law of sines

[zinc flax]

ya

do uk how to use it ??

[cyan totem]

normally, yeah, but multiple triangles makes it kind of confusing for me

[zinc flax]

see the above angle = 42-x

so appplying law of sines we get

8/sin(42-x) = 11/sin(48)

solve for x

[short spruce]

if it makes it easier you can also entirely forget about the smaller triangle for a bit, solve the angle you need in the first triangle then subtract it from 42

[obtuse pebble]

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[frigid oracle]

Let $s_n$ be a sequence that converges. Show that if $s_n \ge a$ for all but finitely many $n$, then $\lim{s_n} \ge a$

[warm shale]

dulg

[frigid oracle]

I looked over the hint for this question, but I want someone to walk me thru this question

The hint goes like this

It's 8.9)a)

But I do not understand why s < a in the hint?

I understand having a finite set $N_0$ $\in \bN$ which leads to $s_n \ge a$ for $n > N_0$ but I do not get why we assume $s < a$??

[warm shale]

dulg

[frigid oracle]

Am I supposed to set up a contradiction? And I don't understand how showing $s_n < a$ is supposed to help answer the question, especially since it seems contradictory

[warm shale]

dulg

[frigid oracle]

help would be really appreciated

[royal basin]

yes, you are going to set up a contradiciton

contradiction*

assume the limit of the sequence exists and is less than a

since lim s_n = s, we know that for every ε > 0 some tail of our sequence will be entirely contained in (s-ε, s+ε)

take ε = (a-s)/2 and observe that (s-ε, s+ε) lies entirely below a

hence our tail has to consist entirely of points which are less than a

why is this impossible?

[frigid oracle]

ah

man this makes so much sense

because we defined s_n >= a, for finitely many n

[royal basin]

no

[frigid oracle]

no?

hmm

[royal basin]

we are given that s_n ≥ a is true for ALL BUT finitely many n

consider that it is in general not possible to remove a word from a mathematical statement and have it still mean the same thing

[frigid oracle]

i think i have a very vague notion that limits are concerned with infinite amount of "n", and so s_n >= a is satisfied

[royal basin]

you're overthinking it.

hence our tail has to consist entirely of points which are less than a
why is this impossible?
the correct answer to this is ||because this would imply there are infinitely many such points, which is known not to be the case||

[frigid oracle]

why would it imply infinitely many points?

[royal basin]

you know what a tail of a sequence is, right

[frigid oracle]

not really

i mean, i suppose i have a vague idea, but not a formal definition

[royal basin]

then why the fuck did you not say it the first time i mentioned the word

[frigid oracle]

chill

[royal basin]

a tail is what you get when you start the sequence at some term other than the 1st

i.e. it is a sequence in its own right

a sequence like $(s_k, s_{k+1}, s_{k+2}, \dots)$

[warm shale]

Ann

[frigid oracle]

other than the 1st term?

[royal basin]

do you object to me saying "at some term other than the 1st" as opposed to "at some term other than the 1st term"?

[frigid oracle]

no i am just asking for a clarification

[royal basin]

i mean

what else could "1st" possibly refer to if not "1st term"

i'm confused at where you could possibly be confused with my wording

[obtuse pebble]

@frigid oracle Has your question been resolved?

[obtuse pebble]

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[earnest vigil]

can anyone help?

[young tartan]

What does the expression approach when t gets veey large?

[earnest vigil]

50

@young tartan

its like a curve

but almost

50

[young tartan]

Thats what question tells

The given expression

[earnest vigil]

but like idk how that translates to C and k

[young tartan]

Just look at the algebraic function

What does it approach as t gets large

[earnest vigil]

i confuse

sub 50 to h?

[young tartan]

Ok what does $-ke^{-0.25t}$ approach as t gets large

[warm shale]

Calm Mango

[earnest vigil]

more negative

so e gets smaller

[young tartan]

Good

The entire expression goes to?

[earnest vigil]

-1/e

ohh

o?

0

[young tartan]

Ye

So h approaches C as t gets large

[earnest vigil]

yes

soo whats k use fort?

ehh so c is 50 right?

[young tartan]

Yes

[earnest vigil]

If k find from where ya

[young tartan]

The second statement

[earnest vigil]

So the function is flipped x axis

bro then what?

[young tartan]

Sorry man. Height at t = 0 is?

We started counting time just after it was planted

[earnest vigil]

Oooo

e is 1

then?

@young tartan

[young tartan]

h = 0

At t = 0

[earnest vigil]

then

[young tartan]

Put them in the function

Solve for k

[earnest vigil]

after that?

[young tartan]

Differentiate the function

To get the function of growth

[earnest vigil]

ohh then done?

[young tartan]

Find the time when growth reaches 2m/y

[earnest vigil]

Ohh okok

so k=50

?

[young tartan]

Put that value in functions of h

Yes

[earnest vigil]

but

the one

reaching 0

is

e^-0.25t

?

nvmm

[young tartan]

?

Thats when t is large

[earnest vigil]

broo

whats ur finall answer?

[young tartan]

Dude calculate it yourself

[earnest vigil]

42?

[young tartan]

Oh you did

[earnest vigil]

yepp

[young tartan]

I am eating

[earnest vigil]

AHAHAHAHHAHA

[young tartan]

Waut 5 min

[earnest vigil]

OKOK

no need

tyyy brooo

.close

[obtuse pebble]

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[young tartan]

No prob

[obtuse pebble]

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[timid silo]

Let $a$ and $b$ be real numbers, where $a < b$, and let $A = (a,a^2)$ and $B = (b,b^2)$. The line $\overline{AB}$ (meaning the unique line that contains the point $A$ and the point $B$) has $x$-intercept $(-3/2,0)$ and $y$-intercept $(0,3)$. Find $a$ and $b$. Express your answer as the ordered pair $(a,b)$.

[timid silo]

i need help

[warm shale]

crabbo

[timid silo]

pls

[lethal sand]

what have you tried

[royal basin]

@timid silo have you worked with equations of straight lines before at all

[timid silo]

i have

yes

[royal basin]

okay

are you able to write down the equation of line AB in terms of a and b?

in any form you like

[timid silo]

i usually do slope-point form

[royal basin]

ok then point-slope form it is

[timid silo]

this may take me some time

[royal basin]

ping me when you're done

[timid silo]

i just need to know

which point is (x_1, y_1) and which is (y_2, y_2)

i know the order matters

@royal basin

[royal basin]

what do you think the order matters for?

finding the slope?

@timid silo

[timid silo]

yes

[royal basin]

no, it doesn't matter.

if you don't screw anything up you will get the same result either way

[timid silo]

ok

$y-3=\frac{b^2-a^2}{b-a}(x-0)$

[warm shale]

crabbo

[timid silo]

i chose to use the y-intercept

[royal basin]

this can be greatly simplified

(b^2 - a^2)/(b - a) can and should be simplified unless you're willing to down in a sea of algebra

and drown you WILL

[timid silo]

is there a way to simplify it?

[spiral maple]

Ann just said there was..

[timid silo]

i dont see it

[spiral maple]

Look harder then

[royal basin]

do you know of any algebraic identities involving two squares being subtracted

[timid silo]

H

let me simplify

$y-3=(b+a)(x)$

[warm shale]

crabbo

[royal basin]

okay good

so you have the equation of your line

y = (a+b)x + 3

you know three things here:
(i) this line passes through (a, a^2)
(ii) this line passes through (b, b^2)
(iii) this line passes through (-3/2, 0)

each can yield an equation in a and b

[timid silo]

it also passes through (0,3)

[royal basin]

you have already made use of this information

[timid silo]

oh right

[royal basin]

trying to use it again will just give you 0=0

a.k.a. an equation that is true but useless

i may be mistaken here, but (i) and (ii) should turn out to be equivalent, giving you two equations in two unknowns as opposed to the initial appearance of three

[timid silo]

we just need to find a and b

which should i choose to use

[royal basin]

all of them

or, if you wish to minimize your workload, pick either (i) and (iii), or (ii) and (iii)

but turning them all into equations will not hurt.

[obtuse pebble]

@timid silo Has your question been resolved?

[timid silo]

(i) -3=ba
(iii) $a=\frac{-3b+6}{3}$

[warm shale]

crabbo

[timid silo]

@royal basin

[royal basin]

(iii) seems more complicated than it needs to be, but is otherwise correct

[timid silo]

let me simplify it

a=-b+2

[royal basin]

okay great

so now you have these equations:

a + b = 2
ab = -3

[timid silo]

system

answer is

a=0, b=2

right?'

doesnt hurt to try

@royal basin

[royal basin]

does a=0, b=2 satisfy ab = -3?

[timid silo]

oh no

a+b-ab=-5

oh well

i dont know

😦

[royal basin]

have you worked with quadratic equations before

[timid silo]

ah yes

[subtle quartz]

+5 not -5

[timid silo]

right

[timid silo]

but it isnt a quadratic

[royal basin]

you can construct a quadratic equation of which a and b will be the roots

you know their sum and product and hence know the coefficients

[subtle quartz]

not exactly

if it was a² + b² - ab then we could

[royal basin]

what do you mean

i'm trying to hint at ||making use of vieta's formulas||

and i don't really see what you're getting at with a^2 + b^2 - ab @subtle quartz

[subtle quartz]

what I mean to say is, that if it was a² - b² - ab we could consider it a quadratic equation

[timid silo]

it has a name

this guys theorem

[subtle quartz]

but that vieta's formula thingy seems to involve heavy algebra

[timid silo]

yes

[subtle quartz]

and there must be a better way

[timid silo]

vietas is simple

[subtle quartz]

I think maybe you could square both sides of a + b = 2

and then add/subtract eqns

and form a quadratic out of that?

oh wow I already kinda got a rough sol in my head

[royal basin]

you both are overthinkingit

[timid silo]

i would have two squares

[royal basin]

y'all please stop

[subtle quartz]

i think i got the sol yeah

[royal basin]

you're severely overthinking it

[timid silo]

god help us

[royal basin]

treat a and b as the roots of a quadratic

i'll use t as the variable of the quadratic so as not to clash with previous notations

(t-a)(t-b) = 0

[timid silo]

(x-a)(x-b)

[subtle quartz]

oh wow

much better sol

[royal basin]

@timid silo we had x in the problem before so i don't want to reuse it

[timid silo]

i got sniped

[royal basin]

(t-a)(t-b) = t^2 - (a+b)t + ab

therefore

[subtle quartz]

my solution was this

[royal basin]

our equation is t^2 - 2t - 3 = 0

which we can solve

[timid silo]

you beat me to it again

[royal basin]

and by construction the roots will be a and b

[timid silo]

t=-1 nor t=3

[royal basin]

nor?

but yes, those are its roots.

so now the final stretch: which is which?

[timid silo]

flip a coin?

[royal basin]

no

no coin required

[timid silo]

both

[royal basin]

you know something about a and b that can and should allow you to disambiguate

go back to the original problem and read it carefully

[timid silo]

they are real numbers?

[royal basin]

go back to the original problem and read it carefully

"they are real numbers" says precisely nothing

we've been working with nothing but real numbers

[timid silo]

how about x-interc ept and y intercept?

@royal basin

how could i use them tho

[royal basin]

let me repost your original problem statement

Let $a$ and $b$ be real numbers, where $a < b$, and let $A = (a,a^2)$ and $B = (b,b^2)$. The line $\overline{AB}$ (meaning the unique line that contains the point $A$ and the point $B$) has $x$-intercept $(-3/2,0)$ and $y$-intercept $(0,3)$. Find $a$ and $b$. Express your answer as the ordered pair $(a,b)$.

[warm shale]

Ann

[royal basin]

Let a and b be real numbers, where a < b, ...

[timid silo]

how can i use that tho

[royal basin]

you know one of a and b is -1 and the other is 3

which of -1 and 3 is the lesser of the two?

[timid silo]

-1

the answer is correct

thank you so much @royal basin

.close

[obtuse pebble]

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[timid silo]

lickkle confused about flipping inequality signs

you flip the sign here because.. why?

say you did

$x < \log_{\frac{1}{2}}{4}$

obviously this is incorrect

because i didn't flip the sign

[warm shale]

The Torrent

[timid silo]

yh so like uhhhhhhhhhhhh why do i flip the sign

[timid silo]

1/2^x = 2^-x

[timid silo]

its way easier than that smh

yessss it is

but what answers my question anyway

[timid silo]

uhh you got it?

i mean

i know that you flip the sign

my question is about the sign

not how to do it

how to know when logging both sides

if your meant to flip the sign or not

i understand you flip the sign if you multiply by a negative

if you are multiplying or dividing -ve, then the sign would flip

[timid silo]

but how do you know if logging both sides in base whatever is gonna be multiplying by -1

x < y

log(some base)(x) < log(some base)(y)

wait a moment

for what values of 'some base' is what i just wrote true

[timid silo]

no you dont, so you cant do that

[timid silo]

why not

[timid silo]

have you solved this? @timid silo

yh

alright

[timid silo]

you mean log? yes you can as we know that both the x are +ve

but

why cant u here

[timid silo]

also

theres only 1 x (in my example)

so like

idk what u mean at all

bruh

bruh

shall i start all over

explaining my question

yeah

x > y

lol

$x > y$
\$log_{z}(x) > log_{z}(y)$ for what values of z

theres my question

[warm shale]

The Torrent

[timid silo]

z>0

thats not right

becauseee

see here

if you log in base 1/2 both sides

$x < log_{\frac{1}{2}}(4)$ FALSE

[warm shale]

The Torrent

[timid silo]

but z is greater than 0, as you just said so?????

[timid silo]

have you even learnt log? :/

CAREFULLY

yes?

wheres the fault in what i just did?

[timid silo]

apply indices here instead of log

but whats wrong with what i just did? i dont understand?

i dont wanna find a different method, i wanna understand flipping the sign with logs

[timid silo]

you cant as you dont know the sign of x here!!!!!!!!!

like if i say

you cant use logs

if you dont know the sign of x?

no 🙂

what has mark scheme done then?

x^2 is always positive 🙂

[timid silo]

wheres x^2

oh i see they just test value

is there no other way

[light geyser]

Is ur question why the sign flipped?

[timid silo]

yes but also the conditions for which the sign flips

[light geyser]

Log(1/2) is negative

[timid silo]

the 2nd method

[light geyser]

I presume here that by log they mean log base 10

[timid silo]

you cant apply log unless you know the sign of x

where you take log in base half

[timid silo]

i maybe? understand this? uhhhhhhhh

but its not like your multiplying both sides by x or anything?

but igg

[light geyser]

The 2nd method requires finding which value of x equals 4 and then (since the less the power the greater the product for numbers less than 0) the inequality is true for all values greater than the value you find

The 2nd method is great for questions like these since it’s just two numbers

[timid silo]

You can just apply this @timid silo

[light geyser]

Also the sign doesn’t flip here so I don’t understand what you find wrong with method 2

[light geyser]

Also this is a pretty intuitive method I wouldn’t have thought of

[timid silo]

why should we take log if we can solve this more easily :/

[light geyser]

Maybe it's in their curriculum but this is some simple playing with indices stuff

[timid silo]

also how can i get the helpers role?

how does this testing thing work

i dont understand how they determine which way the sign is in method 2

[light geyser]

Just go to a general chat and ask, if it's taking too long @ a random staff

[timid silo]

well itts just like mathematical induction, it wont be helping you in complex solving though

but

how determine x bigger

its just like general sense of sensing the range of x

o

i just understood it nevermind

if they tried number smaller than -2

it would be false

yeah

like x-6 should be +ve
so its common sense that x must be greater than 6 to make it positive @timid silo

thanks!! 🙂

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[obtuse pebble]

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[fallow hazel]

The largest dimension of a rectangle exceeds the triple of the smallest by 3.8 dm. Calculate the area of ​​the rectangle knowing that the half perimeter is 45.4 dm.

[fallow hazel]

hlep

[worthy drum]

Fun question, what do you have so far?

[fallow hazel]

idk

[worthy drum]

Ok, no worries. So first let's find the lengths of each side since we know that the sum of all sides is 90.8 (since the problem says 'half perimeter is 45.4')

[unique solstice]

Are they using decimeters?

[fallow hazel]

yes

[worthy drum]

The small side is unknown, so we will call it "x" and the other side is "triple of the smallest" and "exceeds..by 3.8" so that will be 3x+3.8

[fallow hazel]

okay okayy

[worthy drum]

So the parameter will be: x + x + 3x+3.8 + 3x+3.8 (add all sides) and half of that is 45.4

(1/2) (x+x+3x+3x+3.8+3.8) = 45.4
{math}
8x+7.6 = 90.8
x = 10.4

So now that you know the smaller side is 10.4, the larger side is 3*(10.4)+3.8
And the area would just be the multiple of the two

[fallow hazel]

yes

[worthy drum]

Did you get ||364||?

[fallow hazel]

yes

[worthy drum]

Awesome, that should do it!

[fallow hazel]

thank you

have a great day

[worthy drum]

you too

[obtuse pebble]

@fallow hazel Has your question been resolved?

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[formal trout]

can the radius of a circumscribed circle be equal to one of the side of triangle?

[balmy mortar]

im not sure on this

but visualize a right angled triangle ABC. C is the right angle

AB is the diameter

Visualize moving point C

now surely somewhere along the way AC is equal to the radius

it varies between 0 and 2r

[obtuse pebble]

@formal trout Has your question been resolved?

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[formal trout]

come in #math-discussion

[obtuse pebble]

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[woeful musk]

I dont remeber how to find Percentage

Nvm i got it

.close

[obtuse pebble]

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[woeful musk]

.reopen

[obtuse pebble]

✅

[obtuse pebble]

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[stoic osprey]

83

[woeful musk]

Oh thxs

CAn you tell me how you got that

[stoic osprey]

Sure

So do (83x100)/% so it work for all

(83x100) / 83 but its obvious

Ye

[woeful musk]

Bc i accidentally put the wrong screen shot

[stoic osprey]

Exactly number x 100 / percent

[woeful musk]

Oh ok

[woeful musk]

Thxs

.close

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[formal trout]

😂

[woeful musk]

OK I GET IT

.close

.close

[formal trout]

i love ur question btw

[woeful musk]

Thxs

.close

Can someone plz close this

[formal trout]

.close

lol think that the bot is down

[woeful musk]

Oh ok

[stoic osprey]

.reopen

.close

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[upbeat plinth]

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[smoky wraith]

can someone help me confirm my answer for this question, and maybe clarify the difference between a unique solution, no solution and infinite solutions?

[restive acorn]

Are you in linear algebra? Or have access to solving linear systems with matrices?

[timid silo]

For infinite solution, equations must be multiplication of each other

Like this

2x+3y=9
4x+6y=18

X and y have infiniy solutiın here

Because they cancel out

And the only equation remains first one

[smoky wraith]

Im in HL math in IB so we havent done matrices(or maybe not yet). I have cancelled out the variables of x from these equations, and eventually y too. I got to the point where 4m(m-1)z= -2m*2 -12m -4

[timid silo]

The second one is just two times the first equation, so it doesn't give another info

[smoky wraith]

I don't think I understand

I don't think the second equation is two times the first equation

[timid silo]

I gave another example for ininifte many solution

This is something different than your question

[smoky wraith]

ohhh yea I understand what you mean

I understand the basis of the difference between infinite and no solution, but what im having trouble with is the unique solution

like I know (m-2)x=0, if m=2, there is infinite solutions, but if (m-2)x=5(or any other number besides 0), and m=2, there is no solution

[timid silo]

Uhmm

Consider unique solution a situation which is not infinite solution or no solution

[smoky wraith]

so in this equation I just wrote, m = anything besides 2?

[timid silo]

When you put m=2 what happens?

[smoky wraith]

the lefthand side = 0, which means there will be either infinite or no solutions

[timid silo]

Sorry I got to do that on paper

Maybe someone else helps better tho

[smoky wraith]

<@&286206848099549185>

[timid silo]

Man I looked at the question again. The only possibility for the system not to have a solution is when m=1 because if you look at 2nd and 3rd equations, the left side are the same but 2nd one is equal to 2 whereas 3rd equals to -3. This is impossible, so m=1 makes the system no solution. There is also no root for m to make the system infinite solution (you know the conditions.). Summary, any value other than 1 for m makes this system unique solution. Does this make sense?

@smoky wraith

[smoky wraith]

yea I got that if m=1, no unique solution. But I also got that if m=0, no unqiue solution either

[timid silo]

Yeah it makes it no solution

I guess 0 is an exception

[smoky wraith]

<@&286206848099549185>, I got m=1, and m=0 will not give a unique solution. But I also got that if m=(1/2), there will be a unique solution. How do I phrase my answer because I am uncertain when it comes to "parameters"

[smoky wraith]

I was multiplying the 3 equations, and then adding and subtracting them until I got to a point where there was only one variable left which was 4m(m-1)z= something

so if m=0, then the entire left hand side is 0 which does not give a unique solution

[novel knoll]

Why do you say m=0 and m=1 no unique, how does that help you? M can be any real number

You can’t just check numbers 1 by 1

There are an infinite amount

[smoky wraith]

nah I didn't guess, I was multiplying the equations and then adding them up to get rid of x and eventually y. For examples, I multiplied the second equation by 2, and subtracted the first equation by that.

At that point, I got that (1-2m)y - z=-m-2, so I did not have x anymore

[novel knoll]

“I got m=1 and m=0 no unique… but m=1/2 there will be”

You are aware there are gonna be an inf amount of m where there will be an unique solution, right?

[smoky wraith]

yea so my final equation was that 4m(m-1)z = -2m^2 - 12m +4

yes I realize that, but in the book it the answer just says m=1, m=0 & m=0.5 so I got very confused by it, and wanted some clarification

like I understand how they got to these values as I do too, but they say m=1 and m=0, but if m=1 or 0, there is no unique solution

[novel knoll]

Then the answer is wrong

[smoky wraith]

Right so the answer would just be that "m can not = 1 or 0."?

as it is asking the parameters of m so the system has unique solutions

[novel knoll]

Yes

[smoky wraith]

Thank you! This helped clear a lot up!

.close

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[timid silo]

|x-2|=2x-1

[unborn valley]

Solve for x?

[timid silo]

modulus yeah

[unborn valley]

You can sqaure both the sides

To remove modulus

Or take 2 cases (x-2) and (2-x) after removing modulus

Just sqaure both sides

(x-2)^2. =(2x-1)^2

[obtuse pebble]

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[craggy rampart]

Hi guys, can you help me out with this limit? Thanks

[spiral maple]

what have you tried..?

[craggy rampart]

Initially this reminded me of a notable limit (the one elevated to alpha which in this case is 1/2) but I don't have the "+1" and trying to rearrange it seems to mess things even more

[spiral maple]

have you tried rationalizing?

[craggy rampart]

Yes

And even if the numerator has a better look

The denominator still has that 4*x which gives 0...

[spiral maple]

$\frac{(x+9)-9}{4x(\sqrt{x+9}+3)}$

[warm shale]

Mosh

[spiral maple]

after rationalizing

[craggy rampart]

Exatcly but won't the denominator still be 0?

[spiral maple]

what's x+9-9?

[craggy rampart]

x?

[spiral maple]

and what's x/x?

[craggy rampart]

I'm literally blind for the love of god

20 mins on this stupid exercise...

Thanks a lot man

.close

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[nimble steppe]

.close

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[slim temple]

suppose there are three number a , b , c
And if they're in AP , than (a+c) must be always even right ??

My Proof
Since , (a+c) = 2*b . Hence, proved ,** (a+c) must be always even**

[slim temple]

Actually I'm solving a problem and I've used above assertion so just confirming .

So, please reply even if you think it's too obvious

[high lily]

not necessarily

a,b,c and the common difference don't have to be integers

[slim temple]

hmm...

If a,b,c are integers than always yes ?? I think so

[high lily]

yes

[slim temple]

thanks

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[balmy mortar]

any clue?

[signal cobalt]

no

I think need even ai in the odd position

not sure

[balmy mortar]

do you know what f(x) will look like

[signal cobalt]

no?

[balmy mortar]

just a rough idea

uhhh 1 sec

[signal cobalt]

ok

[balmy mortar]

ok so f(x)

the numbers from -100 to 100

are roots, unless the power is 0

at each root you need to worry about if the function changes sign or not

You know when that happens?

[signal cobalt]

no

or maybe but I forgot

[balmy mortar]

another example then

$$f(x) = (x+1)(x-1)^2$$

[warm shale]

Shuri2060

[balmy mortar]

would you know how to sketch this curve?

[signal cobalt]

yes?

find root

and

[balmy mortar]

what happens at -1

and what happens at 1

[signal cobalt]

parabola at x-1

[balmy mortar]

i mistyped

[signal cobalt]

and cross from 1

[balmy mortar]

it touches the axis at x = 1
and crosses at x = -1

but yes

they want you to consider same idea

The important thing is whether the exponent of the root is odd or even

[signal cobalt]

yes

but

how

[balmy mortar]

The sequence has a pattern

maybe write the first few terms

[signal cobalt]

ok

7

9

16

25

41

66

107

173

[balmy mortar]

you only care about the parity, remember

[signal cobalt]

so

Odd

[balmy mortar]

odd or even

[signal cobalt]

Odd

Even

Odd

Odd

Even

Odd
Odd
Even?

[balmy mortar]

yes. you see why from the recursive formula?

[signal cobalt]

odd + odd = even

even + odd = odd

odd + odd = even

...

[balmy mortar]

yes. so now you have the pattern about how the function switches or not

[signal cobalt]

and then

[balmy mortar]

You have all the info you need?

the function is negative to the left of -100

and then you need to figure out what % of it is positive between x = -100 and x = 100

[signal cobalt]

like the exponent

?

[balmy mortar]

if the exponent of a root is even

the sign does not change

if it is odd, it changes

If you're stuck, try drawing a picture for the first few roots

from x = 100

[signal cobalt]

so what I need to do is count the positive exponenent?

[balmy mortar]

no

you need to try to find a pattern

the exponent just tells you the function changes sign or not

[signal cobalt]

of f(x) is pos or neg

[balmy mortar]

start from x = 100

try sketch the graph

for the first few roots

x = 99, 98, ...

[signal cobalt]

?

[balmy mortar]

why is it touching at 100

isnt a1 odd?

[signal cobalt]

i mean

100 is -100

sry

99 is -99

[balmy mortar]

yh thats why i chose x = 100

and to draw backwards

[signal cobalt]

100 is cross

99 cross

98 touch

[balmy mortar]

yes

so from this you should be able to work out pattern

and figure out how much is positive

[signal cobalt]

51?

[balmy mortar]

I havent done q, no paper here

trust yourself 🙂

[signal cobalt]

no

i do not

[balmy mortar]

then double check 🙂

[signal cobalt]

I am still confuse

[balmy mortar]

more reliable than me trying in head

[balmy mortar]

draw a new diagram

so we have odd odd even odd odd even ...

and maybe do a few more than 5 roots so the pattern is obvious

[signal cobalt]

worst drawer

[balmy mortar]

is ok lets see

so there is a pattern of period 3

right?

[signal cobalt]

yes

[balmy mortar]

so roughly 2/3 should be positive

your answer should be roughly 2/3, but this is discrete

so you have to carefully check the ends

[signal cobalt]

201*2/3

=134

[balmy mortar]

just double check the pattern isnt cut short

at -100

(im not sure myself)

uhh 100 is 1 mod 3

-100 is 2 mod 3

so i think there will be an extra bit

[balmy mortar]

it is 200, not 201

200 gaps

[balmy mortar]

So i think you can consider -98 to 100

as one section

then consider -98 to -100

or you can do -100 to 98

and do 98 to 100

if you see what i mean

[balmy mortar]

so we have 198 gaps, 2/3 are positive

[signal cobalt]

132

[balmy mortar]

128. then an extra one. so the answer should be 129/200 probability.
     I might have made mistake

[signal cobalt]

133

129?

[balmy mortar]

198/3 = 66 so yes ur right

133/200 sounds right to me

[signal cobalt]

nice

how about this problem

I got the explicitly form of the sequence

but how inductive proof

[balmy mortar]

whats the formula

[signal cobalt]

a_n = 2^n-1

[balmy mortar]

check the base case

you write that 2^1-1 = 1 = a_1

so base case is good

then you assume your formula is true for k

a_k = 2^n -1

You use this to show it is true for a_{k+1}

(use the recursive formula)

[signal cobalt]

this is what I got

[balmy mortar]

have you seen how a proof by induction works?

[signal cobalt]

no

[balmy mortar]

oh ok, you need to know that

it goes like this

You have a statement P(n) you want to prove for all natural n

so P(1), P(2), ...

uhhh thinking of an example

[signal cobalt]

ok

[balmy mortar]

lets say P(n) is '3n + 3 is divisible by 3'

To do a proof by induction, first you prove the base case

that is P(0) or P(1) depending on if you include 0

so 3*1+3 = 6 which is divisible by 3

so P(1) is true

make sense so far?

[signal cobalt]

yes

[balmy mortar]

Next you prove that P(k) implies P(k+1) for all k

If you manage to prove this, it proves the statement for all natural numbers

because P(1) => P(2) => P(3) => ...

if that makes sense?

[signal cobalt]

kind of

[balmy mortar]

so in this example i would do it like this

Assume P(k) is true

So we assume 3k+3 is divisible by 3

now, 3k+3+3 is also divisible by 3

(some number theory property)

[signal cobalt]

oh

[balmy mortar]

therefore 3(k+1) + 3 is divisible by 3

so we have proven P(k) => P(k+1)

that completes the induction step

[signal cobalt]

so we assume a_n-1 is 2^n-1-1

[balmy mortar]

usually the letter you use is k to be clear

because you want to prove P(n) for all natural n

usually you prove

P(1) and

for all k, P(k) => P(k+1)

[warm shale]

Shuri2060

Shuri2060

[signal cobalt]

ok

I showed

and how should I conclude

the proof

[balmy mortar]

By induction, since P(1) is true, and P(k) => P(k+1) for all k >= 1,

P(n) is true for all n >= 1

Something like that

[signal cobalt]

nicew

ty

.close

[obtuse pebble]

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[timid silo]

Average rate of change over 0 <= x <= 3 is 1 right

Because from 0 to 2 its 2

From 2 to 3 its -1

Or does that make it 1/2 since (2 + -1)/2 = 1/2?

[high lily]

neither

[balmy mortar]

no

[timid silo]

What

Is it just 2

[balmy mortar]

average speed = total distance / total time

even if not a distance time graph, same principle applies

[balmy mortar]

average rate of change = total change in y / total change in x

[timid silo]

So 0

[balmy mortar]

you are interested

in the average rate of change

from 0 to 3

for x

[timid silo]

O right

[balmy mortar]

not from 0 to 4

[timid silo]

1/2 then

Wait

Bruh

2/3

[balmy mortar]

👌

[timid silo]

Holy shit sorry guys

.close

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[timid silo]

Given a graph of f(x), how do i find the x coordinates of each POI?

[timid silo]

I know a POI exists when f''(x) = 0 and has a sign change

[balmy mortar]

hi again?

😄

do you know about concave and convex

with regards to the graph

[timid silo]

lol

concave? Yes

Convex is a term i forgot

[balmy mortar]

its just the opposite

B7rmObB1pPjqAAAAAElFTkSuQmCC.bin

stolen from somewhere

a curve changes between concave and convex at a point of inflection. Can you figure out why?

@timid silo

[timid silo]

no

[balmy mortar]

f'' is 0

this means that the gradient f'

reaches a maximum or minimum

say f' is increasing

[timid silo]

Oh so it changes

[balmy mortar]

that means the gradient is getting steeper and steeper

[timid silo]

Signs

Positive to negative

[balmy mortar]

f'' changes signs yes

[timid silo]

Yes and the slope on f' changes from positive to negative

[balmy mortar]

no

[timid silo]

At that point

[balmy mortar]

oh yes

[timid silo]

What

Ye

[balmy mortar]

slope on f' yes

and this affects the concavity of f

[timid silo]

So at the point where concavity changes on f

Is a POI?

[balmy mortar]

yes

[timid silo]

Sweet

Ty

.close

[obtuse pebble]

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[timid silo]

How do i find area of a trapizoid that has a right angle

[timid silo]

Is it still the same for any kind of trapizoid?

[balmy mortar]

a trapezoid with a right angle is still a trapezoid

but there might be a shortcut

do you mean a trapezium?

split it into a rectangle and triangle. that could help speed things up

[timid silo]

ok

ty

.close

[obtuse pebble]

Channel closed

Closed by @warm gorge

Use .reopen if this was a mistake.

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• Show your work, and if possible, explain where you are stuck.
• 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.

[wary sail]

how do u solve number 4

u can’t determine the equation just from the graph

it’s a quadratic one

not y=mx+b

[balmy mortar]

is 4

related to the other qs

or independent

[wary sail]

i’m pretty sure independent

[balmy mortar]

big hmmmm from me

when the graph doesnt have units on it

are you meant to assume each grid square is 1 by 1

[wary sail]

i think

they’re decimal x’s

[nocturne minnow]

That graph isn't quadratic, just FYI

[wary sail]

o

[balmy mortar]

decimal x ???

[wary sail]

i mean just looking at the points, x’s are decimal