$b=a^{\log_{a}(b)}$
#help-0
surreal crystal
ocean sealBOT
Cycadellic
chrome garnet
x^log_x(4x-3)=x^2?
chrome garnet
b is 2?
surreal crystal
No quite
chrome garnet
or b is 4x-3
surreal crystal
Hint: whats in the base of the power and the base of the log?
surreal crystal
Hey
chrome garnet
or rather what they stand for
chrome garnet
1=a?
ocean sealBOT
Cycadellic
chrome garnet
were focusing on a?
surreal crystal
When the base of the expinent is the base of the log, they cancel
Thats what this is saying right?
This is a rule
chrome garnet
the base of the exponent is the big a here?
chrome garnet
what did you mean when you said whats a?
ocean sealBOT
Cycadellic
ocean sealBOT
Cycadellic
chrome garnet
x^2 is b
surreal crystal
No
chrome garnet
where does the x2 fit in that rule?
surreal crystal
Look at $x^{log_{x}(4x-3)}$
ocean sealBOT
Cycadellic
surreal crystal
$b=a^{\log_{a}(b)}$
ocean sealBOT
Cycadellic
chrome garnet
yeah
surreal crystal
**Cycadellic**
Well, this is a rule
This always works
How can we simplify $x^{log_{x}(4x-3)}$
ocean sealBOT
Cycadellic
surreal crystal
From the rule
chrome garnet
i dont get what im suppose to do
ocean sealBOT
Cycadellic
surreal crystal
Lets call this “cancel the log”
Whenever i say “cancel the log” i mean either this it the other one
So this
$\log_{a}(a^b)=b=a^{\log_{a}(b)}$
ocean sealBOT
Cycadellic
ocean sealBOT
Cycadellic
surreal crystal
What is it
surreal crystal
I have this
chrome garnet
log_x(x^4x-3)??????
vagrant jackal
what grade are you in?
chrome garnet
no
surreal crystal
No, we want to remove the exponent base and the log
chrome garnet
4x-3=x^log_x?
chrome garnet
how?
surreal crystal
Lets put numbers into the rule
chrome garnet
yes
surreal crystal
$34=a^{\log_{a}(34)}$
chrome garnet
i like nymbers better
ocean sealBOT
Cycadellic
surreal crystal
Yeah
chrome garnet
okay
surreal crystal
$50+34=2^{\log_{2}(50+34)}$
ocean sealBOT
Cycadellic
chrome garnet
wait the 2's cancel each other?
surreal crystal
Thats what the rule tells us
chrome garnet
30+34=log^30+34 since the 2 cancels right?
chrome garnet
or do you take the log out entirely
chrome garnet
so itd just be b=b
chrome garnet
is that the entire answer? 4x-3=4x-3?
surreal crystal
Right
So
Lemme ask you
Why can we just say 4x-3?
Just to make sure were on the same page
ocean sealBOT
Cycadellic
ocean sealBOT
Cycadellic
chrome garnet
right
surreal crystal
What can we do from here
chrome garnet
log the 2?
chrome garnet
find x?
chrome garnet
is that x variable or 4 times whatever x is
we can just guess it right
until it matches
chrome garnet
ima assume its multiply
chrome garnet
-x2
chrome garnet
-x^2+4x-3=0
ocean sealBOT
Cycadellic
surreal crystal
This is the eq
chrome garnet
right
surreal crystal
Im gonna give you another rule with variables
Its a little long but straightforward
chrome garnet
x-2 x+2?
surreal crystal
Not quite
surreal crystal
You can but thats work
surreal crystal
We can just plug numbers into a calculator with the quadratic formula
So this will be a rule
a b c will be any number or anything okay?
Number, variable, term, whatever
chrome garnet
a x2 b 4x c3?
ocean sealBOT
Cycadellic
surreal crystal
When we have this equation
There will be two values x can be
We will use the quadratic formula
chrome garnet
quadratic formula is not my friend
surreal crystal
$\frac{-b+\sqrt{b^2-4ac}}{2a}$
ocean sealBOT
Cycadellic
surreal crystal
And
chrome garnet
what the
surreal crystal
$\frac{-b-\sqrt{b^2-4ac}}{2a}$
ocean sealBOT
Cycadellic
surreal crystal
Those are the two values x can be
Lets say x1 and x2
To recap
$x_1=\frac{-b+\sqrt{b^2-4ac}}{2a}$
ocean sealBOT
Cycadellic
surreal crystal
$x_2=\frac{-b-\sqrt{b^2-4ac}}{2a}$
ocean sealBOT
Cycadellic
surreal crystal
Okay?
surreal crystal
Exactly
So we can write it all at once
$x_1,x_2=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
ocean sealBOT
Cycadellic
surreal crystal
If that makes sense
chrome garnet
kinda yeah
surreal crystal
Its just exactly what you said
chrome garnet
ive seen the -+ before
ocean sealBOT
Cycadellic
ocean sealBOT
Cycadellic
surreal crystal
**Cycadellic**
This is the whole point
chrome garnet
right
surreal crystal
Cause we =0
So we can divide a and one of the (x- ) terms
0/anything=0
Very easy to solve x from here
So
Lets use this
$-x^2+4x-3=0$
ocean sealBOT
Cycadellic
surreal crystal
,w solve (-4+sqrt(4^2-4(-1)(-3)))/(-2)
ocean sealBOT
surreal crystal
,w solve (-4-sqrt(4^2-4(-1)(-3)))/(-2)
ocean sealBOT
surreal crystal
1 and 3
ocean sealBOT
Cycadellic
surreal crystal
$-(x-1)(x-3)=0$
ocean sealBOT
Cycadellic
chrome garnet
thats for x^2?
chrome garnet
what happened to the 4?
surreal crystal
Actually wait what am i even going on about weve solved x
Its 1 and 3
Or
Rather
1 or 3
surreal crystal
Actually wait what am i even going on about weve solved x
Quadratic formula tells is this
chrome garnet
-+1 or +-3?
chrome garnet
or just x=1 or 3
ocean sealBOT
surreal crystal
,w solve log_3(4*3-3)
surreal crystal
Good
chrome garnet
yesssss
surreal crystal
1 was undefined so its not a sol
chrome garnet
that felt great
surreal crystal
Trying to undo the square essentially gave us another solution
Cause negative times negative is positive
surreal crystal
Go for it
Ill verify your process
After that i need to go to bed
I have a final tomorrow and its almost 2am
ocean sealBOT
chrome garnet
i understand the 3x3 and 4x3 and -3
but say we didnt have x
is it possible to do in hand?
or just assume were always able to use calculators
like the -(x-1)(x-3)=0
i dont understand that little part
surreal crystal
Usually when you answer you do it closed cause writing log will give the entire solution, if you ever need to calculate it you will have a calculator
chrome garnet
it makes ()() because its x^2 right
surreal crystal
Usually when you answer you do it closed cause writing log will give the entire ...
Before calculators people used tables for it
chrome garnet
right
chrome garnet
and if its not 1 try 2?
surreal crystal
It multiplies the 0, making the whole thing 0
No
Im explaining why we do - the root
The quadratic formula solves the x that make ax^2+bx+c=0
Like
One of the sol was x=1
chrome garnet
right
surreal crystal
,w solve -1^2+4*1-3
ocean sealBOT
chrome garnet
,w 2x^2-4x+2
ocean sealBOT
surreal crystal
chrome garnet
welp
surreal crystal
Im not sure why it integrated it
chrome garnet
thats pretty good in my book
surreal crystal
I guess
chrome garnet
thanks for being super patient man lmfao
surreal crystal
,w sinx
ocean sealBOT
chrome garnet
i know 100 numbers of pi btw
chrome garnet
fun
surreal crystal
I stopped at like 30
chrome garnet
i made the first 69 (43) my phone password
then went to 50
then 100
and yeah
just wanted to share w a math person since nobody else cares
surreal crystal
So you spend 10 mins tryna unlock your phone?
surreal crystal
100?
chrome garnet
43
surreal crystal
Wtf
chrome garnet
100 i can probably do in like 40
surreal crystal
I can kinda see that if you can type fast enough
surreal crystal
Yeah
surreal crystal
Np
chrome garnet
way better than my teacher lmao
surreal crystal
Gn
surreal crystal
.close
chrome garnet
.close
mellow grail
lone urchin
can anyone help me solve this?
lone heartBOT
can anyone help me solve this?
chrome salmon
Factorize denominator
lone urchin
yeh ive done that
chrome salmon
Perform partial fraction then
lone urchin
yeh i tried but i got it wrong an im not sure how to get the right answer
whole shell
show method
chrome salmon
Do you know what (x+2)- (x-3) =
lone urchin
huh?
chrome salmon
Can you solve that instead of doing huh
lone urchin
5?
chrome salmon
So why can't you separate it into two terms when you have 5 in numerator
ocean sealBOT
fäf
lone urchin
ohh
whole shell
well in anycase
theres no need to guess
you can just do normal partial fraction method
A/(x-3) + B/(x+2) = 5x/(x-3)(x+2)
so
A(x+2) + B(x-3) = 5x
(A+B)x + (2A - 3B) = 5x + 0
so
A+B = 5
2A - 3B = 0
solve the simultaneous equation
chrome salmon
what i would do is, 5x = 3(x+2) + 2(x-3)
That's unfortunate man
u sub x²-x-6 is good
lone heartBOT
@lone urchin Has your question been resolved?
keen lava
hi, I would appreciate if some1 can mark these and point me in the right direction for the question I got wrong
lone heartBOT
@hushed ether Has your question been resolved?
echo socket
Yeah, |x - 1| = x - 1 everywhere on the interval
tepid ravine
I have a question in regards to inverse matrix. So suppose that X is a square matrix that is invertible, and we know that C is an inverse of X. Hence, C x X = I. Is this the same as X x C = I, or is it X x C = -I, if it is true that X x C = I, then how would we prove this?
median oar
if it's invertible it must satisfy XC = CX = I
pearl hamlet
I have a question in regards to inverse matrix. So suppose that X is a square ma...
true by definition
tepid ravine
if it's invertible it must satisfy XC = CX = I
Is there a proof for this? I'm not really convinced
median oar
if it's not true it's not invertible
pearl hamlet
that’s literally the definition of invertible
tepid ravine
Okay
median oar
if XC ≠ CX then C is not the inverse of X
tepid ravine
So that would be called one side inverse?
median oar
what od you mean by 1 side inverse
tepid ravine
So is it true that a square matrix has a unique inverse?
pearl hamlet
if it’s invertible yes
pallid scarab
For matrices, having a "right inverse" is equivalent to being invertible
tepid ravine
right cool.
pallid scarab
(square matrices)
pallid scarab
A*A*A*B*B*B ?
tepid ravine
yes
pallid scarab
so how will you proceed?
tepid ravine
is it mathematical valid A*A*B*A*B*B
pallid scarab
A*A*A*B yes
tepid ravine
Or do I take the negative?
pallid scarab
is it mathematical valid A\*A\*B\*A\*B\*B
You have no Idea if A and B commute
Keep it A*A*A*B*B*B
tepid ravine
I thought from an earlier lecture that a particular matrix E,F E*F = -F*E?
is that right?
pallid scarab
for some matrices , E*F = F*E
For other matrices, you have neither E*F = F*E nor E*F = -F*E
tepid ravine
So we have three cases basically, and it is inconclusive right?
pallid scarab
So we have three cases basically, and it is inconclusive right?
stay in the general case
tepid ravine
For other matrices, you have neither E\*F = F\*E nor E\*F = -F\*E
When you can't have E*F = F*E nor E*F = -F*E is if and only if the sizes are different right?
pallid scarab
When you can't have E\*F = F\*E nor E\*F = -F\*E is if and only if the sizes are...
here's an example of counterexample matrices : try looking at $A = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \ 1 & 1 \end{pmatrix}$
ocean sealBOT
rafilou2003
pallid scarab
Try computing A*B and B*A
You will get $AB = \begin{pmatrix} 2 & 1 \ 1 & 1 \end{pmatrix}$, $BA = \begin{pmatrix} 1 & 1 \ 1 & 2 \end{pmatrix}$
tepid ravine
yeah
ocean sealBOT
rafilou2003
tepid ravine
True, it is not in anycase
pallid scarab
Okay, so when we want to simplify lets say A*A*A*B*B*B
To solve this, act like you don't know anything about what A*B is
pallid scarab
matrix product $\neq$ vector product
ocean sealBOT
rafilou2003
pallid scarab
and matrix product is not commutative
tepid ravine
right
cool, also how would we do this?
So I was thinking of AB = AABABB
and before I believed It was commutative so I began rearranging which wasn't right
pallid scarab
Yes
pallid scarab
So I was thinking of AB = AABABB
This is 100% true, you start from ABAB = I, then you multiply by A from the left and B from the right
So AABABB = AB
But then... AABABB = (AA)BA(BB)
rich basin
lone heartBOT
rich basin
I'm not really sure how to find the projection of the one at the bottom
so I tried expanding it out using the formula of projection but struggled to see what to do next?
The question hinted to use the thing above, but not sure how I can apply it
lone heartBOT
@rich basin Has your question been resolved?
rich basin
<@&286206848099549185>
rich basin
<@&286206848099549185>
hybrid sinew
GGbond
rich basin
did you know ive watched all seasons of GG bond
yeah, including the new one
gg bond got alot skinnier
rich basin
<@&286206848099549185>
rich basin
<@&286206848099549185>
lone heartBOT
@rich basin Has your question been resolved?
formal axle
Recall that taking projections is linear and it is being projected onto some line
So you calculate the projection of the second vector onto b, then take the sum of the two projections as your answer
rich basin
So you calculate the projection of the second vector onto b, then take the sum o...
hmm, but how do you know the projection of the vector <614,510>
formal axle
You need to find the value $\lambda b$ such that the distance between $(614,510)$ and $\lambda b$ is minimised
Then take $\lambda b$ as the projection
rich basin
but we don't know what b is?
formal axle
It's just projecting the point onto the line generated by the span of $b$
ocean sealBOT
384920
formal axle
That is true but it's a constant multiple of $(5,13 )$
ocean sealBOT
384920
formal axle
b is in the span of that
rich basin
so we want to find proj (<614,510> - lambda*b) onto b
in which lambda b is lambda * <5,13>
formal axle
No
$$\text{proj}_{\mathbf{b}}\left(\begin{pmatrix} 614 \ 510 \end{pmatrix}\right)$$
is the vector in the span of $\mathbf{b}$ which minimises the distance from itself to $\begin{pmatrix} 614 \ 510 \end{pmatrix}$
rich basin
yes
ocean sealBOT
384920
rich basin
okay
formal axle
Do you know a formula for that?
rich basin
Do you know a formula for that?
i know the projection formula
formal axle
Ok that's good, then you can use that
rich basin
so projection of <614,510> onto b = lambda * b?
formal axle
That just means there is some lambda which satisfies that
rich basin
yeah
formal axle
But yes, that's true for some lambda
You can just project it onto the subspace spanned by (5,13)
or take b=(5,13)
It gives the same projection onto that line
rich basin
oh okay, so afterwards to add it onto <5,13>;
formal axle
Yes as that is the other projection
rich basin
thanks you're a genius
formal axle
You're welcome, hopefully it helps!
rich basin
lone heartBOT
rich basin
From part (b) we can see that Az = a + v*w. Thus, from observing the three position vectors <-5,-5>, <2,5> and <a,b>. Thus, from (a) we have shown that these three vectors are collinear, therefore
This was my explanation
but I'm writing this because I actually don't get how a linear combination of matrices are also collinear points
@formal axle
rich basin
Points which are on the same line?
yes
formal axle
Oh, (a,b) is fixed from part (a) so that those are collinear
The idea is that the image of some "line" under a linear transformation is also some "line"
That linear transformations map lines to lines
You can make it more rigorous if that is required though
But it looks like it is just asking for a brief explanation
Oh part (c) may also be useful for it
rich basin
That linear transformations map lines to lines
interesting
but how does that work with it being a position vector?
formal axle
Which position vector?
rich basin
Which position vector?
we are trying to show the linear combination of these matrices are position vectors
formal axle
What definition does it use for collinear points?
rich basin
What definition does it use for collinear points?
wdym?
formal axle
The question
rich basin
yeah
ocean sealBOT
384920
rich basin
yep
formal axle
So you can choose one point and then use that $A(x+(y-x))=A(x)+A(y-x)$
ocean sealBOT
384920
rich basin
yes
formal axle
Then if $x,y,z$ are collinear, you have $A(z-x)$ and $A(y-x)$ being a scalar multiple of one another
ocean sealBOT
384920
formal axle
One of them being a scalar multiple of the other
So $A(x)$, $A(y)$ and $A(z)$ are all along some same line
ocean sealBOT
384920
formal axle
Unless they all map to the same point
Because
$A(y)=A(x)+A(y-x)$ and $A(z)=A(x)+A(z-x)$
ocean sealBOT
384920
rich basin
okay
I see now
also for this questipon
Could you please explain why we choose q as (157/89)?
I don't really get it
@formal axle
formal axle
Sorry I was AFK
rich basin
algood
formal axle
tanh is a hyperbolic function right?
rich basin
yes
formal axle
Is this to check the answer for this one?
rich basin
Is this to check the answer for this one?
It is just using the comparison test apparantly
rich basin
same
these are the things you can select from
but apparantly 157/89 is the right answer
formal axle
Oh I think it's because of the polynomial to some exponent in the denominator
Because the highest term is $x^{4\cdot\frac{123}{178}}=x^{246/89}$
ocean sealBOT
384920
formal axle
Then introducing $g(x)$ in the denominator brings in $x^{157/89}$ in the numerator
ocean sealBOT
384920
formal axle
And they cancel out to leave $x$ in the denominator
ocean sealBOT
384920
rich basin
why do they want x?
formal axle
I think it's for the limit at the bottom of the first screenshot
rich basin
I think it's for the limit at the bottom of the first screenshot
i'm quite lost
formal axle
This one
$$\lim_{x\to\infty} \left( \frac{\arctan (x)}{x} + \alpha , \text{tanh}(x) \right) $$
ocean sealBOT
384920
formal axle
For this $f$,
$$\lim_{x\to\infty}f(x)=\lim_{x\to\infty}\frac{\dots}{x^{4\cdot\frac{123}{178}}}$$
ocean sealBOT
384920
formal axle
where you have the same terms in the numerator as in f
Because that x term dominates
The choice of q allows you to have that when you consider $f(x)/g(x)$, terms cancel out and you are just looking at the limit for $x$ in the denominator instead
ocean sealBOT
384920
formal axle
**384920**
Then you can apply the existence of this limit
lone heartBOT
@rich basin Has your question been resolved?
formal axle
I have to go to bed very soon
Is there anything you want to check before then?
About the above?
rich basin
Is there anything you want to check before then?
Just thinking
So wouldn’t that Be x* numerator of f
formal axle
No, there would be x in the denominator
If you are talking about the expression for f(x)/g(x)
lone heartBOT
@rich basin Has your question been resolved?
wintry patrol
<@&286206848099549185>
6) a) Copy and complete the table of values for the relation V = -X² + X + 2 for -3 ≤ x ≥ 3.
(b) Using scales of 2 cm to 1 unit on the x-axis and 2 cm to 2 units on the v-axis, draw a graph of the relation
y = -X² + X + 2.
(c) From the graph, find the:
(i)Minimum value of y;
(ii)Roots of the equation X² - x -2 = 0;
(iii)Gradient of the curve at x = -0.5.
worn fox
!15m
lone heartBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
worn fox
You've also killed this channel by deleting your original message you'll have to open a new one
It's also more annoying to delete your pings and ghost ping people
wintry patrol
how do i do it
wintry patrol
oh ok
lone heartBOT
Available help channel!
hybrid sinew
or not
lone heartBOT
lone heartBOT
Available help channel!
raven girder
Guys
raven girder
I need help
sharp thorn
raven girder
let me type 
these channels get reserved too fast
f, g are monotonous increasing
is f \cdot g monotonous increasing too?
I need to prove or disprove this
I think it's false
but not 100% sure
surreal crystal
consider their derivatives
surreal crystal
how do we describe monotonous increasing, then?
raven girder
for all x,y in R with x < y: f(x) <= f(y)
surreal crystal
okay, so then suppose two functions f and g
we have
$\forall x,y \in\mathbb{R}: x<y\implies f(x)\leq f(y)$
ocean sealBOT
Cycadellic
ocean sealBOT
Cycadellic
raven girder
yep
surreal crystal
so we know that
x<y means the same thing for both, right?
raven girder
yep
pallid scarab
Uh, increasing in the english meaning or other meaning?
surreal crystal
consider adding the inequalities together
pallid scarab
Because increasing = (strictly) increasing
surreal crystal
**Cycadellic**
from this definition @pallid scarab
raven girder
Because increasing = (strictly) increasing
without the strict, strict would be <
pallid scarab
from this definition <@246787399991033858>
So non decreasing
surreal crystal
yeah
raven girder
sorry im german
pallid scarab
without the strict, strict would be <
Non-decreasing in english
surreal crystal
yg
pallid scarab
sorry im german
I understand the confusion, same in french
surreal crystal
@raven girder so, do you see how to get what youre looking for out of adding the inequalities?
raven girder
why add?
raven girder
ye
surreal crystal
then multiply the ineq together
make sure to follow ineq for negatives tho
then it should be easy enough to show their product is/isnt monotonic increasing
raven girder
wdym multiply together
surreal crystal
we have a true statement
raven girder
I can multiply two inequalities?
surreal crystal
as long as both statements are true
now granted
if we multiply a negative, we have to swap ineq
but other than that, its just as expected
they just both have to be true to begin with
raven girder
we have f(x) <= f(y) and g(x) <= g(y)
raven girder
now multiply what
raven girder
f(x) * g(x) <= f(y) * g(y)
surreal crystal
right
raven girder
but
raven girder
f(x) > 0, f(y) > 0, g(x) > 0, g(y) > 0
...
raven girder
yeah so f(y) could be positive but f(x) negative right
surreal crystal
but once we have considered these four cases, note that we are multiplying a negative, so we need to swap our ineq
once you get past that
if you get the general solution that f(x)g(x)<=f(y)g(y) in all cases, it too is monotonic increasing
otherwise, its not
make sense?
raven girder
yes but I think we need to check more cases or not?
raven girder
ah
because we multipled the inequality with f(x) on the left and g(y) on the right?
raven girder
but now what about the case f<0, g<0
surreal crystal
both neg
raven girder
ah so 2 swaps
surreal crystal
we care about the g being mult
say
-2>-3
multiply negative one by both sides
2<3
thats what we want to do here
so yeah
since g is negative
it swaps
raven girder
but if we swap once then the inequality is false
surreal crystal
right?
wait do you mean that swapping it contradicts the other ineq, or just in that case, its false?
raven girder
It would contradict the entire thing or not
raven girder
I mean it would show that the initial statement is false
surreal crystal
👍
surreal crystal
right
so
youre saying
f(x)<f(y)
and g(x)<g(y)
given that g is negative
mult g by both sides
f(x)g(x)>f(y)g(y)
is what you should have
which doesnt contradict any of itself
good?
raven girder
wait so we start with g(x) <= g(y) and then turn that into f(x)*g(x) <= f(y)*g(y) right?
so we need to check the sign of f(x) and f(y) right?
or not
surreal crystal
right
raven girder
however if one of them is negative and the other positive then we need to swap sign
surreal crystal
or pos pos
no
we start with f(x)<=f(y)
consider
-2<3
multiply both sides by -1
2>-3
see what i mean?
raven girder
but what we do is multiply one side by a positive and multiply the other side by a negative
or not?
if f(x) is neg and f(y) pos
surreal crystal
right
i think i realize your confusion
so we have
x<y
f(x)<=f(y)
g(x)<=g(y)
now, we need to consider
g(x)>0, g(y)>0
g(x)<0,g(y)>0
g(x)<0,g(y)<0
note that we may not consider
g(x)>0,g(y)<0, because this contradicts our given
i said f
sorry about that
surreal crystal
x<y
f(x)<=f(y)
g(x)<=g(y)
now, we need to consider
g(x)>0, g(y)>0
g(x)<0,g(y)>0
...
this is what we need to check
raven girder
ok what about g(x)<0,g(y)>0
surreal crystal
consider -3<1
raven girder
f(x) * g(x) swaps sign
surreal crystal
for now, just consider the same sign
youll get your answer
if g(x)<0 and g(y)<0
does this even agree with g(x)>0 and g(y)>0?
raven girder
wat
surreal crystal
surreal crystal
if g(x)<0 and g(y)<0
does this even agree with g(x)>0 and g(y)>0?
just look at these two cases
surreal crystal
just look at these two cases
but if you consider these two cases, you should find your solution pretty simply
it just wants to know if all of them agree
do these two cases we can predict agree?
raven girder
yes
surreal crystal
are you sure?
surreal crystal
lets say
-2<2
do *-1
2>-2
weve swapped
but -1<0
now lets see -2<2
lets do *2
-4<4
we havent swapped
right?
raven girder
yes
surreal crystal
does f(x)g(x)<=f(y)g(y) agree with f(x)g(x)>=f(y)g(y)
in both cases, we have
x<y
surreal crystal
does the first one and the second one say the same thing?
raven girder
no
surreal crystal
thats your answer
we have a contradiction
if we assume f(x)g(x) is monotonic increasing
surreal crystal
does f(x)g(x)<=f(y)g(y) agree with f(x)g(x)>=f(y)g(y)
because these definitely do not agree with each other
raven girder
but sir
surreal crystal
consider f(x)=x and g(x)=x
raven girder
what about g(x)<0,g(y)>0
surreal crystal
does f(x)g(x)<=f(y)g(y) agree with f(x)g(x)>=f(y)g(y)
well still have this contradiction
nah
just need to compare their ||
raven girder
ahh
surreal crystal
just consider the graph of y>x and y^2>x^2 and itll make sense
raven girder
still strange
raven girder
yes kek
raven girder
no
raven girder
I feel like an idiot
surreal crystal
what dont you understand
surreal crystal
I feel like an idiot
trust me, nobody is good at math right at the start
we have to make mistakes to truly learn
if we dont make mistakes, we arent learning at our level also
mistakes and misunderstandings are crucial in the learning process
raven girder
so when f(x)<0 and f(y)<0 then we swap the <= to > right
surreal crystal
starting at g, yeah
raven girder
I see
surreal crystal
but by a change of variables, starting at f the mult g necessarily is starting at g then doing f
the change being f->g and g->f
so we only need to consider one in our proof
change of variables is a really funny trick that really shortens proofs
lol
raven girder
idk what you just said but I agree lol
surreal crystal
lol
raven girder
.close
alpine sable
I forgot how to solve this type of questions. I need to calculate the length of x
thick grail
theres a handy dandy theorem named after a guy called Pythagoras
alpine sable
theres a handy dandy theorem named after a guy called Pythagoras
You just ignited a flame in my memory
Thank you so much
thick grail
nw
alpine sable
.close
alpine sable
You buy potato chips for $3.99, pretzels for $1.59, and dip for $2.79. Then you buy a magazine for $2.99 and a newspaper for $.75. You have to pay 2.8% on food items and 7% on non-food items. What is the total amount you paid including the sales tax?
Student Answer:
[(3.99 + 1.59 + 2.79) x 1.028] + [(2.99 + .75) x 1.07] = 12.24062 rounded to 12.24
The total paid including sales tax is $12.24.
alpine sable
I dont see how im wrong
I added food items together and multiplied by the food item tax, then added the non food items together and then multiplied by the non food item tax
1.028 is the total plus 2.8% and same for 1.07
its not asking for only sales tax, it says the total including sales tax
solemn juniper
i input the expression you wrote into a calculator and got a different answer than 12.24, so perhaps there's an arithmetic mistake
try computing it again
alpine sable
I got 12.60
That's disappointing. I feel like my grasp on basic arithmetic is slowly getting worse, I don't understand:C
alpine sable
i input the expression you wrote into a calculator and got a different answer th...
What did you get?
ocean sealBOT
Result:
12.60616
it happens
alpine sable
sigh, thanks for the help though
solemn juniper
np
raven girder
yo
raven girder
[\lim_{x \mapsto 1} \frac{(1-x)^2}{1-x^2} = \lim_{x \mapsto 1} \frac{x^2-2x+1}{1-x^2}]
ocean sealBOT
madmike
raven girder
I am stuck
does anyone know a trick?
if I insert x=1 then I divide by 0
not sure what to do now
I tried pulling out x/x outside the fraction but then I get new fractions
[\lim_{x \mapsto 1} \frac{1+x^2}{1-x^2} - \frac{2x}{1-x^2}]
ocean sealBOT
madmike
raven girder
Can I do this?
ah not sure if that helps
oh
but I am not sure what to write in the numerator then
I never factored out a sum before
how do you do that?
ocean sealBOT
madmike
raven girder
or am I misunderstanding
pallid scarab
Keep it $\frac{(1-x)²}{1-x²}$
ocean sealBOT
rafilou2003
raven girder
oh