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good day students in this clip we are going to be going over part 4 on modeling and optimization
in this particular going to be looking at some minimizing the distance from points to
recur so let's cover the steps again on this is a quick refresher
first of all we going no model the situation to generate a model you need and optimization
function in a constraint function
as long as you well labeled diagram part two you are going to substitute on you isolate
the variable from the constraint and
substitute it into the function that's to be optimize you also state the domain of on
the constraints of the independent
or will you have as resulting equation in part three a potential extreme ends which
include the critical points and the
endpoints in the if the interval is closed with an opening to about you do not need to
consider the endpoints on when finding the
potential extreme as and then you are going to classify the extrema step for a list of
five you going to state the meaning of the results alright
lets go ahead and take a look at the example number four it says what points on the graph
of the function Y equals three
minus X square is closer to the point (0,2) also states the minimum distance right so
let's start with part one where it took
create a model right is accomplished this what i am going to do is try to sketch the
graph of the situation one way to the graph of the situation scares to
current and the point to have her Y axis (x-axis so we have this curve ratio works transformation
does this curve apply to
the standard Y equals X square function you have bothered three MIs exquisitely have a
standards problem facing up
basically means you going to shift three units up in and is minus in front of the x squared
you went to reflect the
parabola downwards To have a standard problem • the origin were going to shift three units
up 123W vertex in SF opening up
our parabola will open downwards can't sell its graph the points of the set of points
here
right in the next point goes
here is to want to argue just a sketch of of the situation and so let's go ahead and
graph the problem facing our generic
something like this in and get a piece through those two points right to leading to problems
we with this problem we have
a point that Tom be calculating the distance limit minimizing the distance from this curve
to ex-wife to the point is
02002 we have 02 right here..025 so what were they doing is able to pick any arbitrary point
is called point ex-wife so we
can take a point here and what we are doing is in minimizing the distance from that point
to the point to zero to select
try to minimize that distance and so what we're going to do is a protocol that distance
D suited to the length of the
segment right here is going to be the all right so what is the distance formula this
is formula unit analytic geometry
remember the equals the square root of X2 minus X1 20 square plus why to minus Y1 quantity
square feet contact included
that this is your distance from you that were going to be using here are excellent some
take a look at the optimization
function first optimization optimization with optimizing away or distance cakes called distance
to the reason distance
formula that's what we're optimizing a looking for a minimum distance right so in order to
use the distance formula we
needs two points a case of the minimize minimize distance distance from where to where distance
from the .02 and any point
ex-wife on the curb that can't so that's the arts about what we going to do is determine
what X1 and why one part and X2
and why to protect to the first point is X1 first point yet why want the second point
is too white to so if we substitute
these values into our distance formula that would give us the optimization function rights
of the equals the square root
them is a real year since it can be a long expression X2 minus X1X2 is asked minus X1
is zero square for this formula was
white to is why minus why one is to quantity square okay so that is the expression that
we're optimizing the minimizing
the distance right here we also need us another function or set of functions known as the
constraint function rights of
funny constraints that optimizes subject to a constraint walk with her constraint here
with constrained by a curve
indicates the curve that with constrained by the vicar of why equals three minus X square
so we have a constraint and
optimization functions are shifts to part two of the problem-solving process will be
going to make a substitution to
substitute if you notice our distance function is in terms of two variables which is hard
to differentiate in one and one
I mentioned so we need to make a substitution here's a what's, substitution Kelly makes
it at this distance function can
be expressed in terms of one variable you see this constraint that the beauty of the
constraint function Y equals three
minus X square selects substitute that into our distance from you so we have distance
equals the square root of as the ridges XX square plus under change
colors here is a winding up its three minus X square that is from my constraint
function -2 quantity square can simplify this before we talk about the domain right so to
simplify this Saudi security
equal to the square root of any 2-3-2 is one so one minus X square square and when you
expand out before it out and have X
square +1 minus 2X square plus X to the force okay so the distance formula to be working
with becoming the distance
function of others working with here and we simplify everything we are going to have X
to the four minus X square +120
constraints are domain is a better domain if you look at this function what is the domain
of Axworthy acceptable inputs
for X the answer is all real numbers in the constraints and exit the every inputs for
next value have a different outfit
so are domain we have no restrictions on our domain to the domain for this problem domain
is from negative infinity to
infinity okay so that's that all right let's look for our potential extreme is important
by perky potential extreme ends
potential extremas right so we know that our potential extreme as are the critical point
and endpoints in this problem our
domain is an open is also our endpoints are not included here are cases who do not have
any endpoints are potential
extreme is in this problem are basically our critical points can't so how do we find the
critical points, find critical
points critical points are when the derivative with doing some deed as a function this time
to wait deprived is equal to
zero or the prying does not exist protect to these are the two by values of X that we
are going to be solving for is the
square root of X in the fourth minus X square +1 to differentiate this remembered the differentiation
shortcut for
radicals DDX of the square root of X is equal to one over two root extract so then apply
that here. The play channel here since
we have an input function okay so let's differentiate the prime equals one over two differentiated
apparent functional
than the inner fourth power polynomial fixed time to derivative of the input function which
is exophoric minus X square
for differentiate that will have for X to the third minus 2XR derivative is for X to
the third minus 2X divided by the
square root of the square root of actually have it to any outside suit to come the square
root of X to the fourth minus X
square +15 so to find a critical point to start with when the d prime is equal to zero
when the time is equal to zero
pages focus on in the Reuters said enumerator to zero okay because we cross multiply the
denominator. All right, it's also
this benefactor to factor out to GCF here which is to ask affect up to ask and then
we have 2X square -1 equals zero
rights away that is said to him the two factors equal to zero fairly have to accept 2X equal
to zero and 2X square -1
equal to zero right so was all this wind up with X equals zero anything so it should end
up with X equals plus or -2 over
two right so of these are the points where dead preservative is equal to zero how about
when the prying does not exist to
find this out without but we going to do is basically said the denominator equal to zero
and sulfur that sold to time the
square root of exophoric minus X square plus one equals 03 divide with size by two and
square root you end up with X the fourth
minus X square plus one equals zero this is a fourth degree polynomial can easily convert
this into a second degree
polynomial by making a nice substitution policy let you equals X a second power to make that
substitution going to have
the quadratic EU square minus you plus one equals zero because solve this using the quadratic
formula where a is one BS -1
NC is one that we do that would have you equals plus or -3 lead to negative B which is positive
one so positive one plus
or minus the square root of the square which is one -4 AC 4×1 has wanted just for divided
by Tuesday at 10 so that uses
one plus or minus the square root of -3/290 examined this terms right here to write code
component units is negative so
our solutions here are going to be nonreal ID and not real solution is not pointing left
on the graph pointed in graph on
the graph right so you know what that means said no real values caused asked to satisfy
the nonexistence of the derivative
so when is the derivative zero dance anyone what is the derivative nonexistent answers
never take so this has no solutions
and now we know what our potential extreme is our potential experience potential extreme
as are in order are negative or
two over 20 and which to overtake okay now the next what's we're going to do is become
a classified and there are two ways
you can do this for which is classified we can use the first derivative by looking at
weighing the function is increasing
and decreasing around the critical points and it was interested in and decreasing there
we know we have a max is
decreasing and increasingly having been Or we can use a quicker method which is the second
derivative right so it is
concave about we know we have committed in this country probably have Iraq's sake is
a second derivative code is faster
right so we know that's right so that their access out of the kind is a call to for X
to the third minus 2X that divided
by 22 times the square root of the square root of the fourth degree polynomial spirit of off X to the fourth minus X
square +1 okay now let's look at this outspoken about thinking about using the second derivative
test here but to
differentiate this using the quotient with all radicals can be a mess I think that she
is a first-rate is testing that
will be much easier to handle content so other than it do here is a snake upon number line
and in and create it about
other than a test case and outdo the definitive answers that we looking for soul are critical
points are negative roots to
over two is approximately -0.72 cal calculator and Italy have zero in Italy have on positive
words to either the potential
extreme is that we have okay and this is approximately .7 O on what I look for the minimum okay we
could have one tomorrow
so desperately looking for now all fortunes about one of the metastasis X equals -1 for
Interpol to the test X equals -1/2
for interlock three US test X equals one half in there for Interpol for this test X equals
1% right so only been about
with the first derivative at these points on in his humor is increasing decreasing in
an aware of vaccinations are intact
so let's starts with them into about 1X equals -1 now if you look at this function the derivative
with this call we care
about is a sign that takes the denominator have the square root can be square written
expression ever yield a negative
outputs the answer is absolutely knocks this is the denominator can never be negative biggest
focus our attention on the
numerator can so let some valid enumerator that before times -1 the sign of the numerator
tells is a sign of the entire
expression K4 times to come -1 the Valley that you have -4 -4 was to which is negative
so it's negative on it about Wanda
Frazier versus negative which tells us something about function that it's decreasing right
is about to the test X equals
-1/2 four times -1/2 just looking at enumerator because that's what tells assigned times two
times -1/2 that's can give us
some negative forward to which is -2 in a few multiply this together is going to be
slight mistake carefully later a quick
correction of this is supposed to be to the third power and one half just to the third
power so the greatest -1/2 to the
third power we going to have -18 occupied out by 4 1/2 -48 -408 and then you can multiply
these two minuses you have plus
one type to negative for ages I -1/2 this one is a positive one half this can be positive
positive content about to the
first derivative price was a sign of the first derivative on the 13th about work X is equal
to positive one half was a
valid enumerator again because that is what God tells us to sign him off the derivative
O4 times one half to the third
power -2×1/2 case so in this case with and have for over eight spotted this time for over
eight innings times -1 a case of
this is what happened is one hour as relate to be negative right so is negative in the
13th about us examine the last
Interpol Interpol for work X is equal to one price of your 4×1 to the third of -2×1 so
you basically have 4-2 which equals
which is positive current service positive available for those positive virtue okay so
what is apt happening in what
Interpol probably have the can clearly see that the function is decreasing on it about
one and decreasing the introductory
And the function is increasing on an football to and increasing on Interpol for two what
does that tell us about the
extreme and we has to to decrease or increase in here this is a man to assist increasing
and decreasing here this is a max
relative Maxx SS is decreasing increasing here this is another relatively accurate silly
have two relative minimize out in
dysfunction cases of the values that are where our minimum minimum distance are case so which
one is that will give us a
small distance out of the two only actually have to confused what the distance is all
for those two values right;
determine how we checked the distances are small to smaller story and if there are equally
small unless right and what we
can do to decipher from the start right here we have a relative max act of X equals zero
Italy have two relative minstrels
in acts of X equals negative or two over two and 62 over two so the question is which is
a switch user smaller distance or
maybe it was equally small to the determine which what the minimum distance is a good
about with the function be we know
what that that that the function is is equal to the square root of X to the fourth minus
X square plus one night to take a
look at it what happens when you put these two into dysfunction get exactly the same
output because this two variables are
recently even power so the negativities of no consequence that takes to what I'm saying
SS is the of questions that
parenthesis should be of negative roots to over two given to be exactly the same thing
as the average tour to because
where we are plugging them into the then be raised to the second power reason even power
which nullifies the negativity
right so let's plug it in said suit negative thinking that maybe the square root of all
discovery of two over two raised
to you the fourth power minus the skirt of two over two races second power +1 Outfeed
to the calculation is he going to
have the first term which two to the fourth power is to test images for two to the fourth
power 16-62 squares to over 4+
can express assess for over four Q1 is a test for before four minus India we have to but
simplify this a little bit more
we are going to have taken divided the 4/16 victory statue one fourth silly have one of
the four LEDs to have pompous
total for which equals the skirt of three before which is reached three over to visit
as a minimum distance right so, what
are the points we know the X Kournikova points Pelosi need to know what coordinates of her
points right so when when never
the question asked for the points that are the minimum distance to zero, to know what
the distance is that we actually
have to figure what the points are so when is from minuteni X is equal to the first value
when X is negative or two or two
but is why will we know the constraint function tells us what why is right silly why is equal
to the three minus X square
right and then sold for plugging this value we can have three minus negative square root
of two over two square prices can
become 3-50 square skirt of tea have to you over for decades in a liquor expressed this
as 12 over four toll for my
assistant of a for which reduces to five over two Anyway X is equal to the square root of
two over two you have the same answer
right because he screamed at the time to the three minus the square root of two over two
square price and now we're ready to
write our conclusion is so conclusion conclusion to start five the minimum distance the minimum
distance from the points
from zero, two on the graph why is equal to three minus X square is what they define the
minimum distances being nasty so
the minimum distance will you protect the X Valley into the reached the distance function
rights to the minimum distance
is reached the river to your distance is reached three over two and your faster points and
the points are X the first one
was on negative or two over two in the why value was actually got which is five over
two and on the second point was
reached to over two and the why value was five over two8 syntax that the tanks are much for taking the time to
watch his
presentation to produce a scratch my telephoto coker could such as this at least once a comments
to let me know what you
think about this presentation appreciated focus of her mother said that complex again
for watching and have a wonderful day