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hello good citizen and my name is batman i will try to make scenes of the easier
for you today
uh... basically after we have talked about the
concept altered riveted we can take stock about what we know about
derivatives and a conceptual knowledge is considerable
i mean we know that the dreaded it this is so fatine declined
we know that positive turn it means that the functions increasingly knows a
native to read creative means that the functions decreasing
and we know that a positive second derivative meets the function skulking
about the native secondary means functions *** now
we know that is zero derivative gives us a critical point
and we know that the coveted pole position function in small city
the one thing that we haven't talked about so far as how to compute those
derivatives more easily uncertainty
that was intentional
beacause working with these difference quotient first of all reminds us a
little bit more *** to whether evidence come from
second of all uh... it
practices some of those algebra skills that i certainly
believe we need to have in order to make things more easy down the road
but at some point in time you really realize that we ought to get you need to
have some kind of shortcuts met with differentiation rolls come in
and first brought the collection
uh... talk about in this module is the power rule which is subject
off uh... this presentation
so well what is the derivative
offer polynomial elasticity blessed removable mode mode only off the floor
makes to the end you've already taken some derivatives using
difference quotient duran is off
square upon those probably also some cuban polynomials
what's the underlying principle
whenever you want to determine the derivative functions that you don't
already know
you have to go back to those difference portions are not going to go away but
the advantages this time we're just going to use it to kurt to the right any
rule and after that
we're not going to mess with them
the definition of the derivative is always a sense of the message goes to
zero expose it reminds of a fixed by h
if you do that with ever picks phoenix to the innuendo expose to the in my
mistakes to the end of my by each
known as h_ goes to zero
and uh... we know from experience novice goes we somehow have to
work out these parentheses and as soon as we have that hopefully the x_-two yen
will cancel that hopefully ph will cancel
if we will end up with
what's accurate it hits
okay how do you multiply out expose to the end well that's what you go to
was called the binomial theorem input you may have seen
uh... and which is called pascal strangle because
basically when you work out a binomial
you
basically right down
you were
ixus in uh... decreasing powers he reaches an increasing power seems
certain dates to the and then you go with ecstasy and muslim towns a chance
to be unwise to attending square and so on
and a half
uh... exquisitely in months to extend sixteen months one each for the end
in the quotations are obtained via pascal strangle
which is something where you put ones on
to diagonals ensign
isri
makes rose entries that are not on these two diagonal stretch s
that some of the two entries
about that so
the first rohan task of trying to respond to one in fact
uh... is
the sequence of quotations for multiplying all expose square dance
expect us to xh
as a tax cut
annex true
yes
execute class three excoriate rose three eighths inch square
h q
uh... hang on just two minutes away just not
and i'm back ok of this
uh... ask us trying to get you the quotations for the expansions
expansion of the when we'll
uh... and salsa fourth round would be one plus three digits four three plus
three kids is six three plus one did you forget
and of course that the arithmetic is terribly exciting you can see that
it really works out as the court fishing ease the
some author
at jason points are about it
um... that keeps going like that and it can be proved that these really are the
coefficients are needed
to expanded binomial
and those are the dates as we extend over the first term will always be extra
vienna
the second term will be a cs to the in minus one
times h times whatever this quote fishing piers and you can see for a
second or the coefficients to
for third party the quotations three for the fourth or the provision for four
fifty or the coefficients five
in someone
insult one well for install the quote reasonable politically in and that is
indeed the case
all the other terms
at the age square at houston it's kind it's square we know that from
the experience with a difference portions most likely it's going to go
away and that's indeed the case so what we can say yes
that this is that if they chose to zero extn
class in that quoting from pascal strangle ties into the in months one h
which is half of the pattern forward
in which we are expanding the binomial
just age squared times more terms
miners extending it
okay well the x_-two the is cancelled so we end up with innovative goes to zero
in next to the m_s_-one each class each square times more terms
now remember better safe than sorry factor out the h
in that was the limit as h goes to zero index to be a muslim trust each times
some workers but they're only finally many of those who solicit rose to zero
this park back here goes
uh... was away
and that means that all rosen ten sixteen pm minus one
okay so that's a poll
other things that we have to deal with especially in poor fishes are some of
the constant multiples well
the derivative of some again we'd just have to write down the difference
quotient
and we realize that you have plenty of expose expense effort to expose trustee
of expose ethnicity of exit sample pics
last year that cindy
pushing the negative sign
if we've restored it then put the s and the geez together we end up with a
message goes to zero advocates for six months of apec's
does give extra say two months to your text about h
and that's two separate difference courses one fourteen f
and one for g
and some of this
that ends up being the critical path considering beautiful g
exponential functions are another
kind of function
that we may have run across party on it
the derivative is fairly easy to work out if we're willing to use graphical
evidence for one of the limits
so the directive of the exponential functions of course in a message goes to
zero
into the experts say it reminds you to the x
divided by h
and remember that if you've got something the exponent
that is just the same as a product in the basis of this incident as h goes to
zero
into the extent speak to the h one z pbx
and this time we cannot be but we can talk about it but the contract ready to
be x
so this is peter the x_ because that doesn't depend on h
time setting the stage goes to zero
each of each one has wonderful diet in fact unit
formerly is actually uh... quite challenging took jing to compute
but graphical evidence suggests that when you've got the function he could be
a transponder by each new one
that then
you end up with a tough one and that is indeed the case
and that means that the derivative of the exponential function as one
okay so that the other end that sums up these different issue rules
first of all we have the powell rule
which tells us that frederick rationally exponent we have that's attributed with
respect to accept next to the are
art on six to the or minus one
now the derivation that we've seen only worked
for positive integer exponent but
exponents but it turns out
we have some more
uh... outbreak finance that were not going to worry about in this
presentation please
uh... you can prove this waiver rational number and then
with someone who she or you can actually proof a pre-trial expo
uh... we also have a theory that talks about how to deal with combinations of
functions here
if you have to functions effigies of differentiable at x
antsy is a real number
wells and the derivative off the sum of some of the derivatives
the directive of the difference is the difference of the derivatives
derivative off a product of the function with a constant is just
the constant times that reparative and finally the directive of the exponential
function is the exponential functions
these
results just summed up here
allow us to
work alcs
uh... derivatives
off all sorts of functions uncertainty first of all
the derivatives of polynomials
intensity relations so far
weren't quite your cup of tea well now it's going to get
really apply and up
that's certainly what were shooting for
mostly in calculus class
okay so
let's first take care of of the uh...
symbolic basics if we are supposed to find that are in a difficult function
like
four x q plus three over exporter minus five times certain graphics
we just say we have to find this derivative but realize that the only
time we can tackle correctly is the four x q
his second term in this third term are written as a power so we need to use
what we know from algebra to realize trio breaks square
is nothing but three times as to the minus two that's the only change that
just happen
and we also know
that five-time sisterhood of fixes nothing else but
five times dates to the third etzioni change that happened in this transition
you can see was that the forex cute
we have three or excluding the form three kazakhstan and too
and we have
the five times through the fix in the form
five-eights to the one-third if you really want to be able to translate
fluently between these notations
that will make you
that much more efficient
as you were doing your computations
what comes now are probably has too many steps in between and we're only going to
do it in this level of detail once but nonetheless
you realize that well if you've got a derivative off at some more a difference
well that's just the derivatives of the individual terms
with the appropriate signs in between
and we know that if you have to be caused him in front of the function
inside that are evident that the constant
can be pulled out
and that can happen for the four x cube that can happen for the three x two main
too
it can happen for the five weeks to the one-third
it now these terms here these three terms
can be killed without rules for the first term is four times
three x squared bracket princy
exponent down that's the three
copy the x
and then subtract one from the expo ninety new investing to expose
you do the same thing here you end up with three times
native to comes from the exponent
uh... you've got the erickson venegas two months ones make three
and finally the x_-two the one-third that's willing to deal with fractions
nato five times exponent comes down one third
plastics to me exponent minus one about one-third minus ones native two-thirds
and uh...
then we can just simplified four times three is twelve swing twelve x where we
end up with native six exterminated three at the end up with
minus five thirds exterminate two-thirds and if we want to go back
to the notation of how the problem west coast
then we would say this is twelve x squared
minus six over x u
minus five over three times for route
orthotics where it gets into non-air because we've got naked exponent
the excess squared because we've got a too
in the numerator of the exponent third route because we've got three binocular
expo
all right now we can find teenagers lines
are truly culpa corny are pretty complicated
functions for example for the polynomial f_ of x_ equals native execute
plus five x plus one day sequence one
it takes a derivative well that bs negative the current above excuses three
x where
the period of a five x
it's five
that's a bummer exit six to the first rights again the basic arithmetic
having one-time sticks to the zeros
which is of course just one
is a terrific up a constant is zero basically
you could say here is that one is one ten sixteen zero three one two zero
comes down twelve and what's your overall
okay how do you get the equation of the clinton plan but you need to soak up the
points that you play a game one
into your terrific to have all that's needed three times one square plus five
it's just too
you also need to point well air force one
is
may have won kyu plus five times one plus one so that should be five private
it now remember that
uh... the equations freight lines always quite possibly next class b
we have and that's the scope that's to yet expects one
and we have wife which is this five so we have to find equals two times one
class b
that gives as b story and that means that
the questions at him that's why you close to expose them
and just me as we have seen earlier presentations
we can just go looking at graph i function
and graph the uh... line that we assumed to be dictated by
as long as that front that tension shined the teachers barely touches and
he's tangential visually
to the function at the point ba point to compute the can't reply
everything's fine which it gets kicked
okay
we can also determine where
fairly complicated functions are increasing
with decreasing such as this polynomial three x two worth
class and execute
trust six x squared minus three
uh... certainly something that yet in principle we can work out the difference
quotient spent to be quite honest
i would not want it
so
what do we have to remember conceptually concept to be a positive during that
means that the functions increasing
and that is the red it means that the function is decreasing that almost
becomes a mantra as you points with this stuff but
will be remembered that it is true in it
uh... we just born want to bondage closely enough to our consciousness
uh... that weekend
rely on that as we do the computations and so will first thing we need a
secretive well
the derivative of three x to the fourth of four comes down and then you have
been executed that should be twelve execute right
that originally execute this weakens down giving us a twenty four
and execute uh... three minus one village to sell
we have twenty four x squared
other dribble sticks its critics who comes down giving us a twelve annex to
the two months one six so we get twelve x
editor at approximately three did zero suffolk at schools went
and uh... well now we need
zero strike because if we want to know where the functions increasing or
decreasing we need to know where it changes it signs
and that means first of all we want to factor on anything that's obviously
these terms you have a twelve x uncommon taken that out
xc request to express one because what's up with a quick break for me out more
this is actually a perfect square and i think i just
accidentally salsa factorization if you see that factorization because
weirded out make sure that it's right extras once where i guess that's expect
us to a plus one
but as soon as you know that you just have to factorization
and you find the zeros from the factorization by setting things he
quickly zero
well first-term zero index is equal to zero the second term zero when exiting
one
and that means we can now sample number line we have to cut points zero
and they have one in zero
and in order to find with agree that it's complicated we need to find
checkpoints one to the left of negative one
one in between
and want to the right let me if one
and i've chosen your negatives too
one
convenient points left for the ride it
well there really is no convenient point between native one in zero they're all
fractions
and that just shows negative one
now we need to find out this sign off the derivative apple these points
and uh... well let's see that
derivative it may have to
admitted to we end up with native to cuba which is negative
times twelve on the int
times twelve here's what that would be great if ninety six
uh... and then you get negatives too
times
twelve here
that gives you may have twenty-four sedated ninety six
uh... from here plus may twenty four for meeting the on twenty
negatives greatest four
well four times twenty fourth ninety six and end up with native ninety-six
deposit ninety six may twenty four
figures negative acne too
and that means a terrific is negative
on the whole interval from these infinity cleaning if one
similarity
at may have one have let me just p conceive of for channel one and then
they have one-half
personally of one-half okay so i think it will hear court and the answer
being negative
well i can plan meeting here
and of that
asap does work out it's it's of course a lot of acrobatics with numbers and
fractions
effect rest by factorization i can also cut it in here what twelve as positive
negative one-half is negative may appoint half plus one point these
credits positive
so we end up with negative
product that's made it different that's of course with these things is here also
and at one will be can go either way it
was easy confident he just blended into the gravity of this twelve plus twenty
forecast well i don't care what that is
it's unknown it's a sum of part of a number so that's going to be
so i really do with the trickiest
this and it looks seems into the derivative
if you're only interested in signing
and i uh... if you don't want to do with all the mental arithmetic it's okay to
plug that into account the guitar
if calculators are allowed on your test otherwise but use a piece of paper and
work it out
it's really not that bad each test methods just number
we can begin chick
uh... or we can now interpret this first of all
there if it is being made it means the functions decreasing for making anything
one
decreasing from may get one to zero
in increasing from zero to take
and we can also check that by just plotting to function
in comparing the plot
what we have to work out in our analysis it here's where it certainly campaigns
to have things
worked out
on a scale because there we can compare correctly and here we see
there was a potential change it make it one bit see there's all this out point
to get a
horizontal changed line but the functions decreasing before
and after because when you get one
instead at zero this
just superimpose it'll do and i knew that
it's kinda hard to see but at zero which include eke recent increase in those
exactly what the draft us so we can
we have practical verification
vast
we have the right resolve to begin this is
where i'm off a little bit preachy but i'm i'm are
not self-conscious about that this is a way to double check your work
without having to look at the back of the book
using an or you're using tools that you will have to as a working scientists
as a working engineer
as we can not on the tuition as a working business individual
uh... in so
that dennis certainly much more powerful than relying on
at the back of the book because well there will be no back of the book to
solve a problem on the job
we can also find out of the extreme off another brother argued to be polynomial
and again well honey five of the extremely first takes a derivative
city code is zero and then find
where you changing for increasing the decreasing with decreasing but
increasing so here
the derivative
of mega six to the forces in eighty four x cubes iterated before x cuba's twelve
x where the derivative of may of two x squared is made for extracurricular to
his zero
that's something that we're going to go over
uh... more more rapidly certain because these computations
should become second nature
we see that there is a four x that is in common to all these terms so and in fact
aimed at opening signs of this is an eighty four x times
exquisite
from here
and we've got a three x and a half weeks remaining from the realtor into one
remaining from the last term note that all the signs change because we talked
about talked about mister
now we want to set an equal to zero and of course when you sent me a four x
equals eureka hero
for the other one it's
definitely desirable to use a quadratic formula because two groups turned out to
be ugly
and remember
if you see a factorization quick the there's no penalty in inviting it down a
sandwich it's it's correct
but if you don't see it quickly
then you might as well use a product format and that's what we'll do it
so first of all
first
the first zero here is zero intent
forex great minus three x plus one equals zero usa clinic for mia
negative be negative negative three plus minus four be squared negative three
square
minus four times date which is one
time see which is one also
divide by two times one that so that ends up giving us three plus minus
square root off
uh... nine
minus for cisco router five divided by two-year replacements for life
divided by two
is that something buckeyes i've never stick out their survey cables were
captives are useful in if i want to know decimal for these numbers
i will go to the calculator and the numbers are
the proximity point three eighty
two zero and two point six points
now there are two possibilities we can use the second derivative test or we can
just do
the sign testing and the signs asking is a bit more reliable
uh... and so that's what i'm going to do here
so we set ourselves up with a number one we've got three cut points
zero
i think the point three eight two zero which is three minus with five over to
u_s_ two point six one eight zero which is three class with five over to you
note that this is not quite as scale because
but i need to squeeze the cops why did you know you can't do that if i
put it but point to
uh... approximately point four would be if this is supposed to be
two point six
but nonetheless i need for pat points here here here in here
convenient point to the left of the city's heroes negative one
convenient point between zero against three months for the fiber which weighs
one tenth of meanness
they're all pretty i'd be very convenient point between these two
difficult ones this product line
any convenient point to the right of two point six would be three
now we need to find out what ex-prime desolate primus pause it at that
negative won and thats
most easily seen by it
just nineteen eighty-nine yes
to make it once you get is a native once you get me to four times make it one
which is five meters one square
out once again dot with
maybe four times one does not
and here's what i think it wants not five it's four
uh... twelve times one instance
twelve cnf with sixteen intent
i don't care if that's made up of positivity five plus four months for in
sixteen customize for ends up being positive right
when we're looking at the one-tenth
well let's see how caffeinate als off one thing that want to look at the fact
reform your
wanted this positive may four th may disagree that i'm a fan of positive
things and we've got
wanted slated which really tiny
minus three tenths plus one well so that's a bunch of tiny stuff plus one so
that ought to be positive also sweetback negative positive positive
event
ends up being negative good
at one well negative four plus twelve minus for that ends up being four that's
positive
and at three
well at three hundred how this thank you is twenty seven
times negative four
ends up being a hundred age
three squared
is nine times
twelve ends up being a hundred nate also so i've made a barney fife articles away
we've gotten it four times three which is made of twelve ends up being
negative begin
in some of the derivatives part of native called negative which means the
function is
increasing decreasing increasing n decreasing again
in that unique and double check
that okay first of all
because we're looking for the relative extrema it's always good to just
answer that with the samples there about the maximize exterior equal zero and it
takes two week whilst
three class route five over to you because those are the places watching
from increasing to decreasing
at zero
and from increasing the decreasing every classroom fine over too
and there's a maximum at excellent equals three months route five over to
you because that's where we go from decreasing
to increase and that's what from trickles down and comes back up
okay we can double-check that again but just plotting the function
in comparing the plot
what we have worked out on that number line i remember this was not quite to
scale back
the point where things change or a zero here that's for a change from increasing
the decrease in net surf
very subtle about that maximum here but we can see that
this is a little bit lower than that so yes there is a subtle
both maximum there
then we've got the three months to fight over to the actually it was two point
three eight
we see there's a very upset about that minimum here if you don't see that well
enough go we have int
throughout this function and zoom in on the serial geographic capital you
computer
and then
at three thousand five over to we've got a very pronounce it very easy to see
uh... maximum and that's really all the extreme exactly half of dysfunction
causa
fourth-quarter polynomial comes from minus infinity
goes to plus and goes back to my sanctity of the leading tours negative
and it can be built around so many turn times daily commute but most three
turnaround points and we found them all
okay
we can also find refund checks concave up or down it's worth the fight second
derivatives and again that's something but wouldn't want to do with difference
portions
and what do you remember conceptually a positive secondary that meets the
functions call uh...
and maybe a second derivative means that the country's *** l
iso
we can take the derivative it's three x squared minus eight x minus one
just from here and that computations not straight
uh... power rule using it with some mental computation
second derivative is six x one isolated secretive
of three expert might say takes months one
and we have to set that equal to zero
which gives us that exists at those meetings over six which is of course for
thirds so this time we only have one cut points made it which is four thirds
check point to the left into the right to zero in
too most likely this time we plug that into the second-rate because we're
interested in
weather functions com gave up or down right
not interested in
rental functions increasing or decreasing
and all the way to remember that typically is that
these checkpoints
need to be cut into exactly the derivative
that gave us the cut points which is two six nine
six x one state in this case well at zero estimated eight which means
the second derivative will be negative primitive infinity to four thirds
intact to we get twelve months eight which is four so the second derivative
will be positive from for starts to infinity
okay that means the functions compete down
afro-americans in eighty four thirty complete thoughts at four thirty finity
if you want you can also write that out of the senate will certainly
and we can double check it again
graph the function and put
what we have computed analytically underneath
we see that the changes at four thirty didn't quite lineup right here that's
quite a pain to actually
make lineup correctly and i
just didn't have the patients spend another
uh... fifteen minutes to half hour to just make sure this absolutely denies a
property maybe it was just a couple of minutes past
it is quite a while to make these flights or didn't do it but we can see
our analytical results certainty is consistent with what the pictures
the functions *** down and then it is called kato backing away it's the
predictive power of mathematics
that we certainly are going to put a good he was in a row occasions account
finally
well begin we also want to work with parameters and and this is a sermon that
you for a couple of times
ultimately will work with problems that involve more than one theory or for
example you will receive this in your physics and chemistry class is
the universal castle
has stopped
five letters four of them of variables when dozens of constant
and if you work with the bondar wall sequence in which the correction to that
more variables to deal with
output is inaccurate more accurate model so we have to work with times
harmonic oscillators which a man singing physics the formula for the frequency
includes the spring concert and the mass which is that these two variables on the
right hand side
and the differential equation which you will see a differential equations class
inordinate advanced physics classes
involves three variables mass friction coefficient
the spring constant trust and unknown viking site which may be a function so
there's a lot of stuff out there
really
we need to be prepared for and that uses more than just one equals affix
terms of working with numbers remember i'd rather work with height and with all
those tickets
in dot
wealth and also in
making arsonists article two more problems
the graph of any quadratic function it describes this in our example actually
will talk about products functions when to stop a small white house
and uh...
in optimization we can optimize the serbs area rectangular box for any fixed
volume
rather than just for one number of that is given
in fact then aching gives us an optimal results that can be adjusted
if you want to have
volumes off
boxes of different volumes
finally
again that's remind ourselves even tho
the routine problems ultimately are sort of our our happy place in mathematics at
the stuff that works
and there's there's nothing wrong with being happy
sold certain problems because i mean what we
consider routine problems at this stage
were not routine problems when we were just looking at it
in terms of alpha finding extreme us not a routine problem in outwit many cases
in fact it is impossible
we are rapidly wrapping up our capabilities here
and so
higgins behavior to solve those problems that's good
it's it's opt not something to be purple to anything like that but it's always
keep our eyes on the fact
that there's more stuff out there
and so we cannot afford to stack
and uh...
edu routine problems and ultimately
we saw them with computers and what we do the program price in the program
okay so and finally we can also be deeper insights into sought by solving
our problems and for example here
this is a problem will resemble find the critical points off the function of a
pic sequels and extract of the experts say you'd you might like to say
i don't really know those problems because the projects for me to tell me
where the for texas
and burgess is the only critical point of these things
so basically another way to formulate these days
this
problem will give us the answer
why is it for kids four we have to work with
a far cry from quadratic is the way it
what we have to do and here's where i personally i'd like to get legal
memorize the critics for minority thread it is not good enough
the derivative opaque script as be extra cds
two-way extras be trying to keep the cost of the period of x_-rays to exceed
it to a x
the derivative all number time six is always just number so we can't just get
to be here and the derivative of a constancy of zero so that was away
gifts at city hall to zero
you end up with exp native the over to a higher the native becomes a within the
department to eight
that critical point is a vortex right away because if we're not doing our site
analysis we've got one qad point where things change
and here's where things get a little bit problematic because with different check
point to the left and check point to the right
and basically what i would do as i would just take may have been over to a minus
one
may have been over to a close one of those are guaranteed to be to the left
into the right
of negative be over to any
and uh...
if we what's on
pushing the wrong button
if we're now looking at the signing of the derivative in those places well to
the left
may be over to a woman to be over two eight
makes
this whole expression equal to zero
and if reasonable minus one in there we end up with
just may have to weigh so it's going to be
negative winning is greater than zero and positive bernie is smaller than zero
similarly to the right if we carry negative be over to any
that will make this one expression equal to zero then cutting the want me to the
extent of to any
into a u_s_ positive forty is greater than zero can't make it winning smaller
than zero insult by the way the derivative changes from negative
positive her face greater than zero in for a positive to negative a small in
zero
and that means it separates either way would change it from decreasing to be
increasing if age present iran from increasing to decreasing
if they s slogan zero that is in fact the pictures that you also have off a
quadratic if you have a negative leading coefficient the singles for stopping
them goes back down if you've got a positive eating coefficient function
goes down and then back up
so what we've got no is we have
a tool wool that allows us to computer images that we
in principle could solve with difference portions but
really i i would not want to do it
and we're going to put that tool to good use we're going to to
talks uh... in two separate presentations about grafting annable
optimization these are important conceptual topics that deserve to be
treated separately and then for the rest of this module we're going to just
derives more differentiation phoniness
demonstrate how they work in action
until then i'm back