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okay well after we have spent a considerable amount of time uh...
burning how to take a religious we're now going to end this module
spent some time learning how to untundi riveted stand and where that at first
may sound
and see if others is mathematics for the sake of just beginning more mathematics
it turns out that as with everything in calculus
their is a physical reality
underlying this process and we're going to see this physical reality definitely
in the last example
off this presentation already even though the example
s with
similar examples enrique
uh... just worked on taking derivatives
is to be artificial because it has to be a little bit passengers and
full reality of the situations that you all to begin concrete physics and
engineering for example
so what's the idea
for indefinite integrals the idea indefinite integrals were easy to read
it is the idea for those things is that
to recover a position function from its philosophy function because to the loss
of use
derivative off the position we must formally undo derivative so in the last
example as well as in the first point on this night
we are talking about
the physical reality
behind the whole thing
and in fact in physics intact in kinematics
one of the standard things that you often have to do is you have to take
most of you who are acceleration data
and recover position functions from zaire
uh... and and uh... as again quite realistic for example for the naked
navigation off a space for this als
uh... you know g_p_s_ healthier uh... you can't draw maps or anything like
that right space born vessel again looking at appropriate
goes to another planet that things just enough warning
uh... and so would you can do with the onboard computer if you can keep track
what kind of acceleration you went through it what stage it's how much
few was ejected at at what velocity to do what kind of change
india
but also the function off the object you can keep track of the gramm patient
feels awful essential bodies that are around that
in front of that
yuki and drop the course
that kind of undoing derivatives on one hand it's it's a quick abstract
uh... problem too
predetermine
and of course of of
odd sale vehicles are being launched into space but also that
to make adjustments
is that the eagle continues to travel
so that means we need to undo the revisions
in simply put this undoing derivatives is what symbolic integration is about
uh... integration has also associated with the computational periods we will
take care of that
in separate modules
off first just the area and one molecule inventor
connection to the editor attitudes and areas
will be taken care of in the market on the fundamental theorem of calculus
uh... so
underway derivatives what does that mean we are given a function and that
function is actually a derivative so it's expect crime known it
in we're supposed to find the original functioning
uh... symbolic integration then also isn't useful
reading skill for applied classes because in unlock off applied texts
you're supposed to solve integrals into text just goes to use the n_ double
here's a solution
in there then of course the wonderful why is that the implant just
being on top of some symbolic integration
will allow you to
reads that kind of thing with a lot more confidence and and get their connection
to the material
uh... so
this is similar to actually after a being a fundamental reading skill
for any kind of mathematics based class
strong rerouted raise and i hope it's
certain because reinforced nicely
through what we've done with differentiation but the scrawny
robberies unless you have to worry about it would be easier if the rest of the
class becomes
uh... presentation here shows the underlying idea and some simple
integrals and offer the presentations will then consider
more complicated functions up where where
pretty much going to verify when teaching system once told me and that is
differentiation is technique
integration oxley is art
so let's see what do we have the definition is that if you have a
function yes
very different branches will function capital as it's called an integer edible
lower case it eternal youth
capitalist runs in our case
what we will see right away yes they will not just be one that you could read
before you can function for example
when you have the function of how they sequels to extend the tron off the
functions info fix equals
xc where capital equipment people think squared
capital g objects being expert class
and capital h ethics being expert months nine all
three of these functions are into jury advocates of two x
and how can we
how can we check that we can just take the director of each one of them
as a derivative of each one of them is to exhort double-checking but always
just being
that we take
derivatives
uh... but that means now
that uh... even though multiple solutions to a problem are not new to us
you've seen that for example that the product for leader
infinitely many solutions may feel name
because after all you can add any number here and the corrective optics quick
lesson number will be to excel
we will have infinity many possible solutions that make you look at me and i
think the only place
in d_o_d_ curriculum where
you may have seen this for sure response solving systems of immunity
he questions
specifically to payment systems of tn of linear equations
so
either way whether we've seen it or not whether we thought we'd like it or not
we need to figure out
how about this and that he would he can be there is a theory that tells us how
bad it can be entities that
if you have a functions it is to find an invalid the interval is definitely need
it but it's just an abstract hypothesis here
uh... and if if capitalism capital g are integrated circuit
then there is a constant sees such that one eighty directly to the or the other
direct anti-competitive
class that constant
for all x
inconvenient reply so
we've seen out here
that the ambiguity in this example was it could be good at constants
this the around down here tells us now that we d
adding archery constance
use all that the ambiguity is going to be
which is typically assuring uncertain
we can define the indicative of a function f and that's supposed to be the
family of all integer disappear
in easton honored by integral f affix d_x_ and uh...
the reason for this notation
will become clear
once we've got the fundamental theorem of calculus but as i said
that will be
another module in this module
we're going to focus specifically on the physical idea of recovering
positions from velocity
if we're going to focus on the techniques how to do that
uh... the function if it's called the integrins
the very latest called the integration variable
and i do think it is the final an interval and capitalism editor and have
a bit then begin definite apple is just given by
the editor of it it
plus see some of the contest
undo
derivatives have been at our tree constantly can find
indefinite integral
and we've got a lot of vocabularies every so often you run into
when you talk to people about this stuff museum to print and the integration
period
okay the constancy is also called the increased constant select couple carry
on this panel
and uh...
before we actually talk about symbolic into derivatives i want to visualized
what disintegration put this into different ciation past
and about visualization to elicit direction at the direction field
consists of little line segments also ethel thanks at our place
at the point ex-wife remembers that when we undo
a derivative so when if it's not clear evidence that if just gives us
slopes off that function capitalists and so
at the point is why it's a function catholic goes to the point xli
it would have to have so careful experience of for just
placed those line segments that we know roughly
how the integrity of should be shipped
the direction if you don't think you'll be used to schedule is indeed a
religious
indulges geometric idea can also be used
to devise a numerical methods and to be precise it's called or dismiss it
which is used to to solve certain differential equations
on directly if you're interested in fact
check out my presentations on differential equations which can also
find all from my home page
uh... within a direction field
we could also visualized on how the ambiguity of prizes but that's another
reason to present
direction kits right here
and so here we go
sketch that russia feel for the phone to ever mexico's eggs in the window native
to to cross next to two
fascinated to any attacks is needed to two on the way axis
and we want to use it to sketch the editor of off the function f
so that capitalist zero is equal to zero
that's getting back and take a while the first set up the axes and then you
sketch the direction field
hears of course where
i'm using a computer and if you've got software thought direction fields
and
it's okay for you to use it as key teacher about that but if it's okay for
you to use it for goodness sake
sketching on both sides by hand
takes quite a while
what does that direction fueled it
but are functioning days ever of a sequel six one one x is equal to zero
that means we get more as well
nine silk zero
isn't increases first of all for every x one of the same soap because assoc sworn
to depend on extract
in finance x increases ourselves with its people one weekend
so one it's the one
and as excuse maker when we get negative slopes that gets deeper and steeper
as we're going to the left side so that's what this looks like and that
sort of already took shows us that
any kind of function that follows these things
and these are really like
uh... velocity vectors every tunnel or so a part of the quick follow these
philosophy
would have to
get this kind of a parabolic shape
if we want this case the integrity of
with efa zero equals zero
physically would really just put archangel right here in ten
code horizontally and realize that well now the slopes are getting more and more
positive
and so we would get a sketch that
goes out like this
toward so that we'd also caught between a low sixty s work one right to left
also calling up like this and we're getting a curve that
looks like that
and in fact well that's what it is and of course
i'll use my computer you can get a better pictures and what i could get
if i bored too
use my moms to cross that bridge quite painful strike that
and i
don't have a tablet pc here which probably wouldn't work for me either
because my hand clapping is quite terrible
let's look at an extension of this problem and simply draw some more into
derivatives
is basically if we start hero
will get
the same kind of parabolic shape because we received the following the same kind
of
directions fran it there has to go to hire a evidence that he would get the
same check with a few week at the same shape with down here here here
and here we always get the same shape and that's a visualization
off the integration concentra mentioned earlier basically
your function gives you the direction field the directive determines the shape
of the integrity of and then the integration constant
just pushes touching pop org
we thought we could not talk about some as an injury
symbolic into derivatives
eighty differentiation formulas are basically differentiation for me as red
backwards
so what's the editor of a vexed indian wells returning to power
is
that same power with the exponent reduced by one well okay so if you under
withering it if you have to add one to the exponent
and that's what happened here
that they have to make sure
wind you differentiate this most essential part of the editor at that you
actually get back with you can't hear integrative affixed to the in plus one
is in class one time thanks to the end
well that's not what we have here so we have to put this additional
multiplicative constant wonder bread cost one here
just like i said
to verify
in integration you just takes a terrific to of the integrated circuitry beautiful
dysfunction videos xd to the incident with sas
isn't something that almost becomes a reflex
typically
that classy here
at
uh... yeah that works for a north equaled one
immediately because he had to look for in not people make it
okay we're back sorry about the interaction and i'm getting better ask
catching my typos and and fixing them from just a quick recompile
this should be
uh... form you know this is a format that works for in not equal to negative
one because of course
if we have to make it one here
we would have a one over zero and
no matter
how much mathematics you know the latter
how sophisticated tools can get by for example done mathematics and post happy
to settle for over twenty years now i still can't divide by zero
and i never will be able to come
all right let's move on and let's hope there aren't any
for other mistakes
if we can do this for a week will make it one well then the next thing to take
europe of course
is
in equals made a point of the remember that that yes
one over x indian and the derivative of the law whereas most natural rhythm is
going over ex-soviet affixed to the negative one bs
annex
classy
uh... the editor of the exponential function with the exponential function
always comes back when you differentiate so the editor of it is just need to be x
plus the integration constant
began to drift of off the site function well let's see the dribble sinus
croissants and red hook wilson is negative sign
technical backwards the editor of double the sign function is the name of course
i'm
nuts because when you take the directive of the native was and you get negative
negative side effects and
immigration consummated
integrated with the course and function is assigned function right real-time
school site
and then we've got a couple that just basically come from the fact that we can
read
integration format is backwards
begin to drip drip of one over square with one minus x squared
that should be the ark signs of sign inversely harks onyx club c_n_n_'s
because
this is the derivative opr sign
the editor of the vote one over one plastic square wells attributed
authority engine is one of the one plastic square through texas
our attention to express the integration constant
and theatre at one of a croissant square ethics
the derivative of attention to sort of abortions critics of this should be
teenage next
schatzi
now the winds
that are typically remember clearly are one through six now that doesn't mean
you're guaranteed that you're never going to encounter those it's just that
arguments waited a little bit more when you encounter those and
that has happened for me too
i can reorganize them now because of course i ve
taught this class a couple of times an ample supply places where you encounter
that the key events
that's these last two are the kinds of formulas that
we'll give you a little bit of a pause because you don't use of that often but
nonetheless we need some tipped
there are also some
formulas that simply come from
how derivatives behave namely the integral to sell mister to some of the
integrals
that comes from the fact that the derivative of the sun is to some of the
greatest
center for differences the end of the difference it's the difference of the
integrals
and if you multiply function with the constant
you could call the constant out of the integral into at the right crisil
these are the
uh... main rules and so you can see the
at this stage it's nothing but
going backwards in differentiation and so when we do that we can now we've got
some examples for example we want to compute the ankle
of to execute
minus three x to the two thirds
classics to the minus one
and again i'm going to put a bunch of stetson u that u probably will not
always right out in fact already backed out but that way
will see that every single step is justified so i think that will offer
some
smash difference here first of all it breaks up into individual integrals
enough to execute what we can also pull out the two
mind is seeing the glove three x to the two thirds or we can also put the three
plus the innova extras and one
and that becomes too much on his eight years we're using this reverse power
albright
industry so one of rent plus one this one fourth
into the entrance one-sixth of the force of the it two times one for the extra
points for the first one
uh... that continues here where of course now when we have to deal with
fractions it gets a little bit of mir
we keep the minus three
and wanted the two
to the two-thirds in the exponent well two-thirds plus minus five thirds and
one of our five thirty three fifths
times x to the for theirs
so that works out with reverse power rule
and then here we could be tempted to ask
wanted negative one if that happens we get zero in it's a warning sign because
we can't
divide by zero ins and hopefully will remember our yeah
aidid redrew accident one is the natural law criminal x so that's what we should
get
remember to put the class sea and air that's a stamp one point deduction on on
many tests
uh... indicates he's just a nuisance factor right
is just not right about the plan c
uh... in the last get then is that we can simplify a little bit will too
tool forces one-half the x_-two the with states saying
three times three-fifths of nine th at six to five thursday to same address the
onyx club scene
also stayed the same inside is called me yet
for this kind of computation
now we can do the same thing when functions are given in a slightly
fancier symbolic form what we've got the info too
time secure ethics class five overruled next month's who signed a cs
is really the first step then just like
in differentiation is that we want to take this phase imitation
intelligent up hours because even though the powers may not be
esprit de
the powers are what we knew how to handle
so this becomes the envelope to thanks to the one-third which is what the key
word is
five exterminated one-half was dividing by square root the division gives us the
negative side of the square root
is an export one-half
the minus four senate support states saying
and now we pull this apart and we do this just term factor and we keep the
two
one-third plus one it's four thirty which is the exponent and one of a four
thirty three floors so we end up with
mister u
odyssey native one half plus one x one half so we keep the five and one of one
half is too as we get five times to engage to the one here
that's the export of the picket protest adding one to the original exponent in
the interactive of negative course annexes
negative sign off it's because the editor of the old wilson's sign
classy
is then again the only thing we can do your simplify a little bit yes three
aspects to the poor thirds
two next to the one-half
minor side effects plus the and if you wish
you could also true this back
into this notation as you can see here i was a quick trip with ideas that are not
obv i didn't and
unless it's specified you might as well be that form i think
uh... so finally
promised example that talks about recovering
positions
from velocity data as i mentioned
in the introduction
well so here's an admittedly made-up problem give credit card moving them
straight road with philosophy view of t_b_ twenty cubed
plus five nights a card at six on a range seriously because
given all city is going up with a cube of the time green at something
cannot be sustained very long
quite sure how he would sustain even for a short amount of time
but it is something we can just do the computation so yes it is my
we think that the position at time t equals zero it's sa zero equals three
and i will tell us a little bit as to how we're using the integration constant
and what we want to know is that
uh... position after
four times units that's assume that all that
times are measured in seconds here that we can talk about
times in a more natural thing
m_-one african interest times units
uh... will
the position is the editor of a possible also days so we need to find the
integral of to teach of this error
class five d t after all the obvious stuff we've done the previous panels
that's not too bad
uh... integral begin to go off teach cuba's one forty two the fourth times to
was one-half to to the one fourth
the in the gulf five
well the into both a constant is just a straight line city of a fight this
feisty
uh... and that's because five years if you want to do that with powell rule
five is five times teachers zero
zero plus one is going to get into the one of the wonderful one uh... just
is once we have to write that down and we have to of course also have the
just seen here so that was simple intentionally
cause this problem to solve this more about what the integration constant
that's because now we know
that three is the position at time t equals zero
and that has to be equal to what are we getting here breaking here's one have to
go through the four th
plus five times the replies c
which gives us that's used equipment three
you are
function here is not zero
point where you have they given then you'd be racial constant need not be
the number of that is given here from the sprawling just
happens to work out that way
by the way the position function is one-half teachers of with
class five t class three
now look at the problem when what i mean the actual task was
to determine the position
at time t equals four
well that would mean we have to find sa four which is one here for to the fourth
class five times four
class three let's see if it works out
for the fourth this foursquare times foursquare
was practiced whispered it's sixteen times sixteen which is two hundred fifty
six
to a fifty six tonight but to listen to any age
five times force twenty slide twenty eight
plus twenty five forty eight class three hundred fifty one
all right i'd instills u
mental arithmetic
that israel should worry
you know when i i i i do this kind of study every so often just two also
reinforce that yet if we can do something mentally it's a good exercise
to do that
remember
ultimately what you want to do is to point what you want to do if you want to
get these problems right
in uh... you can be sure is that
even though i was pretty sure of my wrist and take
i have also has a voice
making peace lies on
pretty sure i remember checking there's also the cabinet
just to be
absolutely on the safe side without we've talked about an entry into
derivatives we've seen geometrically as well as symbolically here
what the integration constant can do
uh... from here on out
it's going to become a bit more challenging one symbolic level because
undoing derivatives
involves a lot more power recognition
then computing derivatives
but we'll get to that in the next couple of presentations
i'll see you