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rhetoric that's one of those topics where a few op are comfortable with
kathy houston about keeping track of which by the surveyor which quantities
are fixed
into the fact that
on challenge comes in theory act of keeping track of once more
in in of some of the geometry that is involved because quite a few days race
problems turn out to be
geometric
obvious just
take a look at a problem that
is part off this
class of problems and then see
how we would solve these problems in general
uh... takes him a little bit
that is being fueled filled with hate helium
at a rate of two cubic feet per second and we want to know at what rate
the radius is changing when the body most of the newness three cubic feet
just make it has inflated to balloon at some point in time that mister
hemophilia
but one of the things
that we do know is when you go up a balloon at first the radius grows rather
rapidly and then it slows down as to the moon gets bigger
or king so
instead of
uh... all other forming ozone quantities in this problem will depend on time so
thats underlying variable that's the first thing
we have to knowledge intense track off
endowed with families of all this week the boy over sphere which is four thirty
hierarchy you'd that's one of those for me to ask you
if you've got to memorize it there there's uh... items now in a good way of
skirting the rising at four minutes so gap memorize it
and he's been ours to raise and be a subordinate doubles depend on time
and
ke insight to raise rates than is that people from tuzla increments
so
veal tea is equal to four third spy are off teach you cracked that's what it
would be if we
roadhouse completely an explicitly and ourselves a certain like what you would
do an implicit differentiation because
in fact implicit differentiation is undermined to address
island we have that
and we take to rehearse on both sides
well let me get through this on both sides
let's see what would you get that would get
dvd tea is four thirty five times three r squared
times the already t
that's the part that he is
always important that's the part that we have to get used to him
implicit differentiation that we have to sell more uh... multiply with the jury
verdict
off the inside and that's the case
okay kentucky ideas into five that that's for our spirit d_r_ detainee
which means really interested in the interest in at the rate
uh... interest in the rate at which the race is changing
so interesting d_r_ g_t_e_ so we sold for the army team that's
one of four pilots there
d g_a_t_t_
enough wealth that means now we can just in the numbers
three cubic feet is the volume grimsby behalf to find the rates too high
copycats if we're looking at work we have here
we need are and we don't have ok well
we can fit they can find
after a two-week feed us the volume volume is four thirty are cute
that means
ninety over four pi cubic feet it is
the q wilson radius and the radio sets the third rudolph
nine oh four point cubic feet which is the third molar
over four point feet
units workout good senate sanity check that our computation ought to be
andy reasonable so far
and therefore
well therefore do you like dat removal units three cubic feet is the already
tearing
radio series
that server at nine oh four four five times feet
which is well before choir radius squares
times twenty-two cubic feet per second and if you work that out that ciated
simplify a little bit
yeah okay well i'd deep cancel that to you
into the third route squared is an exponent two thirds opening it
is up
much more importantly
the square feet in the bottom cans with a cubic feet on the top into the end up
with
feet per second which is west
rate of change when i read it should be
endowed with them we also get a number its point one ninety ok yet that that's
just not
all right so what did we do here
and well determine the independent variable in the past time
we determine which of the involve the ripples for functions all of the
independent variable that was all of them beyond that had volume and ready to
get both payment on time
remark dennis fung shui only do that if
do really want to be sure uncertainty at the beginning at six recommend all that
you can't do that
and then we need to find a place and let's intervals to each other that was
uh... lloyd equation
if possible a picture should help well here you really couldn't draw a picture
will see a geometric example where you can just a little bit
and uh... they differentiate with respect to the in variable yourself for
that as i recall the date which
may well be a rate
uh... ending a substitute values for the quantities with values are known
and well that's all you saw over iraq's problems
uh...
the devil is in the into details that's just my optimization problems u_v_
have to think about
what kinds of phone if you have to use domestic iraq's informing us
if you have to keep thinking often enough to be resent the keys
informix
uh... but also note that
unlike fall termination problems there but he has no obvious typically working
with related rates mean an optimization problem
find the best find the maximum fine the minimum find the optimal that kind of
thing tells us what we have to do
formulated rates you know ki maybe the question that you looking for rages
tip-off
okay so
example let's take a look at sbi example we have one more time
in terms of this framework that we have most of the practice but know that i
spoke with union
at-risk to cuba keeper second at what rate disarray is changing manipal eunice
three feet three cubic feet
uh... in that state to a full thing
independent variable is time to me
dependent variables are
volume depending on timing readies depending on time
the equation is the equation for the volume
that there if it hits keepers what dvd two years
and result for the quantity
is that were interested in we get
any question for the r_t_c_
then we substitute values but note that we have to compute the panic before our
first
so sometimes you have to do some around for a
and that gave us the argentine equals
three cubic feet was ultimately the point one nine eight eight feet per
second week
uh... computed
alright so that was just a regular let's take a look at you metric example
uh... a six foot tall hands walking away from a nine-foot tall them post at a
rate of fight feet per second
how far system and shadow growing
when he is four feet away from the post so obviously that use a
uh... eighty eight
mathematical tolley problem
at the from situation certainty
is realistic
because as you walk away from the city three take your shadow is getting longer
and you quick wonder
uh... at what rate that shadow is getting longer
but it certainly is not my problem
so here's a guy bunch do is is walking away from the lamp post okay maybe he's
not singing arm
what we can see here certainly in this election s
the shadow
is getting longer
anis
guidance walking away from the lamp post because
rays of light
that are
blocked out by them out
hitting any
that uh...
tangential rhetoric and into the top of the year
touching to points
give a flatter and flatter smoke
as we are
looking at that situation
now the other thing that this shows is that you can if you can visualize this
stuff moving
that really doesn't help them so let's draw a picture
uh... so here's the flat ground here's the lamp post here's the guy
and the names of the shadows determined by the first ready off like that
bertie
hits the ground 'cause that's where the showings
in cells and the length of the show is this that it's up to you because it
depends on time
distance from the man to the lamp post that's guilty i guess for post
uh... then
we have to reminder sells minutes walking so that you keep the he is five
feet per second
it's increasing as we have seen
in the last picture
uh... ends
well that total distance from that
post to the end of the child will call that
off key because we want to talk about somethings
uh... sooner triangles
what else do we have we have a good man six-feet on females at the post is nine
feet tall
so now from the definition of the tensions
function we know that nine feet
to ***'s it's the same as six-feet
s right
sixty two s
is the same as ninety two x okay
ninety two x's ninety two px yiv team
it's up to you because you really don't know
how except the changes we just know that its peacekeepers as a team
in so
that's what they can play
calculus with now six feet as its nine p two p plus a s
that means as over six feet disincentives p plans as overnight feet
so that means s
over six feet months as over ninety days p over nine feet we solved
uh... what we brought everything with as to one side
and one six minus one ninth street b one eighteen so
s over eighteen feet east peoria nineteen feet
that means it's oft he is too intense d_o_t_ which also makes sense because
this hasn't been seen it and that has a lengthy list
that means the computation here actually it's very easy
it's the geometry that can get you in some of these problems and some concept
complicated geometry and
uh... well
reasonably complicated computations but
now when is is equal to two times days and e_s_t_ to use two times d_p_ dat
ind are
that use
beacause he's walking it
five feet per second
that de est e tedious
computer settings
and uh...
that's
the end off the presentation
uh... so
that was fairly short and sweet i think that it's similar to what you've also
seen in
optimization problems
technically speaking we haven't heard anything new here
renewed before we do is we knew or did you geometry and people really knew how
to do it as a different issue
but that's exactly the kind of stuff that is often
difficult in cathy's because now we just have to keep
what we know
synthesize it put together
to solve new problems
and then again that's also
exactly why do we need to pay great attention to these kinds of problems
because very creepy russ
for the real-life situation when we
ultimately have a job
because
uh... get training on the job training often is still part of it
but in order to actually make money for your employers the employer
campaign u
you don't get me to let you get paid to you
use what you know
made it into the patent that more content
produce something
that is in the which is exactly
this kind of situation
only that well
unless you're a teacher it's it's unclear whether u
would it take to solve in database problems
you're going to
do homework and then he'll be back for the next one i'll see you there