hello and welcome to my secret math
tutor this video will work on
approximating the instantaneous rate of
change of a function a single point
this time however we'll use our ti-83 or
84 calculator to help us out in that
process so to find something like the
instantaneous rate of change of a
function what you can do is actually
pick two points and then find the
average rate of change between those two
points or you know what we call the
slope now that won't give us the
instantaneous rate of change but if we
move those points closer and closer
together and repeat the process it'll
give us a good idea of what that
instantaneous rate of change actually is
so we will use the calculator to help us
out in this process is we can actually
store our function in we can store in
the slope formula and see what happens
as we're moving one point closer and
closer to the other one this will help
us determine what our limiting value is
and give us a good approximation to that
instantaneous change of a function makes
this a little bit more concrete let's go
ahead and do it with this example
problem so here I have the function x
squared minus 3x minus 4 and I'm curious
what is the instantaneous rate of change
at the point 1 negative 6 so let's go
ahead and grab those calculators and get
started
all right the very first part of this is
we need to get our function into the
calculator itself so go ahead and press
your y equals button it's near the top
here and let's go ahead and type it in
so I have x squared minus 3x minus 4
enter all right so now our function is
stored into the calculator itself and
just for our naming conventions you'll
notice that it stores it under y1 that
will be important for later when we're
making our slope formula so this is
really our y1 function renamed now the
other part of this is we need to
calculate the slope between this point
and some other mystery point you can
consider the slope formula like the
difference between the Y values and the
difference between the X values so I'm
going to use the notation of the
calculator to get a better idea of
what's going on here so wire function is
y1
and I'm going to figure out what is the
difference between the output of the Y
values over the X values well I know
what happens with one of my Y values
that happens when I plug in a 1 into the
function itself and I'll leave the other
one as X because that hasn't been
determined yet it'll be my other point
that I end up choosing and down below
we'll have my X values so this slope
formula is what now I want to program
into the calculator and we'll go ahead
and put that under Y to start this off
I'll go ahead and use a pair of
parentheses and then we need to put in
y1 of 1 to do that you'll go to VARs
arrow over until we get to the Y
variables make sure the first one is
selected function and select y1 so you
can see now that I have parentheses y1
will open up another set of parentheses
and put a 1 on the inside there so what
I'm telling the calculator literally is
to take one and plug it into my function
now let's go ahead and repeat that
process - and we'll put a y1 and we'll
feed X into it so again that's under
variables arrow over to Y variables one
for function and y1 only this time we'll
open up the set of parentheses and just
put an X on the inside all right so I
use the set of parentheses when I start
in this process let's go ahead and put
another set to close the process they'll
take care of everything on the top of my
slope formula now divided by let's go
ahead and open up another set of
parentheses for the bottom here we'll
have 1 minus X close parentheses so
we've programmed in this nice slope
formula we've programmed it in as y2 and
now we can feed it in different values
and see what we get all right so for the
feeding in process we need to get back
to the homescreen let's go ahead and
press 2nd quit and go back to that home
screen
perfect all right so the point that we
are interested in is the point 1
negative 6 and I'll choose other X
values a little bit further away to see
what the slope is through those and here
are some of the other X values we'll end
up choosing and see what we get so to
feed n values into my slope formula now
I need to go back into variables VARs Y
variables one for function and we've
stored that slope formula into y2 so I'm
gonna arrow down make sure y2 is
selected and then press Enter
all right it looks like the first value
I'll feed is to make sure that when you
feed it values that it's also encased
inside parentheses all right so let's
press Enter it looks like we get a value
of zero so what's that really mean it
means if I choose the point at 1 and I
choose the point at x equals 2 and I
find the slope between them the slope is
going to be 0 so now that this is on my
screen let's go ahead and feed in 1.5
repeat those process of variables Y
variables function make sure Y to open
up parentheses 1.5 enter okay so now I'm
getting a different one this is negative
0.5 and you can repeat that process for
the rest of these values now the neat
part is if it's already on the screen
like this I can shortcut the process I
can go 2nd enter and then we can change
this so that I don't have to go through
that variables menu and make a little
bit quicker you know all right this
one's that negative 0.9 second enter 0 1
negative 0.99 and one last time let's
see 0 0 1 close parentheses negative
0.999 so you can see the calculator is
quickly finding the value of the slope
between those two points make it nice
and easy on me and now all I have to do
is look at these slopes and say well
does it look like they're getting close
to anything does it look like they're
getting close to some sort of limiting
value all right so let's take this away
all right for our slopes as we get
closer and closer to 1 the slopes looks
like they're getting closer to negative
1 so I will say that that is my limiting
value and it's my approximation to the
instantaneous change of the function at
that point all right so again the
process for this is really just choosing
two points finding the slope between
those two points and then moving them
closer and closer together and
fortunately with the elbow of the
calculator you can move those points
closer to one another fairly simply
making it nice and quick if you'd like
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