Typically where giving a function f(x) and then from there we take its derivative and
then we could find out lots of things about the original function or lots of things about
the derivative we can graph the derivative and all those sorts of things, so you might
considered that going forward. However, if you�re giving the graph of the derivative
you can actually reveal a lots of things about the original function just based of the graph
of the derivative. Now this is historically something that students struggle with a little
for this reason, the reason is students forget that this is not the graph of F this is the
graph of the derivative and so everything is a little backwards just for example this
is one quick example: right here you wouldn�t think to look at that�s and say that�s
a critical value because it doesn�t look like a critical value at all, however; this
is where the derivative equal 0 so it�s actually is a critical values for the original
function so you just have to be very aware that this is the graph of the derivative and
not the original function. So let�s take a look at some typical question that are asked
giving the graph of the derivative. First step: where is the original function increasing
and decreasing very common question very typical, so looking at this let�s just remind ourselves
what does it mean for a function to be increasing F is increasing if it�s derivative is positive
right. So we�re just going to look at this guy and we are going to try to resist the
temptation to look at where this graph is increasing that not what I�m asking about
I want to know where F the original function is increasing and that where the derivative
function is positive meaning its Y-value are positive. So looking at this graph it look
like it where this yellow highlighted portion is from here to here that where the derivative
is hanging out above the X-axis its hanging out up here not down here. So from here to
here is where it looks like its increasing. Now let�s throw some numbers in here just
so we can actually write some intervals down let�s say this is the interval from we just
make these up maybe -2 to 5 let�s say. So we would say that the function is increasing
F(x) is increasing on the intervals from -2 to 5, now they would give you those numbers
that not something they typically do I just didn�t write them in before I started the
video they will have these X-value plugged in for you. So it increasing on that interval
now where is the original function decreasing well (let me change colors here) that would
be obviously where the derivative is negative so where ever F-prime is negative or less
than 0 then the original function would be decreasing, well the derivative is hanging
out below the X-axis down here and down here, so that will be from � infinity to -2 union
with 5 to infinity, again I got to stress the original function is decreasing. Alright
a couple other typical question where does the original function have maximum and minimum?
Well let�s just think again what does it mean to be a maximum or minimum? Well the
first derivative test which you probably remember says that if your function changes from increasing
to decreasing then this X-value right here in the middle got to be a maximum because
the graph goes up and then down with a critical value in the middle which by the way a critical
value is where the derivative will be 0. So let�s go back to this graph up here and
I think we already identify this part that -2 was a critical value because the first
derivative was 0 there because it went through the X-axis and 5 was a critical value because
the derivative is 0 and then check it out at this first critical value the original
function goes from decreasing to increasing because it�s slope goes from negative to
positive, so decreasing to increasing that mean that this guy will be a relative minimum
because the slope goes from decreasing to increasing, negative slope to positive slope
and then positive derivative to negative derivative indicates that 5 would be a relative maximum.
So we have a relative maximum at X=-2 and a relative minimum at X=, did I get those
right, I�m sorry I got those backward, relative minimum at -2 and a relative maximum at 5,
min at -2 sorry about that, minimum -2 and maximum at 5. So one thing that unfortunately
we can�t do is often times when you want the relative min/max your instructor teacher
will want the X, Y coordinate well we got the X but if this is a graphical type example
there�s really no way to discern what the Y-value for the original function is it -2
all you know is that -2 is a critical value but you don�t know how high is high or how
low is low so typically they will just ask you just for the X coordinate for the relative
maxis and mins. Alright now knowing all that let�s see if we can graph the original function,
so we take what we know so far, we�ve got a maximum at 5 and a minimum at -2 so here
is -2 and here is 5. But remember there�s no really way to discern what the Y-value
are at -2 and 5 so if this is simply just going to be a sketch of the shape of the graph
and so I�m just going to put a minimum just somewhere at -2 and I put a maximum somewhere
at 5 (something like that), so I got a low point and a high point. Now let�s examine
just for a minute before we go any farther because you might be asking this question
in your head, why can we not be more specific than that, well think about it if this is
the derivative then if we can somehow magically find the original function it would be blah,
blah, blah, blah, blah, blah have a bunch of terms and then it may or may not have a
constant on the end could be +7, -2, +3 whatever, what would happen to this constant when you
take the derivative what will happen to this guy, well it would go away, right it will
go to 0 because it a constant. So if this constant happen to have been +7 and your graph
was up here you would not have known that because it went away with the derivative or
your graph could of been here you know could have been like a +5 could have been down here
could have been -3 you know we don�t know. So basically what is going to happen is we
will get the correct shape of the original function but it can be high or low somewhere
in the middle and there is no way to know because with the derivative all those constant
would of gone away that made it fluctuate in height like it did. So we just going to
put a just a generic low point and high point. And let�s see here it decreasing up to -2,
it increasing from -2 to 5 and then it decreasing after 5 okay I buy that. So we go decreasing,
increasing and then after 5 it would be decreasing again okay. Now in hind�s sight this makes
perfectly good sense the blue graph looks roughly like a cubic equation like a degree
3 equation and the derivative look like a quadratic equation and so parabola that upward,
downwards and that quadratic is the derivative of a cubic function so everything seems to
make perfectly good sense. So this blue graph is the original function where the derivative
graph is shown in green. Now that�s pretty much the majority of what you�ll be asked.
Now I do have one extra little thing that since I already got the work shown, so you�re
welcome to go on to the next video but let�s just hang out just for another second. And
then ask what�s would the point of inflection be and can we tell that? Well if you remember
that where our graph changes from concave up to concave down so we can cheat a little
bit and just look at the original function and see where it changes from concave up to
concave down or vice versa. So right there right here would be a point of inflection
on the original graph at this X coordinate. But let�s see if we could of revealed that
from the derivative (let me erase all this stuff now, now that I�m done). What on the
second derivative could of help reveal that we had a point of inflection right here, well
notice that�s where the derivative of the derivative in other word hint, hint the second
derivative that where it equal 0 right, and the first derivative went from increasing
so that the derivative of the derivative in other words the second derivative was positive
indicating that it was concave up and then it change from concave down. Now did I say
that this graph the graph from F-prime change from concave up to down, no that not what
I�m saying, I�m saying the second derivative went from positive to negative indicating
the original function went from concave up to down. So sure enough we found a point of
inflection as well I don�t have that listing as an X-value you know it looks maybe somewhere
right around 2 or something close to that but anyway right here would be a point of
inflection. So anyway, hopefully that reveals kind of explain a little bit how you can read
or reveal things about the original function base off of the graph of F-prime.