BAM! Mr. Tarrou a little birdie told me
you were having trouble sketching
derivative functions from the graph of a
function well my students are struggling
with this too and I'm gonna do so I'm
gonna do an extension to a video I did
years ago I'll have a link to that in
the description I think it was this
little birdie that told me you were
having trouble and we're gonna do five
more examples of looking at a sketch of
a function and trying to draw a
reasonable representation of what the
derivative would look like and remember
a drip well actually the fourth the
fifth example were going to work in
Reverse we're gonna look at the graph of
the derivative and try and sketch what
the function may look like okay so we're
gonna look at some settings that are
more complicated than I did years ago in
my previous lesson and we have to
remember that a derivative tells us the
slope of a function so we're not looking
at just the graphs the function f as a
line or a parabola or a cubic function
like I did years ago we're going to go
into these more complicated settings and
we need so we really need to remember
that not just kind of shortcut this
process and go well he gave me something
that looks like a parabola so the
derivative must be a straight line
because you're going from a second
degree down to a first degree well here
we do have a function that does appear
to look parabolic in shape but then when
we get to x equals two
it just branches off and we have a
linear function so maybe this is the
graph of a piecewise function well what
will the graph of the derivative look
like number one I like to try and
identify any points along the function f
where the slope therefore the derivative
derivative is giving us a value of zero
well certainly if we look at some
tangent lines here and think about the
instantaneous rate of change we have a
horizontal tangent line for this
function at x equals negative 1 so as we
sketch a reasonable representation of
what this derivative should look like
will certainly the derivative which
gives a slope and the slope here is you
know horizontal or the function seems to
have a horizontal tangent line you
better make sure your derivative goes
through the point negative 1 0 now what
happens to the left
of X equals negative 1 while the
function is falling and if the function
is falling that means that it's slope is
negative so the derivative to the left
of negative 1 needs to be below the
y-axis or the excuse me the x-axis given
you a representing that we're getting
negative values of Y because one more
time the answer from a derivative is the
slope and it seems to be helping for me
to you know force my kids to not look at
this as one complete picture but break
it down into pieces so let's say we look
here where X is equal to negative 3 you
know what does it seem like appear that
you know the slope for this graph here
is at x equals negative 3 well if I kind
of just very lightly dot in what might
appear to be a good representation of
what a tangent line would be it seems
like at least at x equals negative 3
that tangent line has a slope of
negative 3 over sort of 1 so at x equals
negative 3 my derivative should be given
me a value of negative 3 now you know if
you say the slope there is negative 2
and 1/2 or negative 2 I'm ok with that
we're looking for a reasonable sketch of
what this derivative may look like so
we're going to be going you know and as
we go here at x equals negative 4 the
functions falling faster giving us rip
you know which B should be showing a
more negative slope and it's negative
and it's still falling so the slope is
still negative it's still falling but
not as fast so the slope is less
negative until of course we get to that
x equals negative at x equals negative 1
where the slope is 0 and then as we move
to the right of negative 1 that function
f starts to rise and of course if a
function is increasing that means its
slope is positive and it's rising just a
little bit and then it's rising a little
bit faster and a little bit faster and
so on and so on so that derivative
should be giving us to the right of
negative 1
ever-increasing positive values and if
there's some symmetry here and I've
drawn this somewhat accurately if I were
to visualize a tangent line here at x
equals negative one maybe we could
approximate that that slope is well my
drawing is not perfect and maybe that
wasn't quite negative three but maybe
two two and a half of course you know I
wanted it does look sort of parabolic in
shapes we're gonna go ahead and let this
derivative the sketch of what a possible
derivative would look like continue to
go up in a linear fashion I'm again
while I'm starting to realize it's
helpful to let my kids ask tell me what
the slope is approximately at certain
points we kind of you know need this
picture to somewhat look neat of course
and we get to this issue here at x
equals negative 2 well it x equals
negative 2 our graph really changes
direction and has a sharp bend well
don't forget you can't find a derivative
at a sharp bend or at a point of
discontinuity so as we approach x equals
positive 2 at that exact value of x
equals positive 2 at that sharp Bend our
derivative is going to be undefined so
you want to indicate with an open dot
that at x equals positive 2 open and say
negative 2 earlier that is undefined now
what happens as you go to the right of
this sharp bend well we have a linear
function we can clearly see it just up 1
over 1 up 1 over 1 which means that the
slope of this piecewise function the
slope to the right of x equals 2 that
stays a consistent value of 1 so I'm
going to come down here and remember
that derivative tells us slope I'm going
to indicate an answer a y-value of 1 and
oh it's 1 all the way so our sketch of
our derivative should look something to
that effect and that is the sketch of
the first derivative now you know do we
have the exact proper you know angle if
you will slant on that oblique section
of the
grant oblique you know it's not
perpendicular not vertical I don't know
but it's a reasonable approximation if a
student you know had a line coming
through here and maybe just going up to
a you know a y-value of three instead of
my y-value of four four and a half I'm
okay with that but making sure that we
are are passing through the x axis at
negative one representing that slope of
zero to the left having those negative
slopes to the right having those
positive slopes having the open dot
showing us our derivative is undefined
at that sharp bend I'm good with that
now if I want to draw another step of
that derivative if this is F and this
really indeed is a you know linear
function it seems like for this portion
of the graph over here our slope is a
little bit less than two over one so the
slope of this derivative this first
derivative to the left of x equals two
seems to be consistent let's say almost
two so I'm going to have it just below
that's Y value of two it's still on it's
still undefined here and you can't take
a derivative at a point of discontinuity
so I'm going to keep that open dot there
and to the right of x equals 2 we have a
horizontal line so the slope will
horizontal line is 0 so our second
derivative is going to look like this so
actually this I just did two examples
there one over here I got a function f
which has a vertical asymptote at x
equals zero and way over here you know x
equals negative 2 4 6 8 ish unless I
miscounted it seems like look that graph
is almost horizontal it's just staying
above the x-axis and rising very very
slowly and let's see I don't really have
a set of markers like it would in my
class but we have a tangent line which
has a slope nearly zero we have a
tangent line which is got a slope of I
don't know maybe one ish a little bit
less than one
I have a tangent line whose slope is
let's say maybe positive three I have a
tangent line whose slope is maybe
positive ten and that that function that
was almost flat having a slope of almost
zero but rising the x-axis is acting as
an asymptote starts to shoot off to
infinity as we approach x equals zero so
that slope is getting ever and ever
greater approaching infinity so our
first derivative here let's keep it in
orange like our first example you know
maybe look something like this I'm gonna
pull it apart just so you can see the
difference between the orange and the
white again this is a reasonable sketch
of what the derivative may look like I'm
not claiming it to be exotic percent
accurate but that function is increasing
giving us of course positive slopes
until we go vertical and that slope is
approaching undefined now over here
though instead of rising and shooting
off to infinity it's like as soon as you
get past that asymptote
boom it's falling very very quickly
meaning we have very very negative
slopes very very negative so maybe like
at 0.5 that slope is like negative 10 or
something like that but that starts to
in it's to the right of this asymptote
that graph never stops falling and if
the graph is always decreasing your
derivative is always negative so you
better make sure your sketch stays below
the x-axis but you can see it's
flattening out and as you have this
graph flattened out and a start
approaching the x-axis as an asymptote
the slope really never becomes zero but
it does approach zero and so this is our
sketch of what f prime the first
derivative may look like and it you know
just make it a reasonable sketch it is
always increasing as X approaches zero
from the left and this function is
always decreasing so we have those
negative slopes as you approach 0 from
the right so let's move on to our next
two examples so for our third example
we've got this sort of wavy function
thing going on here
okay well that looks rather complicated
how are we going to sketch oh and this
is I didn't label it but this is some
kind of function not F prime but this is
some kind of function and we're gonna
sketch its derivative
well like I'd like to do in my class
let's highlight it's kind of if you the
easy points if you will you know or the
key points you know like maybe I see a
point and I go maybe that slope is
negative you know whatever negative
three but it's really negative two point
five so like that there's a little bit
of room for error but look man there is
a horizontal tangent line right there so
at 3 PI over 2 this function it's slope
is equal to zero so we better make sure
that our derivative passes through that
point and like here we have a horizontal
tangent line so let's make sure our
derivative is giving us that answer of
zero that slope of zero and as well here
and as well here okay now I kind of
cheated here a little bit I did not
include some graph paper grid which get
for my handy dandy grid chalk maker
thingy majiggy but at any rate because
of the scale being this PI over 2 pi 3
PI over 2 and so on and pi being you
know 3.14 approximately yadda yadda
yadda and the vertical scale is 1 so
what's happening well what's happening
between this horizontal tangent line
which has a slope of 0 and this
horizontal tangent line with a slope of
0 well between negative PI over two and
positive PI over 2 this function is
rising well the function is rising it
has a positive slope therefore the
derivative have better e giving you
positive answers and where between
negative PI over two and positive PI
over two is this function rising the
fastest where does it seem like it's
giving us the biggest answer for the
slope well it's it's right there because
and you know forgive my drawing I'm not
a computer but it's rising slow
just a little bit to the right of
negative PI over two and it's rising
quicker and quicker and quicker and we
have sort of this greatest icy sort of
like a most steep or the steepest slope
here at x equals zero and then it kind
of starts to come over well that's
because we're going from concave up to
concave down at x equals zero we
actually the point of inflection so at
that steepest visual slope let's make
sure that our derivative is giving us
the largest possible answer and it
doesn't really look like a slope of one
but I kind of happen to know that that
one is so but between negative PI over
two and positive PI over two it's going
it's right it's has a slope of zero
rising slowly and faster and faster and
faster and then that slope starts to
decrease so our derivative is going to
look something like this but I think
someone's at my door let me come right
back
alrighty then so to finish this up yes
zero zero and rising all the way having
that derivative given us positive values
between this x value of zero slope of
zero and between this slope of zero our
function is falling and like over here
instead of rising its fastest rate we
have its most negative slope so this at
this point here where we go from being
concave down to concave up our
derivative is giving us our smallest
value and I just happen to know that
that's going to be equal to negative one
and so as we kind of connect the dots Oh
nope our functions rising again and so
on and so on
we get zero and zero our slopes are
function falling from negative 3 PI over
2 to negative PI over 2 and here it's
falling its fastest rate so falling
means slopes which are negative we get
the graph of our first
derivative so this is our F prime of X
now did you notice a pattern between the
white graph and the orange graph as I
was drawing that I hope so because that
means you know your unit circle this is
f of X as it turns out happens to be
equal to the sine of X and this orange
graph happens to be well what does it
look like at x equals 0 the function is
giving us a value 1 of 1 at PI over 2
that function is giving us a value of 0
at PI this function is giving us a value
of negative 1 actually this derivative
function in case you never really
connected dots before and that's not
really drawn quick let me just change
that a little bit is the cosine of X
right the derivative of sine is cosine
and there you can see it graphically
over here on our fourth example we have
some kind of roots function and this
function at negative 2 4 6 you know 8 or
so negative 9 whatever but we have an
x-axis here that is acting as a
horizontal asymptote again like our
previous example before and this
function is rising but very very slowly
so back here at I counted this but it's
now let's bother me I forgot it was
negative 2 negative 4 negative 6
negative 8 at negative 9 our slope is
just barely positive this this function
is increasing but very very slowly well
okay and it's increasing a little bit
faster and it's increasing a little bit
faster giving us more and more positive
slopes maybe here maybe the slope here
is equal to positive 1 but then it goes
very vertical between the x value of
negative 1 and the x value of 0 and
actually I'm just trying to draw this as
if there was a vertical tangent line at
x equals 0 so the graph of this
derivative from this function
f
is barely increasing increasing faster
and faster and really going vertical
saying that our slope is of course the
slope of a vertical line is it's not
really it's not an asymptote like our
previous example but the slope of a
vertical line is undefined we're
approaching a slope of infinity right
you cannot find a derivative you can't
find the slope of a vertical line or you
can it's sort of it's you can say it's
undefined and then as just as you pass
the other side of x equals 0 you get to
the right of that vertical tangent line
that graph is still increasing very very
rapidly and then slows down and starts
to rise very very slowly meaning the the
slope right this is concave down and
when a function is concave down that
means the first derivative is decreasing
but this function while sorts two starts
to layover and act like there's a
horizontal asymptote out here somewhere
this function is never going to stop
increasing it just increases very very
slowly so just to the right very large
values for the slope we need to as far
as the derivative is concerned see that
there is a vertical asymptote at x
equals zero there's not an asymptote for
the original function but for the
derivative and then we come down here
and make it look something like that
that is a reasonable that orange line
that orange graph is a reasonable sketch
of what the first derivative would be
for this function and again I want to
focus on the graph of the function not
necessarily giving you the equation of
the function how having you find
algebraically the derivative of that
function and then graphing it with a
graphing calculator wants you to see it
and draw that reasonable sketch from a
graph and just using your knowledge you
know of the fact that a derivative is
just simply giving you that slope I got
one more example coming up right now
okay so we have you know I'd basically
draw a parabola and but this time I
wrote in white but then I realized all
my derivatives were in orange for the
previous four examples so let's be
consistent I have a derivative so we're
going to look at the graph of this
derivative and try and sketch what would
be a reasonable approximation of what
the antiderivative or basically function
f is going to look like well one more
time right derivatives give us slope and
what is this derivative telling us about
the original function well it's telling
us that x equals negative three and at x
equals M positive three make a big ol
dot kind of blur that over a little bit
that the slope mean a derivative tells a
slope and the answer we're getting from
this derivative is zero so our original
function on ok.now may be well I could
ever function go through there except
that I don't want you think that the
values from the derivative is that are
the values from the function so if you
know the original function could go
there f could go through there but we
better make sure that our function that
we're about to draw has a horizontal
tangent line you know has a slope of
zero at negative 3 and positive 3
now what's happening between so you kind
of wanted to mark these is sort of like
important points so let me do this here
we go
between those two markers our derivative
is obviously giving us negative values
which means that between negative 3 and
positive 3 our function that we're about
to draw ahead better be what decreasing
because our slopes are that we're
getting from the derivative is negative
so I don't know here so we have want to
kind of visualize a horizontal tangent
line
and I'm going to visualize a horizontal
tangent line there and I just hit my mic
so I hope that static sound wasn't too
bad on the mic between here and here my
function needs to be falling we need to
have horizontal tangent lines and
actually if we can approximate a tangent
line who has a slope of negative 3 that
would be even better so if I maybe here
1 2 3 okay some falling if I want to
drop that down a little warm again it's
just a reasonable sketch but so once for
a period of time to have a tangent line
which looks like it sort of got that
negative 3 slope but zero slope zero
slope most negative slope in the middle
and all negative values from the tangent
excuse me all negative values from the
derivative so our function needs to be
falling something of that effect now
what happens to the left of negative 3
well the answers that we are getting
from our derivative are all positive and
if my derivative is giving us positive
answers then our function needs to be
increasing right if the function is
increasing our derivative is giving us
slopes which are positive and it's got
back here
a very large value from the derivative
so a very positive slope very fast
increase or increasing function and it's
gonna look something like this a little
bit positive more positive more positive
more positive more positive as far as
its increasing so it's increasing slowly
its increasing faster course that seems
weird because I'm doing this in Reverse
because I just want to make a smooth
graph it's increasing very very fast so
let's let's try and visualize this again
so we have a function that's increasing
and increasing very steeply giving us
very positive values of slope and it's
increasing a little slower chalk down
it's increasing a little
it's not increasing the slope is zero
and then to the right of X equals three
we go from a slope of zero now this is
going to kind of flow a little better
because I'm going off to the right of
the graph but slope of zero horizontal
tangent line slope of positive one
so it's starting to whoops that's your
orange starting to swing up and that
those slopes that we are getting are
getting ever bigger as x increases so as
we swing to the right this function
needs to be increasing you know faster
and faster and faster because the slopes
we are getting from the derivative are
you know greater and greater so this
would be maybe a good approximation of
what F looks like now that's not the
only good approximation for the graph of
F by the way you know when you take the
derivative of a cubic function you get a
function with the degree of two so I've
drawn what looks like a cubic function a
third degree and indeed the the
derivative appears to be sort of like a
second degree parabola but what is the
if I say f of X is equal to three and I
asked for f prime of x or say if I just
say well I have a value of three what's
the derivative with respect to X of 3
well that's going to be zero right and
when you take a derivative of a constant
it just kind of goes away so when we
learn we haven't learned it yet but when
we learn how to integrate these
constants are effectively like vertical
shifts like way back in precalc if I say
that we have f of X is some kind of
function sketch f of X plus 3 you would
just take that original parent function
and shift it up three units well when
you take the derivative of that constant
it's zero when you go to integrate
that's like there's nothing in there so
what I'm trying to get across is I could
basically now take this sketch of my
cubic function and just move it up and
down anywhere I like and it would still
be a reasonable sketch because
if there was a vertical shift after you
take a derivative it goes away you don't
know what that vertical shift is in
other words I can say hey at x equals
negative 3 make sure I have a horizontal
tangent line at x equals positive 3 make
sure that slope is 0 make sure that
you're visualizing kind of like a
horizontal tangent line
hey the derivative is negative between
these two values so between negative 3
and positive 3 make sure your function
is falling to the left of negative 3
your derivative is giving you positive
values so make sure your function is
increasing to the right of 3 our
derivative is giving us positive values
so make sure to the right of 3 your
function that you're trying to sketch f
is rising either one of those sketches
of what may be or anti derivative of F
prime of X you know effectively just an
idea of what F may look like is
acceptable because you know through the
derivative process we lose that plus
constant we lose that vertical shift I
hope this helps you now understand how
to take the sketch of a function or even
its a derivative and come up with a
sketch of say a second derivative or in
this case the antiderivative function f
from f prime so it's a challenging I
hope this helps you see how to give
these sketches get these sketches done
with a little bit more accuracy I miss
true ma'am go to your homework
