[MUSIC] Suppose I've got some function 
given by a rule and I want to make a 
graph of that function. 
I wanted to plot say this function f(x) 
equals 2x cubed minus 3x squared minus 
12. 
First thing I might do is just plug in 
some values. 
All right I'll pick. 
Pick some inputs and I'll see what the 
function outputs at those inputs. 
And once I've got this table of values, I 
could then plot those points on a graph. 
The issue is, how do I really know what 
happens between these points that I 
plotted on the graph? How do I know the 
graph isn't doing some crazy wiggling in 
between? How do I know that I've really 
picked enough input points to really get 
a good idea of what this graph is doing? 
We're going to use derivatives to make 
sure that we're really capturing the 
qualitative features of the function. 
I might have been trying to graph a 
function, like f(x) equals sin Pi x, and 
if I just plugged in some whole number 
inputs, the function would always output 
0. 
That might trick me into making a graph 
like this, where I plot 0 as the output 
for all these whole number inputs. 
I might, then, be tempted to just fill in 
this graph by drawing a straight line 
across. 
But that's totally ridiculous, right? 
This graph, you know, actually looks like 
this. 
Not a horizontal straight line. 
There's all kinds of extra wiggling 
that's happening that I missed because I 
chose my in points badly. 
We're going to use derivatives to make 
sure that we're really capturing the 
qualitative features of the function and 
there's a ton of different ways to do 
this. 
So let's work this out in one specific 
concrete example. 
So let's keep working on the graph of 
this function, f(x) equals 2x cubed minus 
3x squared minus 12x. 
First thing I'm going to do is 
differentiate this, 
the derivative is 6x^2-6x-12, 
cause' the derivative of 2x^3 is 6x^2, 
the derivative of minus 3x^2 is minus 6x, 
and the derivative of minus 12x is minus 
12. 
There's a common factor of 6 here which I 
can pull out, and then I'm left with this 
quadratic, and I can factor that 
quadratic into (x+1) times (x-2). 
Now once I've got this nice factorized 
version of the derivative, I can then 
figure out where the derivative is 
positive and negative. 
The derivative is positive when the input 
is more negative than minus and it's 
positive when the input is more positive 
than 2. 
In between -1 and 2, the derivative is 
negative. 
And at the point -1, and at the point 2, 
the derivative is equal to zero. 
Now, since this function is 
differentiable everywhere, the only 
critical points are where the derivative 
is equal to zero. 
These are the critical points, minus 1 
and 2. 
Alright. 
So I found the critical points. 
I found the derivative. 
Now, I'll also find the second derivative 
of this function. 
Which I get by differentiating this 
derivative. 
If I differentiate 6x^2 I get 12x, if I 
differentiate minus 6x I get minus 6, 
and if I differentiate minus 12 I get 
zero. 
Again, I've got a common factor of six so 
I'll pull that out and I'm left with 
2x-1. 
And now I can think about the SIGN of the 
second derivative. 
And what do I know about that? Well, the 
second derivative is negative if I plug 
in an x value which is less than 1/2 and 
the second derivative is positive if I 
plug in an x value which is bigger than 
1/2. 
All right, now I know a lot of 
information about the SIGN of the first 
and the second derivative, so I can use 
this information to say something about 
the function. 
Let me look back to my preliminary graph 
that I made with just plugging in a few 
points. 
All right, so here I plugged in a few 
points and what I'd like to be able to 
say now is where is the function 
increasing and decreasing. 
And by looking at the sign of the first 
derivative I know that the function's 
increasing, decreasing, and then 
increasing. 
Minus 1 and 2 are my critical point and 
in fact, they're local extrema. 
This is a local maximum value, and this 
is a local minimum value down here, and I 
can also see that by considering the 
information given in the sine of the 
second derivative. 
Since the second derivative's negative 
here, the functions concave down. 
And since the second derivative is 
positive over here, the function is 
concave up. 
And that makes this point into a local 
maximum and this point into a local 
minimum. 
Alright, now that I've got all that 
information I can try to just fix the 
graph here filling it in. 
So let's see, so I've got these points 
here and what do I know? I know the 
function is increasing here, and now I 
know that it's decreasing here. 
And I know that it's concaved down in 
this region. 
Over the rest of the graph the rest 
concave up. 
There's an inflection point here when 
x=1/2 and this point over here is a local 
minumum. 
The function's decreasing by looking at 
the sign of the first derivative, until I 
get to two. 
And then when I get to two, the first 
derivative tells me the function's 
increasing. 
So there we go, 
I've drawn a graph of my function. 
The point here is not to capture a 
perfect picture of the function. 
It's like an impressionistic painting, 
the point is to capture all of the 
meaning all of the emotion of the 
function. 
Compare that to a photograph which might 
be a perfectly accurate portrayal, but 
somehow misses everything that's 
essential. 
So here's the graph that I drew in red, 
and here is a more perfect graph 
admittedly, that the soulless robot drew. 
And you'll see that my graph really is 
just as good. 
I mean, it captures all the qualitative 
information which is really what a human 
being cares about. 
Functions increasing, decreasing, 
increasing. 
You can see where it's concave down and 
where it's concave up. 
And you can kind of see roughly where 
this function crosses the x axis. 
Let's summarize the situation. 
There's really 4 basic pieces that you're 
just gluing together when you're doing a 
lot of these curve sketching problems. 
It depends on the SIGN of the first 
derivative, and the SIGN of the second 
derivative. 
If the derivative is positive, and the 
second derivative is positive, then the 
function is increasing, and the slopes of 
the tangent lines are increasing. 
If the function's derivative is negative 
but the second derivative is positive, 
that means although the function's 
decreasing, the slopes of those tangent 
lines are increasing. 
We've got kind of complementary pictures 
over here when the second derivative's 
negative, here the function's increasing 
but the slopes of those tangent lines are 
decreasing, and here both the function is 
decreasing and the slopes of the tangent 
lines are decreasing. 
A lot of the curve sketching problems 
amount to just gluing together these four 
basic pieces in the appropriate way.