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## Calculus, all content (2017 edition)

### Unit 2: Lesson 9

Derivative as a function- The graphical relationship between a function & its derivative (part 1)
- The graphical relationship between a function & its derivative (part 2)
- Connecting f and f' graphically
- Visualizing derivatives
- Connecting f and f' graphically
- Matching functions & their derivatives graphically (old)

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# The graphical relationship between a function & its derivative (part 2)

Given the graph of a function, Sal sketches the graph of its antiderivative. In other words, he sketches the graph of the function whose derivative is the given function. Created by Sal Khan.

## Want to join the conversation?

- Is there any difference between an anti-derivitive and an integral? if not why not just use the term integral?(50 votes)
- No difference: they mean the same thing. Usually, you'll just use the word "integral." Some people like the word "anti-derivative" since an integral is literally taking the reverse of a derivative. Also, since these are instructional videos, the term "anti-derivative" is helpful because it kind of explicitly tells you what's going on (you have some derivative, now you're going back, hence "anti"), while the word "integral" itself doesn't tell you anything about what's going on unless you already know what it means.

In short, it doesn't matter at all. I personally like the term "integral" more, but it's all up to personal preference.(86 votes)

- Is there a conceptual difference between the anti-derivative shown here and the more involved concept of anti-derivatives taught in Integral Calculus?(5 votes)
- No. Sal is merely leading us into a better understanding of derivatives and anti-derivatives by first teaching them abstractly and intuitively.(13 votes)

- So is anti-derivative just another term for the integral? Or is there a difference?(3 votes)
- Strictly speaking, the anti-derivative is one possible value, whereas the indefinite integral gives all possible values.

e.g antiderivative of x^2 is x^3/3 but the indefinite integral of x^2 is x^3/3 + C

Well that's what my calculus for dummies book says anyways.(10 votes)

- what does d/dx mean at0:50?(5 votes)
- Its called the "Derivative operator"! you put it infront of an expression to mean "Take the derivative of this!"

Hes some more info: https://en.wikipedia.org/wiki/Differential_operator(10 votes)

- So why could it be defined in the antiderivative (at2:31) but undefined in the derivative function (when Sal was drawing the derivative of a function in the previous video, he made it the same)?(4 votes)
- well, because of the quirks of derivatives, if there can be multiple possible slopes at a point, then we make the derivative undefined. In this case, the derivative of that 'sharp point' could be 1 if we continued it from the left, but is could also be -2 if we continued it from the right. Also, if you look at https://www.khanacademy.org/math/calculus/differential-calculus/visualizing-derivatives-tutorial/v/intuitively-drawing-the-derivative-of-a-function (2:18)

sal also explains it. To address how the point was defined in this video, you have to note that it is given that the point is at that exact spot. When you take an antiderivative, you don't know what the value is (so you make it undefined), but in that function, they gave you the value, so it is defined there.

great question(5 votes)

- Would the last cusp in the antiderivative to the left of the graph have a defined slope?(4 votes)
- If you are referring to the part of F(x) where the orange and pink lines meet, then yes, the derivative is undefined at this point.

f(x), the derivative of F(x), is discontinuous from the blue part to the pink part and the function does not exist at that point (it has open circles at both the end of the blue part and beginning of the pink part)(3 votes)

- Just making sure I'm understanding this correctly...

We don't know the Original Function, but we know it's derivative. The derivative of the Unknown Function's derivative is the anti-derivative, which is also the Original Function.

Do I have that right?(3 votes) - in this example Sal drew the anti-derivative as one big continuous function...is it okay if I draw the anti-derivative even if the function is not continuous at certain points and has "gaps" like in the derivative function?

I'd really like some help on this...:)(3 votes)- Since differentiability implies continuity, it is not possible for an anti-derivative to be discontinuous at a point where the "derivative-function" is defined, as this would imply that the anti-derivative would not be differentiable at this point (a contradiction).

An anti-derivative may be discontinuous at points where the "derivative-function" is undefined, however, but this is a rather trivial observation.

Observe that a function may have a discontinuous derivative, though. As an example, consider the function`ƒ`

defined on all of

by**R**`ƒ(x) = x²sin(1/x)`

when`x ≠ 0`

, and let`ƒ(0) = 0`

. Then the following holds (see if you can prove all of these claims. In particular, see if you can prove claims`III)`

and`IV)`

):`I)`

`ƒ`

is differentiable everywhere, i.e., differentiable on all of

;**R**`II)`

`ƒ'(x) = 2xsin(1/x) - cos(1/x)`

for`x ≠ 0`

;`III)`

`ƒ'(0) = 0`

;`IV)`

`ƒ'`

is*not*continuous at`0`

.(3 votes)

- Ha! I finally get why we always put + C after calculating the integral of a function. It's because you can shift it up or down by and y-value C and the graph would still be right.(4 votes)
- Why does the derivative of the common point between the parabola and a straight line is undefined?(3 votes)
- Because a sharp angle doesn't have a slope.

Think about what is happening at that point for f(x). Coming from the left, the slope is a constant negative value, and then suddenly becomes positive. But if you were to define it at the common point, what value would you choose? The value coming from the left, or the right, or somewhere in between? Because there's no one value (or any value) that you can assign to the common point, the slope and therefore the derivative is left undefined.

That's my intuition of why a slope can't be defined at a sharp point.

A more mathematically rigorous way to look at it is that f'(x) is a function defined by a limit: the limit of the slope of the secant line between f(x) and f(x+h) as h ->0. At a sharp point, you get two different values depending on whether h approaches 0 from the positive or negative side. Therefore the limit (by definition) doesn't exist at that point, so neither does f'(x)(1 vote)

## Video transcript

In the last video we
looked at a function and tried to draw
its derivative. Now in this video, we're
going to look at a function and try to draw
its antiderivative. Which sounds like
a very fancy word, but it's just saying the
antiderivative of a function is a function whose
derivative is that function. So for example,
if we have f of x, and let's say that the
antiderivative of f of x is capital F of x. And this tends to be the
notation, when you're talking about an antiderivative. This just means
that the derivative of capital F of x, which is
equal to, you could say capital f prime of x, is
equal to f of x. So we're going to try to do
here is, we have our f of x. And we're going to try
to think about what's a possible function that this
could be the derivative of? And you're going to study
this in much more depth when you start looking
at integral calculus. But there's actually
many possible functions that this could be
the derivative of. And our goal in
this video is just to draw a reasonable
possibility. So let's think about
it a little bit. So let's, over here
on the top, draw y is equal to capital F of x. So what we're going
to try to draw is a function where
its derivative could look like this. So what we're
essentially doing is, when we go from what we're
draw up here to this, we're taking the derivative. So let's think about what
this function could look like. So when we look at
this derivative, it says over this interval
over this first interval right over here, let me
do this in purple, it says over this
interval from x is equal to 0 all the way
to whatever value of x, this right over here, it
says that the slope is a constant positive 1. So let me draw a line with a
slope of a constant positive 1. And I could shift
that line up and down. Once again, there's many
possible antiderivatives. I will just pick
a reasonable one. So I could have a line that
looks something like this. I want to draw a slope of
positive 1 as best as I can. So let's say it looks
something like this. And I could make the function
defined here or undefined here. The derivative is
undefined at this point. I could make the function
defined or undefined as I see fit. This will probably be a
point of discontinuity on the original function. It doesn't have to
be, but I'm just trying to draw a
possible function. So let's actually
just say it actually is defined at that
point right over there. But since this is going
to be discontinuous, the derivative is going to
be undefined at that point. So that's that first interval. Now let's look at
the second interval. The second interval, from where
that first interval ended, all the way to right over here. The derivative is a
constant negative 2. So that means over
here, I'm going to have a line of
constant negative slope, or constant negative 2 slope. So it's going to be twice
as steep as this one right over here. So I actually
could just draw it. I could make it a
continuous function, I could just make a negative
2 slope, just like this. And it looks like this
interval is about half as long as this interval. So it maybe gets to
the exact same point. So it could look
something-- let me draw it a little bit
neater-- like this. The slope right over
here is equal to 1, we see that right over
there in the derivative. And then the slope right over
here is equal to negative 2. We see that in the derivative. Now things get interesting. Once again, I could have shifted
this blue line up and down. I did not have to construct a
continuous function like this. But I'm doing it just for fun. There's many possible
antiderivatives of this function. Now what's going on
over the next interval? And I'll do it do it in orange. The slope starts off
at a very high value. It starts off at positive 2. Then it keeps decreasing, and
it gets to 0, right over here. The slope gets to
0 right over there. And then it starts becoming
more and more and more and more negative. So I'll just try to make this
a continuous function, just for fun. Once again, it does
not have to be. So over here the slope
is very positive. It's a positive 2. So our slope is going to be
like, this is negative 2, so it's going to
be a positive 2. And then it gets more and more
and more negative up to this, or, it becomes less and less
and less positive I should say. Even here the slope is positive. And it gets to 0
right over there. So maybe it gets to 0
maybe right over here. And so what we have, we could
have some type of a parabola. So the downward facing parabola. So notice, the slope is
a very positive value. It's a positive 2
right over here, then it becomes less and less
and less and less positive, all the way to 0
right over there. And then the slope
starts turning negative. And so our function
could look something like that over the interval. Let me draw it a
little bit neater. This is symmetric. How ever positive
our slope was here, it's equally negative here. So our curve should probably
be symmetric as well. So let me draw it like
this over that interval. It could look like that. And then finally, over
this last interval, or I guess we could say, it
keeps going, our slope is 0. And once again I don't have
to draw a continuous function. But when the slope
is 0, that just means that I have a
line with slope 0. I have a horizontal line. And I could draw that horizontal
line up here, in which case I'd have to say
its discontinuous. Or I could draw that horizontal
line right over here, and try to make
my antiderivative a continuous function. So once again, I could have
shifted any of these segments up or down, and gotten
the exact same derivative. But then I would not have
gotten a continuous function like this. But we have been
able to construct a possible antiderivative
for f of x. And just as a reminder, that's
just saying this f of x is-- the antiderivative is
a function that f of x could be the derivative of.