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## Calculus, all content (2017 edition)

### Course: 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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# Matching functions & their derivatives graphically (old)

An old video of Sal matching graphs of functions with their derivatives or antiderivatives. Created by Sal Khan.

## Want to join the conversation?

- I can't understand it. The line has a slope of 0, but the purple window shows a negative slope.(14 votes)
- First, I'm very impressed that you're learning Calculus at Age 10!

On Sal's third example at3:02, the green line does have a slope of zero, but since you're trying to match up the anti-derivative to this function, you'll have to visualize this as a derivative function that reveals information about the original function (i.e., the purple line). So, the green line does have a slope of 0, but when visualized as a derivative function this tells you that the slope of the original function (or anti-derivative) has a value of -1 (the y-value of the derivative function) AND that the slope stays constant. That is, when you have a flat line along the derivative graph, this means that the slope of the original function (or anti-derivative) is constant through that interval (i.e., in this case, it was through the x-values of 6-8). Hope this helps.(46 votes)

- Is it still possible to explain harder levels of math like this to people who haven gone through all of the previous steps like algebra and geometry?(4 votes)
- You can certainly explain the rudiments of the calculus to just about anyone willing to hear them. My small children (younger than you) are capable of understanding things like instantaneous velocity and optimization. However, to get any traction with solution methods, a person has to have paid their dues, so to speak. There is no substitute for having a firm grasp on the basics, in my opinion.(7 votes)

- At3:18in Sal's third example, the green function f(x) has a hole in that point, meaning that the slope of F(x) at (8,1) is undefined, but I thought the slope of the purple function F(x) (the anti-derivative of f(x)) at point(8,1) was zero. Am I missing something?

Thanks!(3 votes)- You have a good point. I see why this can be confusing.

derivative=slope=f(x)=green

antiderivative=function=F(x)=purple

In actuality, the antiderivative is just following its pattern. When the slope is constant (horizontal), the function will go down at a constant rate. When the slope is a line (slanted, but straight), the function will become a parabola. The reason the antiderivative isn't undefined at x=8 even though two patterns can be at this point is because the patterns for the points before x=8 and the patterns for the points after x=8 intersect at (8, 1) (The reason they intersect is because Sal made them; the parabola could be way up top, the line could be way down low, and the derivative would still be the same. This is because you're just adding a constant to the antiderivative when you do this, and an added constant doesn't change a function's derivative). You could say that the limit of F(x) as x approaches 8 is 1. This can happen regardless of the limit of f(x) as x approaches 8 being undefined.

I hope this helps!(2 votes)

- What does anti derivative mean?(1 vote)
- There is a long explanation on that but for now:

In these videos, just keep in mind that Anti-Derivate is the actual function. So basically, in this video, you were given the derivate of a function. You were told to find what the function was. That's all.....The word Anti-Derivate just replaced the words 'original function'. In further Calculus, Anti Derivative is also called 'The Indefinite Integral'.....It's all connected, you're not that far to getting there.......So my words, 'just wait to be surprised'.(4 votes)

- There is one thing I cannot understand, when I answered a question, I got it correct, but the purple was a negative slope then a positive slope, and the green window was a positve slope then a negative slope. Was it that, if you switched the green lines around they would match up?

(age 12)(1 vote)- It would have been better if you had taken a screenshot for us to explain. :D(4 votes)

- can decreasing slope be of positive value(1 vote)
- I'm not sure what you're asking but maybe this will help: a function can have a positive value while the slope is negative, and the slope can be positive while it's decreasing.(3 votes)

- I'm very confused with the video... the question asks for the antiderivative and Sal talks about the derivative... is there a glossary of terms or can someone define derivative and antiderivative for me?(2 votes)
- the derivative is f'(x) and is the slope of the tangent of f(x) (the original function)

the antiderivative is the corresponding f(x) for a given f'(x) - Sal uses the terms f(x) to represent the derivative and F(x) to represent the antiderivative in his videos(1 vote)

- where are the exercises on this website I'm lost(1 vote)
- The name of this exercise is Visualizing derivatives. You may click on the bottom right green button for this exercise!(2 votes)

- When do we use the calculator(1 vote)
- The Calculator (TI-81, 82,83,84 and 85) are built to be used for studying functions. You have the graph function available. If you like, it would be good to graph random functions and try to find their anti-derivative or derivative. This Calculator is basically a 'Teaching' Calculator. It is not built to give you answers; it is built to teach you how to be like it. So if you have a calculator, use it to study, not to find answers....

I have done Calculus before.....full.....I had about 11 quizzes and 3 exams. I never had a calculator to help me. I earned 94% in total (not boasting). So if you try hard, you'll certainly get better than me...(1 vote)

- Is the antiderivative of the function the change of the slope of the function?(1 vote)

## Video transcript

The function f of x
is shown in green. The sliding purple window
may contain a section of an antiderivative of
the function, F of x. So, essentially it's
saying, this green function, or part of this green
function, is potentially the derivative of
this purple function. And what we need
to do is-- it says, where does the function
in the sliding window correspond to the
antiderivative of our function? The antiderivative of f of x,
usually, write as big F of x. This is just saying
that, lowercase f of x is just the derivative
of big F of x. So, at what point
could the derivative of the purple
function-- and I'm going to move the purple
function around-- where can the derivative of
that be the green stuff. So let's just focus on
the purple stuff first. So the derivative--
we can just view it as the slope of the tangent
line-- between this point and this point, we see that we
have a constant negative slope, and then we have a
constant positive slope. So let's see, where here do we
have a constant negative slope? Well, now here the slope is
0, and it gets more negative. Here we have a constant
positive slope, not a constant negative slope. Here we have a constant
negative slope, so maybe it matches
up over there. So here we have a
constant negative slope, but then on the purple function,
we have a positive slope, but where the potential
derivative is here, we just have a slope of 0. So, this doesn't
match up either. So it looks like in this case,
there's actually no solution. Let's see if this works out. Yes, correct. Next question. Let's do another one. A function f of x
is shown purple. The sliding green
window may contain a section of its derivative. So now we're trying
to say, at what point of this purple
function might the derivative look like
this green function? So in this green
function, if this is the function's
derivative, here the slope is very negative. It goes to 0, and then
the slope gets positive. So let's think about it. So over here, the slope is
just a constant negative, so that won't work. If we shift it over
here, our slope is very steep in the
negative direction and then it gets
less and less steep in the negative direction,
and it goes all the way, and then over here
the slope is 0. And over here, if this
is the derivative, it seems to match
up, the slope is 0. And then it gets
more and more steep in the positive direction. So this matches up. It looks like over
this interval, the green the function
is indeed the derivative of this purple function. So let's see. Let's check our answer. Correct. Next question. Let's do another one. This is exciting. A function f of x
is shown in green. The sliding purple window
may contain a section of an antiderivative of
the function, F of x. So, now we say, let's match
up this little purple section to its derivative. So the green is the
derivative, the purple function is the thing we're
taking the derivative of. So if we just look
at the purple, we see that we have a
constant negative slope in the first part of
it, then our slope-- so let me just look for where I can
find a constant negative slope. So here, this is a
constant positive slope. This is not a constant slope. This is a constant positive. Here's a constant
negative slope. Let's see if this works. So over this interval, between
here and here, my slope is a constant negative,
and indeed, it looks like a constant negative. And you see it's a
constant negative 1. And over here, you
see the derivative is right at negative
1, and it's constant, so that part looks good. And then when I look
at the purple function, my slope is 0
starting off, then it gets more and more steep
in the negative direction. And so my slope is 0, and it
gets more and more negative, so this is indeed
seems to match up. So, let's check our answer. Yes, got it right. I could keep doing this. This is so much fun.