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### Course: Calculus, all content (2017 edition)>Unit 2

Lesson 9: Derivative as a function

# 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.
• First, I'm very impressed that you're learning Calculus at Age 10!
On Sal's third example at , 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.
• 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?
• 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.
• At in 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!
• 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!
• 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'.
• 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
• 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.
• 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?
• 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!