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Trace or copy the graph of the given function $ f $. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of $ f' $ below it.

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02:30

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

Missouri State University

Campbell University

University of Nottingham

Idaho State University

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

02:41

Trace or copy the graph of…

02:42

06:01

03:21

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it is us that grace or copy the graph of given function. F assume that the access have equal scales. Right? So, now then use the method of example, one to sketch the graph of F dash below it. Right. So, we are just these are the graphs given to us. Right? So, now simply Libya just we are just uhh you can see dropping the arose here. Right. Or you can say they're projecting the grabs. Right. So, we can assume that we can assume that this is yes. All right. So, this function is f for us for everything. Right? For every problem. So, graph to like drop three Graph 4 and five. Right. So here, there is a function F. Okay. So, when f is less than zero then it will be like this. We can see that this function is actually decreasing. Right? So, whenever we differentiate a function differentiate a function then if it is empty, function is decreasing and we are differentiating it. So, it must increase. Right? So, all you can say it will be just flip off this problem or you can say water image kind of thing. But it is not possible for every point. I'm just talking about this particular point. Right? So this is this is started from here. Right, decreasing. But here it will increase. This is attach All right. So here the graph is a dash. Okay, now again the graph is like increasing uh here after that it is starts decreasing. Right? So here there is a constant value. It So for constant valued the differentiation is nothing but it is zero. So that's why we are getting zero year. Okay, after that it is again decreasing. No, it increases from here again when when it when it is start increasing, then it decreases. All right. So this is how we make this, how this is how we plant this. Yeah. For the differentiation of F. So here again, our function is given that is it. Hey! Here it is increasing. So it must decrease from the positive values. Right? So here there is some kind of constant value. Why? Because this is decreasing. Right? So it must be like this and this constant value continues. So it also continues here. Greg. Now again, this is the third part we have. You can say this is A B and C. Right? So she is here, we can say this is going constant. So there will be constant value rate, constant value. Then after that it is decreasing. So it is decreasing. So it must done right and remain constant. It is not increasing. It will also the trees right click. It will remain constantly here after this access. Right? This is what expenses. No, here a semi circle is given to us as a function. Okay, so, so it is increasing in this way. Right? So when we differentiated we are getting a constant value kind of here, right? Similarly like here in option we we have got here similarly, we're got here like this and this is like this, so it will increase, it is increasing all the time, right? In the semi circle, in quarter part of the circle, it is increasing so it must degrees for the quarter part. Right off a circle. We are talking about the semicircle. Alright, so now it will be, it is increasing down the by exist No, here it is decreasing again. So this will be like this and it will start increasing in the direction of quiet. So this is how we solve this problem. This is a F dash again. This is also a gash again. And now for the F this will be a constant value. Thank you. So this is like a semi circle only because half of the access, it is just going constant value. Right? So we can just say that if this is a constant value, obviously we have concluded in this way, so this will be a constant value, right? So uh if part is very easy, it will be constant, simply a constant line going cool. So this is just directly proportional, like you can see. So this is how we solve this problem. I hope your destructive concept. Thanks for watching.

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