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Suppose the graph of $f^{\prime}(x)$ versus $x$ is given in Figure $31,$ sketch a possible graph of $f(x)$ versus $x$ Include concavity in your sketch.

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

04:40

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.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

02:28

Suppose the graph of $f^{\…

01:51

Given the graph of f(x), s…

01:19

Suppose that $f^{\prime}(x…

01:42

Given the graph of f'…

04:49

Finding $f$ from $f^{\prim…

02:13

02:04

00:30

Sketch the graph of $f$

03:26

Sketch a graph of $y=f(x)$…

00:34

Sketch the graph of $f$. <…

00:23

So that's what we're given here. This is F prime X. Asked for a sketch. Uh huh. To fix Okay. Want to indicate con cavity as well. So, let's look at a few things here. First off, is that if it's going from above to below, this is a relative X. It's gonna be real to max. and one that's cyril. You see that it's the opposite. This would be relative men. Okay, then then a negative one. We see that it's also a relative max. So now at this one up here, this is going to be an inflection point. And so is this Okay? So then what we see here is that we're actually going so we're going from decreasing increasing and decreasing here. So, let's take a look at some things here. First off, at negative one. Got a relative max. We're just gonna indicator relative maxim where there? Okay, and Sarah we've got a relative been So just going to put men there then at one I've got a relative max. I just couldn't put a max there. It's been a decade values for .5. All right. So we've got inflection points there. So then from here, but we also see is that too to the left here, it looks like it's going to like flatten out and kind of become like a different almost like it's if we were to continue it all, it kind of looks like it was going to have like another inflection point. So how we can use that to our advantage here. Is that well, we know that we're decreasing up until we get to 8.5 here. Right? But since it looks like it's about to change, so let's do it and thats a relative max. Okay. We know that it's going to go this okay. Up until it reaches .5 and then it becomes switches there. Right? Let's say it does something like that, which is here so that it goes concave down okay. It looks kind of like this except it looks like it's going to It might shift so kind of have it curving like that ever so slightly and so that's what a potential graphic that would look like. And I can see here that it's so concord down here, all up to here. So this can't give up. I think I came down from then on.

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