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Sketch the graph of the function defined in the given exercise. Use all the information obtained from the first derivative.$f(x)=\left(x^{2}-9\right)^{2 / 3}$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

Harvey Mudd College

Baylor University

University of Nottingham

Boston College

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.

01:04

Sketch the graph of the fu…

00:29

Sketch the graph of $f$

03:57

03:00

01:06

02:24

01:48

01:55

find the derivative of this function. And we can do that using chain rule. So first we take the derivative of the outside using the exponents rule and then we take the derivative of the inside. So next we want to set this to zero. And because we have a product, if any of these values of X equals zero, then the whole equation will equal zero. So we set two eggs 20 We get X equals zero. And if we sat x squared minus nine people 20 you look at it Plus or -3 as are zero. All right, Thanks. So next what we want to do is actually plug these values back into our original equation to get our F. Of X values or our wipe out Y coordinates. So when you do that, you will get 4.3 for zero. So those are actually zero. And then help put this here on the wrong spot. Don't get zero for The values of plus and -3. And then you will get 4.3 for the value of zero. So look at this. What we want to find out next is what are these critical points? So we have zero, three and negative three. And what we want to do is pick values slightly above and slightly below each of these zeros. So let's do a negative 41 negative one and four. And we want to see if the derivative is positive or negative. So I will not eyeball it. But looking at you can also plug these values into your derivative equation to figure out what the value is. And if that value is positive or negative for the sake of time We have negative and then at 30 and we have positive At zero, it's the derivative zero. Atnegative fund and we have negative zero and positive. So finally we can graph this and it'll be a rough graph. But using our critical points, we have a minimum At Native 30 and a minimum of 30 And a maximum at zero, 4.3. So our graph will look something like this.

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