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Sketch the graph of the function defined in the given exercise. Use all the information obtained from the first derivative.Exercise 7.

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

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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.

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Sketch the graph of the fu…

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Sketch the graph of each f…

This is Chapter three, Section two. Problem 25 in this problem asks us to sketch the function listed an exercise seven. So I've already written out the function G FX. And to sketch it, we're gonna need to find the critical points. So the critical points are where the derivative of the function is equal to zero. I already went ahead and found the derivative of G of X, which is G prime of X to be 12 times X minus one times X minus three. That's what it factored down to. And that allowed me to find critical points at X equals one and X equals three. From here we can draw our number line, and that will let us find the minimum and maximums. So we know for critical points we have our X component in ry component. So we have two critical points and we know our X component for both. So one is going to be one and one is going to be three to find the y component. We just plug in that one in three into the original function up here. So if I plug in one, I get a Y value of 112. And if I plug in three mhm, I get a value of 96. So these are critical points. So if you have our number line here, we have a critical point at X equals one. We have a critical point. X equals three. We're going to pick a point of X. That's lower than one, a point that's in between one and three in a point that's greater than three. And we're going to plug in each of those values into G prime of X to see if our function is increasing or decreasing over that interval. So for X less than one, let's just choose X equals zero for the middle point. Let's choose X equals two and for X value greater than three. Let's just choose X equals four. Oh, and each of these are going to be plugged into G prime of X. So let's start with X equals zero g Prime of X equals zero. That's gonna give us 12 times zero minus one times zero minus three. That's 12 times negative, one times negative three, and that's going to give us a positive 36. So that means if we have an X value less than one. Our function will be increasing. Let's do this again for X equals two, we'll have 12 times two minus one times two minus three. That's going to be 12 times one times negative. One that's going to give us negative 12. Which means on the interval from X equals 12 X equals three. Our function will be decreasing. And last but not least, X equals four. That will give us 12 times four minus one times four minus three. Yeah, that will give us a positive 36 again. So if we have an X value greater than three, our function will be increasing. Now, from there we can go ahead and draw sketch. Okay, So first, let's find our critical points are critical. Points are going to be 112 which is we can say is up here. And our second point is going to be three 96. So somewhere in there, mhm. Okay, so far X value is less than one. Yeah, our value. Our function is going to be increasing, so it'll look something like that on the interval between X equals one and X equals three, Our function is decreasing. That checks out. And if X is greater than three, our function is increasing again, so our function will increase and our function will look something like that.

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