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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 15.

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

Harvey Mudd College

Baylor University

Idaho State University

Lectures

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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.

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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 graphs of the g…

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This is Chapter three, Section two, Problem 28. And in this problem we are asked to graph the function that's given an exercise 15. So I went ahead and wrote down That function of X is equal to X squared minus two X plus three. So we are supposed to sketch that the first step is to find all of our critical points and are critical. Points are places where are derivative of the function is equal to zero so already went ahead and solved for the derivative f prime of X and I found that to be two X minus two two X minus two is equal to zero when X equals one. Mhm. So for critical points, we have an X component in a Y component. We already have our X component, which we just found to be one to find the y component. We just plug in that one into the original function. So it looks like we have one squared minus two times one plus three. That's just going to be two. So are critical point. Here is 12 Next, we're going to use our number line to determine where our function is increasing in decreasing. So we have a zero at X equals one, and we're going to pick a value that's less than one. And we're gonna pick a value that's greater than one, and we're going to plug in both X equals zero and X equals two into F Prime of X to find out where our function is increasing in decreasing. So first, let's do X equals zero. So we have f prime of X equals zero that's going to give us to time zero minus two. That's going to give us a negative, too, because it's negative. That means that if we have an X value that's less than one, our function will be decreasing. Oh, again for X greater than one. We have f prime of two. We have two times two minus two. That gives us a positive too. Which means that if we use an X value greater than one, our function will be increasing. Oh, From here things are pretty easy to sketch. We have our Y axis and our X axis. Oops, in our critical point is here at 12 We know that any X value less than one are function is going to be decreasing to that point. So it will look something like this on the left side. For X values greater than one, our function is increasing, so it will continue and increase like this. So that's what our function will look like, where we have a minimum at one two.

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