Reproduce the graph of f and then plot a graph of f' on t...

Question

Reproduce the graph of f and then plot a graph of f' on the same axes.

Accepted Answer

Hello. In this video, we are going to be drawing the graph of FX by using the given graph of F of X. So how do we start this process of graphing the derivative of the graph of the given function? Well, the first thing to understand is that the derivative F of X maps the slope at any point of F of X. So what this means is that when we are creating the graph for F of X, what we are really doing is we are mapping the values of all the slopes of F of X. The first thing to label on the given function is the vertical asymptotes. We have two vertical asymptotes, one located at the X value of -3, and the other located at the X value 3. Now, what's important here is that if the original function F of X has restrictions of the domain which are represented by these vertical asymptotes, then its derivative will have the same restrictions. So, for our derivative graph, we are going to start by labeling the vertical asymptotes of the graph. We have one at -3. And we have one at 3. Now, the next thing that we can do is we can label the critical points for the given function. Now the critical points are points of the function where the slope is equal to 0, and wherever the slope is equal to 0, that is going to represent an X intercept for our graph of the derivative. Now, by taking a look at the given graph, we have exactly 1 critical point. This critical point is located at the X value, X is equal to 1. At this value, we have a slope of 0, and that means that for our derivative graph, we have an X intercept located at 1. Now what we need to do is we need to go ahead and graph the behaviors of the slopes of F of X. Let's go ahead and start by viewing the behaviors on the interval from -5 to -3. Notice that the graph is decreasing to negative infinity in this interval. What this means is that for the F function on the interval from -5 to 3, we are going to be below the x axis and decrease to negative infinity. So we can represent that behavior of the graph as the following. Now what we want to do is we want to pay attention to the behaviors of the graph on the interval from -3 to 1. Notice that the graph is slowly increasing. To the value of 1. What this means is that for the F function, we are going to start at positive infinity, and we are going to slowly decrease to the X intercept at 1. So that is going to be the behavior of the graph from -3 to 1. Now, what about the behavior on the interval from 1 to 3? Well, the behavior of the graph is going to switch from increasing to decreasing. So we start off with a small negative slope, and this slope grows over time, going to negative infinity. That means that for the portion of the interval from 1 to 3 of FX, the graph is going to be below the X axis and slowly decrease to negative infinity. Now we need to finally map the values of the slopes from the interval of 3 to 6. Now for this interval, the graph is greatly decreasing from positive infinity all the way to the value of 6, and so we can represent that value as the following. We are going to decrease infinitely. We are going to be below the x-axis since we are decreasing infinitely, and we are slowly increasing to the value of 6. And this is going to be the behaviors of the graph of F of X. Thus, this is going to be the shape of our graph F of X. So, I hope this video helps you in understanding how to graph the derivative, and I will go ahead and see you all in the next video.

Reproduce the graph of f and then plot a graph of f' on the same axes. <IMAGE>

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