Question
Draw F(x)= SSIt 2ldt on a scaled set of axes_
Draw F(x)= SSIt 2ldt on a scaled set of axes_
Answers
Sketch a graph of the derivative of the functions $f$ shown in the figures.
So in this set of problems um because we don't have access to the actual sketch of the graph that's completely fine. However, what we are going to do is just kind of look at a general um understanding of a graph of F. Of X. And then we're going to talk about how we can use a graph to sketch our own graph. That's the directive. So let's say we have some random polynomial um three X cubed minus two, X squared plus X. Um request three X actually was two plus three is a good one. So um if I make this Like this and then we'll actually make this plus one. So with this in mind we want to figure out what the derivatives are going to be. So what the drought of graph is going to look like, well what we see is that if this was for example, um yeah we know that the easiest thing to find is where the slope is going to be zero. So we see that's going to be here and here. So where we have a slip of zero, we're going to make sure that when we draw our derivative graph at this point, 00 is on the graph and that 1.11, is on the graph as well. Then we see that the maximum point of slope is going to be reached right here. We'll soon later call this the inflection point but it's going to be when our slope is changing the largest and right here is where that's occurring, although this is a negative slope, so that's actually going to come down here. The easiest thing to determine is again where we have the slope of zero and also if the slope is going to be negative or positive and based on those, we will be able to either have the derivative value positive if the slope is positive or the derivative value negative if the slope was negative. So using that, let's compare this to the actual f prime of X graph. And sure enough, we see that those zeros are located where we thought we were going to have the slope go down decreasing until it reaches the lowest point, which is right along this line right here where we see the slope is the most negative and then we see that the graph has a positive slope and that positive slope is very, very positive. The farther we go to negative infinity and it's very, very positive. The farther we go to positive infinity. So that is a good explanation of how we can take a general graph and understand what its derivative graph will look like. Just by looking at important points such as the minimums and maximums, as well as points of the greatest negative slope. And seeing the end behavior house slope increases or decreases. Um as exposed to positive and negative infinity
This is indeed a sketch of the graph. Um, between cool sign and co sign in verse. Right. This is positive one. This is negative one, right? You can see that it is symmetric, right? You can rule a line through here. Okay, Y equals x here. And you see that it is symmetric, right? It is symmetric. Um, that is the symmetrically geometrical relation between the two grass. Right? I know that, um, the two glasses are symmetric about a line y equals X right and affection. Uh, this function close and x ray is defined on the restricted domain off zero pie. Right. Um, so this one here zero, right? And then pies gonna be somewhere here, right? So it is is defined on the zero pie, right? That is the restricted domain. Right? And that is also the range off co sign. Ah, close and universe. Right. This was it arranged. This is a domain off clues on X right, restricted domain of Cosa index. That is also the range of co sign the inverse effects, right. And then in every way, you can also see that the range off who's and ex right the range of this close and ex here in the range. It's also negative one and one, which is also did domain, right? The range is negative. One and one, which is also the domain off. Uh, this goes on in verse, right? Negative one. And positive, because this here is negative one, and this is positive one. But I said noticed it too, mate. And then it's also the range for, uh those coincide x so you can see the symmetry, right? What is the domain? Here is the range here. And what is the range here? Is it to me? That is the symmetry you can see there, right? And this is a restricted domain. Because in this domain here in this interval here, co Sinus strictly oneto one right?
So in this set of problems um because we don't have access to the actual sketch of the graph that's completely fine. However, what we are going to do is just kind of look at a general um understanding of a graph of F. Of X. And then we're going to talk about how we can use a graph to sketch our own graph. That's the directive. So let's say we have some random polynomial um three X cubed minus two, X squared plus X. Um request three X actually was two plus three is a good one. So um if I make this Like this and then we'll actually make this plus one. So with this in mind we want to figure out what the derivatives are going to be. So what the drought of graph is going to look like, well what we see is that if this was for example, um yeah we know that the easiest thing to find is where the slope is going to be zero. So we see that's going to be here and here. So where we have a slip of zero, we're going to make sure that when we draw our derivative graph at this point, 00 is on the graph and that 1.11, is on the graph as well. Then we see that the maximum point of slope is going to be reached right here. We'll soon later call this the inflection point but it's going to be when our slope is changing the largest and right here is where that's occurring, although this is a negative slope, so that's actually going to come down here. The easiest thing to determine is again, where we have the slope of zero and also if the slope is going to be negative or positive and based on those, we will be able to either have the derivative value positive if the slope is positive or the derivative value negative if the slope was negative. So using that, let's compare this to the actual f prime of X graph. And sure enough, we see that those zeros are located where we thought we were going to have the slope go down decreasing until it reaches the lowest point, which is right along this line right here where we see the slope is the most negative and then we see that the graph has a positive slope and that positive slope is very, very positive. The farther we go to negative infinity and it's very, very positive. The farther we go to positive infinity. So that is a good explanation of how we can take a general graph and understand what its derivative graph will look like. Just by looking at important points such as the minimums and maximums, as well as points of the greatest negative slope. And seeing the end behavior house slope increases or decreases as exposed to positive and negative infinity
So in this set of problems um because we don't have access to the actual sketch of the graph that's completely fine. However, what we are going to do is just kind of look at a general um understanding of a graph of F. Of X. And then we're going to talk about how we can use a graph to sketch our own graph. That's the directive. So let's say we have some random polynomial um three X cubed minus two, X squared plus X. Um request three X actually was two plus three is a good one. So um if I make this Like this and then we'll actually make this plus one. So with this in mind we want to figure out what the derivatives are going to be. So what the drought of graph is going to look like, well what we see is that if this was for example, um yeah we know that the easiest thing to find is where the slope is going to be zero. So we see that's going to be here and here. So where we have a slip of zero, we're going to make sure that when we draw our derivative graph at this point, 00 is on the graph and that 1.11, is on the graph as well. Then we see that the maximum point of slope is going to be reached right here. We'll soon later call this the inflection point but it's going to be when our slope is changing the largest and right here is where that's occurring, although this is a negative slope, so that's actually going to come down here. The easiest thing to determine is again where we have the slope of zero and also if the slope is going to be negative or positive and based on those, we will be able to either have the derivative value positive if the slope is positive or the derivative value negative if the slope was negative. So using that, let's compare this to the actual f prime of X graph. And sure enough, we see that those zeros are located where we thought we were going to have the slope go down decreasing until it reaches the lowest point, which is right along this line right here where we see the slope is the most negative and then we see that the graph has a positive slope and that positive slope is very, very positive. The farther we go to negative infinity and it's very, very positive. The farther we go to positive infinity. So that is a good explanation of how we can take a general graph and understand what its derivative graph will look like. Just by looking at important points such as the minimums and maximums, as well as points of the greatest negative slope. And seeing the end behavior house slope increases or decreases. Um as exposed to positive and negative infinity