Problem 5.4: Answer the following questions for the graphs below: Compute the largest eigenvalue Amax of the adjacency matrix (feel free [email protected] computer):Calculate t...

$[\mathbf{M}]$ Let $A=\left[\begin{array}{rrr}{-6} & {28} & {21} \\ {4} & {-15} & {-12} \\ {-8} & {a} & {25}\end{array}\right] .$ For each value of $a$ in the set $\{32,31.9,31.8,32.1,32.2\},$ compute the characteristic polynomial of $A$ and the eigenvalues. In each case, create a graph of the characteristic polynomial $p(t)=\operatorname{det}(A-t I)$ for $0 \leq t \leq 3 .$ If possible, construct all graphs on one coordinate system. Describe how the graphs reveal the changes in the eigenvalues as $a$ changes.

Okay, so let's talk about quadratic functions. Um, quadratic functions, like most functions, have grafts. The grafts are parabolas. And any type of graph we could talk about some of the characteristics I have here. We can talk about the intercepts, Um, definitely the range of any function. But, um, for sure you, if you're talking about quadratic functions, you'll have a vertex and a maximum, or it could be a minimum. So let's kind of focus on, you know, just one of these one of these parameters that we have 1234 parabolas. But let's just focus on one. Um, say we look at the graph of the one labeled end, and so this this one is a parabola. I would say this opens down, Um, and then we'll talk about what that means as far as one of these characteristics. But let's look at the start by looking at the intercepts. And so, um, this graph will have X intercepts and y intercepts when it says intercepts, that means where it crosses each axis. And so we have here. We have the X axis and we have the Y axis. So let's start with the X intercepts. So the X i n t the x intercepts. Um, let's see, where does it cross? The X axis is at these points right here. So these X intercepts, what are the coordinates for the X intercepts? Well, it crosses at the X axis at 10 So one comma zero. So the X coordinate is one. The y value is zero, and then the other one is that three. It's a very good three. Do 30 Notice something about the X intercepts. Well, first of all, there are two of them, and that can happen with a parabola. Either has 21 or zero in this case to a crosses the X axis twice, but the Y value. And this is true for any, uh, any function. The X intercepts have a Y value of zero or an output of zero. But let's look at the y intercept. We also have a Y intercept. Where does the graph end? Cross the Y axis. That's the looks like you can't really see it. There we go. That's the Y axis, and it crosses the Y axis right there. So that point right there, the y intercept, Why intercept in this case, it's reversed from the X intercept and that the X coordinate is zero. And in this case, the Y is negative. 30 negative three is the Y intercept of this graph. And while you can have two X intercepts for this type of graph for any graph or function, you can only have one y intercept. Otherwise, it wouldn't be a function because you have, um, the X value of zero are paired with multiple outputs. Or why values? So anyways, there's only ever going to be one y intercept with a function. Okay, so Part two, the Vertex. So the vertex is kind of I would say it's where the There's a lot of things about the Vertex. It's where the axis of symmetry crosses through. You can imagine if this little dotted line it just gets here. If you're able to fold this screen right along the axis of symmetry, the points on either side of it would line up with each other. Um, but the Vertex the axis of symmetry crosses through the Vertex, which is this point right here, which, if I So what are the coordinates of that? I would start at the origin go. 12 is the X value, and one is the Y value. And so the Vertex is the XY two comma one. Okay, so does this graph and have a maximum or a minimum. Well, I would say that this has why Why? Why do I think that this has a maximum is because it yeah, increases from left to right, and it decreases. So when it changes from increasing to decreasing, that is a maximum. It's also you can think of this as the I guess, the highest point on the graph. The greatest point. Um, if we're going to try to identify the minimum, we're going to have trouble is going to have to involve infinity, but at the maximum, they have a maximum value, so the maximum value is actually or the minimum if we're opening up. But this is kind of opening down, so it has a maximum, but the maximum. So we're gonna undo that and notice that the maximum So this definitely is a maximum. It's related to the vertex related to the Vertex. Because if we think of the maximum output, that's the Y. Value is one. You might see the maximum written this way. So when this function is evaluated at two, we get one. So that kind of describes what the maximum is. We think if you think of just the output is one. Um, F F two is another way to think of the maximum. Because when you evaluate this function at two, you get one. So that would be at looks like things kind of get messed around, messed up a little bit here. I don't know if I can fix that. I hope so. Let's see. There we go. That's better. Did that? Okay. Anyways, um, if you evaluate this function, go to the X axis that to the output is one. So that's that's the maximum output. Um, and then we would talk about the range. So the range of any function that is the set of why values and with we're talking about a real number system, you have a lot possible possible y value. So we're not going to say that is Oh, it's just 10 It's negative one. And list them out. Will use, um, special notations. Um, interval notation to right the range. So the set of y values with this being the set of why values We should look at the y axis. So at its maximum, the Y value is one. We learned that in the last problem, it's maximum the Y value is one. So I'm going to make a mark here on the Y axis, the range set of why values are outputs. That's the greatest, the maximum output. And then as far as the minimum will this graph, you know, going and either left or to the right approaches go. These little arrows mean it goes down forever towards negative infinity. So the range, we would say it goes from negative infinity all the way up to that maximum value of one. And that is included because there you take an X value to specifically you get an output of one. Um, we would not this the way I have this written here. Um, we don't put a bracket around infinity, any type of infinity ever because it's not really included as a number. It's just a concept. It's not a number. Okay, so there we have the intercepts X and Y intercepts of this graph and the Vertex the maximum and the range, which now that I think of it might write this in a different way. Um, the set of why values you could use, um, like, inequality notation, like all the Y values that are less than or equal to one. Okay, so anyways, I hope that helps. Thanks for watching.

In problem for the constraints of millennial programming problem Has the physical points of one and 3. Eight and zero, nine and seven. Five and eight. Zero and six. And finally the same 0.1 and three. Again, we want to get the maximum and minimum values of the objective function F equals two X plus five boy in this region. To do so we substitute by each point. In the physical region. We substitute only by the physical points and F here and get the maximum and minimum values. Let's try the first point. We have to buy buy one plus five, multiplied by three. If 17 the second point is two, multiplied by eight plus zero gives 16. Third point is two, multiplied by nine Plus five months. Applied by seven gives 53. Third point gives two, multiplied by five Plus five, multiplied by it gives 50. And the last point gibbs five to multiply by six plus zero gives 30. We can find the minimum value is here. If minimum equals 16 and the maximum value is here then if maximum equals 53. And this is the final answer of our problem.

Okay, so here we are given that V. Is equal to I. R. And we have that the resistance R. Is going to be about between. We have 1.6 is less than or equal to our which is less than equal to 3.6. And were given that i is equal to 2.5. So um we can multiply this inequality here Um with I equal to 2.5. So we have that 2.5 Um Times 1.6. So 2.5 Times 1.6 is going to be um less than or equal to. Um Well 2.5 Um times are so it's less than equal to I times are which is less than or equal to 2.5. Times 3.6. Yeah. Um which gives us here that 2.5 Time. So we had 2.5 Times 1.6 is less than or equal to v. Which is less than or equal to 2.5 um Times 3.6. So we get here that this is just equal of four. So we end up with food, we end up with four is going to be less than or equal to V, which is going to be less than or equal to nine. Uh So yeah, so here we have it, we have v less for less than equal to V, which is less than or equal to nine.