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(6 points) Supporse f' (5) Shoukd expect the estimate of f(53) u-ins linear approximation be bigger than GMa Iler than f(5)? Justily UUL Husier(7 points) certa...

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(6 points) Supporse f' (5) Shoukd expect the estimate of f(53) u-ins linear approximation be bigger than GMa Iler than f(5)? Justily UUL Husier(7 points) certain virils spreading population such that A() TCprCcea the total number of cases of infection in the population_ days from nOw" (0 0is IlOV ) given that the total number of aei increasing Tal- of I0 cases day and the rate of clauge ofthe numlxr o cast> is increasitg At Tal- ol 0) Qais {day" Use the appropriate Taylor polyn

(6 points) Supporse f' (5) Shoukd expect the estimate of f(53) u-ins linear approximation be bigger than GMa Iler than f(5)? Justily UUL Husier (7 points) certain virils spreading population such that A() TCprCcea the total number of cases of infection in the population_ days from nOw" (0 0is IlOV ) given that the total number of aei increasing Tal- of I0 cases day and the rate of clauge ofthe numlxr o cast> is increasitg At Tal- ol 0) Qais {day" Use the appropriate Taylor polynotnia that incorporales this inforqation estiwale the total nuler &f uses ol inlection duys from [ot AssulG tcre M0 cx> ol inlection HUW (5 points) Iu puticular molel of thu spread of coronavirus (wluta f(r) is thue daily nuIer o cA53 And the muber of days since the beginning o inlection) , it is estimated that the daily nuinber cast > js tlie SalLE day and Ou day 25. What do Tol know (ue' erxph of f(r) Wst I bet#tYI 252 Explain using culus. You MUst Halle tle theorem related t0 this Four explanation



Answers

In the spring of $2003,$ SARS (Severe Acute Respiratory Syndrome) spread rapidly in several Asian countries and Canada. Table 4.9 gives the total number, $P$, of SARS cases reported in Hong Kong $^{17}$ by day $t,$ where $t=0$ is March 17,2003. (a) Find the average rate of change of $P$ for each interval in Table 4.9 (b) In early April $2003,$ there was fear that the disease would spread at an ever-increasing rate for a long time. What is the earliest date by which epidemiologists had evidence to indicate that the rate of new cases had begun to slow? (c) Explain why an exponential model for $P$ is not appropriate. (d) It turns out that a logistic model fits the data well. Estimate the value of $t$ at the inflection point. What limiting value of $P$ does this point predict? (e) The best-fitting logistic function for this data turns out to be $$P=\frac{1760}{1+17.53 e^{-0.1408 t}}$$ What limiting value of $P$ does this function predict? Total number of SARS cases in Hong Kong by day $t$ (where $t=0$ is March 17,2003) $$\begin{array}{c|c|c|c|c|c|c|c}t & P & t & P & t & P & t & P \\\hline 0 & 95 & 26 & 1108 & 54 & 1674 & 75 & 1739 \\5 & 222 & 33 & 1358 & 61 & 1710 & 81 & 1750 \\12 & 470 & 40 & 1527 & 68 & 1724 & 87 & 1755 \\19 & 800 & 47 & 1621 & & & & \\\hline\end{array}$$

So hello Strengthen our going. Unless on this question. And I just leave sold here and get because nothing to solve here it is already. They were given everything in the table so it is given in table that no dicks But this I use the table represented Johnson G that compute the number of people here who were infected are plexus. So the this formula we have generated, the people are affected upper x days. So just subject fx from to 61 will get DX. The people affected up accidents. So this is the GX. When people are affected on this the probably using second part and Farsi boards. And if you calculate using the graphing calculator and find a push one why would equal to 1 71 upon one plus 18.6 of the power minus 0.747 x that is that's the difference. The best GXE and using relation B and C using the situation just big FX left unsaid g exciting sex. So this is listening often and the lip replacing gx of it. Why? Because why would antiques both are same. See openness

Given functional apathy equals toe accepted. Three multiplied by 1.0 to wait to the power people and he'll be equals Toe 54,700 in to the power minus B minus 200 all square divided by 75 about judo. Also, we are provided with a table in this question. Here is the table. Now we will change the table. They will put equals to zero equals to zero corresponding toe against 18 14 Now we will change the table accordingly. The new table is as follows here we will like be in their new cases people to zero new cases five Musics at people's to continue your face is 1 to 89 active was six new cases 347 and people 24 new cases 9597 At people's toe, 112 new cases Burnett 817 People's 21 14 New cases is 33,000 and 35. That he was 216 years. Your visit is in cases are 47,001 night. This question consist of three parts. We will start with party in party. We will represent the graphs off the to function and be stable. Cherries are craft. This is our accept things. This is our boxes. This curve represents function fst and this could represent function. Tee off. Now we will do part mhm in bad. We we have to return my in which function better Motors parks Prediction. The function g o t Better models parts production. This is because it's graph is closer to the graph that shows as prediction. Now we will do party in this part. We have to determine the date on this the number of new cases he now considering the graph of the function g o p BC that the number off new cases speak. That equals to 200. This means we have 200 minus 1. 68 equals two 32 days upset January 27. So that date will be February 28. Thank you

So for this given formula, we need to draw the second secret line. Who Slope is average rate of change and infected children over the intervals for six and 12 14 than compute these average rate. So what we'll do is basically a secret line. Is the line drawn between two points? So let's do that for this first interval here or on the Y axis. Let's let's to note that as a point, which is right here and then for six, our wise air here. So we will draw online in between these two points on DH. Then let's do the same for 12 and 14. So 12 is right here. That's it's why and then 14 is right here. That's it's why and so its secret lines from right here. Great. And now let's compute each of these average rates in units of percent per day. So for six, what were simply doing is we need to find the co ordinates before we plug it into a rate of change formula, and the way we do that is we simply plug in these exes back into this function, so when I do that, you should get a value of 12 white, 903 and then for six. It's why coordinate is 20 point or 55 perfect. And now let's plant that into a rate of change equation, which is change of why we're changed bags. And I'm just going. Teo put a green box to help me organized on DH. Now I will subtract both of these y values. So 20 point for by five minus well, 0.9 au three all over six minus six minus four. And what you should get is 3.776 percent per day. That's your answer for the first part of a part. Ay, for the second part, Um, what I'm going to do is always the same exact steps for the interval. 12 14 on DH. When I plug in these X values back into the function to Courtney Hits I get is okay. So for 12 I have eaten point A for five, and then for 14 I have 14. I have, um, 7.2 for nine. And let's plug that into this equation to get our rate of change. So seven point two for nine minus and 0.8 for five all over um 14 minus 12. And he should get a rate of negative 0.798 percent a day. And that is the second answer for part A. So, for part B is a rate of decline. Greater At T equals eight er t equals 16. So what I will do for this part is I'm actually going. Teo, change the color. Let's see if this works this time. Great. So, um, a T equals a What I'm going to do is actually draw a tangent line. Um, in order to help me count Ilia rated change, whether it declines faster, and the steeper it is, the greater the rate of decline. Um, so I'm going, Teo, a race a little bit of this point in orderto make way for Actually, I'll leave it as is, so that we have this visual s o for a We can draw point here. Oops. For eight. We can draw point here. Yeah, the blues working. Okay. Drop point here and then make a small tangent line. And then for 16 which is way out here, we'll draw this tangent line on notice that this is a lot steeper than this line. So we know that the rate of decline is greater. At T equals staying. Um, and then now we're going to estimate the rate of change of end of tea on day 12. So when we estimate the rate of change on a particular day, essentially what we're doing is calculating the slope of the secret line over given Interval. So we're estimating the instantaneous rate of change on day 12. So the interval that I'm going to choose for this problem is going to be a 10 and 14. The reason why I chose 14 is because we already calculated that, um, coordinate, which is right here and now. All I'm going to do is I'm going to calculate the coordinates for 10. So when I plug in 10 back into the function, my wife corner is going to be 11.364 And now, when I plugged this back into the rate of change equation, I get 7.2 for nine oneness, 11.364 all over, uh, 14 minus 10. And my answer is going to be 1.29 Make sure you put that negative over here because we're subtracting a number greater than seven on the top, so negative 1.29 percent each day. So, in sum, all we're doing in this problem is looking at seek it lines and using this change of whatever changes, a formula to calculate average rate of change. And we're also looking at a tangent lines in order to help us compare rates of declining given points. And when we're estimating instantaneous ray of change. What we want to do is essentially calculate the slope of a secret line, using the formula that we mentioned on the very beginning of this problem, which is this change of while retreating six.

All right, we've got a question. Here we have the measles. How to Janet Genesis? A patient has that previous exposure to measles. The immune system responds more quickly. Results in a suppression suppression of the level of virus in the plasma Dorian infection. Suppose that the previous exposure causes the viral density and the platinum 3/5 with, if that in a patient that has no previous exposure. This means that the level of virus in the book in the plasma is given by 3/5 function of t. Right, So here we have FFT given us negative t multiplied by T minus 21 multiplied by t plus form. Alright, for part a of the question we're asked to use up in the rules of within two days and their mid points to estimate the total amount of affection at days 12. Okay, So first of all, we know that the level of the level of virus and plasma is 3/5 function of t. He would say that that function would be equal to 3/5 of the function of T, which is the same thing as saying Brief lifts multiplied by t t minus 21 Chief apostle. All right, So if we're looking to find if we're looking to use the midpoint, it points to estimate the total amount of infection. At days 12, we have to first write out sub intervals. I would say that our change in T is equal to 12 minus here over six, which is equal to we would write out the rose from 0 to 2. 2 to 4. 4 to 6. 68 to turn 10 to 12. All right, so the midpoint here would be 13 579 and 11 between those sub intervals. And then we're gonna plug that in to our function here, and we were plugging each value. And if you plug in one, you'll get 24 plugged in to into that equation. You would get he Excuse me two or three into the equation. You can get a 1 29 6. You plug in five, you will get 2 88 plug in seven. We'll get 4. 70.4, and then plug in nine. You'll get 6 48 and then finally, for 11, you will get 7 92. Okay, so we can calculate our midpoint by setting it equal to the change in T, which is to multiplied by F. One was F three was five plus so on and so forth, all the way to our for a woman, right? Once you plug those answers into your calculator, you will get a final answer that comes out to 4000 0 1706. Yeah, that'll be our answer. Part A and part B were asked that if 7848 cells per million litres X days is the total amount of infection required to develop symptoms, use the sub in the rules two of with two days and their mid points to estimate the day when symptoms will appear. Okay, so this is up until 12 days. So let's take the interval from 12 and 14. It'll be 13. We will calculate for F of 13, multiplied by two and add it to our answer that comes out 8 73 times two 8 73. So F F 13 is the same thing as 8 73.6. We multiply that by two, and that equals to 1747.2 then if we add that to our, uh, previous answer, which was for up until D 12, we would have 47. Our total will come out to be 6000 453 when we get to day 14th. So let's try it until 14 to 16 in the mid point from 14 to 16. We know it's 15. So we saw for two F for 15 and two of the F 15 comes out to be 8 64 8, 64 times to 17 28. If we take 17 28 and add it to our the function of F one f the days at the end of 14 days, which was 64 53 2 we get to or total, which is 8100 8000, 181.1, and we can see that that's clearly larger than 7800 48. Before we can say that at day at the 16th day you will start to develop symptoms, all right, and that will be our final answer there. I hope that clarifies the question. Thank you so much for watching


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