When discussing data and models for the Covid-19 Pandemic, one useful quantity to consideris P(t), the cumulative total number of Covid-19 cases confirmed up to and including day t.Note that we can set t = 0 to correspond to any given date. Although we might think of tas being an integer (i.e "day 1" or "day 10"), for the purpose of graphing and modeling thedata we can think of time as a continuous variable, in which case t = 0 might correspond tomidnight on a particular date.The total number of Covid-19 cases in the US confirmed between March 7, 2020 and March17, 2020 are shown in Table 1 below.1Date in 2020March 7March 8March 9March 10March 11March 12March 13March 14March 15March 16March 17P(t)19025332344460681310861419185024323214Table 1: Total number of Covid-19 cases in US, 2020. March 7 corresponds to t = 0.Covid-19 outbreaks in many countries, at least over short periods of time, appeared toshow an exponential increase in the number of cases. In this case, we would model the growthof P(t) with the differential equationP' (t) = kP(t)where k is a constant. In this assignment, you are asked to investigate whether an exponentialgrowth model is appropriate model for the data given.1. First, graph the data using your favorite graphing tool and describe what you see. Doesit look like exponential growth? Why or why not? (If you don't already have a favoritegraphing tool, I recommend Desmos.) 2. As we know from class, the solution to the differential equation above is P(t) = P₁ekt.Determine a value for the constant k that you think does a good job at describing thedata. Note that there are many different ways to do this, so be sure to discuss in detailwhat you did and why you did it that way. Graph your exponential model along withthe data and comment on how well (or not) it seems to be modeling the data.3. On March 27 (t = 20), there were 58,330 total cases in the US. Compare this number tothe model's prediction for that day. On April 30 (t = 54) there were 988,487 total cases.Compare this number to the prediction for that day. If there are discrepancies betweenthe predicted values and the actual values, comment on possible reasons.4. One way to better understand how close the data are to exponential is to look at thequantity1 dpP dtIf the growth is truly exponential then this would be a constant. On the other hand,if it's not quite constant, that's a clue that growth is not quite exponential. Since thederivative of P can be estimated by the average rate of change over one day, we canapproximate1 dpP dtP(t+1) - P(t)P(t)For the data given, make a graph of the quantity on the right hand side over time. Doesit appear to be constant?

Question

When discussing data and models for the Covid-19 Pandemic, one useful quantity to consideris P(t), the cumulative total number of Covid-19 cases confirmed up to and including day t.Note that we can set t = 0 to correspond to any given date. Although we might think of tas being an integer (i.e &#34;day 1&#34; or &#34;day 10&#34;), for the purpose of graphing and modeling thedata we can think of time as a continuous variable, in which case t = 0 might correspond tomidnight on a particular date.The total number of Covid-19 cases in the US confirmed between March 7, 2020 and March17, 2020 are shown in Table 1 below.1Date in 2020March 7March 8March 9March 10March 11March 12March 13March 14March 15March 16March 17P(t)19025332344460681310861419185024323214Table 1: Total number of Covid-19 cases in US, 2020. March 7 corresponds to t = 0.Covid-19 outbreaks in many countries, at least over short periods of time, appeared toshow an exponential increase in the number of cases. In this case, we would model the growthof P(t) with the differential equationP' (t) = kP(t)where k is a constant. In this assignment, you are asked to investigate whether an exponentialgrowth model is appropriate model for the data given.1. First, graph the data using your favorite graphing tool and describe what you see. Doesit look like exponential growth? Why or why not? (If you don't already have a favoritegraphing tool, I recommend Desmos.) 2. As we know from class, the solution to the differential equation above is P(t) = P₁ekt.Determine a value for the constant k that you think does a good job at describing thedata. Note that there are many different ways to do this, so be sure to discuss in detailwhat you did and why you did it that way. Graph your exponential model along withthe data and comment on how well (or not) it seems to be modeling the data.3. On March 27 (t = 20), there were 58,330 total cases in the US. Compare this number tothe model's prediction for that day. On April 30 (t = 54) there were 988,487 total cases.Compare this number to the prediction for that day. If there are discrepancies betweenthe predicted values and the actual values, comment on possible reasons.4. One way to better understand how close the data are to exponential is to look at thequantity1 dpP dtIf the growth is truly exponential then this would be a constant. On the other hand,if it's not quite constant, that's a clue that growth is not quite exponential. Since thederivative of P can be estimated by the average rate of change over one day, we canapproximate1 dpP dtP(t+1) - P(t)P(t)For the data given, make a graph of the quantity on the right hand side over time. Doesit appear to be constant?

Accepted Answer

Data for covid cases in US is given:
 
(1) To Sketch : The graph of above data using graphing tool.…