MTH 124 Goals Focus Session 3 Average Rate of Change and Pandemics Many real world phenomena are best understood by measuring the change of a function over time Two concepts of change are important the total change as well as the rate of change We use the rate of change to study the 2020 COVID 19 pandemic Practice computing average rate of change and instantaneous rate of change Determine how average rate of change relates to graphs of secant lines Explore the limitations of some models Pre activity Do before class De nitions The average rate of change AROC of a function f t on the interval a t b is AROC Change in f t Change in t f b f a b a A secant line is a line connecting two points on a graph The slope of the secant line of a function over an interval is exactly its AROC over that interval The instantaneous rate of change IROC of f t at t0 can be approximated by the average rate of change of f t over the interval t0 t0 h or t0 h t0 for small h 0 Namely IROC f t0 h f t0 t0 h t0 f t0 h f t0 h Review your class notes for 3 4 3 1 and 3 5 Answer the following questions in your own words What does the average rate of change mean in terms of the graph In terms of a model What does instantaneous rate of change mean in terms of the graph In terms of a model 1 MTH 124 Focus Session 3 Average Rate of Change and Pandemics In Class Activity Below is a graph of the daily new cases in the US from February 1st 2020 to September 30th 2020 Data sets come from https covidtracking com data download Graph available at https www desmos com calculator 3yhoyaho9h Date x days since Feb 1 y new cases Feb 1 2020 Mar 1 2020 Apr 1 2020 May 1 2020 June 1 2020 July 1 2020 Aug 1 2020 Sep 1 2020 0 29 60 90 121 151 182 213 0 24 26271 32808 20101 51046 60679 42242 2 MTH 124 Focus Session 3 Average Rate of Change and Pandemics 1 Looking at the plot on page 2 for daily new COVID 19 cases in the US mark the points corresponding to March 1st 2020 and June 1st 2020 Determine the total change in the daily new cases in the US from March 1st 2020 and June 1st 2020 2 On average how much did the daily new cases change per day from March 1st 2020 and June 1st 2020 3 Given your result from Question 2 how many new cases do you estimate there were on March 3rd 4 In the plot on page 2 sketch a line that intersects March 1st 2020 day 29 and June 1st 2020 day 121 and determine a model equation for the line Verify your answer by graphing your result in Desmos https www desmos com calculator 3yhoyaho9h This line has a special name what is it 5 Determine the AROC of the daily new cases including units a between February 1st and September 1st and b between May 1st and June 1st 3 MTH 124 Focus Session 3 Average Rate of Change and Pandemics 6 Using the full daily data available at https www desmos com calculator 3yhoyaho9h approximate the IROC of the daily new cases on August 1st including units Hint Choose a small value for h Using the data points associated with March 1st and August 1st we can nd the exponential model for the number of new cases per day since February 1st N t 5 4354 1 052545 t 7 Using the model N t above estimate the IROC of the daily new cases on August 1st include units in your solution Hint Choose a small value for h that is h 1 We can use AROC with a small enough interval to estimate the IROC of any function about a point However we must take care to think about if our model makes sense in the rst place If our original model is not a good t then any interpretation such as AROC or IROC is likely to do a poor job explaining the data especially when used for future predictions 8 In Desmos https www desmos com calculator 3yhoyaho9h add the model N t to the graph a Is N t a good model for the number of new cases in August Why or why not b On a new line in Desmos enter N cid 48 182 This number provides the exact IROC of our exponential model on August 1st We ll cover details about this in upcoming sections How does the exact IROC compare to your answer in Question 7 c Consider your answers to Questions 6 and 7 Which one of them is a better approximation of how the number of daily cases is changing around August 1st Why 4