a) Let R(r) be a function that gives the increased risk of an outbreak of the disease in % for a community that has an assortativity index of r. That is, R(r) tells us how much more likely a community with assortativity index r is to have an outbreak than a community with assortativity index 0. In figure 1, we see some data the authors collected for the function R(r). These data show that a person living in a community that has a high assortativity index r (that is, in a community that is opinionated about vaccines) is more likely to be infected with HIN1 than a person living in a community with a lower assortativity index r. The horizontal axis shows how opinionated a community is, as measured by the value r, while the vertical axis gives the increased risk of being infected. For example, a community where the assortativity index is 0.12 is about 200% more likely to have an outbreak of a disease than a community where r = 0. Interpret each of these quantities in a complete sentence that would be understandable to someone who doesn't know calculus (e.g. do not use phrases like \"changing at\" or \"rate of change at\"). For example, when interpreting R(0.06), your response might be: \"R(0.06) is the increased risk of an outbreak in a community that has an assortativity index of 0.06, relative to a community with assortativity index 0. \" i. R~1(150) ii. R'(0.09) iii. (R1)'(600) b) For each of the quantities in Question 1a), use the graph in figure 1 to approximate the quantities. 2. First, let's review the behaviour of exponential functions. Exponential functions are commonly used to model growth, especially of biological phenomena, because they capture the behaviour of cells splitting in half. We used it before in the first tutorial to model cancer cell growth. Let's assume a hypothetical E. coli bacteria replicates every hour (in reality, it can vary a lot depending on conditions, between approximately 20 minutes in a lab to approximately 1 day in the wild). If we start off with one cell, we expect a doubling every hour, giving a) Fill in the Backward, Forward and Average Change columns of the table with the differences between the current row number of cells and the previous, and next row, and the average of the previous and next values respectively. b) What are we estimating when we calculate the change? What do you notice about the Backward and Forward Change columns? Which special type of function has this property? c) Write down the function (f(z)) modeling the Number of Cells over time. Take its derivative, and then use to fill in the last column. What do you notice when comparing these values to the other columns? Why is this the case? 3. Let's now think about the long term behaviour of functions, and how that affects their use in modelling. Let f(z) = 2%, and let g(x) = 22 a) What is lim f(z)? T33O0 b) Referring to the example of cell growth in problem 1, what does this mean? Is this sensible? Does this mean that exponential functions are never a good model for cell growth? ) As a refresher, what is lim g(z)? What is lim f(z) g(z)? T>00 Tr00 d) Quadratic functions are common in physics in situations where the forces acting on an object are constant. For example, the height of a ball dropped from a window 10 meters in the air over time can be modelled by the following equation: B(t) = 4.9t* + 10 What is tlim B(t)? Does this make sense? How do we explain this? 00 e) What are lim f'(r) and lim '(x)? Explain what this means in words. 1 f) Let h(z) =3+ . What is lim h(z)? (Feel free to graph this function). T T00 g) Without calculating the derivative, what can you say about lim h'(z)? rr 00 h) A common building block for mathematical models that does not explode to infinity is the logistic function: S(z) = %. Graph this function using Desmos/Geogebra. What is lim S(z)? 00 el' i) Without calculating S'(z), what is lim S'(z)? j) Calculate and graph S'(z) to check your answer to the previous question. k) The previous questions suggest that if for a function M (z) we know that lim M (z) = C for some Tr0o0 constant C, then lim M'(x) = 0 (this is true for cases where the limit lim M'(x) exists). Is the Tr0o0 Iroo opposite true: Does lim M'(z) = 0 guarantee that M (z) has a horizontal asymptote? Tr00 Hint: you will need to know that % log(z) = % Think about the behaviour of these two functions. Often, the eventual importance and significance of research is not obvious when it is conducted. Rather, some research can gain in significance over time. This has been especially true during the recent pan- demic, where the battle against COVID-19 has been accelerated by research done years ago. The most prominent example of this has been the mRNA vaccines, where years of research allowed companies to cut down development time of a vaccine from years to months. Canadians were fortunate to have ample vaccine available, but we had to deal with a different issue, \"anti-vax\" sentiment. To understand how this sentiment spreads and what effect it may have on future outbreaks of COVID-19, we can also look to previous research. In order to understand the psychology of the anti-vax phenomenon, Marcel Salath and Shashank Khandelwal from the University of Pennsylvania studied the spread of anti-vaccine sentiment through social media by looking at tweets during the 2009-2010 HIN1 (Swine Flu) pandemic. They measured several different communities and considered the \"assortativity\