# Calculus | cal | California State University, Sacramento

In this project, you will be using real COVID data to explore the relationship between rates of vaccination and rates of death for a variety of states in the USA. After you gather some data, you will create models(graph, equations) and then use calculus to assess the accuracy of your models.

1. Go to __Mayo Clinic Vaccination Tracker__.

Just above the map of the United States, select “fully vaccinated.” Then hover your mouse over a state to reveal the percent of that state who are fully vaccinated.

Select 5 states and complete the first two columns of the table below:

**State** **in USA**

**% fully vaccinated**

**Total number of**

**COVID-related deaths**

2. Now go to __COVID Data Tracker__

Use the tool to find the total number of deaths for each of the states you selected in #1. Use the most recent date that is available to complete the third column in the table from #1.

3. Using your complete table, create a Scatter Diagram of the data. In other words, plot your (x,y) data pairs in the Cartesian Plane (x/y coordinate axis system).

• Be sure to scale each axis consistently.• Be sure to label what each axis is measuring, in words.

4. Write the equation of a linear function that “fits” the data in your Scatter Diagram as closely as possible.

• In other words, choose two (x,y) data pairs so that a line passing through those two pairs comes as close to the overall data set as possible. • Write the equation of a line passing through the two (x,y) data pairs you chose. Be sure to show all your work. • Do not just get the equation from software. Find the equation “by hand.”• There is a video posted in the Project #1 Module in Canvas if you need help with this.

5. Next, write the equation of a quadratic function that “fits” the data as closely as possible.

• You will again choose two (x,y) data pairs, or three pairs if you do not have a vertex. You do not need to use the same pairs you used in #4. Choose pairs that you think will make the equation fit the data, keeping in mind this may be very imperfect, depending on the shape of the data. • Show your work in detail. Do not just use software to get the equation.• If you need help with this, check out the video in our Canvas Project #1 Module.

6. Now write the equation of an exponential function that “fits” the data as closely as possible.

• You will again choose two (x,y) data pairs. You do not need to use the same pairs you used in #4 and #5. Choose pairs that you think will make the equation fit the data, keeping in mind this may be very imperfect, depending on the shape of the data. • Show your work in detail. Do not just use software to get the equation.• If you need help with this, check out the video in our Canvas Project #1 Module.

7. Now you are going to test each model (function).

A. Evaluate your linear function at the x-value forone of the states in your table. Compare the y-value of your function to the y-value in the table. How accurate was your function? B. Find the derivative of your linear function at the x-value for the same state you chose in part A. C. By going back to __COVID Data Tracker__, find the actual rate of change in deaths for the two most recent dates available for the state you’ve chosen. Compare this to the value of the derivative from part B. How accurate was your derivative function?D. Repeat parts A, B & C for your quadratic function.E. Repeat parts A, B & C for your exponential function.

8. Based on your work in #7, which of your three models is the most accurate model and why?

9. Using your model that is most accurate, evaluate the limit of the **derivative** of your function as “x” approaches 100 % fully vaccinated. Does the value of that limit seem accurate to you? Explain why or why not.

10. No mathematical model is perfectly accurate. As conditions change, models need to change. Discuss at least 3 conditions that might affect the accuracy of your models? In other words, what are at least 3 factors that might change how many total deaths there are from COVID?