This page is designed to provide helpful information about the laboratory questions. You should start with your Cover Page and be sure that you have all the important contact information about your lab partner for coordination later in the week. On the cover page you begin by typing in the name of each team member and your computer numbers. For your WeBWork answers, it is important that you keep all calculations to 5 or 6 significant figures. Too few digits have been one of the leading causes of errors that we have found when approached by students.

Question 1: This problem is an extension of our problem of throwing a ball in the air. This time the ball is thrown at an angle, so there is an x component to the flight of the ball. The technique for solving this problem is very similar to the throwing a ball example in the lectures, where you find antiderivatives. The primary difference is that you need to find one set of antiderivatives for the x component of the trajectory and another set for the y component. The solution for the y component is exactly the same as before with the minor exception that the upward velocity is v₀sin(a) instead of v₀. But sin(a) is just another constant (though this constant changes with different angles of flight chosen). The solution of the x component is even easier because the acceleration is zero in that direction. (Note that the derivative of a constant is zero, so the antiderivative of the zero function is a constant.) You should get a quadratic function for y(t) and a linear function for x(t).

For graphing this function, you determine the amount of time required until the ball hits the ground. Then in Excel, you divide the time interval from 0 until the ball hits the ground into about 50 even steps. This can be inserted into one column, then the next two columns become the x(t) and y(t) solutions that are graphed. For finding the optimal angle, when you follow the directions in Part c, you should obtain a function in the angle of the ball thrown, a. You will differentiate this function with respect to the angle a and set the derivative equal to zero. This will require the differentiation of trig functions and knowing when they are zero. Your intuition about a ball being thrown should help you understand the results of this problem.

The last part of this problem is easily solved with the help of Maple's fsolve. You have found x(a), so you can set this expression equal to the distance you are throwing and let Maple find the angles a, which give the correct trajectory.

Question 2: This question can be considered an extension of either your work with discrete dynamical systems or with Euler's method. You will want to set up an Excel spreadsheet. In the first column you will put the values for time in your simulation. In the second column, you begin with the initial value of infected females, while in the third column, you begin with the initial value of infected males. In the second row of the second and third columns, you enter the Euler's formula that is given on your lab, inserting the parameter values given and using the values of infected males and females from the first cells. After you have done this, you simply fill down to the desired time, and the simulation is complete. All that remains is selecting particular values and creating a graph like you have done so many times before.

The equilibria can be readily found by taking the system of differential equations, setting the derivatives equal to zero, and solving for the variables x and y (from the right hand side of the differential equations). Maple's fsolve routine easily finds these equilibria. To solve a system of equations using Maple's fsolve, you first define the equations that you want to solve. Perhaps you would enter the following:

> eq1 := f(x,y) = 0;

> eq2 := g(x,y) = 0;

> solve({eq1,eq2},{x,y});

where you have to enter the appropriate functions f(x,y) and g(x,y).

Question 3: This question uses Excel's Solver like you have done several times this semester. This is a slightly more complicated application. The data is provided in your problem, and an Excel sheet is provided to aid your analysis. Data from your problem is entered in Columns B and C. You enter the appropriate guesses for the parameters in Cells J1:J6. Name your variables, then simulate the model with Euler's method in Columns L. M, and N.

In Columns D and E, you place the values from the cells generated in the Euler simulation. Be sure to take only the values from the cells with the same corresponding day. In Columns F and G, you insert the values for the square errors between Columns B and D and Columns C and E. Sum all these squares errors, then have Excel's Solver find the minimum with respect to the parameters in Column J. From here the techniques are similar to the ones you've done before.