SDSU

Math 122 Calculus for Biology II
Fall Semester, 2001
Lab Help

16-Nov-01

San Diego State University


Laboratory Help Page

This page is designed to provide helpful information about the laboratory questions.

Question 1: This problem begins our study of integration. A ball thrown vertically under the influence of gravity satisfies a simple differential equation. Antiderivatives are used to solve the differential equation. The acceleration is the first derivative of the velocity, and it is a constant. Try to think what function which upon differentiation produces a constant and that is the velocity. Since the derivative of a constant is zero, all antiderivatives include an arbitrary constant that can be adjusted to match the initial conditions. Our lectures in lab and and the classroom will help clarify this process of integration.

You can use Maple's int function to find any antiderivative. This is known as integration. Thus, to find the indefinite integral of a function f(x) written as

The Maple command is simply:

> int(f(x), x);

Note that you need to provide the "+C" by hand since MAPLE does not include it.

Question 2: 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 Question 1, 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.