This mathlet provides an interactive and visually attractive illustration of the geometric as well as the practical meaning of the Fundamental Theorem of Calculus.
Barbara Kaskosz is a Professor of Mathematics at the University of Rhode Island.
In the Demo Examples mode, the mathlet displays a graph of the rate of change of a population of bacteria, together with the initial size of the population, and asks the user to sketch the graph of the population itself, using the mouse. The real graph of the population can then be displayed by dragging a slider. Many such examples are provided to illustrate increasing, decreasing, and constant behavior of the population in relation to the shape of the rate of change. This mathlet helps students master the relationship between the rate of change of a quantity and its total change, which is so important in calculus, especially applied calculus.
In the User Defined Function mode, the mathlet allows the user to enter a formula for a function f(x) in simple syntax, basically the same as that used by most graphing calculators. The x- and y-ranges and the initial value F(a) = b for an antiderivative are chosen by the user as well. The mathlet graphs f(x) and allows the user to practice sketching the corresponding graph of the antiderivative. The real graph of F(x) can then be displayed by dragging a slider.
The User Drawn Function mode is similar to the preceding mode except that the user is allowed to draw a graph of a function f(x) whose antiderivative is being sought, rather than enter a formula for f(x). The mathlet will deal with piecewise continuous functions f(x) and is forgiving as to the quality of the graph. Again, the user can practice drawing the anitderivative before displaying its actual graph. This option may be particularly useful for classroom demonstrations to illustrate many different behaviors of f(x) versus F(x) without having to use a specific formula for f(x).
Open Accumulated Change and Antiderivative Plotter in a new window
The mathlet will run on any machine with a generic browser as long as it has Flash Player 6 or higher. The free and small (668KB) Player can be easily downloaded and installed from the Macromedia site -- click on the button at the right. Netscape 6 or higher usually comes with the Player ready to use.