## SecondFundamentalTheorem Applet

SecondFundamentalTheorem illustrates taking the derivative of a definite integral when one or both of the limits of integration depend on the with-respect-to variable. It shows a graph of the initial integrand, a graph of the accumulation function created by the definite integral, and a graph of the derivative of the integral. The third graph is the same as the first graph when the dependent limit is just the variable. When the dependent limit is a function of the variable, the applet illustrates how the result gets multiplied by the derivative of the limit, due to the chain rule.

The applet tag that creates the applet is as follows:

<applet code="net.sourceforge.webcompmath.calculus.SecondFundamentalTheorem" codebase="../../assets"

archive="webcompmath.jar,calculus.jar" height="400" width="640">

<param name="UseLoadButton" value="no" />

<param name="Example1" value="1. A Line;t;0;0;x;-5,5,-5,5;-5,5,-5,5;-5,5,-5,5" />

<param name="Example2" value="2. Sine;sin(t);0;0;x;-6.28,6.28,-2,2;-6.28,6.28,-2,2;-6.28,6.28,-2,2" />

<param name="Example3" value="3. A Line with different b;t;0;0;2x;-5,5,-5,5;-5,5,-5,5;-5,5,-5,5" />

<param name="Example4" value="4. A Line with yet a different b;t;0;0;x^2;-5,5,-5,5;-5,5,-5,5;-5,5,-5,5" />

<param name="Example5" value="5. A Line with a different a;t;0;2x;3x;-5,5,-5,5;-5,5,-5,5;-5,5,-5,5" />

</applet>

This applet takes the following parameters:

Param NameTypeDefaultDescriptionExample1, ...String N/A Each example is a list of strings separated by semicolons. The strings, in order, are:

1. The name of the example, displayed in the example menu.

2. The expression for f(t), the integrand. In this applet, t is used as the variable in the integral, while x is the variable in the limit and in the derivative.

3. The value for x, which controls the location of points and tangents. Also changes the value of any integration limit that depends on x.

4. The expression for a, the lower limit.

5. The expression for b, the upper limit.

6. The xmin, xmax, ymin, and ymax for the left graph, as a comma-separated list of numbers.

7. The xmin, xmax, ymin, and ymax for the middle graph, as a comma-separated list of numbers.

8. The xmin, xmax, ymin, and ymax for the right graph, as a comma-separated list of numbers.