This Geogebra file kindly submitted by Greg Bland dramatically illustrates that the derivative of a cubic function is a quadratic function. Sliders allow you to change the parameters of the cubic function.

Geogebra is a fantastic FREEE dynamic geometry program available from
http://geogebra.org

Students will approximate the area under a curve using Riemann sums. This will be done by utilizing a program that computes the Riemann sum as well as drawing the graphical representation. The activity concludes with students discovering that if enough Riemann sums are used, then the area under a curve can be calculated with the required degree of precision.

http://education.ti.com/educationportal/activityexchange/Activity.do?cid...

Move the red dot on the parabola and the gradient function (a linear function) is graphed below it. Very neat. From Manipula Maths.
http://www.ies.co.jp/math/java/calc/x_diff/x_diff.html

This applet also draws the graph of the 2nd derivative.
http://www.ies.co.jp/math/java/calc/x_2nd/x_2nd.html

Move the red dot on the trig function and the graph of the gradient function is shown below. Very neat! From Manipula Maths.
http://www.ies.co.jp/math/java/calc/sin_diff/sin_diff.html

This applet also shows the graph of the 2nd derivative.
http://www.ies.co.jp/math/java/calc/sin_2nd/sin_2nd.html

A man is riding on the surf. We set f(x) as the curve of the wave . Observe the slope of the surfbord. The trace of the slope is the derivative of f(x). From Manipula Maths.

http://www.ies.co.jp/math/java/calc/doukan/doukan.html

This is a series of interactive pages on distance-time graphs. It is like having a motion sensor when you don't have a motion sensor.

Take Your Dog for a Walk
A clever interactive introduction to rate via distance-time graphs. The full-screen version would be useful if this is done as a whole class activity.
http://nrich.maths.org/public/viewer.php?obj_id=4803

Motion Capture
Motion capture is a similar program but graphs displacement rather than distance, so it may be useful to introduce the concept of first and second derivatives.
http://nrich.maths.org/public/viewer.php?obj_id=4873&part=

Motion Sensor
Motion Sensor tests for understanding - can you describe how a graph was made?
http://nrich.maths.org/public/viewer.php?obj_id=4872&part=

Steady Free Fall
In this rich activity, you drop a ball onto a ramp of your own design and see the graph of vertical distance versus time. Some challenging questions are asked.
http://nrich.maths.org/public/viewer.php?obj_id=4851&part=

From Sean Bird's website: "Put this [TI-Nspire] file in MyLib so that you can access the area approximation methods from any document."
The program will find the approximate areas for the left, right and midpt Reimann sums, as well as the trapezoid, Simpsons and numeric integral areas.
http://cs3.covenantchristian.org/bird/TTT/NspireCalc/area.tns

For other TI-Nspire files from Sean's website, visit:
http://cs3.covenantchristian.org/bird/Nspire.html

Cubic splines are cubic functions which are used in applications such as automobile design. The basic idea is to fit cubic polynomials between two neighboring data points while ensuring that there are "smooth" first and second derivatives at the data points. The program available from this website as a zip file is a useful tool for investigating cubic splines.
http://www.vias.org/simulations/simusoft_spline.html

Distance, time and velocity are an ideal means to understand the concept of derivation. By reducing the time interval until it becomes zero, the average velocity approaches the instantaneos velocity, which is equal to the first derivative of the distance (with respect to time). This is available as a zip file that can be downloaded.
http://www.vias.org/simulations/simusoft_difftangent.html

Choose a function and see the effect on the Riemann sum when you change the slice width. The file itself can be downloaded and installed on your computer or network.
http://www.vias.org/simulations/simusoft_riemannsum.html