## Derivative of a Cubic Function

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

## Derivative of y= x^2

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

## Derivative of Sine and Cosine Functions

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

## Surfing

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

## Take Your Dog for a Walk

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=

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=

## Learning by Simulations - Cubic Splines

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

## Learning by Simulations - Differential Calculus

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

## Average Velocity

Drop the ball in a vacuum under the influence of gravity. Set the green line to start the timer and the red line to stop the timer. From the time taken, estimate the average velocity over this time interval. Bonus: You can also choose your planet!
http://jersey.uoregon.edu/AverageVelocity/

## Newton's Method Animated

This is a really cool interactive applet for demonstrating Newton's Method.
http://www.math.umn.edu/~garrett/qy/Newton.html

And here is one from the same website that shows a 'pathelogical' case, where the iterations don't converge on the nearest solution: