Mean Numbers

Here is a problem for you and your students. The numbers are mean until you get to the end when they all become very nice and well-behaved.

You could give this to your students individually or as a group exploration. Give each person or group a different function and/or different intervals. Choose a function that has several (3 – 5) turning points in the interval. The function should be differentiable on the open interval and continuous on the closed interval.

It is intended that the work be done on a graphing calculator; you will need to carry 6 or 7 decimal places in their work.

Here is a typical problem. A link to the solution is given at the end.

Consider the function $f\left( x \right)=\sin \left( x \right)$ on the closed interval [1, 12].

Write an equation of a line, $y\left( x \right)$ , between the endpoints of the function. Give the decimal value of its slope and give a graph of the function and the line.
Write the equation of a function $h\left( x \right)$ that gives the vertical distance between $f\left( x \right)$ and $y\left( x \right)$ . Since f may be both above and below y this function may have positive and negative values.
Graph h and find its critical values. What are these places with respect to the graph of h?
Calculate the derivative of f at the critical values of h.
Interpret your result graphically.