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 on the closed interval [1, 12].

- Write an equation of a line, , 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 that gives the vertical distance between and . 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.

Click here for the solution.

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