Fun With Tangent Lines Jeff Morgan University of Houston.

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Presentation transcript:

Fun With Tangent Lines Jeff Morgan University of Houston

Before We Start… Shameless Advertisement and Three Challenge Questions

Coming Events at UH AP Calculus Workshop II – 10/21/ Algebra I Workshop II – 10/28/ High School Math Contest – 2/17/2007

1. Single Point Identification There are two rectangles on the right. The original one is depicted in blue. The yellow rectangle is the result of shrinking the blue rectangle in both the vertical and horizontal directions, rotating it, and repositioning it on top of the blue rectangle. Question: Can you show that the rectangles have exactly one common point? The solution requires trigonometry.

2. A Geometric Puzzle The solution requires thought and geometry. This problem was given to a large group of students who had never seen geometry. Many of them solved the problem (although not immediately!!). The Problem: Divide the circle into at least three pieces so that all pieces are the same size and shape, and at least one of the pieces does not touch the center of the circle.

3. Radio Play Joe Smith tunes into the same radio programming for an average length of 30 minutes the same time each day, seven days each week. What he listens to is a pre- recorded program that loops continuously through the 7- day week (meaning it repeats over and over again.) It is easy to see that 3.5 hours is the minimum amount of recording time. Suppose the station decides this just isn’t enough recording time and they want to know if there are other options. What are the other possible recording times which allow Joe to hear a different show every day, while remaining under 24 hours of recording? The solution requires thought and arithmetic.

Fun With Tangent Lines (Using a function and its tangent lines to create a new function.)

Can you make a conjecture? Can you prove your conjecture?

Can you make a conjecture concerning the relationship of critical points and points of inflection from the original function, and the behavior of the resulting function? Can you prove your conjecture?

Make a slight change in this process – Part I

Make a slight change in this process – Part II

Technology Tips Wacom Graphire Tablet Mimio Notebook Flash Animations with Wink.