Fun With Tangent Lines

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Fun With Tangent Lines

• Jeff MorganUniversity of Houston

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Before We StartShameless AdvertisementandTh
ree Challenge Questions
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Coming Events at UH

• AP Calculus Workshop II 10/21/2006
  http//www.HoustonACT.org
• Algebra I Workshop II 10/28/2006
  http//www.EatMath.org
• High School Math Contest 2/17/2007
  http//www.mathcontest.uh.edu

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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.
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2. A Geometric Puzzle
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.
The solution requires thought and geometry.
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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 isnt
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.
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Fun With Tangent Lines(Using a function and
its tangent lines to create a new function.)
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Can you make a conjecture?
Can you prove your conjecture?
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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?
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Make a slight change in this process Part I
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Make a slight change in this process Part II
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Technology Tips

• Wacom Graphire Tablet
• Mimio Notebook
• Flash Animations with Wink.