Fun with graphs and calculators

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Understanding math in calculus is essential, as relying solely on graphing calculators can lead to misconceptions. A practical example involves graphing sin(15*pi*x) using different window sizes: one from -2pi to 2pi and another from -6.1 to 6. The choice of window size significantly affects the representation of the function, with the wider window potentially distorting the graph's appearance. Participants are encouraged to explore these differences using tools like Wolfram Alpha for better clarity. This discussion highlights the importance of grasping mathematical concepts rather than just using technology for calculations.
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I was coming up with examples for my students as to why we demand they actually understand math in calculus class instead of plugging all their functions into their graphing calculators to draw graphs and calculate integrals. I basically hit the motherload... I have confirmed this works on TI-83 and TI-84, so I imagine it works for TI-89s as well. Other graphing calculators probably require a different period depending on the screen size(if you have one post here and let us know what works!)

Draw the graph of sin(15*pi*x) with two different choices for the window size. For the first choice, let xmin=-2pi and xmax=2pi. For the second time, let xmin=-6.1 and xmax=6

Which graph is a worse representation of the function? Discuss.
 
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Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?

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