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Author Archives: Aaron
Sufficient Condition to be a Polynomial?
Appropriate Level: Real analysis. The Question: Suppose is infinitely differentiable and that for each there is some for which the th derivative of at equals 0. Must be a polynomial?
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Cyclotomic Polynomials
Appropriate Level: Abstract Algebra The Question: Recall that the cyclotomic polynomials are defined for so that . Is it true that every coefficient of every cyclotomic polynomial is either 0, 1, or 1?
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Where was this picture taken?
Appropriate Level: Trigonometry Background: Here is a wonderful photograph: This picture was taken by an astronaut onboard the Apollo 17 spacecraft in 1972. This was the last manned spaceflight to the moon and the last time someone was far enough away from … Continue reading
Posted in PreCalculus, Trigonometry
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Coloring the Coordinate Plane
Appropriate Level: Geometry and Beyond The Warmup Question: If every point of the plane is colored either red or blue, do there necessarily exist two points of the same color that are exactly 1 unit apart? The Question: If every point of … Continue reading
Posted in Geometry
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Watching the Sunrise (and figuring the radius of the earth)
Appropriate Level: Trigonometry The Story: I spent a summer in Chicago and early one morning I watched the sun rise over Lake Michigan (the lake is large enough that you cannot see across it, at least from Chicago). I stood … Continue reading
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Cutting a Triangle from a Square
Appropriate Level: Calculus (1 or 2, depending on how your courses are divided). The Question: You start with a unit square (say a piece of paper). Choose two adjacent sides of the square and randomly pick a point on each. … Continue reading
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Power of Exponential Growth
Appropriate Level: Algebra 2 (with logarithms) The Question: Suppose you start with an object that has the mass of a proton. If this object doubles in mass every five minutes, how long will take for it to grow to the … Continue reading
Posted in Logarithms, PreCalculus
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