Imagine a rubber rope one meter long. An inchworm starts at one end and travels along the rope at 1...
Continue reading...analysis
Successive Differences of Powers
List the squares: 0, 1, 4, 9, 16, 25, 36, 49, … Then take their successive differences: 1, 3, 5,...
Continue reading...Suspended Rope Trick
What is the shape of a suspended rope? Is there some function that describes it? Answer: it’s the cosh curve!...
Continue reading...Calculus Limerick
Here’s a calculus limerick: Integral z-squared dz from 1 to the cube root of 3 times the cosine of three pi over 9 equals...
Continue reading...Why an International Date Line?
Can you find a continuous one-to-one function from the circle to the real line? Put it another way: is it possible, for...
Continue reading...Sums of Reciprocal Powers
You have seen that the harmonic series diverges. What about the sum of reciprocal squares? In fact, they converge, and to something...
Continue reading...Reflecting on the Hyperbola
Most calculus students have learned of the “reflecting properties” of the parabola and the ellipse. If a “beam of light” emanates from the...
Continue reading...Bike with Square Wheels
Ride a bike with round wheels and it rolls smoothly on a flat road (i.e., the axle remains level). Can...
Continue reading...One Equals Zero!
The following is a “proof” that one equals zero. Consider two non-zero numbers x and y such that x =...
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