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Electric Vehicles are an attractive option for the environmentally conscious commuter chiefly due to their lack of exhaust pipe. But do EVs have a secret carbon footprint that is hidden away from us? The short answer is yes, but they are still largely better in this regard than traditional internal combustion engine vehicles. If you are considering buying an EV with carbon emissions as your primary concern, going electric is probably your best choice, but there are cases when a hybrid may actually be better (for now)!

Consider a unit square and uniformly pick two points on its perimeter. What is the probability that the line that connects these two points has length less than 1? I first came across this problem in the accompanying youtube video, and decided to investigate it with a bit more depth. If you ever wondered about the sum of squares of uniform distributions, or if you are an avid shape-rotator, this one’s for you.

I claim that the three following definitions of the derivative are equivalent: $$ f’(x) = \begin{cases} \displaystyle \lim_{c \to x} \frac{f(c) - f(x)}{c-x} & \text{(A)} \\[10pt] \displaystyle \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} & \text{(B)} \\[10pt] \displaystyle \lim_{t \to 1} \frac{f(tx) - f(x)}{tx - x} & \text{(C)} \end{cases} $$ Most people who have taken a calculus course will be familiar with form B above, however there are contexts in which forms A and C are more convenient to use. Real analysis courses, for instance, often will prefer form A. In this post, I shall prove the equivalence claim.

One of the first “real” programs I ever wrote would encrypt a message using a substitution cipher and (more impressively) decrypt an encoded message without knowing the encryption key. Perhaps the first thing I had to come up with was an algorithm to take a cipher key that was used to encode a message, and transform it so that it could then be used do decode the ciphertext back into plaintext. I was aware of ROT-13’s property that if applied once to the plaintext, and again to the output ciphertext, it would yield the original plaintext. That is, ROT-13 is its own inverse. However, this is not the case for all possible keys, so I needed to find a more general algorithm.