## What is the carbon footprint of your EV, really?

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)!
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## An Extension of a Probability Puzzle

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## Alternative Definitions of the Derivative

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.

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## On Substitution Ciphers

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.

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