Taylor series

Today, XKCD, with tongue in cheek, is saying that “The Taylor series should have been canceled after the first term”.

After a flying visit to London (more about that shortly), I am reminded again that at least the second term of the Taylor series impinges on real life.

My house in London is in a courtyard entered through double gates hinged at the outside. By entering a code, you can open either a single or a double gate. The gates are rather slow to open, and of course I get impatient.

Assume for simplicity that the width of each gate is one unit of length, and that the gates open at 1 radian per unit of time. Now, if I open a single gate, then after time t, the gap for me to get through is 2 sin t/2, whose Taylor series is t+…; while if I open both gates, the gap is 2(1−cos t), whose Taylor series is t²+…

I have watched people waiting for the double gates to open wide enough for them to slip through, and grumbling about the slowness. Of course I feel a little smug!