A carnival game had a spinning wheel with 20 equal-size sections: some red, some black, and some green. Every time the wheel was
spun, the game attendant recorded the color selected. The table shows the results after 20, 200, and 2000 spins. What is the best estimate of the number of green sections on the wheel?
For to be differentiable, it must be continuous, so we need to have
By its definition, . The one-sided limits are
so we require .
In order for to be differentiable at , we also need to have exist, which requires that also be continuous at . First, compute the derivatives of all pieces of :
is continuous at if
The one-side limits are
so we need to have , and moreover will be differentiable if we set .