The height of a conical sand pile is always twice the radius. if sand is being added to the pile at a rate of 30π cubic centimet
ers per minute, how fast is the height of the pile increasing when the circumference of the base of the sand pile is 120π centimeters?
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For a function to begin to qualify as differentiable, it would need to be continuous, and to that end you would require that is such that
Obviously, both limits are 0, so is indeed continuous at
.
Now, for to be differentiable everywhere, its derivative
must be continuous over its domain. So take the derivative, noting that we can't really say anything about the endpoints of the given intervals:
and at this time, we don't know what's going on at , so we omit that case. We want
to be continuous, so we require that
from which it follows that .
