  Using calculus we show how to prove some combinatorial inequalities of the
type log-concavity or log-convexity. It is shown by this method that binomial
coefficients and Stirling numbers of the first and second kinds are
log-concave, and that Motzkin numbers and secondary structure numbers of rank 1
are log-convex. In fact, we prove via calculus a much stronger result that a
natural continuous ``patchwork'' (i.e. corresponding dynamical systems) of
Motzkin numbers and secondary structures recursions are increasing functions.
We indicate how to prove asymptotically the log-convexity for general secondary
structures. Our method also applies to show that sequences of values of some
orthogonal polynomials, and in particular the sequence of central Delannoy
numbers, are log-convex.
