Background

The title of this talk is the same as the title of the book [1] by Ivan Niven published by the Mathematical Association of America in the Dolciani Series. It is written as exposition at a level for undergraduate students and I recommend it to you if you want to look more closely at the subject of the talk today and related topics. My list of references at the end of the book is not complete. There is a vast list of articles and books compiled in this book by Niven and you should go there first. Although I knew that many problems in calculus had been solved before calculus was invented, I had no idea of the wide variety of problems that one sees in every calculus book among the exercises that can be solved with just a few techniques. I remember solving them as a student and now I assign them as a teacher. However, I do point out to my classes that there are elementary methods that often are easier to implement. Professor Ogilvy has said ``these are `trick methods', each applying solely to its own problem. Usually they cannot be extended, lacking the great generality of the analytic (calculus) methods.'' For example:
Find the coordinates where $$f(x) = \frac{9x^2 \sin^2 x+4} {x \sin x}$$ has a relative maximum on $(0,\pi)$.

Niven on the other hand agrees that while calculus is good ``for solving some problems in maxima and minima, the method is not universal. There are many problems that are awkward, if not impossible, to solve with elementary calculus ... Thus we follow a simple maxim: If a problem can be solved more simply with calculus, leave it to calculus.'' Some examples:

Find the minimum value of $$f(x) = \frac{9x^2 \sin^2 x+4} {x \sin x}$$ on $(0,\pi)$.

Find the quadrilateral with the largest area with a given perimeter.