In the first semester calculus course, we learn how to solve optimization problems such as maximizing the volume of a box or a can given its surface area, i.e., the amount of material, or minimizing the surface area given the desired volume. Whether you are a student or a teacher, have you ever wished you knew the answer to a problem after one glance at it, without doing long calculations? The patterns shown and explained below will enable you to do it for a few “standard” problems, but more excitingly, they will illustrate some very beautiful problem solving techniques and give insights into how different areas of mathematics are connected.

calculus, optimization, maximum area, maximum volume, minimize material, quadratic function.