Pebbles

Calculus means stone or pebble in Latin. Roman numerals were so awkward that ancient Romans carried pebbles to compute their grocery bills and gamble.  As a result, "calculus" came to be known as a method for computing things. The specific calculus is "Differential and Integral Calculus," formerly called "Infinitesimal Calculus."  The fundamental things this calculus computes are rates of change of continuous quantities and accumulation of known rates of change.  There are a lot of technical "pebbles" in the usual course on Calculus, but the subject is important as a language that describes change.

The language of change

Calculus as the language of change really does give us deep insights. Plain English is not as powerful in understanding certain consequences of change.  I like to begin calculus with the question: Why did we eradicate Polio by vaccination, but not measles? because the primary importance of teaching calculus to a broad college audience is to show them that calculus has important things to contribute to solving real problems that they care about.

Why did we eradicate polio by vaccination, but not measles?

The Mathematica Notebook PolioNotMeasles.nb helps YOU answer this question step-by-step.  It illustrates how calculus acts as a language of change and does not require any previous calculus (but does use computing.)  A more detailed approach is BeginningCalculusQuestion.zip.