Introduction: What is Calculus About?

The seeds for the main ideas of calculus go back to ancient times but calculus itself, as we think of it today, was invented (or discovered?--which?) during the 17th century as part of an explosion of interest and discovery in the physical sciences.  Its invention is usually attributed to the English mathematician Sir Isaac Newton and the German mathematician Gottfried Leibniz, working independently. As its power developed, calculus gave scientists a tool to generate remarkable new understandings of the world.  Its creation is considered one of the great intellectual accomplishments of the human mind.  This sweeping assertion is justified not only by the beauty of the subject, but also by the fact that it still retains its fundamental importance, even several centuries after its birth.  In fact, its role has become more important than ever as the use of mathematical models has reached beyond areas such as physics and engineering and into such different fields as biology, economics and business.

Of course there are also other important mathematical tools.  Algebra, discrete mathematics, probability and statistics, topology, and computer science all have roles to play as research areas and as tools in applications.  These diverse parts of mathematics complement each other.  The increasing power and availability of technology enhances their usefulness and doesn't replace the need for any of them.

Graphing calculators are now good enough to be really helpful with numeric calculations and graphical interpretations of what's happening in calculus.  The more powerful calculators (such as the TI-89, TI-92, HP-48 and HP-49) contain a Computer Algebra System (CAS) that can do complex symbolic manipulations.  Computers, of course, can do even better and with much prettier output.  Technology makes it possible to explore calculus numerically and graphically in ways which were impractical even a decade ago, but technology cannot replace understanding the subject.  A calculator or computer is only an assistant that needs an intelligent user.  Otherwise it may be unable to find an answer, or may produce an "answer" which is misleading or even completely incorrect !  The technology needs a user who understands and can tell it exactly what it's supposed to do.

There are lots of details and techniques for us to learn, but in the big picture there are only two "great ideas" in calculus.  Both of them are illustrated on the dashboard of your car.

1)      The first idea concerns "how fast is a quantity changing?"  For example, if you're driving down the highway and s represents the distance you've traveled from home, then you might be interested in how fast s is changing (measured, perhaps, in km/hr).  How fast s is changing at a time t is your velocity v at that time.  

Studying rates of change involves a concept from Calculus I  called the derivative.   The velocity v is the derivative of the position function s.  If we think of  s = f ( t ) as a function of time, then some of the ways of writing the derivative are v = f ' ( t ) or v = ds / dt.  A derivative is analogous to the speedometer on your car.

Of course, instead of how fast s is changing, you might be interested in the rate of change of some other quantity.  For example,

  • y = the size of a population of bacteria
  • y = the amount (mass) of a radioactive isotope in a sample
  • y = the price of a gallon of gasoline.
  • Whether a rate of change occurs in biology, physics, or economics, the same mathematical concept--the derivative--is involved in each case.

    2)     The other idea is "opposite" to the first one.  If you know how fast some quantity is changing, then how big is it at a certain time?  On the highway again, you could imagine trying to figure out how far s you are from home at time t by studying the velocity v (speedometer information).

    This is easy to do if the car moves at constant velocity:  in that case, s = vt because distance = (rate)(time).  But finding s from v is harder if the velocity is variable during the trip. The calculation of a total amount s from a rate v =ds/dt involves a concept from calculus called the integral.   An intergal analogous to your car's trip odometer which (if you set it to 0 when you start out) measures s on your dashboard.

    Instead of a distance, you might be interested in trying to compute the total amount of money in an account if you know how fast it's growing (the interest rate), or the total number of people infected with some disease at time t if you know the infection rate.  The same mathematical concept, the integral, is involved in each case.  The study of the integral and how it relates to the derivative, is the main theme of Calculus II.

      Most of Math 131-132 consists of
    • developing the background for these ideas
    • developing the mathematical concepts of derivative and integral
    • seeing how the two ideas are connected
    • learning techniques for applying the ideas efficiently.

    All of this is hard work.  But what did you expect in learning about "one of the great intellectual accomplishments of the human mind?"  The day-to-day work may seem tedious at times, but it's essential, like finger exercises for the future pianist.  Or, to change the analogy, it's like learning a new language: it can open new vistas but only if you're willing to memorize vocabulary, learn to conjugate verbs, and practice, practice, practice!

    We'll see indications of some of the diverse applications of calculus during the course. But it's not a course in physics, biology, economics, or business.  Many of the most interesting and significant applications you will have to meet elsewhere.  That should be a relief!  It's certainly nice to get ideas about what the material is good for, but students who want "more applications" in math courses often don't realize that applications, generally, are much harder: a little like "story problems," only worse. That's because applying math to a concrete situation involves taking a complicated, messy real-life situation, sorting out what's relevant to the problem and what isn't, creating a mathematical approximation ("model") to reality, and then setting up a mathematical formulation of the problem. Only at that point are you ready to apply the "tools" from calculus. Setting up a mathematical model of a complicated real-world situation is often not easy, and it usually requires detailed knowledge of another subject such as physics, biology or economics.  In a calculus course you learn the tools and see them applied in some "tidy" applications which only hint at the real usefulness of the subject. The biologists, chemists, physicists, engineers, architects, economists, and others who have recommended that you take a calculus course will have to show you the reasons why it's useful in their own fields (please, put them on the spot and ask!! )  For now, try to learn to appreciate the subject itself, its beauty, and how the pieces fit together.