Introduction: What is Calculus About?

The maind ideas of calculus (at least, in seminal versions) go back to ancient times.  But calculus itself, in the form we think of it today, was invented during the 17th century (or, should we say "discovered" ? -- I'll stick with "invented.")  The invention of calculus was part of an explosion of interest and discovery in the physical sciences  The crucial early work on the subject is usually attributed to the English mathematician Sir Isaac Newton and the German mathematician Gottfried Leibniz, working iindependently -- but back then, Newton, Leibniz and their partisans had vigorous and nasty arguments about whose work had priority).

As it developed, calculus gave scientists an powerful tool to generate remarkable new insights into how the physical world works, and its creation is considered one of the great accomplishments of the human mind.  This sweeping assertion is justified not just by the beauty that mathematicians and many students see in the subject, but also by the fact that calculus still has fundamental importance as a scientific tool, even several centuries after its birth.  In fact, its role has become more important than ever as the use of mathematical models now reaches beyond traditional areas like physics and engineering into such different fields as biology, economics, finance and business.

Of course there are a lot of other other important mathematical tools too.  Algebra, discrete mathematics, probability and statistics, topology, and computer science all have their role in applications, and each is a research area in its own right.  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 algebraic manipulations.  And computers, running software like Matlab, Mathematica or Maple can do better still (and produce 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.  This basic understanding is our goal in a beginning study of calculus.

There are lots of techniques and details 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 is about "how fast is some quantity changing?"  For example, if you're driving down the highway
and s = f ( t ) represents your distance (km) from home at time t, then you might be interested in how fast s is changing.  The rate of change of s at a time t (measured, perhaps, in km/hr) is your velocity v at that time and this is shown on the speedometer in your car. 

Studying rates of change uses a concept from Calculus I called the derivative.   The velocity v, it turns out, is the derivative of the position s.  If we think of  s = f ( t ) as a function of time, then some of the ways that people write the velocity (derivative) are v = f ' ( t )  or  v = ds/dt.  At any time, the speedometer on your dashboard shows you the current value of the derivative v = ds/dt.  When you accelerate, the derviative v gets bigger.

Of course, you might be interested in the rate of change of some other quantity.  For example, the rate of change of 

  • 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 quantity is from biology, physics, or economics, the same mathematical concept -- the derivative -- is the tool to talk mathematically about how fast it changes.
     

    2)   The other great 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 you are from home (call this s ) at time t if you are given the velocity information v  (all the speedometer information between leaving  home and time t). 

    To take a simple example: It's very easy to find s if your car has constant velocity: if v  = some constant: 
    then we use the simple formula distance = (rate)(time), that is,     s = vt. 

    But finding s from v is harder if the velocity is changing throughout the trip. From part 1) we know that v = ds/dt and given the values of v;  we want to "think backwards" and recapture s.  Calculating the total distance from home, s, from the rate of change v  = ds/dt involves a concept from calculus called the integral.  An integral is analogous to your car's odometer (tripmeter) which (if you set it to 0 when you start out) shows the current value of s on your dashboard.

    Instead of a distance, you might be interested in trying to compute the total amount of money A in an account if you know its rate of change dA/dt (roughly, this is the interest rate).  Or you might want to figure out the total number I of people infected with a disease at time t if you know the rate of infection dI/dt.  In all these situations, we want to find the "total amount" based on the rate of change. The mathematical tool is the same each time: the integral.

    In Calculus I, we develop the ideas of the derivative and the integral and take a look at how they are related.  These ideas, and a lot of techniques for working with them, are developed further in Calculus II.  Most of Math 131-132 consists of
    • developing the informal concepts behind ideas such as "rate of change"
    • developing the exact mathematical meaning of derivative and integral
    • seeing how derivatives and integrals are connected 
    • learning techniques for use these ideas efficiently.

    All of this is hard work.  But what did you expect in learning about "one of the great 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 and possibilities for you, but only if you're willing to memorize vocabulary, learn to conjugate verbs, and practice, practice, practice!

    We'll get some idea of the diverse applications of calculus during the course. But this is not a course trying to systematically teach physics, biology, economics, or business.  Many of the most interesting and significant applications you will have to learn about somewhere else. That should be a relief !  It certainly is nice to get some 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 then are you ready to use the "tools" you learn in from calculus.

    Setting up a mathematical model of a complicated real-world situation is often not easy, and, except in simple examples, it usually requires detailed knowledge of another subject such as physics, biology or economics.  In a calculus course, everybody can learn the tools and see them applied in some "tidy" or "contrived" applications which are manageable for everybody and which 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.