Introduction: What is Calculus About?

The seeds for the main ideas in calculus 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").  This invention was part of an explosion of renewed interests and discoveries in physical science.  The crucial early developments are usually credited to the English mathematician Sir Isaac Newton and the German mathematician Gottfried Leibniz, working independently -- but back then, Newton, Leibniz and their partisans had vigorous and nasty arguments about whose work had priority.

As it developed, caclulus gave scientists a tool for 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 is very useful even centuries after its birth.  In fact, its role has become more important than ever as the use of mathematical models has grown beyond traditional areas like physics and engineering into such diverse fields as biology, economics, finance and business.

Of course there are also a lot of other other important mathematical tools. Algebra, discrete mathematics, probability and statistics, topology, and computer science all have their role in applications, and each one is a research area in its own right.  These 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.

As computing power has developed and become cheaper, many calculations are possible nowadays that would have been unheard of even 50 years ago. The best calculators can now do complicated graphics and numeric work, as well as symbolic manipulations. And computers, running software like Matlab, Mathematica or Maple can do better still (and produce beautiful output).  Technology makes it possible to explore and use calculus in ways that were impractical even a few decades 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 it may produce an "answer" which is misleading or even completely incorrect!  Technology needs a user who understands and can tell it exactly what needs to be done. This basic understanding is our goal in a beginning study of calculus.

There are lots of details and techniques for us to learn, but if we look at 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 deals with the question: "how fast is some quantity changing?"  For example, if you're driving east along a straight highway and s = f( t ) represents your distance from starting position at time t, then you might be interested in how fast s is changing.  The rate of change of s at a time t is your velocity v at time t -- which is shown on the dashboard in your car. If s is measured in km and t in hrs, then v has units km/hr.

To study rates of change, we use a concept from Calculus I called the derivative.  The velocity v, it turns out, is the derivative of the position function s = f (t ). Some of the ways that people write the velocity (derivative) are v = f ' ( t )  or  v = ds/dt.  At any time, the speedometer in your car shows you the current value of the derivative v = ds/dt.  When you accelerate, the derivative v gets bigger.

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

  • y = f( t ) = the size of a population of bacteria at time t
  • y = f( t ) = the amount (mass) of a radioactive isotope present in a sample at time t
  • V = f( r ) = the volume of gas in an expanding spherical balloon as the radius r grows. The derivative could be written as f ' (r)   or as  dV/dr.  If the radius is measured in cm and the volume in cm^3, then the derivative would have units (cm ^ 3)/cm:  that is (V units)/(r unit) 
  • Whether a quantity is from biology, physics, or economics, the same mathematical tool -- the derivative -- is what we need to talk mathematically about how fast it changes.

    2) The other great idea is "opposite" to the first one: if you know the rate at which some quantity is changing, then by what amount has it changed On the road again, imagine that you are given all the velocity information v -- that is, all the speedometer data starting from time 0 when you departed. Then you might like to calculate the amount that s, the distance from your starting position, has changed after t hours: in other words, how far have you travelled after driving for t hours?  (Note: "distance from the start at time t" could different from "total distance travelled"-- if, for example, you drove to the east part of the time and to west part of the time. "Distance from the starting position" and "total distance travelled" are the same if you always drive in the same direction -- always to the east, say.)

    This calculation is very easy if your car moves at constant velocity: for example, if v is constantly 100 km/hr, then we use the simple formula s = (rate)(time) = 100t. 

    But if your velocity varies during the trip, then finding s from v is harder. Mathematically, we know from part 1)  that v = ds/dt and given the values of v; we want to "think backwards" from v to find s.  Calculating s(t) - s(0) = the change in position from starting position, from the velocity v  = ds/dt, involves a concept from calculus called the integral.  An integral is analogous to your car's odometer (tripmeter) which tells you distance travelled. If you set it to 0 when you start, then at any time t, the odometer tells you the total distance travelled at that moment. 

    Instead of a change in distance, you might be interested in trying to compute the change over one year in the amount of money A in an account given the rate of change dA/dt  (the interest rate).  Or you might want to figure out the change in 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 change" 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.  In fact, most of Math 131-132 consists of
    • developing the informal concepts behind ideas such as "rate of change" and "total 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 that 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 see some ideas illustrating the diverse applications of calculus during the course. But this is not a course trying to 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 !  Although it is nice to get some ideas about what the material is good for, 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 apply the "tools" you learn in calculus.

    Setting up a mathematical model of a complicated real-world situation is often hard, and, except for simple examples, it usually requires detailed knowledge of some other subject such as physics, biology or economics.  In a calculus course, everybody can learn the tools and see them applied in some "tidy" or "over-simplified" 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.