MATH 1441
Technical Mathematics for Biological Sciences
What is Calculus Anyway?
This is a very brief introduction to an enormous topic in mathematics to give you a rough idea of where the
course is going from here, and why.
"Calculus" (in the way in which we will use the term) is a field of mathematics that begins in two broad and
related streams. The development of calculus is usually traced from around 1664 and attributed to Isaac
Newton, though others were working on similar developments before and during that time and made
significant early contributions to the field. It was the development of calculus that made possible much of
the scientific and technological advance that has occurred in the last 300 years.
Differential Calculus involves the study of rates of change. The methods and applications of differential
calculus are extremely important in scientific and technical applications because most of what we know
about how the universe works is information about the way things change. You are familiar with the
importance of rates of change from your study of basic physics where quantities such as speed (the rate of
change of position with time) and acceleration (the rate of change of speed with time) play prominent roles.
We will start by developing a set of methods or formulas for calculating the basic "thing" of differential
calculus  the derivative. The derivative is a formula for the slope the graph of a function. The ability to
calculate the slope of the graph of a function can be exploited to solve a wide variety of practical problems,
from finding the most efficient dimensions for a food container to estimating the propagation of errors from
the known errors in experimentally measured data to the values of more complicated quantities calculated
from those experimental measurements.
Integral Calculus is the second broad branch of elementary calculus and involves the study of bulk
properties such as lengths of curves, areas of surfaces, volumes of solids, and more complex properties of
such figures. The basic "thing" of integral calculus is the integral. Although areas and rates of change don't
seem to have much in common, one of the fundamental discoveries of modern mathematics was the socalled Fundamental Theorem of Calculus which states a very close relationship between the derivative of
differential calculus, and the integral of integral calculus. In fact, the relationship is so close that our
approach to integral calculus in the few weeks we have available in this course would probably more
correctly be termed antidifferential calculus.
Is calculus hard? The only answer can be "yes" and "no". Much of calculus is working with algebraic
expressions, and so in that sense, it is just mostly more of the same sort of thing you've been doing in your
mathematics studies since midway through your basic schooling. On the other hand, whereas in the
expressions of algebra, symbols stand for numbers, in calculus, symbols can also stand for other
expressions, and this introduces a new level of abstraction into our work. Nevertheless, as messy as the
algebra may get, and as strange as the notation may sometimes seem, the ideas and methods of calculus
that we will study in this course are almost completely based on some very simple and easily imagined
properties of graphs, and if you make the effort to understand the visual images, the actual mathematics
should lose much of its initial mystery.
David W. Sabo (1999)
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