Two great problems
The mathematics we will explore in this course arose from real practical problems encountered in day-to day life in the ancient world, as well as from a natural human curiosity about how our world, our solar system, and our universe works. We can identify two separate problems that calculus addresses:
- (1) Understanding Rates:
Rates of change, slopes of graphs, as well as tangents to curves are all interconnected through the idea of the derivative, or the process of differentiation. You may remember that the differentiation involves taking a difference of values of a function (at close by points), i.e. subtracting values.
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f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}
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The derivative formed the key concept in the first semester calculus.
- (2) Understanding integration:
Calculating areas and volumes, adding an incremental process to arrive at a "total", and other operations that smooth or average data are all interconnected through the idea of the integral. We will see that the concept of integration is analogous to addition. The integral will form the main idea underlying what we discuss in this second semester course.
Historically, though we teach about the derivative first in many calculus courses, actually, integration arose historically before differentiation. Some of the problems that spurred the development of the ideas behind integration are mentioned below.
Practical Problems
How much "stuff" does a vessel contain?
This question may seem academic when we look at these old pots from the Late Bronze Age, gathering dust in a museum. But if you were buying olive oil from the Philistine town of Ekron (in the Middle East) or wine from the vineyards of a local merchant, this question would be of great practical interest. Certainly, you would want to be charged a fair price, so knowing how much volume a given vessel contains would be of consequence.
LB I - II pottery collection (Ashmolean Museum)
How much land are you getting?
If you were a nobleman, parcelling out tracts of land to loyal knights who fought on your side of the last Great Battle, you would want to have some idea how much land each one was getting. No point irritating your most ardent fighters by giving them smaller areas than their neighbors ! All your vassals like river-front land, for easy access, and for the rich farms that they can cultivate here. But how, exactly, should you divide up this land? How much area is in one of the river-front plots shown below. (If the river was very straight, the calculation is easy ! But most rivers tend to bend and turn, leading to irregular plots as a result. This makes for a more complicated problem.)
Modern applications of similar ideas
Perhaps you think that problems of this type were
most important long ago, but clearly, similar problems are of
interest today. In our more sophisticated and technological world, we
have pushed the edge of scientific knowlege and technical skill
far what was available, say, in the time of Archimedes, or in
the days when early mathematicians struggled to address these needs.
Yet, today, integral and differential calculus remains one of the
most powerful mathematical tools, used widely in scientific,
social, and technological applications. Here are some of the more
modern applications of the theory of integration which allow us
to be much more decisive about planning for the future, or
assigning measures to quantities we find important:
- (1) Measuring areas, volumes, masses, material properties
of objects to
a very high degree of accuracy - for example for intricate
machine parts accurate to within a single micron, or
less.
- (2) Predicting long-range trends and behaviour of systems
when we know only their rates of change.
The study of differential equations and their solutions is
closely related to integration, and allows us to predict, for example,
if the Hong Kong influenza will pose a threat to world health, and
what measures might be effective at controlling it.
- (3) Smoothing out noisy data, finding average or representative behaviour, or determining how big fluctuations are in the average behaviour. These types of problems form the backbone of statistical methods, and rely heavily on use of the integral calculus.
Though it may seem that the two problems - differentiation and integration - underlying calculus are quite distinct, we will find out that, in fact, a fascinating connection occurs between derivatives and integrals. This discovery, called the Fundamental Theorem of Calculus was one of the greatest pieces of mathematical discoveries ever made. In what follows, we will be arriving at this result and showing its great implications.
For your consideration:
- (1) What other problems of measuring areas or volumes might have been important in antiquity ? How do you suppose people handled such problems before modern mathematical techniques were invented for dealing with these?