# Calculus I

Calculus is usually a major change for math students. This is appropriate, because calculus is the study of change: slope, velocity, growth rate, and other ways that we describe how one quantity changes with respect to another.

Calculus is also perceived as difficult, and historically for the scientific community it was. It took about 300 years of concentrated effort to develop calculus as a usable and well-founded discipline! Fortunately for modern students, we have the benefit of our elders' wisdom. We have their well-laid-out road map, carefully ordered, to lead us gradually toward a grasp of the subject.

Many people contributed to the development of calculus, including (in chronological order),

1. Archimedes (287? - 212 BC)
2. Newton (1642 - 1727)
3. Liebniz (1646 - 1716)
4. Jacob Bernoulli (1654 - 1707)
5. Euler (1707 - 1783)
6. Cauchy (1789 - 1857)
7. Weierstrass (1815 - 1897)
8. Riemann (1826 - 1866)

There are two main branches: the differential calculus and the integral calculus. Derivatives help us to compute rates of change, and integrals help us to calculate the accumulated results of that change. Historically, these two areas developed separately. Thinking in geometric terms, there is no obvious connection between slope and area. Isaac Newton's great theorem, first proved in 1665, relates the two in an astounding equation. This theorem today is known as the Fundamental Theorem of Calculus. Without it, the large field of calculus would be several smaller fields.

Calculus is united by a concept as well as a theorem. The idea of limit was the last major idea to be added to the discipline, but logically it precedes everything else! This is often the way of mathematical history. Mathematicians and scientists were happily computing derivatives and integrals for a couple of centuries before anyone got around to making sure their methods were founded on a consistent, logical base. In fact, there are many subtleties involved, and the limit concept makes it all rigorous.

The function concept is also central to calculus: functions relate the various quantities that change. For example, we can think of the position of a car as a function of time. And the area of a rectangle is a function of its height and width. It is important to understand what a function is, and that can be done from at least four different perspectives.

Descriptively, a function is defined by words:

"The distance traveled by a falling object is proportional to the square of the time it has been falling, with the constant of proportionality equal to half the acceleration of gravity."

Analytically, a function is a rule that relates the variables:

D = 16t2

Numerically, a function is a collection of paired numbers. Thought of in this way, a function is often represented as a table:

Time t Distance D
0 0
1 16
2 64
etc. etc.

Graphically, a function is a geometric picture showing the relation. More precisely, the function is the set of ordered pairs in the tD-plane: We sometimes call this four-pronged approach to understanding functions DANG. Calculus students, presumably overwhelmed by enthusiasm, are often heard to utter this acronym.