Calculus Notes

Section 1.1: A Preview of Calculus

Note: This is the first in a series of notes based on Calculus by Larson/Hostetler/Edwards (8th edition). These notes are intended to serve as a general reference for students as well as a simplified overview of some of the key points covered in the text.

Calculus (often referred to by pedants as "the calculus") is a branch of mathematics that deals with continuously varying quantities. Its invention is traditionally ascribed to two 17th century mathematicians: Gottfried Wilhelm Leibniz and Isaac Newton. However, many of the key concepts of the calculus had been explored well before their times. For instance, the basic idea behind the integral had been studied by Archimedes in the third century BCE. Newton and Leibniz have the distinction, however, of being the first to synthesize the two major halves of calculus: the differential calculus and the integral calculus.

Differential calculus is related to the problem of finding the slope of a curve at a particular point, while the idea of integral calculus came from efforts to calculate the area under curves. Newton and Leibniz showed that these apparently unrelated ideas are in fact intimately related by a statement known as the fundamental theorem of calculus. (This very important concept is introduced in section 4.4.) Pages 43 and 44 show several concepts which can be analyzed using calculus, but for which algebraic techniques are insufficient.