- 2.1: Newton and Leibniz Get Started
- The rules for calculus were ﬁrst laid out in Gottfried Wilhelm Leibniz’s 1684 paper Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales, quantitates moratur, et singulare pro illi calculi genus (A New Method for Maxima and Minima as Well as Tangents, Which is Impeded Neither by Fractional Nor by Irrational Quantities, and a Remarkable Type of Calculus for This).
- 2.2: Power Series as Infinite Polynomials
- Applied to polynomials, the rules of diﬀerential and integral calculus are straightforward. Indeed, diﬀerentiating and integrating polynomials represent some of the easiest tasks in a calculus course. Unfortunately, not all functions can be expressed as a polynomial. A standard technique is to write such functions as an “inﬁnite polynomial,” what we typically refer to as a power series. Such “inﬁnite polynomials” are much more subtle object than mere polynomials, which by deﬁnition are ﬁnite.
Thumbnail: Engraving of Gottfried Wilhelm Leibniz. (Public Domain; Pierre Savart)