precalculus

In mathematics education, precalculus is a course, or a set of courses, that includes algebra and trigonometry at a level which is designed to prepare students for the study of calculus. Schools often distinguish between algebra precalculus and trigonometry as two separate parts of the coursework.

For students to succeed at finding the precalculus derivatives and antiderivatives of calculus, they will need facility with algebraic expressions, particularly in modification and transformation of such expressions. Leonhard Euler wrote the first precalculus book in 1748 called Introductio in analysin infinitorum (Latin: Introduction to the Analysis of the Infinite), which "was meant as a survey of concepts and methods in analysis and analytic geometry preliminary to the study of differential and integral calculus."[2] He began with the fundamental concepts of variables and functions. His innovation is noted for its use of exponentiation to introduce the transcendental functions. The general precalculus logarithm, to an arbitrary positive base, Euler presents as the inverse of an exponential function.

Then the natural precalculus logarithm is obtained by taking as base "the number for which the hyperbolic logarithm is one", sometimes called Euler's number, and written e {\displaystyle e} e. This appropriation of the significant number from Gregoire de Saint-Vincent’s calculus suffices to establish the natural logarithm. This part of precalculus prepares the student for integration of the monomial x p {\displaystyle x^{p}} x^{p} in the instance of p = − 1 {\displaystyle p=-1} {\displaystyle p=-1}.

Today's precalculus text computes e {\displaystyle e} e as the limit e = lim n → ∞ ( 1 + 1 n ) n {\displaystyle e=\lim _{n\rightarrow \infty }\left(1+{\frac {1}{n}}\right)^{n}} {\displaystyle e=\lim _{n\rightarrow \infty }\left(1+{\frac {1}{n}}\right)^{n}}. An exposition on compound interest in financial mathematics may motivate this limit. Another difference in the modern text is avoidance of complex numbers, except as they may arise as roots of a quadratic equation with a negative discriminant, or in Euler's formula as application of trigonometry. Euler used not only complex numbers but also infinite series in his precalculus. Today's course may cover arithmetic and geometric sequences and series, but not the application by Saint-Vincent to gain his hyperbolic logarithm, which Euler used to finesse his precalculus.