Category Archives: Calculus

Math Introduction to Infinitesimals

Overview Using infinitesimal quantities to approximate measurement of any item is an ancient way to determine the size and shape of irregular objects. Although it was very controversial in the 17th and 18th century Europe, the practical aspects of using infinitesimal quantities in calculations led to advances in science, engineering, and technology, along with the […]

Math Introduction to Integrals

Overview Finding integrals, or integration, is the opposite of finding the derivative in calculus. The concept is closely connected with functions, and was independently discussed by both Newton and Leibniz in the Fundamental Theorem of Calculus. Review of Functions Suppose the values of x are {1, 2, 3, 4, 5} using set notation, and the […]

Math Introduction to Derivatives

Overview A derivative of a function describes its rate of change at a particular point on the function. The rate of change doesn’t have to be constant, so it can be approximated along any point of a curve. Derivatives in calculus have many applications in quantitative sciences such as physics and chemistry. Geometric Definition Not […]

Math Review of Real, Complex, and Hyperreal Numbers

Overview Real numbers are the rational and irrational numbers that people deal with in everyday life. Hyperreal numbers include numbers that are infinitely large, infinitely small, or infinitesimal, along with the reals. Surreal numbers include the reals, the hyperreals, and other constructs in advanced mathematics that sometimes behave like numbers and sometimes do not. Rational […]

Math Introduction to Sets and Logic

Overview Thinking about sets of objects and numbers is as simple as counting and as complex as infinity and transfinite numbers. Sets are an essential underlying concept of mathematics and logic. Roster or Description Mathematicians use the language of set theory to describe sets. For example, small, finite sets can be defined using the roster […]

Solving Integrals Through the Use of Integration by Parts

Overview Suppose we have an integral which appears unsolvable by ordinary means, such as ∫xsin(x)dx In many cases, we can do what is called integration by parts, which is where we split the equation we are taking the integral of into two parts, with the goal of simplifying the equation such that we can reduce […]