Category Archives: Pre-Calculus

The Physics of Spinning Objects

Overview Any spinning object rotates around a central point called an axis. Tops, dreidels, gyroscopes, and spinning eggs rotate, rise, and seem to defy gravity, as long as they are moving. When they stop moving, they fall. Angular Momentum Angular momentum refers to movement in a circle rather than a straight line. Imagine the movement of […]

Math Review of Roulettes and Cycloids

Overview Roulettes are special types of curves that are made when one curve is rotated around another by a fixed point. They are in Euclidean space, and can be described by calculus equations. Roulettes, cycloids, and other variations of circular movement have applications to pendulums, gears and subatomic particles. General Definition of a Roulette A […]

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 […]