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Math Review of Leibniz, Newton, and the Development of Calculus 150 150 Deborah

Math Review of Leibniz, Newton, and the Development of Calculus

Overview One of the biggest controversies in science in the early 18th century was around the development of a new mathematical tool called calculus. In Europe, the mathematician, philosopher, and scientist Gottfried Leibniz held the attention of the scientific community. The most famous scientist of the day, Sir Isaac Newton, was the champion of Great Britain.…

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The AC Method of Factoring Polynomials when A Is Not Equal to 1 150 150 Deborah

The AC Method of Factoring Polynomials when A Is Not Equal to 1

Overview The AC-method, also known as factoring by grouping, is often used to factor polynomials when the a coefficient is equal to 1. It can also be used when the a coefficient is not equal to 1, similar to the FOIL method. Questioning the Polynomial Expression The first step to defining the polynomial expression, before…

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Math Review of Roulettes and Cycloids 150 150 Deborah

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…

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Math Introduction to Infinitesimals 150 150 Deborah

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…

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Math Introduction to Integrals 150 150 Deborah

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…

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Math Introduction to Derivatives 150 150 Deborah

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…

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Math Review of Real, Complex, and Hyperreal Numbers 150 150 Deborah

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…

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Math Review of Platonic Solids and Polyhedra 150 150 Deborah

Math Review of Platonic Solids and Polyhedra

Overview There are infinitely many regular polygons. However, there are a fixed number of regular polyhedra, called Platonic solids. Mathematicians use the definitions of the regular polygons and the characteristics of solid objects to arrive at their conclusions. Characteristics of Regular Polyhedra Regular polygons, such as equilateral triangles, squares, equilateral pentagons, hexagons, heptagons, octagons, and…

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Math Review of More Properties of Circular Functions 150 150 Deborah

Math Review of More Properties of Circular Functions

Overview The unit circle has as its center the origin point of the Cartesian coordinates x and y, and has a radius 1. The trigonometric functions are also called circular functions, because they describe relationships between angles on the unit circle. We use different ways to describe trigonometry in order to see how the relationships…

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Math Review of Applications of Circular Functions 150 150 Deborah

Math Review of Applications of Circular Functions

Overview Circular trigonometric functions can be applied to situations in physical, biological, and social sciences involving data that follows a pattern that is not linear. Many of those patterns are periodic, and can be modeled by approximations of sine, cosine, or other functions. The Sine and Cosine Waves The numerical value of t around the…

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