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Calculus

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

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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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Math Introduction to Infinitesimals

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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

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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

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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

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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 Introduction to Sets and Logic

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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…

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Solving Integrals Through the Use of Integration by Parts

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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…

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How to Use the Chain Rule to Find Derivatives

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Overview Occasionally, you will have what are called composite functions; that is, functions that are composed of multiple functions and thus cannot be differentiated easily. In reality, although these may look tricky, they are actually fairly straightforward. Function composed of two functions Suppose we have an equation written as f(g(x)). Then d/dx f(g(x)) = f’(g(x))g’(x)…

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U Substitution as a Method of Solving Integrals

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Overview Sometimes, a derivative is done using the chain rule, and it leaves with an equation that, at first glance, can look intractable when we are attempting to integrate it. In those cases, although the problem may look difficult or often will look like something that can only be solved through the use of Integration…

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Distance, Velocity, and Acceleration

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There is a relationship between distance, velocity and acceleration. When you understand how they interact, and their equations; they are much easier to grasp. The relationship between distance and velocity is proportional. Distance = velocity x time If acceleration is involved in the question, the equation becomes Distance = v0 x t + 0.5 a…

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