# Mathematics Articles

## Extreme Value Theorem (Finding Extrema)

Relative and absolute extrema are studied. Finding a function’s critical numbers and using the derivative of the function to understanding its behavior is the crucial step in finding relative and absolute extrema. We illustrate these ideas with several examples. Relative Extreme Values If a function is defined on an open interval and if at some …

## Related Rates (Applying Implicit Differentiation)

In this topic we show how implicit differentiation and the chain rule can be used to calculate the rate of change of one variable in terms of the rate of change of another variable (which may be more easily measured). Introduction to Related Rates The procedure of solving a related rates problem is to find …

## Derivatives of Inverse Functions

We discuss the Implicit Function Theorem and demonstrate its importance through examples. We derive differentiation formulas for exponential, logarithmic, and inverse trigonometric functions. Derivatives of Inverse Functions In this section we state the derivative rules for the natural exponential function and the general exponential function. We also go over several examples of the chain rule …

## Implicit Differentiation (and Logarithmic Differentiation)

The procedure of implicit differentiation is outlined and many examples are given. Proofs of the derivative formulas for the inverse trigonometric functions are provided and several examples of using them are given. Also detailed is the logarithmic differentiation procedure which can simplify the process of taking derivatives of equations involving products and quotients. Implicit Differentiation …

## Derivative Examples (The Role of the Derivative)

We discuss the difference between average rate of change and instantaneous rate of change. We work through several examples demonstrating how the derivative can be used in understanding the motion of objects in the macro world. Average Rate of Change In general, suppose an object moves along a straight line according to an equation of …

## The Chain Rule (Examples and Proof)

With a lot of work, we can sometimes find derivatives without using the chain rule either by expanding a polynomial, by using another differentiation rule, or maybe by using a trigonometric identity. The derivative would be the same in either approach; however, the chain rule allows us to find derivatives that would otherwise be very …

## Derivatives of Trigonometric Functions

Formulas for finding the derivative of the six trigonometric functions are given. We assume that the trigonometric functions are functions of real numbers (angles measured in radians) because the trigonometric differentiation formulas rely on limit formulas that become more complicated if the degree measurement is used instead of radian measure. Derivative Formulas Theorem. The trigonometric …

## Product Rule and Quotient Rule

We work through several examples illustrating how to use the product rule (also known as “Leibniz‘s rule”) and the quotient rule. Several examples are given at the end to practice with. The Product Rule We being with the product rule for find the derivative of a product of functions. The rule can be generalized to …

## Differentiation Rules (with Examples)

Computing the limit of the difference quotient can be tedious and require ingenuity; fortunately for a large number of common functions there is a better way to compute the derivative. In this section, we detail the power rule and the linearity rule for differentiation. These rules greatly simplify the task of differentiation. We also give …

## Derivative Definition (The Derivative as a Function)

We begin with the definition of the derivative as a limit of a difference quotient. We then give several examples of how to find the derivative of a function using this definition. Finding an equation of the tangent line is then considered, and after several examples of this, we then give examples of how a …

## Quantifiers and Predicate Logic

Variables in mathematical statements can be quantified in different ways. First, the symbol $\forall$ is called a universal quantifier and is used to express that a variable may take on any value in a given collection. For example, $\forall x$  is a symbolic representation for any of the following  $\qquad$ “For any $x,\ldots$”,   $\qquad$ …

## Propositional Logic (Truth Tables and Their Usage)

By a mathematical statement (or statement, or proposition) we mean  a declarative sentence that can be classified as either true or false, but not both. For example, the sentences $$1+3=4, \quad 1+3=5, \quad \text{July is not a month}$$ can be accepted as statements. The first one is true, and the second two are false. It is currently …

## Mathematical Proofs (Using Various Methods)

In this section we discuss valid arguments, inference rules, and various methods of proof including direct proofs, indirect proofs, proof by contrapositive, and proof by cases.  Valid Arguments An argument is defined as a  statement $q$ being asserted as a consequence of some list of statements $p_1, p_2, \ldots,p_k.$ The statements $p_1, p_2, \ldots,p_k$ are …

## Logical Discourse Using Rules of Inference

Logical Discourse In this section we discuss axiomatic systems and inference rules for quantified statements. To give example, we carry out a simple logical discourse for incidence geometry involving points, lines, and incidence.    Axiomatic Systems An axiomatic (or formal) system must contain a set of technical terms that are deliberately chosen as undefined, called undefined …

## Rate of Change and Tangent Lines

The importance of the tangent line is motivated through examples by discussing average rate of change and instantaneous rate of change. We place emphasis on finding an equation of a tangent line especially horizontal line tangent lines. At the end we consider relative rates of change. Average Rate of Change We begin with the average …

## Horizontal Asymptotes and Vertical Asymptotes

We discuss limits that involve infinity in some way. First we study unbounded growth of functions using infinite limits and then the long term behavior of functions using limits at infinity. We also consider vertical asymptotes and horizontal asymptotes. Infinite Limits Infinite limits are used to described unbounded behavior of a function near a given …

## Continuous (Its Meaning and Applications)

A function is called continuous whenever sufficiently small changes in the input results in arbitrarily small changes in the output. We discuss continuous functions, one-sided and two-sided continuity, and removable continuity. The infamous Intermediate Value Theorem is considered at the end. Definition. A function is continuous at a point $c$ means $f(c)$ is defined, \$\lim_{x\to …

## Find the Limit (Techniques for Finding Limits)

We demonstrate calculating limits using Limit Theorems. We place importance on examples, especially examples with trigonometric and rational functions. We also discuss rationalization, limits of piecewise defined functions, and the Squeeze Theorem. Using Limit Theorems In this topic we concentrate not on the formal definition of a limit of a function of one variable but …

## Limit Definition (Precise Definition of Limit)

The history of calculus is interconnected with the history of philosophy and scientific discover. In the 16th Century, French philosopher Rene Descartes combined principles of algebra and geometry by mapping lines and objects using horizontal and vertical planes. By applying numbers to different points on geometric shapes, Descartes created what we now refer to as …

## Limits (Calculus Starts with Limits)

Limits are used to study the behavior of quantities under a process of change. For example, limits can be used to describe the behavior of a function on its domain. Here we study one-sided limits and two-sided limits with emphasis on graphs. We discuss unbounded behavior and oscillating behavior with many examples given. An Intuitive …

## What is Calculus? (An Introduction)

Calculus In Its Simplest Terms: Introduction Calculus might simply be described as a set of clever tricks. That may seem a bit glib, but it’s not far off! You might lose your money on the street when the person operating the shell game shows you the impossible. The ball appears under a different shell, and …

## Calculus (Start Here) – Enter the World of Calculus

Welcome, calculus enthusiasts, and enthusiasts-to-be! This page is your introduction to the world of calculus. You may be studying it for the first time, or maybe for the first time since high school. You may be using it in your work, or maybe have just always wondered what all the fuss was about. However you …

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