## 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

## 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

## 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

## 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

## 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

## 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

## 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

## 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

## 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

## Mathematical Proofs (Using Various Methods)

Are you someone who relies on logic and evidence for solving problems? Mathematical proofs will help you refine and take advantage of this valuable way of thinking as it applies to mathematics, and potentially other areas such as philosophy and

## 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

## 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

## 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

## 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

## 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

## 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

## 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

## 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

## 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

## 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

## Data in the Modern World (Summary)

Unless you’ve been cloistered in a medieval monastery for the past decades, you know that data science is an important emerging field of study. It is driven by our increasing ability to process data and the proliferation of the ways

## Building Blocks of Data Science

Data science has been around for a while, but today it’s really starting to grow into a vital part of industry and business. It consists of a number of skills and activities that take sets of numbers (data) and turn

## What is Computer Science? (Historical Summary)

What is Computer Science? Computer science, in simple terms, is the study of computers, computer technology and computational systems. It includes both software and hardware. Unlike computer and electrical engineers, computer scientists mostly deal with software as well as software

## Applications For Artificial Intelligence

The 2004 reboot of the classic television show Battlestar Galactica explored a crucial aspect of what it means to be human. If machines can think and learn for themselves, then what differentiates us from them? It’s hard to decide where