When research takes place in biology, two experts can be working on two different things. That’s because the field of biology and its current events cover a wide variety of topics and subjects. Read about the various ways biology is in the news and what the latest findings have to say about the future of living things.

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

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

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

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

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

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

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

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 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

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

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

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

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

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

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

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

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

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

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

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

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

When we’ve heard a phrase over and over in our lives, it can sometimes go in one ear and out the other without us realizing what it really means. Civil engineering might be one of those phrases, especially for young

Civil engineers create our streets, buildings, transportation hubs, water supply structures, and sewage treatment plans. Overall, the job is one of the most technical and highly skilled in the world. Civil engineering is a rapidly growing, with new developments and discoveries in innovative construction technology occurring every day. From self-healing concrete to industrial exoskeletons, learn more about these current events in civil engineering.

Civil engineers design our streets, buildings, transportation hubs, water supply structures, and sewage treatment plans. Outstanding civil engineers are in high demand with a projected growth of up to 20 percent by 2022. Learn about these outstanding civil engineers and draw inspiration from their accomplishments.