Definition and Notation If y = f(x) and f(x) is smooth and continuous (differentiable) at x then:

“There have been several occasions in the history of mathematics when an important discovery was made independently and almost simultaneously by two individuals … To the extent that we can rationally explain these remarkable events, it must be on the basis of ideas already “in the air,” of conditions becoming…

For centuries mathematicians had found ways to describe and manipulate the perfect geometries they used to represent phenomenon of the natural world. Orbits were round. Length and amount were always positive numbers. There was no such thing as the square root of a negative number. In 1572, in his book…

When taking the anti-derivative of a polynomial, the result is a set of functions that differ by a constant. We call this set the general anti-derivative. Here is a graph of x² (green) and it’s anti-derivative x³/3 + C (red). We can think of the integrand x² as the curve…

Calculating the area of a two dimensional object or the volume of a three dimensional object has been a subject of study for mathematicians and natural philosophers since antiquity. When the enclosed area or volume is a rectangle or a cube the calculations are simple enough. However, when the enclosed…

Many quantities that change over time demonstrate exponential growth or exponential decay. For example, populations grow exponentially as long as there are enough resources to carry the population as its numbers increase. Continually compounded bank balances grow exponentially at a rate that depends on the interest rate. On the other…

The Normal distribution is a Gaussian probability distribution. Gaussian probability distributions are functions designed to reflect principles of the central limit theorem which states that a population sample will tend towards the expected value with a sufficiently large random sample and that values farther away from the expected value will…

Differentiation is the process used to find a derivative. It involves finding a function whose value at every point is equal to the slope of some other function . By performing one or more of a few simple operations on the initial function, we can acquire this second function —…

Like so many others, I felt a slight tinge of inspiration while watching the launch of the James Web Space Telescope this week. The launch gave me the sense that, in spite of all the horrid episodes of the past three to five years, at least some small segment of…