Let \(f_0\) and \(f_1\) be two functions such that \(f_1\) represents the rate at which \(f_0\) is changing with regards to some independent variable \(t\). (If you prefer to read an example first, jump down to that section.) This relationship is typically represented by saying that \(f_1(t)\) is the derivative of \(f_0(t)\), that is: \[f_1(t)=f_0^\prime(t)\]…
Category: Calculus
Rate of Change and the Power Rule
I’ll keep this one short. Also, it’s on calculus, for whatever that’s worth. The Power Rule for Differentiation says that the derivative of a monomial \(ax^b\) is \(abx^{b-1}\). Last night I noticed a way to derive this for positive integers that I believe I’ve seen before (so I’m not claiming originality), but which is different…
Maria Agnesi and the Second Derivative
I’m currently reading selected parts of “Analytical Institutions”, the 1801 edition of John Colson’s translation of Maria Gaetana Agnesi’s 1748 text. Near the beginning of her second book, she presents the following theorem. Consider the diagrams: In each diagram, H, I, and M are points along the x-axis and are equally spaced. A, B, and E…
Deriving Euler’s Identity
Euler’s Identity has been called “the most beautiful equation” in mathematics. It neatly encapsulates five key values and three operators into a true equation: \[e^{\pi i} – 1 = 0\] But why is it true? In this entry, I’m going to take it apart. Fully understanding the equation involves looking at various parts of algebra…
The Natural Base \(e\): Thoughts on Teaching
In “Burn Math Class”, Jason Wilkes spends quite a few pages deriving the value of \(e\). I did not notice him at any point mentioning compound interest. Since we’re currently wrapping up the chapter on exponential functions and logarithms in the Algebra II classes I’m teaching, I was already thinking about the best way to…
0.999… = 1 and Zeno’s Paradox
Overview One surprisingly difficult concept for many students of mathematics is understanding that 0.999… (more properly depicted as \(0.9\overline{9}\)), that is, a decimal with an infinite number of 9s, is equal to 1. There are various proofs of it, and various arguments against it. Below, I’m going to present a discussion of this problem in…