# Month: December 2018

## Why I Became a Math Doctor

While I was setting up The Math Doctors site a year ago, I ran across the following email I received in 1998 inviting people to become Math Doctors. It illustrates well the ethos of the team: In the fall of 1994 the Math Forum at Swarthmore College (then the Geometry Forum) started an email program …

## Finding the Range of a Tricky Rational Function

I previously wrote about finding the range of various kinds of functions. The examples there were relatively easy. A recent question raised the level of difficulty, bringing up some interesting issues. Here is the initial question: Hi, I am trying to calculate the domain and range of this function f(x)= (x^2 – 3x + 2)/(x^2 …

## Finding the Median of Grouped Data

(An archive question of the week) Last time we looked at a formula for approximating the mode of grouped data, which works well for normal distributions, though I have never seen an actual proof, or a statement of conditions under which it is appropriate. We have also received questions about a much more well-known, and …

## Finding the Mode of Grouped Data

The mode of a list of data values is simply the most common value (or values … if any). When data is grouped (binned) as in a histogram, we normally talk only about the modal class (the class, or group, with the greatest frequency), because we don’t know the individual values. But some sources teach …

## A Bad Geometry Proof Problem: SSA

(A new question of the week) It’s surprising how many questions we get that end up being about problems that are poorly worded or simply wrong. But these can be as illuminating as good problems, by showing ways to catch the error. This is one simple in itself, but will lead us into a common …

## Max and Min of a Cubic Without Calculus

Last time we looked at various ways to find tangent lines to a parabola without using calculus. Another standard calculus task is to find the maximum or minimum of a function; this is commonly done in the case of a parabola (quadratic function) using algebra, but can it be done with a cubic function? Yes, …

## Tangents Without Calculus

I always like solving advanced problems with basic methods. For example, many problems that we usually think of as “algebra problems” can be solved by creative thinking without algebra; and some “calculus problems” can be solved using only algebra or geometry. Using simple tools for a big job requires more thought than using “the right …

## A Bad System of Nonlinear Equations

(A new question of the week) Sometimes a problem that looks complicated turns out to have a simple answer. And sometimes that simple answer turns out to be too simple for its own good. Today’s problem is an example of this. A nonlinear system Here is the question, which came in last month: Sir, What …

## Division by Zero and the Derivative

(An archive question of the week) The indeterminate nature of 0/0, which we looked at last time, is an essential part of the derivative (in calculus): every derivative that exists is a limit of that form! So it is a good idea to think about how these ideas relate. Here is a question from 2007: …

## Zero Divided By Zero: Undefined and Indeterminate

Back in January, I discussed the issue of division by zero. There is a special case of that that causes even more trouble, in every field from arithmetic to calculus: zero divided by zero. I’ll look at several typical questions that we answered at different levels. Conflicting rules for division? Let’s start here: Zero Laws …