## Finding a Function Value Recursively

(A new question of the week) May was a particularly good month for interesting questions! Here is one requiring us to find one value of a function, based on an unusual property: If $$a+b=2^x$$, then $$f(a)+f(b)=x^2$$. The problem turned out to be not as hard as it looked, yet the function itself is quite interesting …

## Separable Differential Equations

(A new question of the week) We received a couple different questions recently about solving differential equations by separation of variables, and why the method is valid. We’ll start with a direct question about it, and then look at an attempt at an alternate perspective using differentials.

## Long Division with Zero, Revisited

(A new question of the week) One of our first posts, in 2018, was about zeros in long division. But we still get many questions about this issue, and it’s time to dig in deeper. We’ll look here at two of them, answering the twin questions, “When do you put a zero in the quotient …

## Two Integration Puzzlers

Two recent questions (that came to us within two hours) dealt with apparent contradictions in integration. The first seems to give a result of zero that is clearly wrong; the second seems to give two different results for the same integral.

## Graphing a Reciprocal Function

There are a number of standard techniques for graphing functions, such as transforming simple functions, or finding asymptotes and holes for rational functions, and using calculus to find slopes. What if you have a rational function of a trig  function, and can’t yet use calculus to figure it out? We’ll look at how we can …

## A Random Walk on a Graph

(A new question of the week) It seems that most of the interesting questions recently have been about relatively advanced topics, though commonly in introductory classes. Here, we’ll help a student think through a problem introducing the idea of a random walk on a graph. (“Graph” here doesn’t mean the graph of an equation, which …

## Proving Two Groups are Isomorphic

(A new question of the week) Two weeks ago, in Proving Certain Polynomials Form a Group, we joined a beginner in learning about groups. Here we will pick up where that left off, learning how to prove that the group we saw there, a subset of polynomials, is isomorphic to a group of matrices. As …

## Filling a Cistern: Three Pipes, No Numbers

(A new question of the week) Today we’ll look at a classic algebra word problem: Finding how long it takes to fill a cistern through two pipes, with a drain open. But usually these problems are given with specific numbers, as a simple exercise in algebra. What if it’s all variables? the discussion provides some …

## What Happened to Ask Dr. Math?

In January and February of this year, the entire Math Forum site, including Ask Dr. Math, was first moved from http://mathforum.org/ to https://www.nctm.org/tmf/, and then made inaccessible to non-members, by its current owner, the National Council of Teachers of Mathematics. We are hoping that at least some of this material (the pages we often link …

## Proving Certain Polynomials Form a Group

Abstract algebra can be a huge leap for many students, who may know algebra well, but are not used to abstraction – generalizing the concept of numbers so we can invent new kinds of “numbers” and “operations” and comparing their properties. Here we will look at a question from a student beginning the study of …