# Calculus

## Integration by Substitution

(An archive question of the week) Last time, we looked at a method of integration, namely partial fractions, so it seems appropriate to find something about another method of integration (this one more specifically part of calculus rather than algebra). We will look at a question about integration by substitution; as a bonus, I will …

## Partial Fractions: How and Why

I have often noted that calculus class is where you really learn algebra. Certain techniques in calculus demand algebraic skills that either were not taught in algebra classes (because they are not needed until you get to calculus), or have been forgotten. Chief among these is the method of partial fractions. I have here put …

## Integration: Partial Fractions and Substitution

(New Question of the Week) Many of the questions we answer are primarily about “how do you solve this problem?”, but at the same time ask deeper, more general questions: “How do you solve problems like this, and are there alternatives?” Today’s question is a good example of this, and raises an interesting point or …

## Open or Closed Intervals? It Depends

(Archive Question of the Week) Students commonly expect that textbooks all say the same thing (in fact, some think they can ask us about “Theorem 6.2” and we’ll know what they’re talking about!). The reality is that they can even give conflicting definitions, depending on the perspective from which they approach a topic. Here, I …

## Ranges of Inverse Trig Functions

(Archive Question of the Week) We have had a number of questions over the years about inverse trig functions and their ranges. For today’s question, I have chosen one from 2011, which will link to a number of others that I will not quote in detail.

## Finding the Range of a Function

Recall that the domain of a function is the set of all valid input values (x), and the range is the set of all possible output values (y). It is reasonably easy to find the domain: look for what could make it impossible to evaluate, such as dividing by zero or taking the square root …

## Arrhenius Equation: Which Graph is Right?

(New Question of the Week) Occasionally we get questions challenging the correctness of a textbook problem, or of a test grade. And sometimes we get questions about mathematics used in sciences like physics or chemistry, which lead us to explore unfamiliar fields. The most interesting question this week is one of these. It is one …

## Why Do People Treat dy/dx as a Fraction?

(New question of the week) Here is a recent question from Fida, another long-time “patient” of ours at Ask Dr. Math:

## Why Can’t You Divide by Zero?

Last time, I talked about students’ difficulties carrying out divisions that involve a  zero, which  reminded me of another issue that sounds almost the same, but is quite different: division of a number by zero. Students often either forget the rule they were taught, or they don’t believe the rule, or they just wonder about …