Calculus

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 …

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What’s the Point of Limits?

(An archive question of the week) Many calculus courses start out with a chapter on limits; or they may be introduced in a “precalculus” course. But too often the concept is not sufficiently motivated. What good are limits? Why did they have to be invented? Are they as simple as they seem? Why is an epsilon-delta …

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A Tank with a Conical End

(An archive question of the week) Last time I surveyed what we have said about the volume of liquid in various kinds of tanks. One more special case I ran across deserved more detailed attention, because it demonstrates in detail how to do the calculations without much knowledge of calculus. The problem Here is the …

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Numerical Approximation Methods: When Algebra Doesn’t Work

The problems students see in class are usually only those that can be solved by the methods they have been taught. Too many students conclude that algebra can solve anything! But the reality is that if you just wrote an equation at random, it probably could not be solved algebraically. When students ask us about …

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Derivative of Arcsin: From the Definition

(A new question of the week) In Monday’s post about fallacies in calculus, one of them used the definition of the derivative (or rather, misused it). Today we’ll look at a short question about applying that same definition, that came in last month.