Calculus

Minimizing a Function of Two Variables: Multiple Methods

(A new question of the week) A recent question from a student working beyond what he has learned led to an interesting discussion of alternative methods for solving a minimization problem, both with and without calculus. The problem The question came from Kurisada a couple months ago: f(x, y) = x2 – 4xy + 5y2 – 4y …

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More on 0.999…

(An archive question of the week) In collecting questions and answers about 0.999… for the last post, there were two that were too long to include, but that dig more deeply into issues that some of the standard answers tend to gloss over. So here, I want to look at those two answers, both of …

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Frequently Questioned Answers: 0.999… = 1

Having looked at two common questions in probability that are often challenged, let’s turn to the realm of numbers. Non-terminating decimals are inherently problematic, and one particular example causes difficulty for many, even after they fully accept the mathematics of it. Our FAQ page on this topic, at 0.9999… = 1, is very brief, and …

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L’Hôpital’s Rule: What and Why

The next few posts will look at a powerful technique for finding limits in calculus, called L’Hôpital’s Rule. Here, we’ll introduce what it is, and why it works. In the next post we’ll examine some harder cases. Indeterminate forms The method we will be discussing is used to find limits that have an indeterminate form. …

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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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