Alternatives

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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Equivocal Function Transformations

The last two posts were about transformations of functions (shift, stretch, reflect) and their effect on a graph, first individually and then in combination. The next thing to look at will be how to determine the transformations when you are given a graph; but before we take that challenge in general, we need to see …

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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 …

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How Many Different Meals Are Possible?

(An archive question of the week) While gathering combinatorics questions, there were several that stood out. This one will serve well to summarize the topic, showing multiple methods for counting, and contrasting other kinds of problems. The problem The question, from 2007, relates to an Arby’s promotion: How Many Different Dinners Can Be Ordered? I …

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Distances to an Arc: Exact and Approximate Formulas

(A new question of the week) It can be an interesting challenge to be presented with a formula and asked how it was derived. This becomes a bigger challenge when the formula is only approximate, so we have to figure out how to arrive at this particular approximation. But it is impressive when several different …

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Greatest Common Divisor: Extending the Definition

Having just talked about definition issues in geometry, I thought a recent, short question related to a definition would be of interest. We know what the Greatest Common Divisor (GCD, also called the Greatest Common Factor, GCF, or the Highest Common Factor, HCF) of two numbers is; or do we? Negative GCD? Here is the …

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Rolling a 6 on Three Dice

(An archive question of the week) One of the discussions we looked at last time involved rolling three dice and getting at least one six. I didn’t go into detail on the calculation there; but I found another place where we discussed it at length. We’ll look at that here. A wrong way and a …

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