A question came in just this month, closely related to the last question in the post When Is an Improper Integral Not an Improper Integral? last April. There, we considered an integral where substitution led to a new integral with limits from 0 to 0, and after some discussion we decided that, while odd, the …
Last time, we saw several problems where we were given remainders when an unknown polynomial was divided by two or three linear polynomials, and had to find the remainder on division by the product of those divisors. Here we’ll look at a recent question, and an older one, that add a twist: we’ll be given remainders …
Polynomial Remainder Problems: Changing the Conditions Read More »
In searching for questions about polynomial division, I ran across several about problems where you are given the remainders when an unknown polynomial is divided by two or three different small polynomials, and have to find the remainder when it is divided by a different, but related, polynomial (typically the product of the others). We’ll …
Finding a Polynomial Remainder, Given Other Remainders Read More »
Last time we introduced standard deviation. Here we’ll look into why two formulas (namely, population and sample standard deviation) are different, and why several different formulas for either are equivalent. We’ll also discover how to update the standard deviation when a new value is added. In doing so, we’ll see some different perspectives than we …
Formulas for Standard Deviation: More Than Just One! Read More »
We’ve had a number of questions about “measures of dispersion”, such as standard deviation, which tell us how much data spreads out, as opposed to “measures of central tendency”, which tell us where the middle of the data is (as we discussed in Three Kinds of “Average” and Mean, Median, Mode: Which is Best?). Why …
A recent question illustrates well the different ways to solve problems in combinatorics. We’ll see an easy way, another easy way, and a … less suitable … way to solve a set of problems.
Let’s look at a nice little challenge: to find a cubic function with maximum and minimum at given locations – without using calculus. We’ll explore how to solve it with graphing software, and using algebra in a couple ways, and finally with calculus. And, surprise! They all give the same answer, though the results look …
Having looked at improper integrals last time, let’s look at some questions we’ve had involving integrals that either look improper but aren’t, or are improper but were missed, or that have other issues with their interval of integration.
Last week we examined three probability problems that had problems. Looking further back, I find that Jonathan, who asked the first of those questions, asked a group of questions about rolling multiple dice in 2022. They provide some additional lessons about easy mistakes to make.
It’s been a while since we’ve looked at probability. Here, we’ll look at three questions that we received last year. In each case, we have to detect an error! They’re good examples of what can go wrong, and what to do when your answer appears to be wrong.