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.
Why
Significant Digits: Digging Deeper
We’ve looked at the basic concept of significant digits, then at how they interact with operations, which is one reason for defining them. This time I want to look a little closer at why they are defined as they are, which will involve considering some special cases.
Significant Digits: Operations
Last time, we introduced what we mean by significant digits (or figures), and touched on why they are defined as they are. Here we will look at how significant digits and decimal places differ, and how they are affected by operations (primarily addition and multiplication). This is another aspect of why they are defined at …
Combining Function Transformations: Order Matters
Last time we looked at questions about how to shift, stretch, or flip a graph by changing the equation of a function. All our examples involved only a single transformation. Now we can look at cases where two or more transformations are combined. As we do this, we will develop a deeper understanding of how …
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Shifting and Stretching Graphs
A common topic in algebra courses is how to transform functions and their graphs. In the series starting today, we’ll start with the basics of how and why a graph is moved or stretched, then combine transformations and look at various special cases and other transformations, ending up with graphing trigonometric functions.
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.
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 …
The Method of False Position: Old and New
(An archive question of the week) Last time, as part of our series on estimation, we looked at some numerical methods for solving equations approximately. I mentioned the Method of False Position, but when I looked for more detailed expositions in our archive, I realized that in a sense it is really two different things, …
Estimation vs. Other Ways to Check Subtraction
(An archive question of the week) Last time I looked at reasons for learning to estimate. In searching for answers on that topic, I ran across a question that touches not just on reasons for estimation, but on other ways to check an answer, and on some of the specific ideas we will be looking …
Estimation: The Value of Imprecision
Many questions we have received have been about various aspects of estimation. Often this topic has been downplayed, because we tend to think of math as being all about precision; but it is essential in many applications, sometimes because there is nothing else to do, and other times because exactness wastes effort. I am starting …