Tangent lines to the sine function with rational slope

Today I was wondering the following thing (I won’t bore you with how I ended up with this question): Are there any rational values of for which the line is tangent to the graph of Clearly the answer is yes: But my gut feeling was that this was the only such After some head scratching,…

Gabriel’s paper horn

I just returned from the eleventh Gathering for Gardner. One of the many special things about this unusual conference is that the attendees are strongly encouraged to participate in the “gift exchange.” We were each asked to bring a physical exchange item (one for each of the 350 conference-goers) or to submit a written contribution….

How do you place incoming mathematics students?

Our department is looking for a better method of placing incoming students in mathematics courses. Currently we have a placement exam that determines whether a student should begin in a calculus I course or in a calculus/precalculus hybrid course (our lowest-level math class). The exam consists of 25 precalculus questions. It does a pretty good…

Parametric curve project for multivariable calculus

I’m teaching two sections of Multivariable Calculus this semester. Each class has 3 hours of lecture and a 1 hour 20 minute lab each week. Last week the students were learning about parametric equations. So in lab I wanted to give them some hands-on experience with 2-dimensional parametric curves. Their assignment was to create a…

Applet to illustrate the epsilon-delta definition of limit

Here’s a GeoGebra applet that I made for my Real Analysis class. It can be used to explore the definition of limit: Definition. The limit of as approaches is , or equivalently if for any there exists such that whenever , it follows that .

Volumes of n-dimensional balls

We all know that the area of a circle is and the volume of a sphere is , but what about the volumes (or hypervolumes) of balls of higher dimension? For a fun exercise I had my multivariable calculus class compute the volumes of various balls using multiple integrals. The surprising results inspired this post….

Making a hyperboloid out of skewers and rubber bands

George Hart, of the Museum of Mathematics, writes a weekly column at Make Magazine called “Math Monday.” A few weeks ago he showed how to make a hyperboloid of one sheet out of 32 shish kabob skewers and 176 hair rubber bands. (Here is a direct link to the instructions.) We just finished talking about…

Why do mirrors reverse right and left but not up and down?

[I apologize to those of you who have been reading my blog for more than a year. I’m reposting something I wrote last year at this time. I was then, and am now, teaching Calculus III, and we just finished discussing the cross product. I ended the conversation by telling my classes how the cross…

Three applets for linear algebra or multivariable calculus

This semester I’m teaching two sections of Calculus III (multivariable calculus) and I happen to be teaching the first four weeks of Linear Algebra. The first couple weeks of both courses cover properties of vectors in Rn. (Of course, just to confuse the instructor and the students who happen to be in both classes, the…

Indeterminate form in The New Yorker

In Calculus II we teach our students about a variety of indeterminate forms: , , , etc. I was reminded of another indeterminate form when reading Malcolm Gladwell’s thought-provoking (negative) review of the book Free: The Future of a Radical Price, by Chris Anderson (editor of Wired). The review appears in The New Yorker (that you can read…

Pentagrams and quartic polynomials

I’m still enjoying my new-found freedom that comes with the end of the semester. I’ve gotten some research done and I’ve been able to catch up on some reading. One article that I found particularly interesting was “Quartic Polynomials and the Golden Ration,” by Harland Totland, from the June 2009 issue of Mathematics Magazine. This…

Computing integer sums using l’Hôpital’s rule

Now that the busy semester is over, I’ve been able to catch up on some reading. Yesterday I read William Dunham’s article “When Euler Met l’Hôpital,” in the February 2009 issue of Mathematics Magazine. The aim of the article is to showcase some of Euler’s applications of l’Hôpital’s rule in his Institutiones calculi differentialis (1755)….