Category Archives: calculus

A-thinkin’ and a-wonderin’ all the way down the road

A thought I had on a roadtrip not too long ago, which is not exactly true, but close enough:

If the white line on the side of a road is real analytic- then every twist and turn on the road should be discernable by investigating just an arbitrarily small neighborhood of a line.  Thus, all 3,365 miles of the path that Route 20 takes from Kendall Square in Boston to Newport, Oregon could be deduced if one measured, precisely, a 1-foot section of the white line in Sioux City, Iowa.

What I am leaning on here is the definition of a real analytic function (and in light of my recent post on functions, I’ll specify that I am talking about real-valued functions on the real line): that a function has a convergent power series in a neighborhood of any point.  One should also worry about the difference between an analytic function, and one which is merely smooth.  A good example to remember is the function

f(x) = \left\{ \begin{array}{rl} e^{-\frac{1}{x}} & x > 0 \\ 0 & x \leq 0 \end{array}\right.

One must take my word that this function is smooth (though it is easy enough to believe- note that the exponential function and all of its derivatives will be equal to 0 at 0, making a sort of smooth splice).  However, note that the Taylor series at x= 0 is identically zero (as we just pointed out), so the Taylor series does not converge to the function for x>0.  Hence f is a smooth, but non-analytic, function.

I mentioned that this statement was not exactly true.  Mathematically, I meant it would be hard to verify that all 3,365 miles were real analytic, unless you could write down the formula for the road, or knew a priori that paint only dried in a smooth fashion.  I prefer my brother’s objection when I told him the story though: exits.  When I moved my story to the center line, he moved his to passing lanes.  Ah well.

Generating research, interest

I have taken Stephen Semmes‘s topics course for each of the past six semesters, and think he has a fantastic teaching style, partly because I could never see myself using the same one.  During a typical class, he will make a definition or prove a theorem, and then ask “Does anyone have anything to say about this?”  The path the class takes is then largely controlled by answers to this question.  I am constantly impressed by how many answers to this question (often early in the morning!) Dr. Semmes and the students in the class have.  More specifically, I am interested in the process of formulating such answers.

This is, of course, a huge topic (that is to say, how are scientific ideas/good research questions generated).  Just to put down a few observations of good questions to ask:

Statement of a definition (“this is an X“):

  • An example of the object/phenomena being described (“x is an X”)
  • A characterization of the object/phenomena (“x is an X if and only if x is a Y,Z,…”
  • The object/phenomena’s relationship to previously defined objects (“every object that is X is also Y” or “every object that is Y is also X“)

Statement of a theorem (“If A, then B“):

  • Is the converse true? (“If B, then A“)
  • What is true with weaker hypotheses? (“If a, then b“)
  • What is true with stronger hypotheses? (“If A, then B“)
  • Is the converse true with stronger hypotheses? (“If B and C, then A“)
  • Is there a stereotypical example of the situation described? (“X is A, so X is also B“)
  • Why is this surprising or interesting? (“If almost A, then not B“)

A note on functions

A quick word on functions that I have noticed often confuses students.

A function is (roughly) a rule for associating objects in one set with objects in another.  The calculus sequence focuses almost exclusively on real-valued functions on the real line.  That is to say, functions that eat a real number, and give back a real number (one writes f:\mathbb{R} \to \mathbb{R}).  In a more general world, we might like to study functions f: \mathbb{R}^m \to \mathbb{R}^n (where m and n might not be 1).  Indeed, some common courses will study functions where either m or n is not equal to 1.

In multivariable calculus, one often moves to functions that eat a vector and give back a real number.  A typical starting place might be functions f:\mathbb{R}^2 \to \mathbb{R}, interpreted as a surface, later looking at functions T:\mathbb{R}^3 \to \mathbb{R}, which might be thought of as, say, temperature at a point in space.  In reality, once the student understands the ideas behind partial differentiation and integration over a region, the only to not proceed immediately to functions f:\mathbb{R}^n \to \mathbb{R} is concreteness.  Indeed, I would argue that students who take linear algebra before calculus would benefit from this approach.  Note that the range of the function remains one-dimensional, which allows us to continue talking intuitively about those calculus stalwarts, rates of change and area under a graph.

In a differential geometry class (and maybe the last few weeks of a calculus course), we also deal with functions whose range has a higher dimension.  The function might then be called a vector-valued function, or parametrized line/surface/hypersurface/region etc.  That is to say, a function \alpha: \mathbb{R} \to \mathbb{R}^2 could be interpreted as a line in the plane, and would write \alpha(t) = (x(t),y(t)), so that at any moment in time, the curve is at some point in the plane.  Similarly, a function f: \mathbb{R}^2 \to \mathbb{R}^3 could be construed as a parametrization of a surface, and we would write f(u,v) = (x(u,v),y(u,v),z(u,v)).  Discussing proper notions of “derivative” and “integral” for these functions is a topic for another time, but it is worth it to realize that this might not be entirely straightforward.

A very general setting for theorems in calculus is for functions f: \mathbb{R}^m \to \mathbb{R}^n, which, notationally, we can write

\mathbf{f}(x_1,\ldots,x_m) = (f^1(x_1,\ldots,x_m),\ldots,f^n(x_1,\ldots,x_m)).

I’m sure I will discuss calculus in this context in the future.  I guess the point for now is, as usual, is to mind your m‘s and n‘s.

Also, happy New Year!

Generalizations of the Mean Value Theorem

One of my hobbies here is running, and I recently ran 3km on the track, which led me to the following thought:

Was there a mile (~1600m) which I ran at least as fast as my average pace?

This can be thought of as a generalization of the mean value theorem, which says that a differentiable curve f:[a,b] \to \mathbb{R} has at least one point where the derivative is equal to the “average” slope.  However, this asks whether there exist some (connected) interval where the derivative is less than the average slope.

This statement can be seen to be false, and we might as well use the running example.  Suppose the 3km was run at an average pace of 5:00min/mile, and in the race, for (counter-)example, I rode a bike to the 1,599m mark (which is still considered cheating so far as the IAAF is concerned), and got there after 1:00.  I stay there for 5:01, then ride my bike to the finishline (taking another minute).  I’ve just completed a 7:01 3k (WR, by the way) but there was never a 5minute period where I covered a mile.  With faster and faster bikes (or cars!) my “moving” time could be reduced until it takes me 5:00+\epsilon to run the 3km.  Of course, this is the best I can do without covering a mile in under 5minutes, since as soon as cover 3km in under 5 minutes, I must have (by continuity) covered a mile in under 5 minutes.

Now, since I hate to be a downer, there’s a positive result to share, which can be thought of as a generalization of the above discussion, but does not require any differentiability, or even continuity (unless there are philosophical objections to the discontinuous crossing of a mile).  I’ll state the theorem for the unit interval, then translate that to the running example:

Suppose  f:[0,1] \to \mathbb{R} has average speed M (i.e., f(1)-f(0) = M) and n is fixed.  Then for any p \in (0,1] with p \leq 1/n, there is an interval with length p where the average speed is at least M/n.

To see why this is true, just split up the interval into n equal subintervals, and observe that the “average speed” on at least one of them must be at least M/n, or else the total change would not be great enough.

Hence, in the running example, if I ran a 9:00 3k, then I must have covered at least one of the kilometers in 3minutes or less, even if I didn’t cover any mile in under 5minutes!