Tag Archives: limits

Series Comfort

I ‘blogged a couple days ago about a way to use statistical regressions to develop Maclaurin Series in a way that precalculus or algebra II students could understand. In short, that approach worked because the graph of y=e^x is differentiable–or locally linear as we describe it in my class.  Following is a student solution to a limit problem that proves, I believe, that students can become quite comfortable with series approximations to functions, even while they are still very early in their calculus understanding.

This year, I’m teaching a course we call Honors Calculus.  It is my school’s prerequisite for AP Calculus BC and covers an accelerated precalculus before teaching differential calculus to a group of mostly juniors and some sophomores.  If we taught on trimesters, the precalculus portion would cover the first two trimesters.  My students explored the regression activity above 2-3 months ago and from that discovered the basic three Maclaurin series.

\displaystyle e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\ldots
\displaystyle cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\ldots
\displaystyle sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\ldots

Late last week, we used local linearity to establish L’Hopital’s Rule which I expected my students to invoke when I asked them on a quiz two days ago to evaluate \displaystyle\lim_{x\to 0}\frac{x\cdot sin(x)}{1-cos(x)}.  One student surprised me with his solution.

He didn’t recall L’Hopital’s, but he did remember his series exploration from January.

Had he not made a subtraction error in the denominator, he would have evaluated the limit this way.

\displaystyle\lim_{x\to 0}\frac{x\cdot sin(x)}{1-cos(x)}=\lim_{x\to 0}\frac{x*(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\ldots)}{1-(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\ldots)}

\displaystyle=\lim_{x\to 0}\frac{x^2-\frac{x^4}{3!}+\frac{x^6}{5!}-\ldots}{\frac{x^2}{2!}-\frac{x^4}{4!}+\ldots}

And dividing the numerator and denominator by x^2 leads to the final step.

\displaystyle=\lim_{x\to 0}\frac{1-\frac{x^2}{3!}+\frac{x^4}{5!}-\ldots}{\frac{1}{2!}-\frac{x^2}{4!}+\ldots}

=\displaystyle\frac{1-0+0-\ldots}{\frac{1}{2}-0+0-\ldots}=2

Despite his sign error, he came very close to a great answer without using L’hopital’s Rule at all, and showed an understanding of series utility long before most calculus students ever do.

Advertisements

Learning to listen

Listen to and learn from your students; they hold the key to deeper understanding.

My precalculus class yesterday was exploring graphical behavior of rational functions.  We started with \frac {1}{(x-2)^3(x+3)^{6}} which the group easily handled with their understanding of even and odd vertical asymptotes (VAs) from the previous class.

The curve approaches 0^+ on the right, the VA at x=2 is odd so the rational function’s graph “passes through infinity” there, and “bounces off infinity” at x=-3.

We hadn’t explored what happened when rational functions had variable expressions in their numerators, and they had never seen holes in these curves, so I had no idea what they would do when I asked for the graphical behavior of \frac {x-2}{(x-2)^3(x+3)^{6}} at x=2.  As a student, I was taught to perform algebraic simplifications, but I thankfully remained silent.

My students were initially bothered by the \frac{0}{0}  form of the function at x=2, and various clusters were working toward different solutions when student NC declared, “at x=2, the denominator has the dominant exponent, so there still is a VA at that point.  It might be a different kind of VA, but it’s a VA.”  In an instant, everyone in the room understood what was happening.  Changing the degree of the numerator’s factor in later examples ultimately led to holes on and off the x-axis, but NC’s dominance argument yielded far deeper and lasting understanding than my plans for an algebraic approach ever would have.

OK, I admit that you can get to the same place by doing algebraic simplifications, but my deliberate silence allowed my students to develop their own subtly different understanding that exceeded what I had planned to offer.  After years of being the information dispenser in my classes, I’m learning the uncomfortable lesson that it’s often better to set them up and then shut up.