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