Reminding students about Taylor series (Part 1)

At my university, Calculus II covers approximately the same topics covered in an AP Calculus BC course: integrals and derivatives with logarithms and exponential functions, various techniques of integration (including integration by parts and trigonometric substitutions), and convergence of infinite series.

In my opinion, the single most important of these topics is Taylor series (or, if you prefer, Maclaurin series), as these approximations to transcendental functions like $e^x$ and $\sin x$ are used over and over again in higher mathematics.

$\bullet$ A good working knowledge of Taylor series is necessary for computing series solutions of ordinary differential equations.

$\bullet$ In physics, elementary approximations like $\sin x \approx x$ are used over and over again. For example, the governing differential equation for the motion of oscillating pendulums is

$\displaystyle \frac{d^2 \theta}{dt^2} + \frac{g}{\ell} \sin \theta = 0$,

where $g$ is the acceleration due to gravity and $\ell$ is the length of the pendulum. This differential equation cannot be solved exactly, and its solution is very complex.

However, for small angles, we may use the approximation $\sin \theta \approx \theta$, so that the differential equation becomes

$\displaystyle \frac{d^2 \theta}{dt^2} + \frac{g}{\ell} \theta = 0$,

By eliminating the $\sin \theta$ term, we now have a second-order differential equation with constant coefficients, which can be solved in a straightforward manner using standard techniques from differential equations. If $\theta(0) = \theta_0$ and $\theta'(0) = 0$ (i.e., the pendulum is pulled a small angle $\theta_0$ and is then released), the solution is

$\theta(t) = \theta_0 \cos\left(t \sqrt{\displaystyle \frac{g}{\ell}} \right)$.

In other words, the pendulum exhibits sinusoidal behavior. (FYI, for an amazing display of kinetic art, see this demonstration of pendulum waves.)

$\bullet$ The primary way that students interface with Taylor series is through their calculators. When a calculator computes $\cos 1000^o$, it doesn’t draw a unit circle, trace out an angle of $1000^o$ in standard position, and find the $x-$coordinate of the terminal point. Instead, the calculator converts $1000^o$ into radians and adds the first few terms of the Taylor series expansion for $\cos x.$

The calculator may use a few tricks to accelerate convergence. For this example, using some trigonometric identities, $\cos 1000^o= \cos 280^o= \cos 80^o= \sin 10^o$, and (as I’ll discuss) the Maclaurin series for $\sin x$ at $x = 10^o$ converges much faster than the Maclaurin series for $\cos x$ at $x = 1000^o$.

I’ve argued the importance of Taylor series in higher-level courses in both mathematics and physics. Sadly, at least at my university, Taylor series is probably the topic that is least retained by students years after taking Calculus II. They can remember the rules for integration and differentiation, but their command of Taylor series seems to slip through the cracks.

In my opinion, the reason for this lack of retention is completely understandable from a student’s perspective: Taylor series is usually the last topic covered in a semester, and so students learn them quickly for the final and quickly forget about them as soon as the final is over.

Of course, when I need to use Taylor series in an advanced course but my students have completely forgotten this prerequisite knowledge, I have to get them up to speed as soon as possible. Over the next few posts, I will present the sequence of examples that I use to accomplish this task. Covering this sequence usually takes me about 30-40 minutes of class time, depending on the class.

I should emphasize that, as much as possible, I present this sequence inductively and in an inquiry-based format: I ask leading questions of my students so that the answers of my students are driving the lecture. In other words, I don’t ask my students to simply take dictation. It’s a little hard to describe a question-and-answer format in a blog, but I’ll attempt to do this below.

Beginning with the next post, I’ll describe this sequence.

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