Since moving to my current school, this is my first time teaching Edexcel’s FP2 syllabus. It is interesting to see how different exam boards prioritise different topics, or how far they develop them. In the past I’ve taught both MEI and OCR and, if I remember rightly, neither pushes Taylor series as far as Edexcel. I think it is a particularly satisfying section of work, developing from Maclaurin to a more general Taylor series and then seeing how these can be used to approximate solutions to differential equations.
Maclaurin or Taylor?
From the scant history I’ve read, it would seem that Brook Taylor developed this theory a little earlier (and more extensively) than Colin Maclaurin, and that Maclaurin made much more important contributions to maths in other areas. This was all taking place in the first half of the eighteenth century when British mathematicians were developing Newton’s work on calculus.
The wow factor
I love to begin this topic by showing students some graphs. The key questions are:
- What are the equations of these graphs?
- What is your evidence?
Here is my Desmos page with them drawn.
The title is a deliberate red herring, but I still want students to explain their evidence using correct mathematical terminology. They should notice intercepts, periodicity, the fact that the blue graph appears to have vertical asymptotes etc.
Of course, the excitement comes when we zoom out a little:
Now again is an important time to ask students what they notice and what they think. (Can they develop the ideas that the curves might be polynomials? The red one of odd degree, whilst the orange is of even degree?)
Naturally, the key motivating question here is how did we create polynomials that are such a good likeness for sin x, cos x and tan x, at least near the origin?
A good match
We need to talk about what it means for a polynomial to be a ‘good match’ to another function. I like to impress upon students that we only focus on a single point on the graph, eg where x=0. We want the polynomial p(x) to have the same value as the function f(x) that we are approximating. Then develop the idea that we would also like p'(0)=f'(0) and p”(0)=f”(0) etc. Students can work with a general quartic, and apply those conditions to determine an approximation for . (Following through this differentiation process helps the students understand why the Maclaurin series has the pattern it does: they see the factorials build up as they differentiate repeatedly, for example.)
Maths is a game of information
I also like to use this lesson to discuss the concept of information. How much information do we need to specify the equation of a straight line? What form could it be given in? (Two pieces: eg coordinates of two points, or a y-intercept and gradient.) What about for a quadratic? Cubic? The conditions we apply to develop a Maclaurin series are simply alternative forms of supplying the same amount of information.
Acupuncture or Earthquake
I surprised myself by coming out with an analogy to acupuncture this year, but there is an incredibly powerful concept behind specifying the derivatives of a function at a single point. Purely by specifying f(0), f'(0), f”(0) and f”'(0) the effect of these values shapes the curve over a much wider domain – their effect ripples out like shockwaves from an earthquake.
Set them exploring
It’s great for the students to be able to derive the famous series expansions themselves. Set them up with Maclaurin’s formula and then ask them to find series for cos x, ex and tan x. Ensure you precede the statement of Maclaurin’s expansion with the need for f(x) and all of its derivatives to exist at x=0.
Level up: Taylor series
Taylor series develops simply as a translation of the concept which we’ve been calling a Maclaurin series. Here is another Desmos sheet (not written by me) that allows students to really see the effect of increasing the degree of the polynomial approximation, or translating the point at which we approximating f(x).
Unlike the Edexcel textbook (which provides two forms), I always teach Taylor as:
This is all the students need and it is the easiest to understand in terms of translation away from the origin. I have even gone to the extent of rewriting solutions to some textbook examples (in particular, Example 10) using this simpler approach. The documents are in this Dropbox folder. (I have also rewritten an example of approximating a solution to a DE with clearer notation.)
Advice for students
- Focus on the part of the question asking for “powers of …”. This will let you know the value of needed to create the approximation.
- Separate your working into two clear columns: one for the algebra of differentiation, and one for the numerics of evaluating the derivatives.
- Simplify your notation as much as possible: writing f”(0) is much simpler than .
- When differentiating tan x and similar functions, use a recursive strategy to simplify the process of finding higher derivatives.
- Enjoy Taylor series! They are so powerful and not too difficult to find!