Teacher page: Maclaurin series (AA 5.19)

The one thing to do with the figure

Add terms one at a time and say nothing.

The polynomial grips the curve outward from the origin, and the readout says how far the match extends. Each new term pushes that number out.

This is the single most useful thing on the page: students who have only seen the series written down do not know it is LOCAL, and will happily use three terms of sin at x = 5.

Start from a known series, almost always

Deriving from scratch means repeated differentiation and is slow and error-prone. Nearly every question wants a substitution, product, differentiation or integration applied to one of the six standard series.

Saying which one you started from, and what you did to it, is a method mark.

The answers

e^x	1 + x + x²/2! + x³/3! + …; four terms at x = 1 give 2.667 against e = 2.718.
sin x	x − x³/3! + x⁵/5! − …; odd powers only.
cos x	1 − x²/2! + x⁴/4! − …; even powers only.
ln(1 + x), arctan x	Plain denominators, not factorials. A frequent mix-up.
1, 2, 3	−1/6; 2.5; B, substitute x² into the e^x series.

Where the marks go

1 markNaming the standard series you started from and the operation applied.

1 markCorrect coefficients, with factorials in the right places.

1 markStopping at the required power, and saying so.

1 markAny numerical evaluation, to the accuracy asked for.

What each wrong answer tells you

They give	What it means
Factorials in the ln series	The ln and arctan series have plain denominators. A very common slip under pressure.
Even powers in sin	sin is odd, so they vanish. If one appears, something has gone wrong earlier.
Squaring the e^x series for e^x²	That gives e^2x. The argument changes, not the function.
Too many terms	Harmless mathematically, but a question asking up to x⁴ is testing whether they stop.

Other things they will say

"Does it work everywhere?" No, and the animation shows it. These are series about zero, accurate nearby and progressively worse further out. Radius of convergence is beyond this course but the idea is visible.

"Why factorials?" Because the nth coefficient is the nth derivative divided by n!. Differentiating xⁿ n times produces exactly n!, so they cancel.

"Can I get a series from a differential equation?" Yes, and the guide expects it: differentiate the equation repeatedly, evaluate at zero, and build the coefficients.

A possible order

	What is happening
1	The animation on sin x. Add terms and watch the grip spread.
2	Where the coefficients come from, building e^x from scratch once.
3	The six standard series, with the odd and even patterns.
4	Building new series by substitution, product, differentiation and integration.
5	Questions 1 to 3.
6	From a differential equation, if there is time.

Two things not to say

Do not derive every series from scratch. One is enough; the rest come from the standard list and transformations, which is what is actually examined.

Do not leave the local nature implicit. A student who uses a short series far from zero gets a wrong answer and no warning.

Questions to set

Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.

1Fluency

The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.

Write the seriesState the first four terms of the Maclaurin series for e^x.
1 + x + x²/2 + x³/6
Use itUse those four terms to estimate e^0.5.
1 + 0.5 + 0.125 + 0.0208 = 1.6458.

2In context

The same skill inside a real situation, where the first job is working out what is being asked.

Measure the errorThe exact value of e^0.5 is 1.6487. State the error and say what reduces it.
0.0029 low. More terms, or a value of x closer to zero. The series is built at 0 and is most accurate there.
A trigonometric oneUse x − x³/6 + x⁵/120 to estimate sin 0.3, and compare.
0.2955, and the exact value is 0.2955 to four decimal places. Three terms is already enough at this size of x.

3Challenge

Reasoning, working backwards, or spotting an error. These are where the top grades are decided.

Why it grips near zeroExplain why a Maclaurin polynomial fits well near x = 0 and badly far from it, in terms of what the construction matches.
Each term is chosen so one more derivative agrees with the function at x = 0. Matching value, gradient, curvature and so on pins the polynomial tightly at that single point, and nothing in the construction controls behaviour away from it.
Find a limit with itUse series to evaluate the limit of (e^x − 1 − x)/x² as x tends to 0.
The numerator is x²/2 + x³/6 + ..., so dividing gives 1/2 + x/6 + ..., which tends to 1/2. L'Hopital twice gives the same answer, and the series shows why.

Practicalities

Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.