Maths IA · Calculus · topic 14 of 40
Maclaurin series: how many terms are enough? · IB Maths IA idea: guide, sample graph and step-by-step plan
all about the maths: great if you like algebraYour calculator never 'knows' the value of sin 0.3. It builds it from a handful of terms. How many does it need, and how do you know?
- ✓ you like seeing how an approximation improves
- ✓ you enjoy algebra and are happy with careful working
- ✓ you want a topic about the maths, not about data
The big idea
reason, don't just plotA Maclaurin series replaces a function by a polynomial built from its derivatives at zero. Take more terms and, near zero, the polynomial hugs the function. Far from zero, it may not. Your job is to choose a function, build the series yourself, and measure the error as terms increase. Find where the approximation works and where it fails. Then explain why, with reasoning and not just a picture. The syllabus gives standard series, so copying them is routine. The marks are in what you do with them.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Something like the inverse tangent, a square root, or e to the minus x squared. Build its series from derivatives.
→ You do real derivative work and can justify each coefficient. The standard functions are everyone's topic.Test numerically where the approximation breaks down, and compare with what a ratio test says.
→ The ratio test is beyond the course, so introduce it carefully. You link numerical evidence to proof, which is the sort of sophistication HL rewards.Estimate or bound the error after n terms, using an alternating-series estimate or the remainder term.
→ You can say how many terms are enough for a stated accuracy, with a reason, not just a comparison of values.The pendulum period, relativistic kinetic energy, or how a calculator computes a function.
→ The error has a meaning for the context, so 'how many terms' becomes a decision.Build the series about a point other than zero and compare with the Maclaurin version.
→ A better approximation near a different point, and a clear reason for choosing where to expand.A rational approximation, or interpolation through points on the curve.
→ You judge the polynomial against a rival on accuracy and effort, which makes the evaluation sharper.Your Maths IA research question: run the check
Write your own question first. Then make sure it ticks every box:
- Names the function and the range of x you care about, and says why.
- Says what 'good enough' means (a stated accuracy, or a stated error) so 'enough terms' has an answer.
- Says how error will be measured and over which values of x.
- Includes a part where you build or justify something, not only compute values.
- Is narrow enough for 12 to 20 pages and goes beyond the standard series.
The picture to draw
The method as a chain, and a sketch of a function against its polynomial. Draw yours with your function and your range.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
Why this function, which range of x matters, and what accuracy you will aim for. A prediction of where the approximation will fail.
Derivatives of your function at zero, the coefficients, and the first few terms, in proper notation. State the general term if you can find it.
A table of error against number of terms at chosen values of x, with a clear definition of error and a worked example.
Where the series works and where it does not. A reason, such as a ratio test, and a check against your numbers.
A decision for a stated accuracy, with justification. A graph of function and polynomials together.
Back to the aim: how many terms, for which values of x, and why.
Strengths and limits of the approach, a comparison with another approximation, and what a student could try next.
Sources for the series and any theorem you use.
What your Maths IA graph could look like
show where it FAILS tooMade-up form: a function against one of its polynomials. Yours will look different. example only
- The function and the polynomial share axes, so the reader sees exactly where they part.
- Axes cover the range you care about, including the region where it fails. Hiding the failure would be a weakness.
- Say in the text what the graph shows and what it does not: a picture cannot prove convergence.
- Add more polynomials (a lower and a higher degree) to show the pattern as terms increase.
- Another graph to consider: the error itself, on a log scale, against the number of terms at a few chosen values of x.
How fast does the error fall?
then try a bigger x!The same function at one value of x, with more and more terms. A chart like this, with your own function, shows how many terms it takes to reach the accuracy you set.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Choose a function and predictPick one that is not in the formula booklet. Predict where you expect the approximation to work and fail.
- Learn the constructionKnow why the coefficients are what they are. Try the method on a function whose series you already know.
- Build your own seriesDifferentiate repeatedly, evaluate at zero, and find a pattern. Check the first terms numerically.
- Trial run (1 lesson)Compute the error at one value of x for a few terms. Is your definition of error clear and the arithmetic right?
- Systematic error tableChoose values of x, including near the boundary, and record the error for increasing terms.
- Study convergenceUse a ratio test or another method to say where the series works. Compare with your table.
- Decide how many termsSet the accuracy you want, and show how many terms reach it at different x. Give a reason, not just a table.
- Compare or extendTry an expansion about another point, or a different approximation, and judge it against yours.
- Reflect as you goAfter each result, say what it tells you about the function. HL: say what you proved and what you only observed.
- Cut, check, then sendCheck notation (no calculator syntax), that every claim is justified, and that the conclusion answers your aim. Give your teacher one draft.
Maths you'll need
Traps to avoid in this Maths IA
- Copying the standard series from the booklet and plotting them. That is routine at HL and will cap criterion E.
- Calling the approximation good because the graph looks close, with no error measured.
- Ignoring convergence: a series can fit well near zero and be useless a little way off.
- Writing the series with calculator notation such as x^3 in the body text.
- Using equals where you mean approximately equals.
- A conclusion of 'more terms are more accurate' without saying how many, for what x, or why.
- Rounding too early, so the error table is noise.
More Maths IA ideas like this
- Draining tank: predict the time to empty●●●
- Pursuit curves: does the chaser ever catch up?●●●
- Cans: the least metal for a given volume●○○
This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.