Maths IA · Calculus · topic 11 of 40

When is the rate of change greatest in real data? · IB Maths IA idea: guide, sample graph and step-by-step plan

gentle, but only if you test it

Real data is jagged, and you cannot differentiate a jagged line. So how do you say when it was changing fastest?

Difficulty●○○ Gentle
KitA real data set (shares, scores, growth), GDC or spreadsheet
TimeAbout 10 to 12 hours
FitsAA and AI, SL and HL
  • ✓ you follow a team, a share price or a trend
  • ✓ you can download and tidy a data set
  • ✓ you want a topic where interpretation matters

The big idea

the curve you choose IS the answer

The rate of change is the derivative, but data is not a formula. To use calculus you must first choose a smooth curve that represents the data. That choice is the heart of the IA. A curve that is too smooth hides real events, and one that is too wiggly turns noise into fake bursts of speed. Find where the rate is greatest, then ask whether the answer survives a different choice of curve. Explain what your result means for the sport, the company or the trend, not only the number.

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 →

Choose the data for a reason

A team's cumulative points, a share price around an event, a game's player count, or temperature through a day.

→ The interpretation of 'fastest' changes with the context, and your conclusion can say something real about it.
Compare smoothers

Fit a logistic or other model, a moving average and a spline to the same data.

→ You compare where each says the rate peaks. If they disagree, that is a finding, and your reflection has real evidence.
Differentiate two ways

Estimate the rate by finite differences on the raw data, and by differentiating the fitted formula.

→ You compare a noisy estimate with a smooth one, and explain why they differ.
Look at the second derivative

Find where the rate itself changes fastest, or where the curve turns from speeding up to slowing down.

→ You use the second derivative with meaning, linking the maths to momentum or trend changes.
Test for overfitting

Hold back part of the data, fit to the rest, and see how well the curve predicts it.

→ An honest measure of whether your curve is a model or just a tracing. A strong point in evaluation.
Event study

Mark a known event (a signing, an announcement) and ask whether the rate really changed around it.

→ Your analysis has a question with an answer you can test, not just a description.

Your Maths IA research question: run the check

Write your own question first. Then make sure it ticks every box:

  • Names the data, the period and the quantity whose rate you want, and why those choices.
  • Says what 'greatest rate' will mean in the context (fastest rise? fastest change in either direction?).
  • Says how you will smooth the data and how you will check that the smoothing is not driving the answer.
  • Includes a test of the result, such as a second smoother or held-back data.
  • Is narrow enough for 12 to 20 pages and links the result back to the real situation.

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The picture to draw

Noisy data, a smooth curve, and its rate of change underneath. The peak of the lower graph is the answer to your question. Draw yours with your own data.

ytdy/dtnoisy data + a smooth curve you chosetangent: slope = rategreatest rate: the peak

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.

Aim and context~1 page
Your data and why. What 'fastest' will mean in this context, and a prediction of when it happens.
The data~2 pages
Source, how you chose the period, what you removed or kept and why, and a graph of the raw data.
Choosing a smooth curve~3 pages
At least two candidate curves with reasons, how each was fitted, and how you judged them beyond one number.
Finding the rate~2.5 pages
The derivative of your chosen curve, worked by hand, the point where it is greatest, and a check that it is a maximum.
Testing the answer~2 pages
What happens when you change the curve or leave data out. A table of where each version puts the peak.
Conclusion~1 page
Back to the aim: when was the rate greatest, how sure are you, and what does it mean in context?
Evaluation~2 pages, the biggest part!
Overfitting, the choice of period, whether the data is complete, and what a different method would change.
Bibliographynot counted
Data source with date accessed, and any method you used.

What your Maths IA graph could look like

show the overfit on purpose

Made-up form: noisy data and two curves. Yours will look different. example only

01020304050600102030405060708090100110Trading day (counted from the start)Share price / $ (made-up)smooth curve (a logistic model)overfitted curve: chases the noisemade-up daily pricessteepest point of the smooth curve:where is it in YOUR data, and does itmove if you change the smoother?ILLUSTRATIVE · MADE-UP DATA
  • Raw data and the fitted curve share the same axes, so the reader sees how much smoothing was done.
  • The wiggly curve shows what overfitting looks like. Its slope jumps around, and so would the rate you calculate from it.
  • Say why you chose your curve: from the shape of the data, the situation, or a test, not because the software offered it.
  • Mark the steepest point and write the rate there, with units, in the text.
  • Another graph to consider: the rate of change itself, as in the diagram, for two or three different smoothers.

Your game plan

break it on purpose in step 7

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Choose the data and predictPick data you care about. Say when you expect the fastest change and what you think caused it.
  2. Clean and lookPlot the raw data. Note gaps, jumps and events. Decide what to keep, with a reason.
  3. Pick candidate curvesThink about what shape the data suggests and why. Choose at least two types of curve.
  4. Trial run (1 lesson)Fit one curve and differentiate it. Does the rate make sense in context? Is the peak sharp or broad?
  5. Fit and compareFit each curve, look at residuals, and decide which one you trust and why.
  6. Differentiate by handWork out the derivative of your chosen formula in full, and find where it is greatest.
  7. Stress the answerChange the smoother, drop some data, or move the period. Record where the peak goes.
  8. InterpretSay what the result means in the real situation, including how uncertain the timing is.
  9. Reflect as you goAfter each step, say what it assumes. HL: say why your chosen curve is justified, not just fitted.
  10. Cut, check, then sendCheck axes, units, notation and that the conclusion answers your aim. Give your teacher one draft.

Maths you'll need

Traps to avoid in this Maths IA

  • Picking the curve with the highest R² and calling it the model. High-degree polynomials fit anything.
  • One smoother only, so you never learn whether the answer depends on it.
  • Dumping a year of stock data with no reason for the dates.
  • Differentiating a raw, noisy data set and reading a peak that is only noise.
  • A result in the wrong units, or a rate of 'per index point' with no context.
  • At HL: no justification for the model beyond the software offering it.
  • Reflection that only says the result 'matched expectations'.

More Maths IA ideas like this

Make sure it's yours

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.

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