4.13 Non-linear regression questions for IB Maths AI

53 questions on Non-linear regression, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Applications and Interpretation
Part of 4 Statistics and probability
HL only

Questions by paper and level

Paper	Questions
Paper 1	18
Paper 2	31
Paper 3	4
Every paper	53

Counted on 10 November 2026

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Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AI-00576Paper 1 · 7 marks

The number of subscribers, $y$ thousand, to a new podcast $x$ months after its launch is shown in the table.

$x$ (months)	1	2	3	4	5	6
$y$ (thousand)	9.2	11.0	14.3	18.9	25.0	33.1

The podcast's producers consider two models.

Model A: the least squares regression line of $y$ on $x$ .

Model B: $y = 7.0 (1.29)^{x}$ .

(a)
For Model A, write down the equation of the regression line and its sum of the squares of the residuals, $S S_{r e s}$ .
2 marks
(b)
Find $S S_{r e s}$ for Model B.
2 marks
(c)
State which model fits the data better. Give a reason for your answer.
2 marks
(d)
Give one reason why Model B should not be used to predict the number of subscribers 5 years after the launch.
1 mark

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Question 2 · AI-00246Paper 2 · 15 marks

Dr Okafor grows a bacterial culture in a petri dish. She measures the number of bacteria, $N$ thousand, $t$ hours after the start of the experiment.

$t$ (hours)	0	2	4	6	8	10	12	14
$N$ (thousands)	2.1	5.8	14.2	27.6	39.5	46.0	48.7	49.9

(a)
Dr Okafor first fits an exponential model, $N = a b^{t}$ , to the data for $0 \leq t \leq 6$ only. Use exponential regression on these four data points to find the value of $a$ and the value of $b$ .
3 marks
(b)
Use this exponential model to predict the number of bacteria when $t = 14$ . Hence comment on the suitability of the exponential model for $t = 14$ .
3 marks
(c)
She then fits a logistic model of the form $N (t) = \frac{L}{1 + C e^{- k t}}$ to all the data. Use logistic regression to find the value of $L$ , of $C$ and of $k$ .
3 marks
(d)
Interpret the value of $L$ in the context of the experiment.
1 mark
(e)
Use the logistic model to find the time at which the number of bacteria first reaches 45 thousand.
2 marks
(f)
Find the time at which, according to the logistic model, the number of bacteria is increasing most rapidly, and the rate of increase at that time.
3 marks

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Question 3 · AI-01461Paper 3 · 27 marks

The purpose of this question is to model the cooling of bread so that a bakery knows when its loaves can be sliced.

At Brot und Mehl bakery, loaves are taken out of the oven with a core temperature of 96 °C and left on racks in a cooling room kept at 22 °C. Newton's law of cooling states that the rate at which the temperature of an object decreases is proportional to the difference between its temperature and the temperature of its surroundings.

Let $T$ °C be the core temperature of a loaf $t$ minutes after it is taken out of the oven.

(a)
Write down a differential equation for $T$ , using a positive constant $k$ .
2 marks
(b)
By solving your differential equation, show that $T = 22 + A e^{- k t}$ , where $A$ is a constant.
4 marks
(c)(i)
Write down the value of $A$ .
1 mark
(c)(ii)
After 20 minutes the core temperature of a loaf is 61 °C.
Show that $k = 0.0320$ , correct to three significant figures.
2 marks
(d)
A loaf can be sliced once its core temperature has fallen to 35 °C.
Use the model to find how long after leaving the oven a loaf can be sliced. Give your answer to the nearest minute.
2 marks
(e)
Find the rate at which the core temperature is changing at the moment the loaf leaves the oven.
2 marks
(f)(i)
The head baker, Jonas, thinks that $k$ should be estimated from more than one reading. He records the core temperature of a loaf every 10 minutes.
$t$ (minutes)
0
10
20
30
40
50
60
$T$ (°C)
96
75
61
51
43
37
33
$y = \ln (T - 22)$
4.304
3.664
3.367
2.708
Complete the table, giving each value of $y$ correct to three decimal places.
2 marks
(f)(ii)
Find the equation of the regression line of $y$ on $t$ .
2 marks
(f)(iii)
Hence find Jonas's estimates for $k$ and for $A$ .
3 marks
(g)
Write down the value of the coefficient of determination, $R^{2}$ , for the regression in part (f)(ii), and interpret it in context.
2 marks
(h)(i)
On busy days the ovens warm the cooling room. Jonas models the room temperature as $22 + 0.15 t$ °C and uses the differential equation
$\frac{d T}{d t} = - 0.032 (T - 22 - 0.15 t), T (0) = 96.$
Use Euler's method with a step length of 5 minutes to estimate the core temperature of a loaf after 40 minutes on a busy day.
4 marks
(h)(ii)
On a busy day, the measured core temperature of a loaf after 40 minutes is 45.2 °C. Suggest how Jonas could use Euler's method to obtain a more accurate estimate.
1 mark

$t$ (minutes)	0	10	20	30	40	50	60
$T$ (°C)	96	75	61	51	43	37	33
$y = \ln (T - 22)$	4.304		3.664	3.367		2.708

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Question 4 · AI-00616Paper 1 · 5 marks

Leilani photographs a ceramic vase lying on its side and places axes on the photograph, with the origin at the centre of the base of the vase. The upper edge of the vase on the photograph is modelled by the cubic regression curve

y = 0.02 x^{3} - 0.3 x^{2} + 1.4 x + 2, 0 \leq x \leq 8,

where $x$ and $y$ are measured in cm. This curve has a local maximum point at $(3.71, 4.09)$ .

The real vase is an enlargement of the image on the photograph, with scale factor 3 about the origin.

(a)
Find the equation of the curve that models the upper edge of the real vase.
3 marks
(b)
Write down the coordinates of the corresponding local maximum point on the edge of the real vase.
2 marks

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Question 5 · AI-00253Paper 2 · 12 marks

Ana pours a cup of tea in a room where the temperature is a constant 21 °C. She records the temperature of the tea, $T$ °C, $t$ minutes later.

$t$ (minutes)	0	2	4	6	8	10	15	20
$T$ (°C)	88	79.5	72.0	65.6	60.2	55.4	46.2	39.5

Ana believes that the temperature of the tea will approach the temperature of the room.

(a)
Let $y = T - 21$ . Use exponential regression to find a model of the form $y = a b^{t}$ .
3 marks
(b)
Hence write the model in the form $T = 21 + a e^{- k t}$ , giving the value of $k$ .
2 marks
(c)
Find the time taken for the difference between the temperature of the tea and the temperature of the room to halve.
2 marks
(d)
Ana drinks her tea once its temperature falls below 60 °C. Use the model to find the time at which she starts to drink it.
2 marks
(e)(i)
Ana's friend Bruno fits an exponential model directly to the values of $T$ and obtains $T = 85.0 \times {0.961}^{t}$ .
Find the temperature that Bruno's model predicts after 2 hours.
2 marks
(e)(ii)
Hence explain why Ana's model is more appropriate for large values of $t$ .
1 mark

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Question 6 · AI-01644Paper 3 · 25 marks

This question compares models for the power output of a wind turbine.

An engineer, Tomasz, records the wind speed, $v$ m s⁻¹, and the power output, $P$ kW, of a small wind turbine at a coastal site.

$v$ (m s⁻¹)	4	5	6	7	8	9	10	11	12
$P$ (kW)	58	102	190	285	441	612	862	1120	1475

(a)(i)
Find the equation of the regression line of $P$ on $v$ .
2 marks
(a)(ii)
Write down the coefficient of determination, $R^{2}$ , for this linear model.
1 mark
(a)(iii)
Calculate the residuals of the linear model at $v = 4$ , $v = 8$ and $v = 12$ , and hence explain why a linear model is not appropriate, despite the high value of $R^{2}$ .
3 marks
(b)(i)
Tomasz next considers a power model $P = a v^{b}$ .
Show that this model gives a linear relationship between $\ln P$ and $\ln v$ .
2 marks
(b)(ii)
Use your graphic display calculator to find the values of $a$ and $b$ .
2 marks
(b)(iii)
For a model $P = f (v)$ , the sum of square residuals is $S S_{r e s} = \sum_{i = 1}^{9} (P_{i} - f (v_{i}))^{2}$ .
Find the value of $S S_{r e s}$ for the power model.
2 marks
(c)(i)
Physical theory suggests that the power output is proportional to the cube of the wind speed, so Tomasz now considers the model $P = k v^{3}$ . He chooses $k$ to minimize
$S (k) = \sum_{i = 1}^{9} {(P_{i} - k v_{i}^{3})}^{2} .$
Show that $S (k)$ is minimized when $k = \frac{\sum v_{i}^{3} P_{i}}{\sum v_{i}^{6}}$ .
4 marks
(c)(ii)
Hence find the value of $k$ .
2 marks
(d)
For the model $P = k v^{3}$ , $S S_{r e s} = 516$ , correct to three significant figures.
Tomasz notices that this one-parameter model has a smaller value of $S S_{r e s}$ than the power model in part (b), which has two parameters. Explain why this is possible, and state which of the two models fits the data better.
3 marks
(e)
On one day, the wind speed at the site $t$ hours after midnight is modelled by $v = 7 + 3 \sin (\frac{π t}{12})$ , for $0 \leq t \leq 24$ .
The energy, in kWh, produced by the turbine in one day is $\int_{0}^{24} P d t$ . Use the model $P = k v^{3}$ to find the energy produced that day.
3 marks
(f)
The turbine's maximum power output is 1500 kW. Comment on using the model $P = k v^{3}$ to predict the power output when the wind speed is 15 m s⁻¹.
1 mark

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