5.1 Limits and the derivative questions for IB Maths AI

23 questions on Limits and the derivative, 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 5 Calculus
SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

Paper	SL and HL	All
Paper 1	11	11
Paper 2	12	12
Every paper	23	23

Counted on 10 November 2026

Questions to try

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-01153Paper 1 · 5 marks

Consider the function $f (x) = 0.5 x^{3}$ . The point $P (2, 4)$ lies on the graph of $f$ .

A second point $Q (x, f (x))$ also lies on the graph of $f$ . The following table shows the gradient of the chord $P Q$ for different values of $x$ .

$x$ -coordinate of $Q$	2.5	2.1	2.01	2.001
Gradient of $P Q$	7.625	$p$	6.03005	$q$

(a)
Find the value of $p$ and the value of $q$ . Give your answers correct to five decimal places where necessary.
3 marks
(b)
Use the table to estimate the gradient of the tangent to the graph of $f$ at $P$ .
1 mark
(c)
Write down the value of $f^{'} (2)$ .
1 mark

The markschemeCheck this one with a free account

Question 2 · AI-01163Paper 2 · 16 marks

Matteo switches on an oven in his pizzeria. The temperature inside the oven, $T^{\circ} C$ , $t$ minutes after it is switched on, is modelled by

T (t) = 220 - 200 e^{- 0.08 t}, t \geq 0.

(a)
Write down the temperature inside the oven when it is switched on.
1 mark
(b)
Complete the following table, giving your answers correct to one decimal place.
$t$ (minutes)
30
60
90
$T$ (°C)
201.9
2 marks
(c)(i)
Write down the value that $T$ approaches as $t \to \infty$ .
1 mark
(c)(ii)
Interpret this value in the context of the question.
1 mark
(d)
Find the average rate of change of the temperature during the first 10 minutes.
3 marks
(e)(i)
Use your graphic display calculator to find $T^{'} (10)$ .
2 marks
(e)(ii)
Interpret your answer to part (e)(i) in the context of the question.
1 mark
(f)
Explain why your answers to part (d) and part (e)(i) are different.
1 mark
(g)
Matteo puts a pizza in the oven when the temperature is increasing at a rate of $5^{\circ} C$ per minute.
Find the temperature of the oven at this time.
3 marks
(h)
Suggest one reason why the model may not be valid once the pizza has been put in the oven.
1 mark

$t$ (minutes)	30	60	90
$T$ (°C)	201.9

The markschemeCheck this one with a free account

Question 3 · AI-01154Paper 1 · 6 marks

A laboratory culture of bacteria contains $N$ bacteria, $t$ hours after it is first observed. The number of bacteria can be modelled by $N (t) = 500 \times 3^{t}$ , $t \geq 0$ .

The average rate of increase of the number of bacteria between $t = 0$ and $t = h$ is given by

g (h) = \frac{500 (3^{h} - 1)}{h}, h > 0.

(a)
Write down the reason why $g (0)$ cannot be calculated.
1 mark
(b)
Complete the following table. Give your answers correct to one decimal place.
$h$
1
0.1
0.01
0.001
$g (h)$
1000
2 marks
(c)
Hence estimate the rate at which the number of bacteria is increasing at the instant $t = 0$ .
2 marks
(d)
State the units of $g (h)$ .
1 mark

$h$	1	0.1	0.01	0.001
$g (h)$	1000

The markschemeCheck this one with a free account

Question 4 · AI-01164Paper 2 · 16 marks

A tram travels in a straight line between two stops. It leaves the first stop at time $t = 0$ and arrives at the second stop at $t = 20$ , where $t$ is measured in seconds. The distance, $s$ metres, that the tram has travelled after $t$ seconds is modelled by

s (t) = 1.8 t^{2} - 0.06 t^{3}, 0 \leq t \leq 20.

(a)
Find the distance between the two stops.
2 marks
(b)
Find the average speed of the tram between the two stops.
2 marks
(c)
The table below shows the average speed of the tram between $t = 5$ and $t = 5 + k$ for some values of $k$ .
$k$ (s)
1
0.1
0.01
0.001
Average speed (m s⁻¹)
14.34
$a$
13.508994
13.500900
Find the value of $a$ , and hence estimate the speed of the tram at $t = 5$ .
3 marks
(d)
Find $\frac{d s}{d t}$ .
2 marks
(e)
Use your answer to part (d) to show that the speed of the tram at $t = 5$ is $13.5$ m s⁻¹.
2 marks
(f)
Find the values of $t$ at which the speed of the tram is equal to its average speed between the two stops.
3 marks
(g)
Explain why the tram's speed at $t = 20$ is consistent with the context.
2 marks

$k$ (s)	1	0.1	0.01	0.001
Average speed (m s⁻¹)	14.34	$a$	13.508994	13.500900

The markschemeCheck this one with a free account

Question 5 · AI-01155Paper 1 · 6 marks

Amara pours a cup of coffee and leaves it on her desk. The temperature of the coffee, $T^{\circ} C$ , $t$ minutes after it is poured, can be modelled by

T (t) = 22 + 68 e^{- 0.045 t}, t \geq 0.

(a)
Calculate the average rate of change of the temperature of the coffee during the first 10 minutes.
3 marks
(b)
Use your graphic display calculator to find $T^{'} (4)$ .
2 marks
(c)
Interpret your answer to part (b) in the context of the question.
1 mark

The markschemeCheck this one with a free account

Question 6 · AI-01165Paper 2 · 16 marks

In this question, give all answers to two decimal places where necessary.

A company makes electric scooters. The monthly cost, $C$ US dollars (USD), of making $x$ scooters is modelled by

C (x) = 0.025 x^{2} + 15 x + 2400, 0 \leq x \leq 700.

(a)
Find the cost of making 100 scooters in a month.
2 marks
(b)
Find the extra cost of making the 101st scooter.
2 marks
(c)(i)
Find $C^{'} (x)$ .
2 marks
(c)(ii)
Hence find the marginal cost when $x = 100$ .
1 mark
(c)(iii)
Explain why your answers to parts (b) and (c)(ii) are close but not equal.
1 mark
(d)
Each scooter can be sold for $(63 - 0.035 x)$ USD, where $x$ is the number of scooters made and sold in a month.
Show that the monthly profit, $P$ USD, is given by $P (x) = - 0.06 x^{2} + 48 x - 2400$ .
2 marks
(e)
Find $P^{'} (200)$ and interpret its value in the context of the question.
2 marks
(f)
State what the sign of $P^{'} (500)$ tells the company.
1 mark
(g)
Find the number of scooters the company should make each month to maximize its profit, and the maximum monthly profit.
3 marks

The markschemeCheck this one with a free account

Practice all 23 in Augment

Free with an account. Every question has its markscheme, and Augment keeps the ones you got wrong so you can come back to them.

Start free Log in

The notes on 5.1 Limits and the derivative

Check your answers in Augment

A free account opens all 23 questions on Limits and the derivative with their markschemes, and keeps a record of what you have marked.

Start free

Questions by paper and level

Questions to try

Practice all 23 in Augment

More questions in Calculus

Check your answers in Augment