The cost, $C$ in euros, of producing $x$ items is given by $C(x) = 1500 + 12x - 0.05x^2$ . Calculate the average rate of change of cost when production increases from $100$ to $120$ items. Find the instantaneous rate of change of cost (marginal cost) when exactly $100$ items are produced.

$C(100) = 1500 + 1200 - 500 = 2200$ . $C(120) = 1500 + 1440 - 720 = 2220$ . (M1) Average rate $= \frac{C(120) - C(100)}{120 - 100} = \frac{2220 - 2200}{20} = \frac{20}{20} = \text{€}1\text{ per item}$ . A1 Use CG50 d/dx . Enter $\frac{d}{dx}(1500 + 12x - 0.05x^2) \big|_{x=100}$ . (M1) Result $= \text{€}2\text{ per item}$ . A1

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IB Maths AI SL · Unit 5: Calculus

IB Maths AI SL Limits, Derivatives & Rates of Change Questions

Updated October 2026

Exam-style IB Maths AI SL limits, derivatives & rates of change questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

23 questions
Paper 1: 17
Paper 2: 6
5 easy
9 medium
6 hard
3 very hard
3 worked examples

Practise Limits, Derivatives & Rates of Change questions →

🖩 How do I do this on my calculator? Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire.

What you need to know

SL AI differentiation is limited to power rule, sum/difference, and the interpretation of dy/dx as a rate. Always link the derivative to the real-world quantity in the question. Basic differentiation and rates of change overview →

What's examined in AI SL limits, derivatives & rates of change

The question bank covers these limits, derivatives & rates of change question types (number of questions in brackets):

Function Analysis & Limits (11)
Instantaneous Rate of Change (8)
Average Rate of Change (4)

Key formulas

Derivative of xⁿ: $f (x) = x^{n} ⟹ f^{'} (x) = n x^{n - 1}$

In the same notation as the IB formula booklet. All AI SL formulas →

Limits, Derivatives & Rates of Change worked examples

Worked example 1: Average rate of change · easy

A particle's displacement is $s (t) = 3 t^{2} - 2 t + 1$ metres. Find the exact average velocity between $t = 1$ and $t = 4$ seconds.

Solution

5 steps · 1 to fill in

Examiner tip: Average rate of change is the gradient of the secant joining two points. Instantaneous velocity requires differentiation instead.

Worked example 2: Instantaneous rate of change via GDC · medium

Profit $P (x) = - 0.5 x^{3} + 12 x^{2} - 10 x + 50$ euros for $x$ units sold. Use the GDC to find the instantaneous rate of change when $x = 10$ .

Solution

5 steps

Examiner tip: The AI syllabus expects the GDC's numerical derivative for this — much faster and safer than differentiating by hand.

Worked example 3: Estimating a limit numerically · hard

$f (x) = \frac{x^{2} - 16}{x - 4}$ . Explain why $f (4)$ is undefined and estimate $lim_{x \to 4} f (x)$ using a GDC table.

Solution

5 steps

Examiner tip: AI SL treats limits conceptually. A GDC table with values like

x = 3.999

and

x = 4.001

is a valid method when direct substitution fails.

Try these IB Maths AI SL limits, derivatives & rates of change questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 4 marks · Paper 1

Consider the function $f (x) = \frac{x^{2} - 25}{x - 5}$ .

Explain why $f (5)$ is mathematically undefined.
Use your graphic display calculator’s Table function to evaluate $f (x)$ for $x = 4.9, 4.99, 5.01,$ and $5.1$ . Hence, estimate the limit of $f (x)$ as $x \to 5$ .

Attempt it and see the mark scheme →

Question 2 · medium · 4 marks · Paper 1

The cost, $C$ in euros, of producing $x$ items is given by $C (x) = 1500 + 12 x - 0.05 x^{2}$ .

Calculate the average rate of change of cost when production increases from $100$ to $120$ items.
Find the instantaneous rate of change of cost (marginal cost) when exactly $100$ items are produced.

Full worked solution Free, no sign-in

$C (100) = 1500 + 1200 - 500 = 2200$ .
$C (120) = 1500 + 1440 - 720 = 2220$ . (M1)
Average rate $= \frac{C (120) - C (100)}{120 - 100} = \frac{2220 - 2200}{20} = \frac{20}{20} = € 1 per item$ . A1
Use CG50 d/dx. Enter $\frac{d}{d x} (1500 + 12 x - 0.05 x^{2}) |_{x = 100}$ . (M1)
Result $= € 2 per item$ . A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 6 marks · Paper 2

The volume of water in a draining tank, $V$ in litres, after $t$ minutes is modelled by $V (t) = 500 (1 - \frac{t}{20})^{2}$ for $0 \leq t \leq 20$ .

Find the volume of water in the tank at $t = 0$ and $t = 10$ .
Calculate the average rate at which water drains from the tank during the first 10 minutes.
Using your graphic display calculator, find the exact rate at which water is draining at $t = 10$ .
Explain why the answers to part (b) and part (c) are different.

Attempt it and see the mark scheme →

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FAQ

How many IB Maths AI SL limits, derivatives & rates of change questions are there?

There are 23 exam-style limits, derivatives & rates of change questions in the AI SL question bank (Paper 1: 17 · Paper 2: 6), graded 5 easy, 9 medium, 6 hard, 3 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is limits, derivatives & rates of change on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 17 · Paper 2: 6. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI SL Unit 5 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

What else is free here

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IB Maths AI SL Limits, Derivatives & Rates of Change Questions

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What you need to know

What's examined in AI SL limits, derivatives & rates of change

Key formulas

Limits, Derivatives & Rates of Change worked examples

Worked example 1: Average rate of change · easy

Solution

Worked example 2: Instantaneous rate of change via GDC · medium

Solution

Worked example 3: Estimating a limit numerically · hard

Solution

Try these IB Maths AI SL limits, derivatives & rates of change questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 4 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 6 marks · Paper 2

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