IB Maths AI SL · Unit 5: Calculus
IB Maths AI SL Limits, Derivatives & Rates of Change Questions
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 the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 16 questions
- Paper 1: 13
- Paper 2: 3
- 3 easy
- 5 medium
- 5 hard
- 3 very hard
- 3 worked examples
Practise Limits, Derivatives & Rates of Change questions →
AI SL formula booklet
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):
- Instantaneous Rate of Change (8)
- Function Analysis & Limits (5)
- Average Rate of Change (3)
Key formulas
- Derivative of x^n
- \(f(x) = x^n \implies f'(x) = n x^{\,n-1}\)
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 $s(t) = 3t^2 - 2t + 1$ metres. Find the exact average velocity between $t = 1$ and $t = 4$ seconds.
1. $s(1)$: $3 - 2 + 1 = 2\text{ m}$.
2. $s(4)$: $48 - 8 + 1 = 41\text{ m}$.
3. Formula: $\bar{v} = \frac{\Delta s}{\Delta t} = \frac{s(4)-s(1)}{4-1}$.
4. Substitute: $\frac{41-2}{3} = \frac{39}{3}$.
5. State: $\mathbf{13\text{ m s}^{-1}}$.
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 $P(x) = -0.5x^3 + 12x^2 - 10x + 50$ euros for $x$ units sold. Use the GDC to find the instantaneous rate of change when $x = 10$.
1. Recognise: need $P'(10)$.
2. Open: numerical derivative tool ($\tfrac{d}{dx}$).
3. Enter: $\tfrac{d}{dx}(-0.5x^3+12x^2-10x+50)\big|_{x=10}$.
4. Execute: GDC returns exactly $80$.
5. State: $\mathbf{\text{€}80\text{ per unit}}$.
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.
1. Substitute $x=4$: $\frac{0}{0}$ — undefined (division by zero).
2. Enter: $Y_1 = \frac{x^2-16}{x-4}$ in the table menu.
3. Test: $f(3.99) = 7.99$, $f(4.01) = 8.01$.
4. Observe: values converge from both sides.
5. State: $\lim_{x \to 4} f(x) = \mathbf{8}$.
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.
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 \(x \to 5\).
Question 2 · medium · 4 marks · Paper 1
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.
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 \(0 \le t \le 20\).
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.
All 16 limits, derivatives & rates of change questions with mark schemes →
FAQ
How many IB Maths AI SL limits, derivatives & rates of change questions are there?
There are 16 exam-style limits, derivatives & rates of change questions in the AI SL question bank (Paper 1: 13 · Paper 2: 3), graded 3 easy, 5 medium, 5 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: 13 · Paper 2: 3. 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.
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