A company’s daily profit, $P$ in euros, from manufacturing $x$ items is modelled by the quadratic function $P(x) = -2x^2 + 120x - 500$ . Find the marginal profit function, $P'(x)$ . Determine the number of items that must be manufactured to maximise the daily profit. Calculate the maximum daily profit.

$P'(x) = 2(-2)x + 120 = -4x + 120$ . A1A1 Set $P'(x) = 0$ to find the maximum. (M1) $-4x + 120 = 0 \implies 4x = 120 \implies x = 30$ items. A1 (Alternatively, use G-Solv -> MAX on the CG50). Substitute $x = 30$ into $P(x)$ : $P(30) = -2(30)^2 + 120(30) - 500 = -1800 + 3600 - 500 = \text{€}1300$ . A1

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IB Maths AI SL · Unit 5: Calculus

IB Maths AI SL Tangents & Normals Questions

Updated October 2026

Exam-style IB Maths AI SL tangents & normals questions with worked solutions. Start with the 5 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.

28 questions
Paper 1: 16
Paper 2: 12
3 easy
9 medium
9 hard
7 very hard
5 worked examples

Practise Tangents & Normals questions →

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

What's examined in AI SL tangents & normals

The question bank covers these tangents & normals question types (number of questions in brackets):

Gradient Properties & Optimization (17)
Equation of Normal (6)
Equation of Tangent (5)

Tangents & Normals worked examples

Worked example 1: Horizontal tangent locations · easy

$y = - x^{3} + 6 x^{2} - 5 x + 4$ . Use the GDC to find the two $x$ -coordinates of horizontal tangents (3 s.f.).

Solution

5 steps · 1 to fill in

Examiner tip: Turning points, stationary points, local extrema, and horizontal tangents all mean the same thing — set

f^{'} (x) = 0

Worked example 2: Normal at a local minimum · medium

$C (x) = x^{2} - 6 x + 14$ has a local minimum. Find the exact equation of the normal at that minimum.

Solution

6 steps · 1 to fill in

Examiner tip: Do NOT use

m_{n} = - \frac{1}{m_{t}}

when

m_{t} = 0

— it divides by zero. The normal to a horizontal tangent is vertical.

Worked example 3: Building a curve from properties · hard

$y = a x^{2} + b x + c$ has a horizontal tangent at $(2, - 5)$ and $y$ -intercept $(0, 3)$ . Find $a, b, c$ exactly.

Solution

6 steps · 1 to fill in

Examiner tip: Deal with the

y

-intercept

(0, c)

first — it instantly collapses a three-parameter problem to two.

Worked example 4: Equation of a tangent line · easy

$f (x) = x^{2} - 4 x + 5$ . Find the exact tangent line at $x = 3$ in the form $y = m x + c$ .

Solution

6 steps · 1 to fill in

Examiner tip: Always rearrange your answer into the requested form (usually

y = m x + c

) — leaving it in point-slope form can cost a mark.

Worked example 5: Gradient of a normal via GDC · medium

$g (x) = 0.5 x^{3} - 2 x + 1$ . Use the GDC numerical derivative to find the exact normal gradient at $x = 2$ .

Solution

6 steps · 1 to fill in

Examiner tip: The derivative gives the TANGENT gradient. You must flip AND negate for the normal gradient — a very common trap.

Try these IB Maths AI SL tangents & normals 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 $g (x) = x^{3} - 3 x^{2} - 9 x + 2$ .

Graph the function on your graphic display calculator and use the G-Solv feature to find the coordinates of the local maximum point.
Find the coordinates of the local minimum point.

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 1

A company’s daily profit, $P$ in euros, from manufacturing $x$ items is modelled by the quadratic function $P (x) = - 2 x^{2} + 120 x - 500$ .

Find the marginal profit function, $P^{'} (x)$ .
Determine the number of items that must be manufactured to maximise the daily profit.
Calculate the maximum daily profit.

Full worked solution Free, no sign-in

$P^{'} (x) = 2 (- 2) x + 120 = - 4 x + 120$ . A1A1
Set $P^{'} (x) = 0$ to find the maximum. (M1)
$- 4 x + 120 = 0 ⟹ 4 x = 120 ⟹ x = 30$ items. A1
(Alternatively, use G-Solv -> MAX on the CG50).
Substitute $x = 30$ into $P (x)$ :
$P (30) = - 2 (30)^{2} + 120 (30) - 500 = - 1800 + 3600 - 500 = € 1300$ . A1

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

Question 3 · hard · 7 marks · Paper 2

A function is defined by $f (x) = p x^{3} + q x^{2} + 5 x$ . The curve has a local maximum point at $x = 1$ and passes through the point $(1, 2)$ .

Using the fact that the curve passes through $(1, 2)$ , write down an equation in terms of $p$ and $q$ .
Find $f^{'} (x)$ .
Using the fact that the local maximum occurs at $x = 1$ , write down a second equation in terms of $p$ and $q$ .
Solve the system of equations to find the values of $p$ and $q$ .

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FAQ

How many IB Maths AI SL tangents & normals questions are there?

There are 28 exam-style tangents & normals questions in the AI SL question bank (Paper 1: 16 · Paper 2: 12), graded 3 easy, 9 medium, 9 hard, 7 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is tangents & normals on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 16 · Paper 2: 12. 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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IB Maths AI SL Tangents & Normals Questions

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What's examined in AI SL tangents & normals

Tangents & Normals worked examples

Worked example 1: Horizontal tangent locations · easy

Solution

Worked example 2: Normal at a local minimum · medium

Solution

Worked example 3: Building a curve from properties · hard

Solution

Worked example 4: Equation of a tangent line · easy

Solution

Worked example 5: Gradient of a normal via GDC · medium

Solution

Try these IB Maths AI SL tangents & normals questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 5 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 7 marks · Paper 2

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