5.4 Tangents and normals questions for IB Maths AI

34 questions on Tangents and normals, 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	HL only	All
Paper 1	16	0	16
Paper 2	17	1	18
Every paper	33	1	34

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-01034Paper 1 · 6 marks

Consider the function $f (x) = 2 x^{3} - 5 x^{2} + x + 4$ .

(a)
Find $f^{'} (x)$ .
2 marks
(b)
Write down the value of $f (2)$ .
1 mark
(c)
Find the equation of the tangent to the graph of $y = f (x)$ at the point where $x = 2$ . Give your answer in the form $y = m x + c$ .
3 marks

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Question 2 · AI-00188Paper 2 · 14 marks

The profile of a mound on a BMX track is modelled by two curves joined at $x = 4$ :

y = {\begin{cases} 0.1 x^{2}, & 0 \leq x \leq 4 \\ a x^{2} + b x + c, & 4 < x \leq 10 \end{cases}

where $x$ is the horizontal distance, in metres, from the start of the mound and $y$ is the height in metres. The two curves meet at $x = 4$ and have the same gradient there, so that the track is smooth. The second curve passes through the point $(10, 2.2)$ .

(a)
Find the height of the track and the gradient of the track when $x = 4$ .
2 marks
(b)
Show that $8 a + b = 0.8$ .
2 marks
(c)
Write down two further equations in $a$ , $b$ and $c$ .
2 marks
(d)
Find the values of $a$ , $b$ and $c$ .
2 marks
(e)
Find the maximum height of the mound.
2 marks
(f)
After $x = 10$ the track continues in a straight line which is a tangent to the second curve at $x = 10$ . Find the horizontal distance from the start of the mound to the point where this straight section reaches the ground ( $y = 0$ ).
4 marks

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Question 3 · AI-01035Paper 1 · 7 marks

A curve has equation $y = 0.5 x^{2} + \frac{8}{x}$ , $x \neq 0$ . The point P on the curve has $x$ -coordinate 4.

(a)
Find $\frac{d y}{d x}$ .
2 marks
(b)
Find the gradient of the normal to the curve at P.
2 marks
(c)
Find the equation of the normal to the curve at P. Give your answer in the form $a x + b y + d = 0$ , where $a$ , $b$ and $d$ are integers.
3 marks

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Question 4 · AI-00268Paper 2 · 13 marks

The edge of a lake is modelled by the curve

y = f (x) = - 0.05 x^{3} + 0.6 x^{2} - 1.3 x + 4, 0 \leq x \leq 10,

where $x$ and $y$ are measured in hundreds of metres. The point P $(2, 3.4)$ lies on the edge of the lake.

(a)
Find $f^{'} (x)$ .
2 marks
(b)
Show that the gradient of the curve at P is $0.5$ .
1 mark
(c)
A straight path is built along the tangent to the curve at P. Find the equation of the tangent in the form $y = m x + c$ .
2 marks
(d)
The path meets the edge of the lake again at the point Q. Find the coordinates of Q.
2 marks
(e)
Find the length of the path from P to Q, in metres.
2 marks
(f)
A straight jetty is to be built from P along the normal to the curve at P. Find the equation of the normal, giving your answer in the form $a x + b y + d = 0$ , where $a, b, d \in Z$ .
3 marks
(g)
The line $y = 0$ represents a straight road. Write down the $x$ -coordinate of the point where the line of the jetty, if extended, would meet the road.
1 mark

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Question 5 · AI-01036Paper 1 · 7 marks

Consider the curve $y = x^{3} - 3 x^{2} - 5 x + 1$ and the line $L$ with equation $y = 4 x + 7$ .

(a)
Find $\frac{d y}{d x}$ .
1 mark
(b)
Find the $x$ -coordinates of the two points on the curve at which the tangent is parallel to $L$ .
3 marks
(c)
One of these tangents is at the point A, where $x > 0$ . Find the equation of the tangent at A.
3 marks

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Question 6 · AI-01167Paper 2 · 14 marks

A footpath goes over a small hill in a park. A cross-section of the path is modelled by

y = - 0.02 x^{3} + 0.3 x^{2} + 4, 0 \leq x \leq 15,

where $x$ is the horizontal distance, in metres, from the start of the path and $y$ is the height, in metres, above a reference level. The point $P$ on the path has $x = 5$ .

(a)
Find the height of the path at $P$ .
1 mark
(b)
The point $Q$ also lies on the path. Complete the following table, which shows the gradient of the chord $P Q$ for different $x$ -coordinates of $Q$ .
$x$ -coordinate of $Q$
6
5.5
5.1
5.01
Gradient of $P Q$
1.48
3 marks
(c)
Hence estimate the gradient of the path at $P$ .
1 mark
(d)
Find $\frac{d y}{d x}$ .
2 marks
(e)
Use your answer to part (d) to confirm your estimate in part (c).
2 marks
(f)
A straight wooden handrail is to be fixed along the tangent to the path at $P$ .
Find the equation of the tangent to the path at $P$ . Give your answer in the form $y = m x + c$ .
3 marks
(g)
Find the coordinates of the highest point of the path.
2 marks

$x$ -coordinate of $Q$	6	5.5	5.1	5.01
Gradient of $P Q$	1.48

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