5.3 Differentiating polynomials questions for IB Maths AI

65 questions on Differentiating polynomials, 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	27	1	28
Paper 2	30	7	37
Every paper	57	8	65

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-00068Paper 1 · HL only · 6 marks

The cross-section of a skateboard ramp is modelled by

h (x) = {\begin{cases} 0.1 x^{2}, & 0 \leq x \leq 4 \\ p x + q, & 4 < x \leq 10 \end{cases}

where $h$ is the height in metres at a horizontal distance $x$ metres from the start of the ramp, and $p$ and $q$ are constants. The two pieces meet at $x = 4$ with no jump and no sharp corner, so that $h$ is continuous and its gradient is continuous at $x = 4$ .

(a)
Find the value of $p$ and the value of $q$ .
4 marks
(b)
Find the horizontal distance from the start of the ramp at which the height of the ramp is 5 m.
2 marks

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Question 2 · AI-00154Paper 2 · HL only · 17 marks

Inés uses a rotating sprinkler to water a lawn. The sprinkler waters a region in the shape of a sector of a circle with radius $r$ m and angle $θ$ radians.

Inés first sets the sprinkler so that $r = 12$ and $θ = 2.3$ .

(a)
Find the area of lawn watered.
2 marks
(b)
Find the perimeter of the watered region.
2 marks
(c)
Write down the value of $θ$ in degrees.
1 mark
(d)
Inés now plans a new flower bed in the shape of a sector with radius $r$ m and angle $θ$ radians. The area of the flower bed must be $90 m^{2}$ .
Show that $θ = \frac{180}{r^{2}}$ .
2 marks
(e)
The whole perimeter of the flower bed will be edged with metal strip. Show that the length of strip needed, $P$ m, is given by
$P = 2 r + \frac{180}{r} .$
2 marks
(f)
Find $\frac{d P}{d r}$ .
2 marks
(g)
Hence find the value of $r$ that minimizes $P$ , and the minimum length of strip needed.
3 marks
(h)
Find the corresponding value of $θ$ .
1 mark
(i)
The metal strip costs 14.50 euros (EUR) per metre. Find the minimum cost of the strip. Give your answer correct to two decimal places.
2 marks

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Question 3 · 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 4 · 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 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-00189Paper 2 · 14 marks

A coach models the distance, $d$ metres, run by a sprinter $t$ seconds after the start of a 100 m race by

d (t) = a t^{3} + b t^{2} + c t, t \geq 0.

Video analysis shows that the sprinter has run 10.76 m after 2 seconds, 35.0 m after 5 seconds and 64.64 m after 8 seconds.

(a)
Explain why the model has no constant term.
1 mark
(b)
Write down three equations in $a$ , $b$ and $c$ .
2 marks
(c)
Find the values of $a$ , $b$ and $c$ .
2 marks
(d)
Find the time taken by the sprinter to finish the race, according to the model.
2 marks
(e)
The velocity of the sprinter is $v = \frac{d d}{d t}$ . Find an expression for $v$ in terms of $t$ .
2 marks
(f)
Find the maximum velocity of the sprinter during the race and the time at which it occurs.
3 marks
(g)
The coach suggests using the model to describe the sprinter slowing down after the finish line. Find the time when the model predicts that the sprinter stops, and comment on the coach's suggestion.
2 marks

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