5.1 Limits and the derivative questions for IB Maths AA

24 questions on Limits and the derivative, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Analysis and Approaches
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	9	2	11
Paper 2	11	2	13
Every paper	20	4	24

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 · AA-00639Paper 1 · 6 marks

Let $f (x) = 3 x - x^{2}$ , for $x \in R$ . The point $P (1, 2)$ lies on the graph of $f$ .

(a)
Show that $f (1 + h) = 2 + h - h^{2}$ .
2 marks
(b)
The point $Q (1 + h, f (1 + h))$ , where $h \neq 0$ , also lies on the graph of $f$ .
Show that the gradient of the chord $[P Q]$ is $1 - h$ .
2 marks
(c)
Hence find the equation of the tangent at P.
2 marks

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Question 2 · AA-00637Paper 2 · 5 marks

Consider the curve $y = 2^{x}$ , where $x \in R$ . The point $P (1, 2)$ lies on the curve.

The point $Q (1 + h, 2^{1 + h})$ , where $h > 0$ , also lies on the curve. The gradient of the chord $[P Q]$ is $m$ .

The following table shows some values of $h$ and the corresponding values of $m$ , correct to five significant figures.

$h$	$1$	$0.1$	$0.01$	$0.001$
$m$	$2.0000$	$1.4355$	$p$	$q$

(a)
Use your GDC to find $p$ and $q$ .
2 marks
(b)
Hence write down an estimate for the gradient of the tangent to the curve at P.
1 mark
(c)
Use your answer to part (b) to find the equation of the tangent to the curve at P. Give your answer in the form $y = a x + b$ .
2 marks

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Question 3 · AA-00640Paper 1 · 6 marks

A water tank is being drained. The volume of water in the tank, $V$ litres, $t$ minutes after draining starts is modelled by

V (t) = 2000 - 40 t + 0.2 t^{2}, 0 \leq t \leq 100.

(a)
Write down the volume of water in the tank when draining starts.
1 mark
(b)(i)
Find $\frac{d V}{d t}$ .
2 marks
(b)(ii)
Find the value of $\frac{d V}{d t}$ when $t = 25$ .
1 mark
(c)
Interpret your answer to part (b)(ii) in the context of the question.
2 marks

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Question 4 · AA-00638Paper 2 · 5 marks

Let $f (x) = \frac{\sin 3 x}{x}$ , for $- π \leq x \leq π$ , $x \neq 0$ , where $x$ is in radians.

The following diagram shows the graph of $f$ .

y = f(x)

(a)
Use the graph to write down an estimate of $\lim_{x \to 0} f (x)$ .
1 mark
(b)(i)
Find the value of $f (0.1)$ .
1 mark
(b)(ii)
Find the value of $f (0.01)$ .
1 mark
(c)
Hence, by choosing a suitable value of $x$ , find an estimate for $\sin 0.06$ .
2 marks

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Question 5 · AA-00642Paper 1 · 6 marks

The following diagram shows the curve $y = f (x)$ , for $x > - 1$ . The line $y = 4$ is a horizontal asymptote to the curve.

The line $L$ is tangent to the curve at the point $A (1, 1)$ and passes through the point $B (3, 4)$ .

y = f(x)
L
y = 4

(a)
Write down the value of $\lim_{x \to \infty} f (x)$ .
1 mark
(b)
Find the value of $f^{'} (1)$ .
2 marks
(c)
Let $g (x) = 2 f (x) + x$ . Find the equation of the tangent to the graph of $g$ at the point where $x = 1$ .
3 marks

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Question 6 · AA-00641Paper 2 · 15 marks

A new mobile app is released. The number of times it has been downloaded, $N$ thousand, $t$ weeks after its release is modelled by

N (t) = \frac{60}{1 + 14 e^{- 0.35 t}}, t \geq 0.

(a)
Find the number of downloads predicted by the model 10 weeks after release.
2 marks
(b)(i)
Find the value of $N^{'} (4)$ .
2 marks
(b)(ii)
Explain what your answer to part (b)(i) means in this context.
2 marks
(c)(i)
Write down the value of $\lim_{t \to \infty} N (t)$ .
1 mark
(c)(ii)
Interpret this value in the context of the question.
1 mark
(d)
Find the time at which the number of downloads is increasing most rapidly, and the rate of increase at this time.
3 marks
(e)
Write down the value of $\lim_{t \to \infty} N^{'} (t)$ .
1 mark
(f)
Find the length of time for which the number of downloads is increasing at a rate of more than $4000$ per week.
3 marks

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