IB Mathematics AI HL 5.2 Calculus Question Bank

Practise IB Mathematics AI HL 5.2 by combining advanced calculus, technology, differential models, numerical methods and optimisation in extended contexts.

Exam points

Analyse derivatives, related rates, higher-order behaviour and parameterised models with technology support.
Apply integration, differential-equation ideas and numerical methods to accumulation, motion, growth and optimisation.
Evaluate approximation error, domains, sensitivity and model assumptions in extended real-world evidence.

Covered syllabus points

AHL 5.9 (HL)—Further differentiation AHL 5.10 (HL)—Second derivative and concavity AHL 5.11 (HL)—Further integration AHL 5.12 (HL)—Areas and volumes AHL 5.13 (HL)—Kinematics AHL 5.14 (HL)—Differential equation modelling AHL 5.15 (HL)—Slope fields AHL 5.16 (HL)—Euler's method AHL 5.17 (HL)—Phase portraits AHL 5.18 (HL)—Second-order differential equations

[Maximum number: 3]

This question is about applying ideas from logarithms, calculus and probability to an unfamiliar mathematical theory called information theory.
Claude Shannon developed a mathematical theory called information theory to measure the information gained when random events occur. He defined the information, I, that is gained when an event with probability p occurs as

I=-\ln p

where $0<p \leq 1$ . For example, no information is gained ( I=0 ) when an event is certain to $\operatorname{occur}(p=1)$ .

Show, using calculus, that I is a decreasing function of p.

$\frac{\mathrm{d} I}{\mathrm{~d} p}=-\frac{1}{p}$

Note: Award A1 for $\frac{\mathrm{d} I}{\mathrm{~d} p}$ or equivalent (e.g. $I^{\prime}$ ) and A1 for $-\frac{1}{p}$ . Do not accept $\frac{\mathrm{d} y}{\mathrm{~d} x}$ for the first A1 unless followed by $-\frac{1}{x}$ , which can earn A1 A0.
(for $0<p \leq 1$ ) we therefore have $\frac{\mathrm{d} I}{\mathrm{~d} p}<0$
hence the function is decreasing

[Maximum number: 3]

Kailash manufactures drink containers in the shape of a cuboid. The container has a square top and a square base of length, $l\,\mathrm{cm}$ . Its height, $d\,\mathrm{cm}$ , is three times the length of the base.

diagram not to scale

(a)

Find $\frac{\mathrm{d}^{2} A}{\mathrm{~d} r^{2}}$ .

[ 1 ]

$\left(\frac{\mathrm{d}^{2} A}{\mathrm{~d} r^{2}}=\right) 4 \pi+\frac{1500}{r^{3}}$

(b)

Hence determine whether the graph of A is concave-up or concave-down for r>0. Justify your answer.

[ 2 ]

As r>0, $\frac{\mathrm d^2A}{\mathrm d r^2}>0$ .
Hence the graph is concave-up.
Note: marks in part (ii) require an expression seen in part (i).

[Maximum number: 7]

(a)

Find the indefinite integral $\int x e^{-x^{2}} d x$ .

[ 4 ]

A valid/coherent attempt to use substitution:

u=-x^2,\quad \frac{du}{dx}=-2x

The integral becomes

-\frac12\int e^u\,du=-\frac12e^{-x^2}+c

Accept integration by inspection using $\frac{d}{dx}(e^{-x^2})=-2xe^{-x^2}$ .
Note: Award M1A1M1A0 for $-\frac12e^{-x^2}$ with c omitted.

(b)

Hence find the area bounded by the x-axis, the curve $\mathrm{y}=x e^{-x^{2}}$ and the line x=k. Give your answer in terms of k.

[ 3 ]

Recognition that 0 is the lower limit and substitute both 0 and k into the previous expression.

\int_0^k xe^{-x^2}\,dx=\left[-\frac12e^{-x^2}\right]_0^k=\frac12-\frac12e^{-k^2}

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Covered syllabus points

5.2 Calculus - AHL content question 1

5.2 Calculus - AHL content question 2

Question (a)

Question (b)

5.2 Calculus - AHL content question 3

Question (a)

Question (b)