IB Maths AI HL 5.2 Calculus Ahl Content Questions

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

Exam points

Apply chain/product/quotient and higher-order derivatives to related rates, concavity, inflexion and advanced extrema.
Use substitution and standard antiderivatives for further integrals, enclosed areas and volumes of revolution.
Use displacement, velocity and acceleration relationships to solve advanced kinematics and total-distance problems.
Set up and solve separable differential equations and interpret general/particular solutions and slope fields.
Apply Euler/numerical methods to first- or second-order and coupled systems, reporting approximation effects.

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

Learn this topic

Study 5.2 Calculus - AHL content conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

[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}

All question bank results loaded

Covered syllabus points

Learn this topic

Question 1

Question 2

Question (a)

Question (b)

Question 3

Question (a)

Question (b)