5.2 Calculus - AHL content Topic Practice

IB Maths AI syllabusMaths AI SL/HLFirst assessment 2025

Students practise advanced calculus techniques including chain/product rules, differential equations, volumes of revolution, and numerical methods.

Exam points

use product and chain rules together to differentiate composite functions like x√(3x−2)
separate variables in dy/dx = k(1−y) and integrate both sides to derive exponential model y = 1−e⁻ᵏᵗ
apply Euler’s method with step h=0.1: iterate xₙ₊₁ = xₙ + h·f(xₙ,yₙ) for given differential equation

[Maximum number: 18]

In this question researchers are trying to find the most accurate model to use when modelling a population of wolves.
Historically, a population of wolves in an area had a stable size of 200 . After some years of disruption, the population was reduced to 40 wolves. At this point, the area became a protected space and the population began to grow again.
Researchers in the area wish to model the size of the wolf population, x, as a function of t, where t is the time, in years, since the area became protected.

(a)

An alternative model for population growth is called the Gompertz model. When applied by the researchers to the wolf population, this model satisfies the differential equation

\frac{\mathrm{d} x}{\mathrm{~d} t}=a x \ln \left(\frac{200}{x}\right), a \in \mathbb{R}^{+} .

[ 13 ]

(i)

Write down the value of $\frac{\mathrm{d} x}{\mathrm{~d} t}$ when x=200.

[ 1 ]

(ii)

Interpret your answer to part (b)(i) in context.

Consider the function $f(x)=\ln (\ln 200-\ln x)$ , where 0<x<200.

[ 1 ]

(iii)

Show that $f^{\prime}(x)=\frac{-1}{x \ln \left(\frac{200}{x}\right)}$ .

[ 2 ]

(iv)

Hence, use separation of variables to show that the general solution of

\frac{\mathrm{d} x}{\mathrm{~d} t}=a x \ln \left(\frac{200}{x}\right), \text { where } 0<x<200

can be written as

\ln x=\ln 200-A \mathrm{e}^{-a t}

where A is an arbitrary positive constant.

[ 5 ]

(v)

Use the size of the wolf population at t=0 to find the value of A. Give your answer in the form $A=\ln p$ , where $p \in \mathbb{Z}^{+}$ .

[ 2 ]

(vi)

Use the size of the wolf population at t=5, given in part (a), to show that a=0.0855, correct to three significant figures.

[ 2 ]

(b)

Use Euler's method, with a step size of 0.5 years and an initial value of $x_{0}=70$ when t=5, to find an estimate for the size of the wolf population when t=10. Give your answer correct to the nearest whole number.

[ 4 ]

(c)

Comment on your answer.

[ 1 ]

[Maximum number: 3]

The function f is defined by $f(x)=\frac{6}{2 x^{2}+5 x+4}$ .

Hence or otherwise, find the x-coordinates of the points of inflexion of f.

[Maximum number: 17]

A financial analyst models the change in value of one share, x dollars at time t minutes, after a report is released. She uses the differential equation

\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+4 \frac{\mathrm{~d} x}{\mathrm{~d} t}+3 x=0

This equation can be written as the coupled differential equations

\frac{\mathrm{d} x}{\mathrm{~d} t}=y

\frac{\mathrm{d} y}{\mathrm{~d} t}=-3 x-4 y

(a)

Find the general solution for x.

Initially x=0 and $\frac{\mathrm{d} x}{\mathrm{~d} t}=-1$ .

[ 5 ]

(b)

Find an expression for x in terms of t.

[ 6 ]

(c)

Use Euler's method with a t-interval of 0.1 to predict the value of x when t=1.

[ 6 ]

[Maximum number: 14]

Rea is investigating how the number of healthy skin cells changes following infection by a virus. At the beginning of the investigation, she inserts 27000 virus particles into a sample of 1350 healthy skin cells.
Let h be the number of healthy skin cells at time t, where t is the number of hours after the investigation has begun. Let v be the number of virus particles, in thousands, at time t.
Rea finds that the rate of change, $\frac{\mathrm{d} h}{\mathrm{~d} t}$ , is proportional to $\frac{h}{v}$ .

(a)

Given that $\frac{\mathrm{d} h}{\mathrm{~d} t}=-5$ at time t=0, show that $\frac{\mathrm{d} h}{\mathrm{~d} t}=-\frac{h}{10 v}$ .

Rea models the number of virus particles, in thousands, after t hours as v=27+0.3 t.

[ 3 ]

(b)

Find an expression for the number of healthy skin cells at time t.

Rea asks her colleague Artem to attempt the same investigation. Artem's models differ from those of Rea.

Artem models the number of virus particles, in thousands, after t hours as v=27+0.28 t and finds that $h=4380(27+0.28 t)^{-0.36}$ .

[ 7 ]

(c)

Using Artem's models, predict how many hours it will take for the number of virus particles to be at least 100 times the number of healthy skin cells.

[ 4 ]

[Maximum number: 6]

Consider the system of coupled differential equations given by

\begin{aligned} & \frac{\mathrm{d} x}{\mathrm{~d} t}=2.2 x-2.6 y \\ & \frac{\mathrm{~d} y}{\mathrm{~d} t}=3.4 x-2.2 y . \end{aligned}

When t=0, x=5 and y=2.

(a)

Find the value of $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at t=0.

The eigenvalues for this system are $\pm 2 \mathrm{i}$ .

[ 3 ]

(b)

On the following phase portrait, sketch the trajectory that passes through the point (5,2). Clearly indicate the direction of this trajectory.

[ 3 ]

[Maximum number: 8]

Jun Ho models the cross-section of a bowl, in order to calculate its volume.
His model for part of the cross-section is $y=\frac{x^{6}}{65536}$ , where $0 \leq x \leq 8$ , as shown in the following graph. One unit represents one centimetre.

Let R be the region enclosed by this graph, the line y=4 and the line x=0.
He obtains the volume of the bowl by rotating $R, 2 \pi$ radians about the y-axis.

(a)

Find the volume of the bowl.

Jun Ho pours $250 \mathrm{~cm}^{3}$ of water into the bowl.

[ 5 ]

(b)

Find the depth of the water in the bowl.

[ 3 ]

Younsue rides on a Ferris wheel. As the wheel rotates, her height above the ground, h metres, can be modelled in terms of the angle, $\theta$ radians, that the wheel has rotated, using $h=16-15 \cos \theta$ . The value of $\theta$ is measured from Younsue's starting position at the bottom of the wheel.

(a)

Find the angle rotated by the Ferris wheel between t=0 and t=4.5.

(b)

Find the value of $\frac{\mathrm{d} h}{\mathrm{~d} t}$ , when t=4.5.

0 selected

Exam points

Question 2

Question 2(b)

Question 2(b)(i)

Question 2(b)(ii)

Question 2(b)(iii)

Question 2(b)(iv)

Question 2(b)(v)

Question 2(b)(vi)

Question 2(d)(i)

Question 2(d)(ii)

Question 10(b)

Question 6

Question 6(a)

Question 6(b)(i)

Question 6(d)

Question 6

Question 6(a)

Question 6(b)

Question 6(c)

Question 14

Question 14(a)

Question 14(b)

Question 15

Question 15(a)

Question 15(b)

Question 17

Question 17(b)(i)

Question 17(b)(ii)