IB Mathematics Applications & Interpretation Topic Calculus Questionbank (SL/HL)

IB Mathematics Applications & Interpretation Topic Calculus Questionbank (SL/HL) | RevisionDojo

Practice IB Mathematics Applications & Interpretation (AI) Topic Calculus with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for Calculus and mirrors Paper 1, 2, 3 style where relevant.

Get instant solutions, detailed explanations, and build confidence with questions aligned to IB examiner expectations.

Paper

Level

Question 1

HLPaper 2

A model for the spread of a rumor involves susceptible ( $x$ ) and informed ( $y$ ) individuals:

$\frac{d x}{d t} = - 0.4 x y, \frac{d y}{d t} = 0.4 x y - 0.1 y$

where $x$ and $y$ are in thousands and $t$ is in days.

Show that the system has a line of equilibrium points on the $x$ -axis (i.e. $y = 0$ ).

[3]

For the equilibrium point $(0, 0)$ , find the Jacobian matrix and its eigenvalues, and classify the stability.

[5]

Given initial conditions $x (0) = 8, y (0) = 1$ , use Euler's method with a step size of $0.1$ days to estimate the populations after $0.3$ days.

[5]

Sketch the phase portrait on $0 \leq x \leq 10$ , $0 \leq y \leq 3$ . Show the equilibrium line(s) and at least three trajectories with direction arrows.

[4]

Question 2

HLPaper 1

The slope field for

\frac{d y}{d x} = x y

, for

- 2 \leq x \leq 2, - 2 \leq y \leq 2

, is shown below. The green curve should be ignored; the point

A (2, - 1)

is marked.

Write down the equations of the curves where $\frac{d y}{d x} = 0$ .

[2]

Sketch the lines $x = 0$ and $y = 0$ on the slope field.

[2]

Sketch the solution curve passing through $A (2, - 1)$ .

[2]

Determine the concavity at $A (2, - 1)$ of the solution curve from Part 2.

[3]

Question 3

HLPaper 2

The motion of a particle is given by:

$\overset{x}{¨} + 5 \overset{x}{˙}^{2} = 2 sin (t),$

with $x$ in metres, $t$ in seconds $(t \geq 0)$ , and $sin (t)$ using radian measure.

Convert to a system of first-order differential equations.

[2]

With $x (0) = 1, \overset{x}{˙} (0) = - 1$ , use Euler's method with step size 0.2 to estimate $x$ and $\overset{x}{˙}$ at $t = 0.6$ .

[4]

Hence estimate the acceleration at $t = 0.6$ .

[2]

Suggest one limitation of Euler's method in this context.

[2]

Question 4

HLPaper 1

Consider the function $f (x) = \frac{x ^{3}}{x ^{2} + 4}$ for $x \in R$ .

Find $f^{''} (x)$ .

[3]

Determine the interval(s) of $x$ for which the graph of $f$ is concave down.

[2]

Find the coordinates of any points of inflection and verify by showing that $f^{''} (x)$ changes sign at each point.

[3]

Sketch the graph of $f^{''} (x)$ , indicating points where $f^{''} (x) = 0$ .

[2]

Question 5

HLPaper 2

A model for the populations of two symbiotic algae species, $x$ and $y$ (in thousands per $m^{2}$ ), is given by:

$\frac{d x}{d t} = x + 2 y - 10, \frac{d y}{d t} = 2 x + y - 15$

Time $t$ is measured in arbitrary time units.

Find the equilibrium point.

[3]

Use the substitution $X = x - \frac{20}{3}, Y = y - \frac{5}{3}$ to transform the system into a homogeneous system, and find its eigenvalues.

[5]

Find the general solution of the original system, given the eigenvalues and eigenvectors from Part 1.

[6]

Given initial conditions $x (0) = 5, y (0) = 3$ , find the particular solution and determine the limit of $\frac{y}{x}$ as $t \to \infty$ .

[4]

Sketch the phase portrait for $0 \leq x \leq 6, 0 \leq y \leq 6$ , showing the equilibrium point and trajectories.

[4]

IB Mathematics Applications & Interpretation Calculus Questionbank

Unlock all 345 questions with a free account

IB Mathematics Applications & Interpretation Calculus Questionbank

IB Mathematics Applications & Interpretation (AI) Topic Calculus Questionbank

Unlock all 345 questions with a free account