IB · Mathematics: Applications & Interpretation SL & HL · Topic

Calculus: IB Mathematics: Applications & Interpretation questions and answers

10 free IB Mathematics: Applications & Interpretation Calculus questions written in the style of IB Mathematics: Applications & Interpretation papers, each with its answer and mark scheme. 3 of them are A* challenge questions, built to catch the mistakes strong students still make.

This topic covers Differentiation & tangents, Optimisation, Integration & areas, Trapezoidal rule, Kinematics, Differential equations & Euler’s method, Volumes of revolution, Coupled systems & phase portraits. Work through the questions below, then generate a full Calculus paper with a fresh set every time.

Generate a full Calculus paper →

Calculus sub-topics

Differentiation & tangentsOptimisationIntegration & areasTrapezoidal ruleKinematicsDifferential equations & Euler’s methodVolumes of revolutionCoupled systems & phase portraits

Calculus practice questions

Question 1Structured · 7 marksA* challenge

A pond is 50 m long. Its width, w metres, is measured every 10 m along its length.

Distance (m)01020304050
Width (m)0102010140
Table 1.1
(a)

Use the trapezoidal rule to estimate the surface area of the pond.

[3]
(b)

The pond has a constant depth of 1.2 m. One litre of treatment is needed for every 40 m3 of water. Find the number of litres needed, and explain whether this is likely to be enough.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)540 m2
h = 10 (A1)
trapezoidal rule (M1)
540 m2 A1
3
(b)17 litres
volume 648 m3 (A1)
÷ 40 (M1)
17 litres (round up) A1
the edges are curved outwards, so the rule may underestimate; buy a little extra R1
4
Question 2Structured · 7 marksA* challenge

A cyclist’s velocity is v(t) = 5te−0.5t m s−1, for t ≥ 0 seconds.

(a)

Find the maximum velocity and when it occurs.

[3]
(b)

Find the distance travelled in the first 4 seconds.

[2]
(c)

Find the acceleration at t = 4.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)3.68 m s−1 at t = 2.00
v′(t) = 0 (M1)
t = 2.00 A1
3.68 m s−1 A1
3
(b)11.9 m
∫04v dt (M1)
11.9 m A1
2
(c)−0.677 m s−2
v′(4) (M1)
−0.677 m s−2 A1
2
Question 3Structured · 6 marksA* challenge

Consider the system dxdt = −2x − 2y, dydt = 2x − 2y.

(a)

Find the eigenvalues of the coefficient matrix.

[3]
(b)

Describe the trajectories near the origin, and their direction.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)−2 ± 2i
det(M − λI) = 0 M1
(−2 − λ)2 + 4 = 0 A1
λ = −2 ± 2i A1
3
(b)stable spiral
real part negative: spiral inwards R1
stable spiral A1
at (1, 0), dydt = 2 > 0 so anticlockwise R1
3
Question 4Structured · 6 marks

The table shows values of f(x) = 10x + 1, correct to four significant figures.

x01234
f(x)1053.3332.52
Table 4.1
(a)

Use the trapezoidal rule with 4 intervals to estimate ∫04f(x)dx.

[3]
(b)

Find the percentage error in this estimate.

[2]
(c)

State, with a reason, whether the estimate is an overestimate or an underestimate.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)16.8
h = 1 (A1)
12(10.00 + 2(…) + 2.000) (M1)
16.8 A1
3
(b)4.59%
exact value 16.1 from GDC (M1)
4.59% A1
2
(c)overestimate
overestimate, the curve is concave up so the trapezoids lie above it R1
1
Question 5Structured · 4 marks

Consider the differential equation dydx = x − 0.5y, with y = 2 when x = 0.

Use Euler’s method with step length 0.2 to estimate y when x = 0.8.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
1.54
use of Euler’s formula (M1)
y1 = 1.80 (A1)
y2 = 1.66 (A1)
y ≈ 1.54 A1
4
Question 6Structured · 5 marks

A closed cylindrical can holds 500 cm3.

(a)

Show that the surface area is A = 2πr2 + 2Vr.

[2]
(b)

Find the radius that minimizes the surface area, and the minimum area.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)shown
h = V/(πr2) M1
substitution A1
2
(b)r = 4.30 cm; 349 cm2
A′(r) = 0 or GDC (M1)
r = 4.30 A1
349 A1
3
Question 7Structured · 6 marks

Let f(x) = x3 + 2x2, x ≠ 0.

(a)

Find f′(x).

[2]
(b)

Find the equation of the tangent to the graph of f at the point where x = 2.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)3x2 − 4x−3
use of x−2 (M1)
3x2 − 4x−3 A1
2
(b)y = 232x − 292
f(2) = 8.50 (A1)
f′(2) = 11.5 (A1)
y − 8.50 = 11.5(x − 2) (M1)
y = 11.5x − 14.5 A1
4
Question 8Structured · 6 marks

A particle moves in a straight line with velocity v(t) = t2 − 5t + 4 m s−1, for 0 ≤ t ≤ 6.

(a)

Find the times when the particle changes direction.

[2]
(b)

Find the total distance travelled.

[2]
(c)

Find the displacement at t = 6.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)t = 1, t = 4
v(t) = 0 with a sign change (M1)
t = 1, t = 4 A1
2
(b)15.0 m
∫06|v|dt (M1)
15.0 m A1
2
(c)6.00 m
∫06v dt (M1)
6.00 m A1
2
Question 9Structured · 5 marks

The rate of growth of a population P is proportional to P, so that dPdt = kP. Initially P = 500, and after 10 years P = 824.

(a)

By solving the differential equation, show that P = 500ekt.

[3]
(b)

Find the value of k.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)shown
separating variables: ∫1PdP = ∫k dt M1
ln P = kt + c A1
using P(0) = 500 AG A1
3
(b)0.0500
824 = 500e10k (M1)
k = 0.0500 A1
2
Question 10Structured · 5 marks

The gradient function of a curve is f′(x) = 3x2 − 4x + 3. The curve passes through the point (1, 2).

Find f(x).

[5]
Show answers and mark scheme
PartAnswer and marking pointsMarks
f(x) = x3 − 2x2 + 3x
attempt to integrate (M1)
x3 − 2x2 + 3x + C A2
substituting the point (M1)
C = 0 A1
5

Other IB Mathematics: Applications & Interpretation topics

Number & algebraFunctionsGeometry & trigonometryStatistics & probability

Frequently asked questions

What is in IB Mathematics: Applications & Interpretation Calculus?

In the SL & HL syllabus, Calculus covers Differentiation & tangents, Optimisation, Integration & areas, Trapezoidal rule, Kinematics, Differential equations & Euler’s method, Volumes of revolution, Coupled systems & phase portraits.

Where can I get Calculus questions with mark schemes?

This page has 10 Calculus questions with answers. For more, the free generator on mentorriyaz.in builds a new Calculus paper with a full mark scheme each time you click.

Want a mentor, not just a paper?

Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.

Meet Riyaz →

Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in