IB · Mathematics: Analysis & Approaches SL & HL · Topic

Calculus: IB Mathematics: Analysis & Approaches questions and answers

10 free IB Mathematics: Analysis & Approaches Calculus questions written in the style of IB Mathematics: Analysis & Approaches papers, each with its answer and mark scheme.

This topic covers Differentiation, Tangents & normals, Stationary points & optimisation, Integration, Areas, Kinematics, Implicit, related rates & first principles, Integration techniques, Volumes of revolution, Differential equations, Maclaurin series & limits. Work through the questions below, then generate a full Calculus paper with a fresh set every time.

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Calculus sub-topics

DifferentiationTangents & normalsStationary points & optimisationIntegrationAreasKinematicsImplicit, related rates & first principlesIntegration techniquesVolumes of revolutionDifferential equationsMaclaurin series & limits

Calculus practice questions

Question 1Structured · 13 marks

A particle moves in a straight line. Its velocity v m s−1 at time t seconds is given by v(t) = 5e−t/4sin t, for 0 ≤ t ≤ 2π.

(a)

Find the times when the particle is at rest.

[2]
(b)

Find the maximum speed of the particle, and the time at which it occurs.

[3]
(c)

Find the acceleration of the particle when t = 2.

[2]
(d)

Find the total distance travelled by the particle.

[3]
(e)

Find the displacement of the particle at t = 2π and explain why it differs from the distance travelled.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)t = 0, π, 2π
v = 0 ⇒ sin t = 0 (M1)
t = 0, π, 2π A1
2
(b)t = 1.33 s, 3.48 m s−1
use of graph or solving v′(t) = 0 (M1)
t = 1.33 A1
3.48 m s−1 A1
3
(c)−1.95 m s−2
v′(2) (M1)
−1.95 m s−2 A1
2
(d)9.98 m
∫02π|v(t)|dt (M1)(A1)
9.98 m A1
3
(e)3.73 m
∫02πv(t)dt (M1)
3.73 m A1
the particle changes direction at t = π, so displacement < distance R1
3
Question 2Structured · 5 marks

A ladder of length 25 m leans against a vertical wall. The bottom of the ladder slides away from the wall at a constant 0.4 m s−1.

Find the rate at which the top of the ladder is sliding down the wall when the bottom is 7 m from the wall.

[5]
Show answers and mark scheme
PartAnswer and marking pointsMarks
0.117 m s−1
x2 + y2 = 625 (A1)
2xdxdt + 2ydydt = 0 M1A1
y = 24 when x = 7 (A1)
dydt = −0.117, so sliding down at 0.117 m s−1 A1
5
Question 3Structured · 5 marks

Using the Maclaurin series for standard functions, find the Maclaurin series for cos x ln(1 + 2x), up to and including the term in x3.

[5]
Show answers and mark scheme
PartAnswer and marking pointsMarks
2x − 2x2 + 53x3
writing down the standard series (to x3) A1
attempt to multiply the series M1
2x − 2x2 + 53x3 A3
award A1 for each correct non-zero term
5
Question 4Structured · 7 marks

A particle moves in a straight line. Its velocity v m s−1 after t seconds is given by v(t) = (t − 2)(t − 5)e−0.3t, for 0 ≤ t ≤ 6.

(a)

Find the times when the particle is at rest.

[2]
(b)

Find the total distance travelled by the particle for 0 ≤ t ≤ 6.

[3]
(c)

Find the displacement of the particle at t = 6, relative to its starting position.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)t = 2, t = 5
v = 0 (M1)
t = 2, t = 5 A1
2
(b)9.21 m
∫06|v(t)|dt (M1)(A1)
9.21 m A1
3
(c)6.00 m
∫06v(t)dt (M1)
6.00 m A1
2
Question 5Structured · 7 marks

Let f(x) = x3 + ax2 + bx − 1. The graph of f has stationary points at x = 0 and x = 2.

(a)

Find the value of a and the value of b.

[4]
(b)

Find the coordinates of the local maximum point, justifying that it is a maximum.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)a = −3, b = 0
f′(x) = 3x2 + 2ax + b A1
f′(0) = 0 and f′(2) = 0, or f′(x) = 3(x)(x − 2) M1
a = −3 A1
b = 0 A1
4
(b)(0, −1)
f″(x) = 6x − 6 (A1)
f″(0) = −6 < 0, so a maximum R1
(0, −1) A1
3
Question 6Structured · 5 marks

Find limx→0sin 3x − 3xx3.

[5]
Show answers and mark scheme
PartAnswer and marking pointsMarks
−92
recognising the 00 indeterminate form R1
use of L’Hôpital’s rule or a Maclaurin series M1
correct derivatives or series A1
repeat (or simplify) to remove the indeterminate form M1
−92 A1
5
Question 7Structured · 3 marks

The region enclosed by y = 2√x, the x-axis and x = 4 is rotated through 2π about the x-axis. Find the volume.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
101
π∫y2 dx (M1)
correct integral (A1)
101 A1
3
Question 8Structured · 5 marks

Let f(x) = e0.5x − 3 and g(x) = 2 − x2. The graphs of f and g intersect at x = p and x = q, where p < q are the two smallest solutions of f(x) = g(x).

(a)

Find the value of p and the value of q.

[2]
(b)

Find the area of the region enclosed by the graphs of f and g between x = p and x = q.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)p = −2.16, q = 1.65
p = −2.16 A1
q = 1.65 A1
2
(b)10.3
integral of the difference of the functions (M1)
correct limits (A1)
10.3 A1
3
Question 9Structured · 5 marks

Use differentiation from first principles to show that the derivative of x2 + 5x is 2x + 5.

[5]
Show answers and mark scheme
PartAnswer and marking pointsMarks
shown
f′(x) = limh→0(x + h)2 + 5(x + h) − x2 − 5xh M1A1
2xh + h2 + 5hh A1
2x + h + 5 A1
as h → 0, f′(x) → 2x + 5 AG R1
5
Question 10Structured · 6 marks

Find the exact value of ∫1exln x dx.

[6]
Show answers and mark scheme
PartAnswer and marking pointsMarks
e2 + 14
attempt at integration by parts with u = ln x M1
x22ln x − ∫x2dx A1A1
x22ln x − x24 A1
substituting limits (M1)
e2 + 14 A1
6

Other IB Mathematics: Analysis & Approaches topics

Number & algebraFunctionsGeometry & trigonometryStatistics & probability

Frequently asked questions

What is in IB Mathematics: Analysis & Approaches Calculus?

In the SL & HL syllabus, Calculus covers Differentiation, Tangents & normals, Stationary points & optimisation, Integration, Areas, Kinematics, Implicit, related rates & first principles, Integration techniques, Volumes of revolution, Differential equations, Maclaurin series & limits.

Where can I get Calculus questions with mark schemes?

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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