IB Mathematics Analysis and Approaches Topic SL 5.1—introduction of Differential Calculus Questionbank (SL/HL)

IB Mathematics Analysis and Approaches Topic SL 5.1—introduction of Differential Calculus Questionbank (SL/HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic SL 5.1—introduction of Differential Calculus with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 5.1—introduction of Differential 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

SLPaper 1

A right circular cylinder of radius $r$ and height $h$ is inscribed in a hemisphere of fixed radius $R$ , such that the base of the cylinder lies on the base of the hemisphere.

Show that the volume $V$ of the cylinder can be expressed as $V = π h (R^{2} - h^{2})$ .

[2]

Find the ratio of the height $h$ to the radius $R$ that maximizes the volume of the cylinder.

[4]

Hence, find the exact maximum volume in terms of $R$ .

[2]

Question 2

SLPaper 1

The function $g (x) = \frac{2 x + 3}{x ^{2} + 3 x - 4}$ models the electric potential $g (x)$ (in volts) at a point with coordinate $x$ metres on a straight sensor line.

Identify the vertical and horizontal asymptotes of $g (x)$ .

[3]

Describe the behavior of the electric potential $g (x)$ as $x \to \infty$ and $x \to - \infty$ , and interpret its significance.

[2]

Sketch the graph of $g (x)$ , marking key points and asymptotes, to illustrate the potential distribution.

[3]

Question 3

SLPaper 2

The function $f (x) = \frac{3 x + 1}{x ^{2} - 4}$ models the stress $f (x)$ (in MPa) on a structural beam as a function of position $x$ (in metres) along the beam.

Identify the vertical and horizontal asymptotes of $f (x)$ .

[3]

Describe the behavior of the stress $f (x)$ as $x \to \infty$ and $x \to - \infty$ , and interpret its significance.

[2]

Sketch the graph of $f (x)$ , marking key points and asymptotes, to illustrate the beam's stress distribution.

[3]

Question 4

SLPaper 2

The 'SkyEye' Ferris wheel at an amusement park models the vertical height of a passenger cabin using the function $H (t) = 14 sin (0.52 (t - 2.5)) + 16$ , where $H$ is the height in metres above the ground and $t$ is the time in minutes since the ride started. At its lowest point, the cabin is $2$ m above the ground, and at its highest point, it reaches $30$ m.

The cabin reaches a height of $26$ metres for the first time when the time elapsed is $m$ minutes and $s$ seconds. Find the value of $s$ to the nearest integer, given that $m = 4$ .

[3]

During one complete revolution, find the total time, in minutes, for which the cabin is at a height of at least $25$ metres.

[2]

Determine the vertical velocity of the cabin when $t = 4$ , giving your answer in metres per minute.

[2]

The height of a cabin on a second Ferris wheel, the 'Junior Wheel', is modelled by $h (t) = a cos (b (t - c)) + d$ , where $t$ is the time in minutes. For this wheel, the minimum height is $2$ m and the maximum height is $12$ m. The cabin is first at its minimum height at $t = 1$ and first reaches its maximum height at $t = 5$ . Determine the values of $a, b, c,$ and $d$ .

[6]

Find the first time $t > 6$ when the cabins on both wheels are at the same height.

[2]

Question 5

SLPaper 1

The function $f (x) = \frac{x + 1}{x ^{2} + x - 6}$ models the pressure variation $f (x)$ (in kPa) in a pipeline as a function of position $x$ (in metres), measured relative to a monitoring station.

Identify the vertical and horizontal asymptotes of $f (x)$ .

[3]

Describe the behavior of the pressure variation $f (x)$ as $x \to \infty$ and $x \to - \infty$ , and interpret its significance.

[2]

Sketch the graph of $f (x)$ , marking key points and asymptotes, to illustrate the pipeline pressure variation.

[3]

IB Mathematics Analysis and Approaches SL 5.1—introduction of Differential Calculus Questionbank

Unlock all 7 questions with a free account

IB Mathematics Analysis and Approaches SL 5.1—introduction of Differential Calculus Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 5.1—introduction of Differential Calculus Questionbank

Unlock all 7 questions with a free account