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IB Maths AI HL 5.2 Increasing and decreasing functions Question Bank

Practise IB Maths AI HL 5.2 by analysing derivative behaviour, curve shape and increasing or decreasing model predictions.

Syllabus
First assessment 2021
Course
Mathematics: applications and interpretation HL
Level
HL

Exam points

  • Analyse derivative signs and curve shape to determine increasing or decreasing behaviour.
  • Evaluate model predictions using gradients, intervals and contextual constraints on the function.

SL 5.2—Increasing and decreasing functions question 1

[Maximum number: 2]

This question uses differential equations to model the maximum velocity of a skydiver in free fall.
In 2012, Felix Baumgartner jumped from a height of 40000 m . He was attempting to travel at the speed of sound, 330 m s−1330 \mathrm{~m} \mathrm{~s}^{-1}, whilst free-falling to the Earth.
Before making his attempt, Felix used mathematical models to check how realistic his attempt would be. The simplest model he used suggests that

dv dt=g\frac{\mathrm{d} v}{\mathrm{~d} t}=g

where v m s−1v \mathrm{~m} \mathrm{~s}^{-1} is Felix's velocity and g ms−2g \mathrm{~ms}^{-2} is the acceleration due to gravity. The time since he began to free-fall is t seconds and the displacement from his initial position is s metres.

Throughout this question, the direction towards the centre of the Earth is taken to be positive and v is a positive quantity.

When s=0, it is given that Felix jumps with an initial velocity v=10.

To test the model

dv dt=g\frac{\mathrm{d} v}{\mathrm{~d} t}=g

Felix conducted a trial jump from a lower height, and data for v against t was found.

If the model is correct, describe the shape of the graph of v against t.

Felix's data are plotted on the following graph.

Figure for Question SL 5.2—Increasing and decreasing functions question 1 — IB Maths AI HL
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