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IB Maths AI HL Sl 5 2 Increasing and Decreasing Functions Questions

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

  • Use the sign of f′(x) to identify intervals where a function is increasing or decreasing.
  • Interpret f′(x)=0 and changes in derivative sign from a graph, table or model.
  • Explain the monotonic behaviour over the stated domain and context.

IB Maths AI HL Sl 5 2 Increasing and Decreasing Functions Questions 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 IB Maths AI HL Sl 5 2 Increasing and Decreasing Functions Questions question 1 — IB Maths AI HL
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