IB Maths AA HL Topic 5: Calculus questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

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Question 1

Paper 2, calculatorMedium12 marks

A liquid is cooling in a room where the temperature remains at $20\,^\circ\mathrm C$ The measured temperatures are shown in Figure 1. Temperatures were recorded to the nearest $1\,^\circ\mathrm C$

For $0\leq t\leq 15$ where $t$ is measured in minutes and $T$ is measured in degrees Celsius, the cooling is modelled by

\frac{dT}{dt}=-k(T-20), \qquad k>0,

where $k$ is constant.

(a)

Write down the measured temperatures at $t=0$ and $t=10$ minutes, as shown in Figure 1.

[2]

(b)

Solve the differential equation and use the measurements in Figure 1 to determine $k$ giving your answer to four significant figures with units.

[4]

(c)

Determine estimates for $T$ at $t=5$ and $t=10$ minutes using Euler’s method with step length $h=5$ minutes. Give both estimates to three significant figures.

[3]

(d)

Hence, use the solution from part (b) to determine the model’s predicted temperature at $t=15$ minutes. Give your answer to the nearest $0.1\,^\circ\mathrm C$ and compare it with the measurement in Figure 1.

[3]

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