IB Maths AA HL AA 5.4 Differential equations and Maclaurin series 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
Paper 1, no calculatorPaper 2, calculator
Difficulty
EasyMediumHard

Question 1

Paper 1, no calculatorMedium6 marks

The positive solution yy of

dydx=2xy\frac{dy}{dx}=\frac{2x}{y}

satisfies y(1)=1y(1)=1 for 1≤x≤21\leq x\leq 2 An Euler estimate is made using a step length of 12\frac12

(a)

Write down the Euler estimate for y(32)y\left(\frac32\right)

[1]
(b)

Hence, write down the Euler estimate for y(2)y(2)

[1]
(c)

Determine the exact value of y(2)y(2) by solving the differential equation.

[3]
(d)

Hence, state whether the estimate in (b) is an overestimate or an underestimate of the exact value. Justify your answer.

[1]

Question 2

Paper 2, calculatorMedium12 marks

A liquid is cooling in a room where the temperature remains at 20 ∘C20\,^\circ\mathrm C The measured temperatures are shown in Figure 1. Temperatures were recorded to the nearest 1 ∘C1\,^\circ\mathrm C

For 0≤t≤150\leq t\leq 15 where tt is measured in minutes and TT is measured in degrees Celsius, the cooling is modelled by

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

where kk is constant.

05101520304050607080t / minT / °C804334
(a)

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

[2]
(b)

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

[4]
(c)

Determine estimates for TT at t=5t=5 and t=10t=10 minutes using Euler’s method with step length h=5h=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=15t=15 minutes. Give your answer to the nearest 0.1 ∘C0.1\,^\circ\mathrm C and compare it with the measurement in Figure 1.

[3]

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