5.16 Euler's methodIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   A function $y$ satisfies $\frac{dy}{dx}=2x-y$, with $y=3$ when $x=0$. Euler's method with step length $h=0.5$ is used to approximate $y$.

   (a)

   Find the approximation to $y$ when $x=0.5$.

   [1 mark]

   • A$2$
   • B$0$
   • C$-1.5$
   • D$1.5$

   (b)

   Find the approximation to $y$ when $x=1$.

   [1 mark]

   • A$1.75$
   • B$1.25$
   • C$1$
   • D$0.5$

   (c)

   Find the approximation to $y$ when $x=1.5$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The temperature $T\,^\circ$C of a cup of tea, $t$ minutes after it is poured, satisfies $\frac{dT}{dt}=-0.1(T-20)$, and $T=90$ when $t=0$. Euler's method with step length $h=2$ minutes is used to estimate $T$.

   (a)

   Find the estimate of $T$ when $t=2$.

   [1 mark]

   • A$83$
   • B$72$
   • C$76$
   • D$-14$

   (b)

   Find the estimate of $T$ when $t=4$.

   [1 mark]

   • A$64.8$
   • B$70.4$
   • C$62$
   • D$87.2$

   (c)

   Continue the method to estimate $T$ when $t=6$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The number of fish $P$ in a lake, $t$ years after the lake is stocked, is modelled by $\frac{dP}{dt}=0.4P\left(1-\frac{P}{500}\right)$, with $P=100$ when $t=0$. Euler's method with step length $h=1$ year is used. A GDC or spreadsheet may be used.

   (a)

   Use Euler's method to estimate the number of fish after 1 year and after 2 years.

   [3 marks]

   (b)

   Use your GDC or a spreadsheet to find the first year in which the estimate of $P$ exceeds 300. Write down the estimate for the end of each of the last two years you use.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A nature reserve has $x$ rabbits and $y$ foxes at time $t$ months. Treating $x$ and $y$ as continuous, the populations are modelled by $\frac{dx}{dt}=0.5x-0.02xy$ and $\frac{dy}{dt}=0.01xy-0.4y$, with $x=60$ and $y=15$ when $t=0$. Euler's method with step length $h=0.5$ months is used.

   (a)

   (i) Find the value of $\frac{dx}{dt}$ and of $\frac{dy}{dt}$ when $t=0$.
   (ii) Hence use Euler's method to estimate $x$ and $y$ when $t=0.5$.
   (iii) Interpret the value of $\frac{dy}{dt}$ when $t=0$ in context.

   [6 marks]

   (b)

   (i) Find the values of $x$ and $y$, both non-zero, for which both populations are constant.
   (ii) Use your GDC or a spreadsheet to estimate $x$ and $y$ when $t=2$, giving your answers to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

End of questions