5.16 Euler's methodIB Maths: Applications and Interpretation HL: Flashcards

What these 13 flashcards ask

State the recurrence for Euler's method for \frac{dy}{dx}=f(x,y).
What is the step length?
What does Euler's method give?
What is an initial condition?
Which gradient is used in each step?
How do you make Euler's method more accurate?
When does Euler's method underestimate?
\frac{dy}{dx}=2x-y, y(0)=3, h=0.5: find y1.
How do you carry out many steps quickly?
Euler's method for \frac{dx}{dt}=f1, \frac{dy}{dt}=f2?
In a coupled step, which values are used on the right-hand side?
Predator-prey: \frac{dx}{dt}=ax-bxy, \frac{dy}{dt}=cxy-dy. Non-zero equilibrium?
In a predator-prey model, what does the term -bxy represent?

Exam questions on 5.16 Euler's method

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 $y$ .
Find the approximation to $y$ when $x=1.5$ .2 marks
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 $T$ .
Continue the method to estimate $T$ when $t=6$ .2 marks
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.
Use Euler's method to estimate the number of fish after 1 year and after 2 years.3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).