Further Mathematics · AQA 7367 · Topic J

Numerical methods

Approximating integrals with the mid-ordinate and Simpson's rules, and solving differential equations step by step with Euler's methods.

3 sub-topics
Taught in Year 13
33 practice questions
3 animated lessons

Practice questions (33) Lessons Worksheets All AQA Further Mathematics topics

What the specification says

Every sub-topic AQA can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

J1Mid-ordinate rule and Simpson's ruleY13
J2Euler's step-by-step method for first-order differential equationsY13
J3Improved Euler methodY13

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Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 33 questions in this topic, free, no account needed.

Further numerical methods

Easy

One step of Euler's method · Year 12 · 1 mark

$\dfrac{\mathrm{d}y}{\mathrm{d}x}$ ${}= xy$ and $y = 2$ when $x = 1$ . Using Euler's method with step length 0.2, what is the estimate of $y$ at $x = 1.2$ ?
(1)

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$y_1 = y_0 + h\,f(x_0, y_0)$ .A: 2.4

Total1

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Further numerical methods

Medium

Euler's method · Year 12 · 3 marks

y' = x + y

y = 1

x = 0

Use Euler's method with $h = 0.1$ to estimate $y$ at $x = 0.2$ .
(3)

Check your answer before you look at the solution

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$y_{n+1} = y_n + h f(x_n, y_n)$
$y_1 = 1.1$
$y_2 = 1.22$

Total3

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Further numerical methods

Hard

Numerical solution of a differential equation · Year 12 · 6 marks

\dfrac{\mathrm{d}y}{\mathrm{d}x}

{}= 2x + \frac{y^2}{2}

, with

y = 1

x = 0

(a)
Use Euler's method with step length 0.1 to find an estimate of $y$ at $x = 0.1$ .
(2)
(b)
Use the approximation $\displaystyle y_{r + 1}$ ${}\approx y_{r - 1}$ ${}+ 2h\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)_r$ with your answer to (a) to estimate $y$ at $x = 0.2$ and at $x = 0.3$ , giving answers to 3 decimal places.
(4)

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Part (a)2 marks

$y_1$ ${}= y_0$ ${}+ h\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)_0$ ${}= 1 + 0.1 \times 0.5$
$y_1 = 1.05$

Part (b)4 marks

Use y₁ from (a). $\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)_1$ ${}= 0.7512$
Keep the unrounded value for the next step. $y_2$ ${}= 1 + 0.2 \times 0.7512$ ${}= 1.150$
$\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)_2$ ${}= 1.0615$
$y_3$ ${}= 1.05 + 0.2 \times 1.0615$ ${}= 1.262$

Total6

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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What the specification says

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Part (a)2 marks

Part (b)4 marks

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