Further Mathematics · OCR H245 · Topic 4.08

Further calculus

Maclaurin series, improper integrals, volumes of solids of revolution, mean values, partial fractions, inverse trigonometric and hyperbolic functions, further integration.

7 sub-topics
1 in Year 12, 6 in Year 13
140 practice questions
7 animated lessons

Practice questions (140) Lessons Worksheets coming soon All OCR Further Mathematics topics

What the specification says

Every sub-topic OCR 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.

4.08a-bMaclaurin seriesY13
4.08cImproper integralsY13
4.08dVolumes of solids of revolutionY12
4.08eMean valuesY13
4.08fPartial fractionsY13
4.08gInverse trigonometric and hyperbolic functionsY13
4.08hFurther integrationY13

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Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 140 questions in this topic, free, no account needed.

Further algebra and functions

Not needed for Year 12Easy

Using standard series · Year 13 · 2 marks

Use the series for eˣ to find the first three terms of $\mathrm{e}^{2x}$ .
(2)

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Replace $x$ by kx. $1 + (2x) + \frac{(2x)^2}{2}$
$1 + 2x + 2x^2$

Total2

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 algebra and functions

Not needed for Year 12Medium

Maclaurin series from first principles · Year 13 · 4 marks

Find the Maclaurin series of $\mathrm{e}^{2x}$ up to the term in $x^3$ .
(4)

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Differentiate three times and evaluate at $x = 0$ .
$f(0)$ ${}+ f'(0)x$ ${}+ \frac{f''(0)}{2!}x^2$ ${}+ \frac{f'''(0)}{3!}x^3$
$1 + 2x + 2x^2 + \tfrac{4}{3}x^3$

Total4

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 algebra and functions

Not needed for Year 12Hard

Product of series · Year 13 · 3 marks

Find the series of $\mathrm{e}^{x}\sin 2x$ up to $x^3$ .
(3)

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Multiply truncated series. $(1 + x + \tfrac{x^2}{2})(2x - \tfrac{8x^3}{6})$
$2x + 2x^2$
Collect the $x^3$ terms. $-\tfrac{1}{3}x^3$

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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What the specification says

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