Further calculus questions

Practice questions covering further calculus from this specification. Attempt the questions below; real-paper references for this topic are being imported.

Questions to attempt

6 questions on Further calculus, each with its mark scheme and worked method. Reveal the answer when you have had a go.

FP1-2.1 · Derivation and use of Taylor series.
Q1
ORIGINAL
Year 2
Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.
1.
Find the Taylor expansion of $\cos x$ in ascending powers of $\left(x-\tfrac{\pi}{3}\right)$ as far as the term in $\left(x-\tfrac{\pi}{3}\right)^2$ .
(4)
(Total for Question 1 is 4 marks)
FP1-2.1 · Derivation and use of Taylor series.
Q2
ORIGINAL
Year 2
Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.
2.
Find the Taylor expansion of $\mathrm{e}^{x}$ in ascending powers of $(x-1)$ as far as the term in $(x-1)^2$ .
(3)
(Total for Question 2 is 3 marks)
FP1-2.2 · Use of series expansions to find limits.
Q3
ORIGINAL
Year 2
Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.
3.
Use series expansions to evaluate $\lim_{x\to0}\dfrac{x-\sin x}{x^3}$ .
(3)
(Total for Question 3 is 3 marks)
FP1-2.3 · Leibnitz's theorem.
Q4
ORIGINAL
Year 2
Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.
4.
Use Leibnitz's theorem to show that the $n$ th derivative of $x^2\mathrm{e}^{x}$ is $\mathrm{e}^{x}\left(x^2+2nx+n(n-1)\right)$ .
(4)
(Total for Question 4 is 4 marks)
FP1-2.4 · L'Hospital's Rule.
Q5
ORIGINAL
Year 2
Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.
5.
Use L'Hospital's Rule to evaluate $\lim_{x\to0^{+}}x\ln x$ .
(4)
(Total for Question 5 is 4 marks)
FP1-2.5 · The Weierstrass substitution for integration.
Q6
ORIGINAL
Year 2
Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.
6.
Use the substitution $t=\tan\tfrac{x}{2}$ to find $\displaystyle\int\frac{\mathrm{d}x}{5+4\cos x}$ .
(5)
(Total for Question 6 is 5 marks)

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Printable question pack — FP1-2 Further calculus (Further Pure 1)All 50 questions across 5 specification points — 44 more than the 6 above — with the full mark scheme. Guest accessible · browser print.

FP1-2 Further calculus (Further Pure 1) revision notes

Real-paper references

Real-paper references for this module are being imported — the questions above are the site's own, exam-faithful, marked.

Reference rows contain series, paper, question number, marks and classification metadata only. Pearson's question papers open on Pearson's own website.

Questions to attempt

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