Standard Maclaurin seriesAQA A-Level Further Maths: Subtopic test

1. 1
   Let $f(x)=\mathrm{e}^{3x}$.

   (a)

   What is the coefficient of $x^2$ in the Maclaurin series of $f(x)$?

   [1 mark]

   • A$\frac92$
   • B$9$
   • C$\frac32$
   • D$3$

   (b)

   For which values of $x$ is the series for $f(x)$ valid?

   [1 mark]

   • A$|x|<\frac13$
   • B$|x|<1$
   • C$-1<x\le1$
   • Dall real $x$

   (c)

   Use the first three terms of the series, with $x=0.1$, to estimate $\mathrm{e}^{0.3}$ to 3 decimal places.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $g(x)=\ln(1+2x)$.

   (a)

   What is the coefficient of $x^3$ in the Maclaurin series of $g(x)$?

   [1 mark]

   • A$\frac23$
   • B$8$
   • C$\frac83$
   • D$-\frac83$

   (b)

   For which values of $x$ is the series for $g(x)$ valid?

   [1 mark]

   • A$-1<x\le1$
   • B$-\frac12<x\le\frac12$
   • C$-2<x\le2$
   • D$-1<x<1$

   (c)

   Use the series for $g(x)$, and for $g(-x)$, to find the first two non-zero terms in the series for $\ln\left(\dfrac{1+2x}{1-2x}\right)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(x)=\mathrm{e}^{x}\sin x$.

   (a)

   Use standard series to find the expansion of $f(x)$ in ascending powers of $x$, up to and including the term in $x^3$.

   [3 marks]

   (b)

   Hence estimate $\int_0^{0.2}\mathrm{e}^x\sin x\,\mathrm{d}x$, giving your answer to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $h(x)=\dfrac{1}{\sqrt{1+4x}}$.

   (a)

   (i) Find the expansion of $h(x)$ in ascending powers of $x$ up to and including the term in $x^3$.
   (ii) State the range of values of $x$ for which the expansion is valid.
   (iii) Use $x=0.01$ in your answer to (i) to estimate $\frac{1}{\sqrt{1.04}}$ to 5 decimal places.

   [6 marks]

   (b)

   Hence find the expansion of $\dfrac{\cos x}{\sqrt{1+4x}}$ up to and including the term in $x^3$, and state the range of values of $x$ for which it is valid, giving a reason.

   [6 marks]

   Total for question 4: 12 marks

End of questions