Maclaurin seriesEdexcel International A Level Further Maths: Subtopic test

1. 1
   A function is defined by $f(x)=\mathrm{e}^{3x}$.

   (a)

   Find the value of $f'''(0)$.

   [1 mark]

   • A$9$
   • B$27$
   • C$81$
   • D$3$

   (b)

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

   [1 mark]

   • A$27$
   • B$9$
   • C$\frac92$
   • D$\frac{27}{8}$

   (c)

   Write down the Maclaurin series of $f(x)$ up to and including the term in $x^3$, and use it with $x=0.3$ to estimate $\mathrm{e}^{0.9}$ to 3 significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A function is defined by $f(x)=\cos2x$.

   (a)

   Find the value of $f''(0)$.

   [1 mark]

   • A$-4$
   • B$4$
   • C$-2$
   • D$0$

   (b)

   What is the coefficient of $x^4$ in the Maclaurin series of $f(x)$?

   [1 mark]

   • A$16$
   • B$\frac83$
   • C$\frac43$
   • D$\frac23$

   (c)

   Use the identity $\cos2x=1-2\sin^2x$ and the Maclaurin series of $\cos2x$ up to the term in $x^4$ to find the first two non-zero terms of the Maclaurin series of $\sin^2x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A function is defined by $f(x)=\ln(1+x)$, for $-1<x\le1$.

   (a)

   Use differentiation to find the Maclaurin series of $f(x)$ up to and including the term in $x^3$.

   [3 marks]

   (b)

   By replacing $x$ with $-x$ in your series, find the series for $\ln\left(\frac{1+x}{1-x}\right)$ up to the term in $x^3$. Hence estimate $\ln2$ to 3 significant figures by choosing a suitable value of $x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A function is defined by $y=\tan x$, for $-\frac{\pi}{2}<x<\frac{\pi}{2}$.

   (a)

   Show that $\frac{dy}{dx}=1+y^2$, and by differentiating this result repeatedly find the Maclaurin series of $\tan x$ up to and including the term in $x^5$.

   [6 marks]

   (b)

   (i) By differentiating $\ln(\cos x)$ and using your series from (a), find the Maclaurin series of $\ln(\cos x)$ up to and including the term in $x^6$.
   (ii) Hence find $\lim_{x\to0}\frac{\ln(\cos x)+\frac12x^2}{x^4}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions