Maclaurin series of a functionAQA A-Level Further Maths: Subtopic test

1. 1
   Let $f(x)=\ln(\cos x)$ for $-\frac{\pi}{2}<x<\frac{\pi}{2}$.

   (a)

   What is $f'(x)$?

   [1 mark]

   • A$\tan x$
   • B$-\cot x$
   • C$-\tan x$
   • D$-\sec^2x$

   (b)

   What is the value of $f''(0)$?

   [1 mark]

   • A$0$
   • B$-1$
   • C$1$
   • D$-\frac12$

   (c)

   Given that $f'''(0)=0$ and $f^{(4)}(0)=-2$, find the first two non-zero terms of the Maclaurin series of $f(x)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $f(x)=\dfrac{1}{1-2x}$.

   (a)

   What is the value of $f'(0)$?

   [1 mark]

   • A$2$
   • B$1$
   • C$-2$
   • D$4$

   (b)

   What is the general term of the Maclaurin series of $f(x)$?

   [1 mark]

   • A$2x^r$
   • B$\frac{x^r}{2^r}$
   • C$\frac{2^rx^r}{r!}$
   • D$2^rx^r$

   (c)

   Deduce an expression for $f^{(r)}(0)$ in terms of $r$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(x)=\tan x$.

   (a)

   Show that $f'''(0)=2$.

   [3 marks]

   (b)

   Hence find the Maclaurin series of $f(x)$ up to and including the term in $x^3$, and use it to estimate $\tan0.2$ to 4 decimal places.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $y=\mathrm{e}^{\sin x}$.

   (a)

   (i) Show that $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}=\left(\cos^2x-\sin x\right)y$.
   (ii) Find the value of $\dfrac{\mathrm{d}^3y}{\mathrm{d}x^3}$ when $x=0$.

   [6 marks]

   (b)

   (i) Write down the Maclaurin series of $y$ up to and including the term in $x^3$.
   (ii) Verify your series by substituting the Maclaurin series for $\sin x$ into the standard series for $\mathrm{e}^u$.

   [6 marks]

   Total for question 4: 12 marks

End of questions