Taylor seriesEdexcel International A Level Further Maths: Subtopic test

1. 1
   A function is defined by $f(x)=\ln x$, for $x>0$. Its Taylor series about $x=2$ is to be found.

   (a)

   Find the value of $f'''(2)$.

   [1 mark]

   • A$\frac12$
   • B$-\frac14$
   • C$2$
   • D$\frac14$

   (b)

   What is the coefficient of $(x-2)^3$ in the Taylor series of $f(x)$ about $x=2$?

   [1 mark]

   • A$\frac14$
   • B$\frac{1}{24}$
   • C$\frac{1}{12}$
   • D$-\frac18$

   (c)

   Use the Taylor series up to and including the term in $(x-2)^3$ to estimate $\ln2.1$ to 5 significant figures, given that $\ln2=0.693147$ to 6 significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A function is defined by $f(x)=\sqrt{x}$, for $x>0$. Its Taylor series about $x=4$ is to be found.

   (a)

   Find the value of $f''(4)$.

   [1 mark]

   • A$\frac14$
   • B$-\frac{1}{16}$
   • C$-\frac{1}{32}$
   • D$\frac{3}{256}$

   (b)

   What is the coefficient of $(x-4)^2$ in the Taylor series of $f(x)$ about $x=4$?

   [1 mark]

   • A$-\frac{1}{64}$
   • B$-\frac{1}{32}$
   • C$\frac{1}{64}$
   • D$-\frac{1}{128}$

   (c)

   The Taylor series about $x=4$ up to the term in $(x-4)^3$ is $2+\frac14(x-4)-\frac{1}{64}(x-4)^2+\frac{1}{512}(x-4)^3$. Use it to estimate $\sqrt{4.2}$ to 4 decimal places.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A function is defined by $f(x)=\sin x$. Its Taylor series about $x=\pi$ is to be found.

   (a)

   Find the Taylor series of $f(x)$ in ascending powers of $(x-\pi)$ up to and including the term in $(x-\pi)^3$.

   [3 marks]

   (b)

   (i) Use the series to find $\lim_{x\to\pi}\frac{\sin x}{x-\pi}$.
   (ii) Use the series to estimate $\sin3$ to 5 decimal places.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A function is defined by $f(x)=\tan x$. Its Taylor series about $x=\frac{\pi}{4}$ is to be found.

   (a)

   Find the Taylor series of $f(x)$ in ascending powers of $\left(x-\frac{\pi}{4}\right)$ up to and including the term in $\left(x-\frac{\pi}{4}\right)^3$.

   [6 marks]

   (b)

   Let $h=x-\frac{\pi}{4}$. (i) Differentiate your series in (a) to find the series for $\sec^2x$ up to the term in $h^2$.
   (ii) Square your series for $\tan x$, keeping terms up to $h^2$, and use $\sec^2x=1+\tan^2x$ to show that you obtain the same series for $\sec^2x$.

   [6 marks]

   Total for question 4: 12 marks

End of questions