Taylor seriesEdexcel International A Level Further Maths: Mind map

Exam questions on Taylor series

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