Edexcel IAL Mathematics FP2.6.3 Taylor series

Practise forming Taylor expansions about non-zero centres by evaluating derivatives and writing powers of x-a with exact coefficients.

Exam points

evaluate y, y' and y'' at the given centre to form a Taylor series in powers of x-a

[Maximum number: 3]

Given that $y=\tan ^{2} x$

Hence determine the Taylor series expansion about $\frac{\pi}{3}$ of $\tan ^2 x$ in ascending powers of $x-\frac{\pi}{3}$ up to and including the term in $\left(x-\frac{\pi}{3}\right)^3$ , giving each coefficient in simplest form.

$(y)_{\pi/3}=3,\quad (y')_{\pi/3}=8\sqrt3,\quad (y'')_{\pi/3}=80,\quad (y''')_{\pi/3}=352\sqrt3$
M1

$y=3+8\sqrt3\left(x-\frac{\pi}{3}\right) +\frac{80}{2!}\left(x-\frac{\pi}{3}\right)^2 +\frac{352\sqrt3}{3!}\left(x-\frac{\pi}{3}\right)^3+\cdots$
M1

$y=3+8\sqrt3\left(x-\frac{\pi}{3}\right) +40\left(x-\frac{\pi}{3}\right)^2 +\frac{176\sqrt3}{3}\left(x-\frac{\pi}{3}\right)^3+\cdots$
A1
(3)

All question bank results loaded

FP2.6.3 - Derivation question 1