Maclaurin series for products/secant

Questions asking to find Maclaurin series by differentiation for products like e^x·tan(x) or sec(x) where the function structure requires product/quotient rule techniques.

2 questions

AQA FP3 2011 January Q7
7
  1. Write down the expansions in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\) of:
    1. \(\cos x + \sin x\);
    2. \(\quad \ln ( 1 + 3 x )\).
  2. It is given that \(y = \mathrm { e } ^ { \tan x }\).
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = ( 1 + \tan x ) ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x }\).
    2. Find the value of \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) when \(x = 0\).
    3. Hence, by using Maclaurin's theorem, show that the first four terms in the expansion, in ascending powers of \(x\), of \(\mathrm { e } ^ { \tan x }\) are $$1 + x + \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 } x ^ { 3 }$$
  3. Find $$\lim _ { x \rightarrow 0 } \left[ \frac { \mathrm { e } ^ { \tan x } - ( \cos x + \sin x ) } { x \ln ( 1 + 3 x ) } \right]$$
AQA Further Paper 2 2022 June Q8
8
  1. The function f is defined as \(\mathrm { f } ( x ) = \sec x\) 8
    1. Show that \(\mathrm { f } ^ { ( 4 ) } ( 0 ) = 5\)
      8
  3. (ii) Hence find the first three non-zero terms of the Maclaurin series for \(\mathrm { f } ( x ) = \sec x\)
    8
  4. Prove that $$\lim _ { x \rightarrow 0 } \left( \frac { \sec x - \cosh x } { x ^ { 4 } } \right) = \frac { 1 } { 6 }$$