Maclaurin series for composite exponential/root functions

Questions asking to find Maclaurin series by differentiation for functions involving roots or exponentials of trigonometric/exponential expressions, such as √(a+e^x), √(a+sin x), or e^(sin x).

6 questions

Edexcel FP2 2014 June Q3

3. $$y = \sqrt { 8 + \mathrm { e } ^ { x } } , \quad x \in \mathbb { R }$$ Find the series expansion for $y$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, giving each coefficient in its simplest form.

OCR FP2 2009 June Q3

Given that $\mathrm { f } ( x ) = \mathrm { e } ^ { \sin x }$, find $\mathrm { f } ^ { \prime } ( 0 )$ and $\mathrm { f } ^ { \prime \prime } ( 0 )$.
Hence find the first three terms of the Maclaurin series for $\mathrm { f } ( x )$.

AQA FP3 2008 June Q7

Write down the expansion of $\sin 2 x$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$.
1. Given that $y = \sqrt { 3 + \mathrm { e } ^ { x } }$, find the values of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ when $x = 0$.
2. Using Maclaurin's theorem, show that, for small values of $x$, $$\sqrt { 3 + \mathrm { e } ^ { x } } \approx 2 + \frac { 1 } { 4 } x + \frac { 7 } { 64 } x ^ { 2 }$$
Find $$\lim _ { x \rightarrow 0 } \left[ \frac { \sqrt { 3 + \mathrm { e } ^ { x } } - 2 } { \sin 2 x } \right]$$

AQA FP3 2009 June Q6

6 The function f is defined by $$\mathrm { f } ( x ) = ( 9 + \tan x ) ^ { \frac { 1 } { 2 } }$$

1. Find $f ^ { \prime \prime } ( x )$.
2. By using Maclaurin's theorem, show that, for small values of $x$, $$( 9 + \tan x ) ^ { \frac { 1 } { 2 } } \approx 3 + \frac { x } { 6 } - \frac { x ^ { 2 } } { 216 }$$
Find $$\lim _ { x \rightarrow 0 } \left[ \frac { f ( x ) - 3 } { \sin 3 x } \right]$$

AQA FP3 2013 June Q6

6 It is given that $y = ( 4 + \sin x ) ^ { \frac { 1 } { 2 } }$.

Express $y \frac { \mathrm {~d} y } { \mathrm {~d} x }$ in terms of $\cos x$.
Find the value of $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$ when $x = 0$.
Hence, by using Maclaurin's theorem, find the first four terms in the expansion, in ascending powers of $x$, of $( 4 + \sin x ) ^ { \frac { 1 } { 2 } }$.
(2 marks)

OCR MEI Further Pure Core 2023 June Q4

1. Given that $\mathrm { f } ( x ) = \sqrt { 1 + 2 x }$, find $\mathrm { f } ^ { \prime } ( x )$ and $\mathrm { f } ^ { \prime \prime } ( x )$.
2. Hence, find the first three terms of the Maclaurin series for $\sqrt { 1 + 2 x }$.
Hence, using a suitable value for $x$, show that $\sqrt { 5 } \approx \frac { 143 } { 64 }$.