Maclaurin series for ln(trigonometric expressions)

Finding Maclaurin series for logarithmic functions involving trigonometric expressions such as ln(cos x), ln(cos ax), ln(sin x + cos x), ln(1 + sin x), or ln(1 + tan x). These require using trigonometric derivatives and often involve finding non-zero terms.

8 questions

OCR FP2 2008 January Q1

1 It is given that $\mathrm { f } ( x ) = \ln ( 1 + \cos x )$.

Find the exact values of $f ( 0 ) , f ^ { \prime } ( 0 )$ and $f ^ { \prime \prime } ( 0 )$.
Hence find the first two non-zero terms of the Maclaurin series for $\mathrm { f } ( x )$.

OCR FP2 2012 January Q1

1 Given that $\mathrm { f } ( x ) = \ln ( \cos 3 x )$, find $\mathrm { f } ^ { \prime } ( 0 )$ and $\mathrm { f } ^ { \prime \prime } ( 0 )$. Hence show that the first term in the Maclaurin series for $\mathrm { f } ( x )$ is $a x ^ { 2 }$, where the value of $a$ is to be found.

AQA FP3 2008 January Q7

1. Write down the expansion of $\ln ( 1 + 2 x )$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$.
2. State the range of values of $x$ for which this expansion is valid.
1. Given that $y = \ln \cos x$, find $\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ and $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$.
2. Find the value of $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }$ when $x = 0$.
3. Hence, by using Maclaurin's theorem, show that the first two non-zero terms in the expansion, in ascending powers of $x$, of $\ln \cos x$ are $$- \frac { x ^ { 2 } } { 2 } - \frac { x ^ { 4 } } { 12 }$$
Find $$\lim _ { x \rightarrow 0 } \left[ \frac { x \ln ( 1 + 2 x ) } { x ^ { 2 } - \ln \cos x } \right]$$

AQA FP3 2012 January Q6

Given that $y = \ln \cos 2 x$, find $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }$.
Use Maclaurin's theorem to show that the first two non-zero terms in the expansion, in ascending powers of $x$, of $\ln \cos 2 x$ are $- 2 x ^ { 2 } - \frac { 4 } { 3 } x ^ { 4 }$.
Hence find the first two non-zero terms in the expansion, in ascending powers of $x$, of $\ln \sec ^ { 2 } 2 x$.

AQA FP3 2011 June Q5

Given that $y = \ln ( 1 + 2 \tan x )$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
(You may leave your expression for $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ unsimplified.)
Hence, using Maclaurin's theorem, find the first two non-zero terms in the expansion, in ascending powers of $x$, of $\ln ( 1 + 2 \tan x )$.
(2 marks)
Find $$\lim _ { x \rightarrow 0 } \left[ \frac { \ln ( 1 + 2 \tan x ) } { \ln ( 1 - x ) } \right]$$ (4 marks)

AQA FP3 2012 June Q6

6 It is given that $y = \ln ( 1 + \sin x )$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Show that $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \mathrm { e } ^ { - y }$.
Express $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }$ in terms of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and $\mathrm { e } ^ { - y }$.
Hence, by using Maclaurin's theorem, find the first four non-zero terms in the expansion, in ascending powers of $x$, of $\ln ( 1 + \sin x )$.

AQA FP3 2014 June Q7

4 marks

It is given that $y = \ln ( \cos x + \sin x )$.
1. Show that $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 2 } { 1 + \sin 2 x }$.
2. Find $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$.
1. Hence use Maclaurin's theorem to show that the first three non-zero terms in the expansion, in ascending powers of $x$, of $\ln ( \cos x + \sin x )$ are $x - x ^ { 2 } + \frac { 2 } { 3 } x ^ { 3 }$.
2. Write down the first three non-zero terms in the expansion, in ascending powers of $x$, of $\ln ( \cos x - \sin x )$.
Hence find the first three non-zero terms in the expansion, in ascending powers of $x$, of $\ln \left( \frac { \cos 2 x } { \mathrm { e } ^ { 3 x - 1 } } \right)$.
[0pt] [4 marks]
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OCR Further Pure Core 2 2019 June Q10

Use differentiation to find the first two non-zero terms of the Maclaurin expansion of $\ln \left( \frac { 1 } { 2 } + \cos x \right)$.
By considering the root of the equation $\ln \left( \frac { 1 } { 2 } + \cos x \right) = 0$ deduce that $\pi \approx 3 \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }$. \section*{END OF QUESTION PAPER}