Maclaurin series for ln(exponential expressions)

AQA FP3 2013 January Q6

14 marks Challenging +1.2

6

It is given that $y = \ln \left( \mathrm { e } ^ { 3 x } \cos x \right)$.
1. Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 - \tan x$.
2. Find $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }$.
Hence use Maclaurin's theorem to show that the first three non-zero terms in the expansion, in ascending powers of $x$, of $\ln \left( \mathrm { e } ^ { 3 x } \cos x \right)$ are $3 x - \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 12 } x ^ { 4 }$.
(3 marks)
Write down the expansion of $\ln ( 1 + p x )$, where $p$ is a constant, in ascending powers of $x$ up to and including the term in $x ^ { 2 }$.
1. Find the value of $p$ for which $\lim _ { x \rightarrow 0 } \left[ \frac { 1 } { x ^ { 2 } } \ln \left( \frac { \mathrm { e } ^ { 3 x } \cos x } { 1 + p x } \right) \right]$ exists.
2. Hence find the value of $\lim _ { x \rightarrow 0 } \left[ \frac { 1 } { x ^ { 2 } } \ln \left( \frac { \mathrm { e } ^ { 3 x } \cos x } { 1 + p x } \right) \right]$ when $p$ takes the value found in part (d)(i).

AQA FP3 2010 June Q5

13 marks Standard +0.8

5

Write down the expansion of $\cos 4 x$ in ascending powers of $x$ up to and including the term in $x ^ { 4 }$. Give your answer in its simplest form.
1. Given that $y = \ln \left( 2 - \mathrm { e } ^ { x } \right)$, find $\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ and $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$.
  (You may leave your expression for $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$ unsimplified.)
2. Hence, by using Maclaurin's theorem, show that the first three non-zero terms in the expansion, in ascending powers of $x$, of $\ln \left( 2 - \mathrm { e } ^ { x } \right)$ are $$- x - x ^ { 2 } - x ^ { 3 }$$
Find $$\lim _ { x \rightarrow 0 } \left[ \frac { x \ln \left( 2 - \mathrm { e } ^ { x } \right) } { 1 - \cos 4 x } \right]$$

AQA FP3 2007 June Q6

15 marks Standard +0.8

6

The function f is defined by $$\mathrm { f } ( x ) = \ln \left( 1 + \mathrm { e } ^ { x } \right)$$ Use Maclaurin's theorem to show that when $\mathrm { f } ( x )$ is expanded in ascending powers of $x$ :
1. the first three terms are $$\ln 2 + \frac { 1 } { 2 } x + \frac { 1 } { 8 } x ^ { 2 }$$
2. the coefficient of $x ^ { 3 }$ is zero.
Hence write down the first two non-zero terms in the expansion, in ascending powers of $x$, of $\ln \left( \frac { 1 + \mathrm { e } ^ { x } } { 2 } \right)$.
Use the series expansion $$\ln ( 1 + x ) = x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } - \ldots$$ to write down the first three terms in the expansion, in ascending powers of $x$, of $\ln \left( 1 - \frac { x } { 2 } \right)$.
Use your answers to parts (b) and (c) to find $$\lim _ { x \rightarrow 0 } \left[ \frac { \ln \left( \frac { 1 + \mathrm { e } ^ { x } } { 2 } \right) + \ln \left( 1 - \frac { x } { 2 } \right) } { x - \sin x } \right]$$

AQA Further AS Paper 1 2021 June Q7

3 marks Standard +0.3

7 Show that the Maclaurin series for $\ln ( \mathrm { e } + 2 \mathrm { e } x )$ is $$1 + 2 x - 2 x ^ { 2 } + a x ^ { 3 } - \ldots$$ where $a$ is to be determined.

Edexcel FP1 2023 June Q6

Challenging +1.2

6. $$y = \ln \left( \mathrm { e } ^ { 2 x } \cos 3 x \right) \quad - \frac { 1 } { 2 } < x < \frac { 1 } { 2 }$$

Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 - 3 \tan 3 x$$
Determine $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }$
Hence determine the first 3 non-zero terms in ascending powers of $x$ of the Maclaurin series expansion of $\ln \left( \mathrm { e } ^ { 2 x } \cos 3 x \right)$, giving each coefficient in simplest form.
Use the Maclaurin series expansion for $\ln ( 1 + x )$ to write down the first 4 non-zero terms in ascending powers of $x$ of the Maclaurin series expansion of $\ln ( 1 + k x )$, where $k$ is a constant.
Hence determine the value of $k$ for which $$\lim _ { x \rightarrow 0 } \left( \frac { 1 } { x ^ { 2 } } \ln \frac { \mathrm { e } ^ { 2 x } \cos 3 x } { 1 + k x } \right)$$ exists.

OCR FP2 2009 January Q1

6 marks Standard +0.3

Write down and simplify the first three terms of the Maclaurin series for $e^{2x}$. [2]
Hence show that the Maclaurin series for $$\ln(e^{2x} + e^{-2x})$$ begins $\ln a + bx^2$, where $a$ and $b$ are constants to be found. [4]

SPS SPS FM Pure 2023 February Q14

7 marks Challenging +1.3

Use differentiation to find the first two non-zero terms of the Maclaurin expansion of $\ln\left(\frac{1}{2} + \cos x\right)$. [4]
By considering the root of the equation $\ln\left(\frac{1}{2} + \cos x\right) = 0$ deduce that $\pi \approx 3\sqrt{3 \ln\left(\frac{3}{2}\right)}$. [3]