4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n

3

1. 1. Write down the expansion of \(\ln ( 1 + 2 x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 4 }\).
   2. Hence, or otherwise, find the first two non-zero terms in the expansion of $$\ln [ ( 1 + 2 x ) ( 1 - 2 x ) ]$$ in ascending powers of \(x\) and state the range of values of \(x\) for which the expansion is valid.
2. Find \(\lim _ { x \rightarrow 0 } \left[ \frac { 3 x - x \sqrt { 9 + x } } { \ln [ ( 1 + 2 x ) ( 1 - 2 x ) ] } \right]\).

2

1. Write down the expansion of \(\sin 2 x\) in ascending powers of \(x\) up to and including the term in \(x ^ { 5 }\).
2. It is given that the first non-zero term in the expansion of $$\sin 2 x - 2 x \left( 1 - p x ^ { 2 } \right) \left( 1 - x ^ { 2 } \right) ^ { - 1 }$$ in ascending powers of \(x\) is \(q x ^ { 5 }\).
   Find the values of the rational numbers \(p\) and \(q\).

6

1. Find and simplify the first five terms in the Maclaurin series for \(\mathrm { e } ^ { 2 x }\) 6
2. Hence, or otherwise, write down the first five terms in the Maclaurin series for \(\mathrm { e } ^ { - 2 x }\) 6
3. Hence, or otherwise, show that the Maclaurin series for \(\cosh ( 2 x )\) is $$a + b x ^ { 2 } + c x ^ { 4 } + \ldots$$ where \(a\), \(b\) and \(c\) are rational numbers to be determined.

9

1. Find the Maclaurin series of \(( \ln ( 1 + x ) ) ^ { 2 }\) up to and including the term in \(x ^ { 4 }\). The diagram below shows parts of the graphs of the curves with equations \(y = ( \ln ( 1 + x ) ) ^ { 2 }\) and \(y = 2 x ^ { 3 }\). The curves intersect at the origin, \(O\), and at the point \(A\). \includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-4_663_906_831_248} \section*{(b) In this question you must show detailed reasoning.} Use your answer to part (a) to determine an approximation for the value of the \(x\)-coordinate of \(A\). Give your answer to \(\mathbf { 2 }\) decimal places.

10

1. Use differentiation to find the first two non-zero terms of the Maclaurin expansion of \(\ln \left( \frac { 1 } { 2 } + \cos x \right)\).
2. 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}

10 Let \(\mathrm { f } ( x ) = \sin ^ { - 1 } ( x )\).

1. 1. Determine \(\mathrm { f } ^ { \prime \prime } ( x )\).
   2. Determine the first two non-zero terms of the Maclaurin expansion for \(\mathrm { f } ( x )\).
   3. By considering the first two non-zero terms of the Maclaurin expansion for \(\mathrm { f } ( x )\), find an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer correct to 6 decimal places.
2. By writing \(\mathrm { f } ( x )\) as \(\sin ^ { - 1 } ( x ) \times 1\), determine the value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer in exact form.

10 In this question you must show detailed reasoning.

1. By using an appropriate Maclaurin series prove that if \(x > 0\) then \(\mathrm { e } ^ { x } > 1 + x\).
2. Hence, by using a suitable substitution, deduce that \(\mathrm { e } ^ { t } > \mathrm { e } t\) for \(t > 1\).
3. Using the inequality in part (b), and by making a suitable choice for \(t\), determine which is greater, \(\mathrm { e } ^ { \pi }\) or \(\pi ^ { \mathrm { e } }\).

3. (a) Expand \(( 1 + 3 x ) ^ { - 2 } , | x | < \frac { 1 } { 3 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each term.
(b) Hence, or otherwise, find the first three terms in the expansion of \(\frac { x + 4 } { ( 1 + 3 x ) ^ { 2 } }\) as a series in ascending powers of \(x\).

5 Using the Maclaurin series for \(\cos 2 x\), show that, for small values of \(x\), \(\sin ^ { 2 } x \approx a x ^ { 2 } + b x ^ { 4 } + c x ^ { 6 }\),
where the values of \(a , b\) and \(c\) are to be given in exact form.

4

1. 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 \(\sqrt { 1 + 2 x }\).
2. Hence, using a suitable value for \(x\), show that \(\sqrt { 5 } \approx \frac { 143 } { 64 }\).

10

1. Write down the first three terms of the Maclaurin series for \(\ln \left( 1 + x ^ { 3 } \right)\).
2. Use these three terms to show that \(\ln ( 1.125 ) \approx \frac { n } { 1536 }\), where \(n\) is an integer to be determined.
3. Charlie uses the same first three terms of the series to approximate \(\ln 9\) and gets an answer of 147, correct to 3 significant figures. However, \(\ln 9 = 2.20\) correct to 3 significant figures. Explain Charlie's error.

13

1. Using exponentials, prove that \(\sinh 2 x = 2 \cosh x \sinh x\).
2. Hence show that if \(\mathrm { f } ( x ) = \sinh ^ { 2 } x\), then \(\mathrm { f } ^ { \prime \prime } ( x ) = 2 \cosh 2 x\).
3. Explain why the coefficients of odd powers in the Maclaurin series for \(\sinh ^ { 2 } x\) are all zero.
4. Find the coefficient of \(x ^ { n }\) in this series when \(n\) is a positive even number.

5

1. Use a Maclaurin series to find a quadratic approximation for \(\ln ( 1 + 2 x )\).
2. Find the percentage error in using the approximation in part (a) to calculate \(\ln ( 1.2 )\).
3. Jane uses the Maclaurin series in part (a) to try to calculate an approximation for \(\ln 3\). Explain whether her method is valid.

1. (a) Write down the Maclaurin series of \(\mathrm { e } ^ { x }\), in ascending power of \(x\), up to and including the term in \(x ^ { 3 }\) (b) Hence, without differentiating, determine the Maclaurin series of

$$\mathrm { e } ^ { \left( \mathrm { e } ^ { x } - 1 \right) }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), giving each coefficient in simplest form.

4. $$f ( x ) = x ^ { 4 } \sin ( 2 x )$$ Use Leibnitz's theorem to show that the coefficient of \(( x - \pi ) ^ { 8 }\) in the Taylor series expansion of \(\mathrm { f } ( x )\) about \(\pi\) is $$\frac { a \pi + b \pi ^ { 3 } } { 315 }$$ where \(a\) and \(b\) are integers to be determined. The Taylor series expansion of \(\mathrm { f } ( \mathrm { x } )\) about \(\mathrm { x } = \mathrm { k }\) is given by $$f ( x ) = f ( k ) + ( x - k ) f ^ { \prime } ( k ) + \frac { ( x - k ) ^ { 2 } } { 2 ! } f ^ { \prime \prime } ( k ) + \ldots + \frac { ( x - k ) ^ { r } } { r ! } f ^ { ( r ) } ( k ) + \ldots$$

1. The Taylor series expansion of \(f ( x )\) about \(x = a\) is given by

$$f ( x ) = f ( a ) + ( x - a ) f ^ { \prime } ( a ) + \frac { ( x - a ) ^ { 2 } } { 2 ! } f ^ { \prime \prime } ( a ) + \ldots + \frac { ( x - a ) ^ { r } } { r ! } f ^ { ( r ) } ( a ) + \ldots$$ Given that $$y = ( 1 + \ln x ) ^ { 2 } \quad x > 0$$

1. show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 2 \ln x } { x ^ { 2 } }\)
2. Hence find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\)
3. Determine the Taylor series expansion about \(x = 1\) of $$( 1 + \ln x ) ^ { 2 }$$ in ascending powers of ( \(x - 1\) ), up to and including the term in \(( x - 1 ) ^ { 3 }\) Give each coefficient in simplest form.
4. Use this series expansion to evaluate $$\lim _ { x \rightarrow 1 } \frac { 2 x - 1 - ( 1 + \ln x ) ^ { 2 } } { ( x - 1 ) ^ { 3 } }$$ explaining your reasoning clearly.

1. The Taylor series expansion of \(f ( x )\) about \(x = a\) is given by

$$f ( x ) = f ( a ) + ( x - a ) f ^ { \prime } ( a ) + \frac { ( x - a ) ^ { 2 } } { 2 ! } f ^ { \prime \prime } ( a ) + \ldots + \frac { ( x - a ) ^ { r } } { r ! } f ^ { ( r ) } ( a ) + \ldots$$

1. (a) Use differentiation to determine the Taylor series expansion of \(\ln x\), in ascending powers of ( \(x - 1\) ), up to and including the term in \(( x - 1 ) ^ { 2 }\) (b) Hence prove that $$\lim _ { x \rightarrow 1 } \left( \frac { \ln x } { x - 1 } \right) = 1$$
2. Use L'Hospital's rule to determine $$\lim _ { x \rightarrow 0 } \left( \frac { 1 } { ( x + 3 ) \tan ( 6 x ) \operatorname { cosec } ( 2 x ) } \right)$$ (Solutions relying entirely on calculator technology are not acceptable.)

1. Use L'Hospital's rule to show that

$$\lim _ { x \rightarrow 0 } \left( \frac { 1 } { \sin x } - \frac { 1 } { x } \right) = 0$$ (6)

8 You are given that \(\mathrm { f } ( x ) = ( 1 - a \sin x ) \mathrm { e } ^ { b x }\) where \(a\) and \(b\) are positive constants. The first three terms in the Maclaurin expansion of \(\mathrm { f } ( x )\) are \(1 + 2 x + \frac { 3 } { 2 } x ^ { 2 }\).

1. Find the value of \(a\) and the value of \(b\).
2. Explain if there is any restriction on the value of \(x\) in order for the expansion to be valid.

10

1. Using the Maclaurin series for \(\ln ( 1 + x )\), find the first four terms in the series expansion for \(\ln \left( 1 + 3 x ^ { 2 } \right)\).
2. Find the range of \(x\) for which the expansion is valid.
3. Find the exact value of the series $$\frac { 3 ^ { 1 } } { 2 \times 2 ^ { 2 } } - \frac { 3 ^ { 2 } } { 3 \times 2 ^ { 4 } } + \frac { 3 ^ { 3 } } { 4 \times 2 ^ { 6 } } - \frac { 3 ^ { 4 } } { 5 \times 2 ^ { 8 } } + \ldots .$$

3 You are given that \(\mathrm { f } ( x ) = \ln ( 2 + x )\).

1. Determine the exact value of \(\mathrm { f } ^ { \prime } ( 0 )\).
2. Show that \(\mathrm { f } ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 4 }\).
3. Hence write down the first three terms of the Maclaurin series for \(\mathrm { f } ( x )\).

7

1. Use the Maclaurin series for \(\sin x\) to work out the series expansion of \(\sin x \sin 2 x \sin 4 x\) up to and including the term in \(x ^ { 3 }\).
2. Hence find, in exact surd form, an approximation to the least positive root of the equation \(2 \sin x \sin 2 x \sin 4 x = x\).

1

1. Write down and simplify the first three non-zero terms of the Maclaurin series for \(\ln ( 1 + 3 x )\).
2. Hence find the first three non-zero terms of the Maclaurin series for $$\mathrm { e } ^ { x } \ln ( 1 + 3 x )$$ simplifying the coefficients.

4

1. Use the series expansion $$\ln ( 1 + x ) = x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } - \frac { 1 } { 4 } x ^ { 4 } + \ldots$$ to write down the first four terms in the expansion, in ascending powers of \(x\), of \(\ln ( 1 - x )\).
2. The function f is defined by $$\mathrm { f } ( x ) = \mathrm { e } ^ { \sin x }$$ Use Maclaurin's theorem to show that when \(\mathrm { f } ( x )\) is expanded in ascending powers of \(x\) :
   1. the first three terms are $$1 + x + \frac { 1 } { 2 } x ^ { 2 }$$
   2. the coefficient of \(x ^ { 3 }\) is zero.
3. Find $$\lim _ { x \rightarrow 0 } \frac { \mathrm { e } ^ { \sin x } - 1 + \ln ( 1 - x ) } { x ^ { 2 } \sin x }$$ (4 marks)

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

1. 1. Find f'''(x).
   2. Using Maclaurin's theorem, show that, for small values of \(x\), $$\mathrm { f } ( x ) \approx 1 + x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 } x ^ { 3 }$$
2. Use the expansion of \(\mathrm { e } ^ { x }\) together with the result in part (a)(ii) to show that, for small values of \(x\), $$\mathrm { e } ^ { x } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } \approx 1 + 2 x + x ^ { 2 } + k x ^ { 3 }$$ where \(k\) is a rational number to be found.
3. Write down the first four terms in the expansion, in ascending powers of \(x\), of \(\mathrm { e } ^ { 2 x }\).
4. Find $$\lim _ { x \rightarrow 0 } \frac { \mathrm { e } ^ { x } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } - \mathrm { e } ^ { 2 x } } { 1 - \cos x }$$ (4 marks)