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

5

1. Prove that, if \(y = \sin ^ { - 1 } x\), then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\).
2. Find the Maclaurin series for \(\sin ^ { - 1 } x\), up to and including the term in \(x ^ { 3 }\).
3. Use the result of part (ii) and the Maclaurin series for \(\ln ( 1 + x )\) to find the Maclaurin series for \(\left( \sin ^ { - 1 } x \right) \ln ( 1 + x )\), up to and including the term in \(x ^ { 4 }\).

3

1. Find the first three terms of the Maclaurin series for \(\ln ( 2 + x )\).
2. Write down the first three terms of the series for \(\ln ( 2 - x )\), and hence show that, if \(x\) is small, then $$\ln \left( \frac { 2 + x } { 2 - x } \right) \approx x$$

1

1. A curve has polar equation \(r = a \mathrm { e } ^ { - k \theta }\) for \(0 \leqslant \theta \leqslant \pi\), where \(a\) and \(k\) are positive constants. The points A and B on the curve correspond to \(\theta = 0\) and \(\theta = \pi\) respectively.
   1. Sketch the curve.
   2. Find the area of the region enclosed by the curve and the line AB .
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { 3 + 4 x ^ { 2 } } \mathrm {~d} x\).
3. 1. Find the Maclaurin series for \(\tan x\), up to the term in \(x ^ { 3 }\).
   2. Use this Maclaurin series to show that, when \(h\) is small, \(\int _ { h } ^ { 4 h } \frac { \tan x } { x } \mathrm {~d} x \approx 3 h + 7 h ^ { 3 }\).

1

1. Fig. 1 shows the curve with polar equation \(r = a ( 1 - \cos 2 \theta )\) for \(0 \leqslant \theta \leqslant \pi\), where \(a\) is a positive constant. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{43b4c7ed-3556-4d87-8aef-0111fe747982-2_529_620_577_799} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Find the area of the region enclosed by the curve.
2. 1. Given that \(\mathrm { f } ( x ) = \arctan ( \sqrt { 3 } + x )\), find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ^ { \prime \prime } ( x )\).
   2. Hence find the Maclaurin series for \(\arctan ( \sqrt { 3 } + x )\), as far as the term in \(x ^ { 2 }\).
   3. Hence show that, if \(h\) is small, \(\int _ { - h } ^ { h } x \arctan ( \sqrt { 3 } + x ) \mathrm { d } x \approx \frac { 1 } { 6 } h ^ { 3 }\).

2

1. Use de Moivre's theorem to find the constants \(a , b , c\) in the identity $$\cos 5 \theta \equiv a \cos ^ { 5 } \theta + b \cos ^ { 3 } \theta + c \cos \theta$$
2. Let $$\begin{aligned} C & = \cos \theta + \cos \left( \theta + \frac { 2 \pi } { n } \right) + \cos \left( \theta + \frac { 4 \pi } { n } \right) + \ldots + \cos \left( \theta + \frac { ( 2 n - 2 ) \pi } { n } \right) \\ \text { and } S & = \sin \theta + \sin \left( \theta + \frac { 2 \pi } { n } \right) + \sin \left( \theta + \frac { 4 \pi } { n } \right) + \ldots + \sin \left( \theta + \frac { ( 2 n - 2 ) \pi } { n } \right) \end{aligned}$$ where \(n\) is an integer greater than 1 .
   By considering \(C + \mathrm { j } S\), show that \(C = 0\) and \(S = 0\).
3. Write down the Maclaurin series for \(\mathrm { e } ^ { t }\) as far as the term in \(t ^ { 2 }\). Hence show that, for \(t\) close to zero, $$\frac { t } { \mathrm { e } ^ { t } - 1 } \approx 1 - \frac { 1 } { 2 } t$$

1

1. A curve has polar equation \(r = 1 + \cos \theta\) for \(0 \leqslant \theta < 2 \pi\).
   1. Sketch the curve.
   2. Find the area of the region enclosed by the curve, giving your answer in exact form.
2. Assuming that \(x ^ { 4 }\) and higher powers may be neglected, write down the Maclaurin series approximations for \(\sin x\) and \(\cos x\) (where \(x\) is in radians). Hence or otherwise obtain an approximation for \(\tan x\) in the form \(a x + b x ^ { 3 }\).
3. Find \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 1 - \frac { 1 } { 4 } X ^ { 2 } } } \mathrm {~d} x\), giving your answer in exact form.

1

1. Given that \(\mathrm { f } ( x ) = \arccos x\),
   1. sketch the graph of \(y = \mathrm { f } ( x )\),
   2. show that \(\mathrm { f } ^ { \prime } ( x ) = - \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\),
   3. obtain the Maclaurin series for \(\mathrm { f } ( x )\) as far as the term in \(x ^ { 3 }\).
2. A curve has polar equation \(r = \theta + \sin \theta , \theta \geqslant 0\).
   1. By considering \(\frac { \mathrm { d } r } { \mathrm {~d} \theta }\) show that \(r\) increases as \(\theta\) increases. Sketch the curve for \(0 \leqslant \theta \leqslant 4 \pi\).
   2. You are given that \(\sin \theta \approx \theta\) for small \(\theta\). Find in terms of \(\alpha\) the approximate area bounded by the curve and the lines \(\theta = 0\) and \(\theta = \alpha\), where \(\alpha\) is small.

7 \includegraphics[max width=\textwidth, alt={}, center]{72a1330a-c6dc-4f3a-9b0e-333b099f4509-4_782_1065_251_500} The diagram shows the curve \(y = \frac { 1 } { x }\) for \(x > 0\) and a set of \(( n - 1 )\) rectangles of unit width below the curve. These rectangles can be used to obtain an inequality of the form $$\frac { 1 } { a } + \frac { 1 } { a + 1 } + \frac { 1 } { a + 2 } + \ldots + \frac { 1 } { b } < \int _ { 1 } ^ { n } \frac { 1 } { x } \mathrm {~d} x$$ Another set of rectangles can be used similarly to obtain $$\int _ { 1 } ^ { n } \frac { 1 } { x } \mathrm {~d} x < \frac { 1 } { c } + \frac { 1 } { c + 1 } + \frac { 1 } { c + 2 } + \ldots + \frac { 1 } { d }$$

1. Write down the values of the constants \(a\) and \(c\), and express \(b\) and \(d\) in terms of \(n\). The function f is defined by \(\mathrm { f } ( n ) = 1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \ldots + \frac { 1 } { n } - \ln n\), for positive integers \(n\).
2. Use your answers to part (i) to obtain upper and lower bounds for \(\mathrm { f } ( n )\).
3. By using the first 2 terms of the Maclaurin series for \(\ln ( 1 + x )\) show that, for large \(n\), $$f ( n + 1 ) - f ( n ) \approx - \frac { n - 1 } { 2 n ^ { 2 } ( n + 1 ) } .$$

2 It is given that \(\mathrm { f } ( x ) = \ln \left( 1 + x ^ { 2 } \right)\).

1. Using the standard Maclaurin expansion for \(\ln ( 1 + x )\), write down the first four terms in the expansion of \(\mathrm { f } ( x )\), stating the set of values of \(x\) for which the expansion is valid.
2. Hence find the exact value of $$1 - \frac { 1 } { 2 } \left( \frac { 1 } { 2 } \right) ^ { 2 } + \frac { 1 } { 3 } \left( \frac { 1 } { 2 } \right) ^ { 4 } - \frac { 1 } { 4 } \left( \frac { 1 } { 2 } \right) ^ { 6 } + \ldots .$$

2 It is given that \(\mathrm { f } ( x ) = \ln ( 1 + \sin x )\). Using standard series, find the Maclaurin series for \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 3 }\).

7

1. 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.
2. 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 $$- \frac { x ^ { 2 } } { 2 } - \frac { x ^ { 4 } } { 12 }$$
3. Find $$\lim _ { x \rightarrow 0 } \left[ \frac { x \ln ( 1 + 2 x ) } { x ^ { 2 } - \ln \cos x } \right]$$

6 The function f is defined by \(\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } ( 1 + 3 x ) ^ { - \frac { 2 } { 3 } }\).

1. 1. Use the series expansion for \(\mathrm { e } ^ { x }\) to write down the first four terms in the series expansion of \(\mathrm { e } ^ { 2 x }\).
   2. Use the binomial series expansion of \(( 1 + 3 x ) ^ { - \frac { 2 } { 3 } }\) and your answer to part (a)(i) to show that the first three non-zero terms in the series expansion of \(\mathrm { f } ( x )\) are \(1 + 3 x ^ { 2 } - 6 x ^ { 3 }\).
2. 1. Given that \(y = \ln ( 1 + 2 \sin x )\), find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   2. By using Maclaurin's theorem, show that, for small values of \(x\), $$\ln ( 1 + 2 \sin x ) \approx 2 x - 2 x ^ { 2 }$$
3. Find $$\lim _ { x \rightarrow 0 } \frac { 1 - \mathrm { f } ( x ) } { x \ln ( 1 + 2 \sin x ) }$$

2

1. Given that \(y = \ln ( 4 + 3 x )\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Hence, by using Maclaurin's theorem, find the first three terms in the expansion, in ascending powers of \(x\), of \(\ln ( 4 + 3 x )\).
3. Write down the first three terms in the expansion, in ascending powers of \(x\), of \(\ln ( 4 - 3 x )\).
4. Show that, for small values of \(x\), $$\ln \left( \frac { 4 + 3 x } { 4 - 3 x } \right) \approx \frac { 3 } { 2 } x$$

4

1. Write down the expansion of \(\sin 3 x\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\).
2. Find $$\lim _ { x \rightarrow 0 } \left[ \frac { 3 x \cos 2 x - \sin 3 x } { 5 x ^ { 3 } } \right]$$

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 $$\lim _ { x \rightarrow 0 } \left[ \frac { \mathrm { e } ^ { \tan x } - ( \cos x + \sin x ) } { x \ln ( 1 + 3 x ) } \right]$$

2 Find $$\lim _ { x \rightarrow 0 } \left[ \frac { \sqrt { 4 + x } - 2 } { x + x ^ { 2 } } \right]$$ (3 marks)

6

1. Given that \(y = \ln \cos 2 x\), find \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\).
2. 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 \(- 2 x ^ { 2 } - \frac { 4 } { 3 } x ^ { 4 }\).
3. Hence find the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln \sec ^ { 2 } 2 x\).

2

1. Write down the expansion of \(\mathrm { e } ^ { 3 x }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\).
2. Hence, or otherwise, find the term in \(x ^ { 2 }\) in the expansion, in ascending powers of \(x\), of \(\mathrm { e } ^ { 3 x } ( 1 + 2 x ) ^ { - \frac { 3 } { 2 } }\).
   (4 marks)

6

1. 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 \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\).
2. 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)
3. 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 \(x ^ { 2 }\).
4. 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).

7

1. 1. Write down the first three terms of the binomial expansion of \(( 1 + y ) ^ { - 1 }\), in ascending powers of \(y\).
   2. By using the expansion $$\cos x = 1 - \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 4 } } { 4 ! } - \ldots$$ and your answer to part (a)(i), or otherwise, show that the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(\sec x\) are $$1 + \frac { x ^ { 2 } } { 2 } + \frac { 5 x ^ { 4 } } { 24 }$$
2. By using Maclaurin's theorem, or otherwise, show that the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\tan x\) are $$x + \frac { x ^ { 3 } } { 3 }$$
3. Hence find \(\lim _ { x \rightarrow 0 } \left( \frac { x \tan 2 x } { \sec x - 1 } \right)\).

7

1. Write down the expansion of \(\sin 2 x\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\).
2. 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 \(x\), $$\sqrt { 3 + \mathrm { e } ^ { x } } \approx 2 + \frac { 1 } { 4 } x + \frac { 7 } { 64 } x ^ { 2 }$$
3. Find $$\lim _ { x \rightarrow 0 } \left[ \frac { \sqrt { 3 + \mathrm { e } ^ { x } } - 2 } { \sin 2 x } \right]$$

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

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

5

1. 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.
2. 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 $$- x - x ^ { 2 } - x ^ { 3 }$$
3. Find $$\lim _ { x \rightarrow 0 } \left[ \frac { x \ln \left( 2 - \mathrm { e } ^ { x } \right) } { 1 - \cos 4 x } \right]$$

5

1. 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.)
2. 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)
3. Find $$\lim _ { x \rightarrow 0 } \left[ \frac { \ln ( 1 + 2 \tan x ) } { \ln ( 1 - x ) } \right]$$ (4 marks)

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. Show that, for some value of \(k\), $$\lim _ { x \rightarrow 0 } \left[ \frac { 2 x - \sin 2 x } { x ^ { 2 } \ln ( 1 + k x ) } \right] = 16$$ and state this value of \(k\).