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

9 Let \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 7 x - 6 } { ( x - 1 ) ( x - 2 ) ( x + 1 ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Show that, when \(x\) is sufficiently small for \(x ^ { 4 }\) and higher powers to be neglected, $$f ( x ) = - 3 + 2 x - \frac { 3 } { 2 } x ^ { 2 } + \frac { 11 } { 4 } x ^ { 3 } .$$

6 The integral \(\mathrm { I } _ { \mathrm { n } }\), where \(n\) is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).

1. Find the exact value of \(I _ { 1 }\).
2. By considering \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \mathrm { x } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \right)\), or otherwise, show that $$\mathrm { nl } _ { \mathrm { n } + 2 } = 2 ^ { \mathrm { n } - 1 } 3 ^ { - \frac { 1 } { 2 } \mathrm { n } } + ( \mathrm { n } - 1 ) \mathrm { I } _ { \mathrm { n } } .$$
3. Find the exact value of \(I _ { 5 }\) giving the answer in the form \(k \sqrt { 3 }\), where \(k\) is a rational number to be determined. \includegraphics[max width=\textwidth, alt={}, center]{20e14db3-0eb0-4954-91cf-027e16f8bf14-11_78_1576_336_321}

6 The integral \(\mathrm { I } _ { \mathrm { n } }\), where \(n\) is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).

1. Find the exact value of \(I _ { 1 }\).
2. By considering \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \mathrm { x } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \right)\), or otherwise, show that $$\mathrm { nl } _ { \mathrm { n } + 2 } = 2 ^ { \mathrm { n } - 1 } 3 ^ { - \frac { 1 } { 2 } \mathrm { n } } + ( \mathrm { n } - 1 ) \mathrm { I } _ { \mathrm { n } } .$$
3. Find the exact value of \(I _ { 5 }\) giving the answer in the form \(k \sqrt { 3 }\), where \(k\) is a rational number to be determined. \includegraphics[max width=\textwidth, alt={}, center]{1de67949-6262-4ade-b986-02b6563ae404-11_78_1576_336_321}

2

1. Find the coefficient of \(x ^ { 2 }\) in the Maclaurin's series for \(- \ln \cos x\).
2. Find the length of the arc of the curve with equation \(\mathrm { y } = - \operatorname { Incos } \mathrm { x }\) from the point where \(x = 0\) to the point where \(x = \frac { 1 } { 4 } \pi\).

1

1. Find the Maclaurin series for \(\sin ^ { - 1 } x\) up to and including the term in \(x ^ { 3 }\).
2. Deduce an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 5 } } \frac { 1 } { \sqrt { 1 - u ^ { 2 } } } \mathrm {~d} u\), giving your answer as a fraction.

2 Find the Maclaurin's series for \(\mathrm { e } ^ { 1 + x ^ { 2 } } + \mathrm { e } ^ { 1 - x }\) up to and including the term in \(x ^ { 2 }\).

2 The curve \(C\) has parametric equations $$x = \cosh t , \quad y = \sinh t , \quad \text { for } 0 < t \leqslant \frac { 3 } { 5 }$$ The length of \(C\) is denoted by \(s\).

1. Show that \(s = \int _ { 0 } ^ { \frac { 3 } { 5 } } \sqrt { \cosh 2 t } \mathrm {~d} t\). \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-04_2714_37_143_2008}
2. By finding the Maclaurin's series for \(\sqrt { \cosh 2 t }\) up to and including the term in \(t ^ { 2 }\) ,deduce an approximation to \(s\) .

1 Find the Maclaurin's series for \(\ln \left( 1 + \mathrm { e } ^ { x } \right)\) up to and including the term in \(x ^ { 2 }\).

3

1. A curve has equation \(\mathrm { y } = \mathrm { e } ^ { \mathrm { x } } + \frac { 1 } { 4 } \mathrm { e } ^ { - \mathrm { x } }\), for \(0 \leqslant x \leqslant 1\). Find, in terms of \(\pi\) and e , the area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. Using standard results from the list of formulae (MF19), or otherwise, find the Maclaurin's series for \(\mathrm { e } ^ { x } + \frac { 1 } { 4 } \mathrm { e } ^ { - x }\) up to and including the term in \(x ^ { 2 }\).

3 Find the first three terms in the Maclaurin's series for \(\tanh ^ { - 1 } \left( \frac { 1 } { 2 } e ^ { x } \right)\) in the form \(\frac { 1 } { 2 } \ln a + b x + c x ^ { 2 }\), giving the exact values of the constants \(a , b\) and \(c\).

1 Find the Maclaurin's series for \(\ln ( x + 2 ) + \ln \left( x ^ { 2 } + 5 \right)\) up to and including the term in \(x ^ { 2 }\).

11. (b) Hence find the Maclaurin series expansion of \(\mathrm { e } ^ { x } \cos x\), in ascending powers of \(x\), up to and including the term in \(x ^ { 4 }\).
(Total 11 marks)

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.

6. (a) Find the Taylor series expansion about \(\frac { \pi } { 4 }\) of \(\tan x\) in ascending powers of \(\left( x - \frac { \pi } { 4 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 4 } \right) ^ { 3 }\).
(b) Deduce that an approximation for \(\tan \frac { 5 \pi } { 12 }\) is \(1 + \frac { \pi } { 3 } + \frac { \pi ^ { 2 } } { 18 } + \frac { \pi ^ { 3 } } { 81 }\)

4. $$y = \ln \left( \frac { 1 } { 1 - 2 x } \right) , \quad | x | < \frac { 1 } { 2 }$$

1. 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. Hence, or otherwise, find the series expansion of \(\ln \left( \frac { 1 } { 1 - 2 x } \right)\) about \(x = 0\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient in its simplest form.
3. Use your expansion to find an approximate value for \(\ln \left( \frac { 3 } { 2 } \right)\), giving your answer
   to 3 decimal places.

3. (a) Given that \(y = \ln ( 1 + 5 x ) , | x | < 0.2\), find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\).
(b) Hence obtain the M aclaurin series for \(\ln ( 1 + 5 x ) , | x | < 0.2\), up to and including the term in \(x ^ { 3 }\).

1

1. A curve has polar equation \(r = a ( \sqrt { 2 } + 2 \cos \theta )\) for \(- \frac { 3 } { 4 } \pi \leqslant \theta \leqslant \frac { 3 } { 4 } \pi\), where \(a\) is a positive constant.
   1. Sketch the curve.
   2. Find, in an exact form, the area of the region enclosed by the curve.
2. 1. Find the Maclaurin series for the function \(\mathrm { f } ( x ) = \tan \left( \frac { 1 } { 4 } \pi + x \right)\), up to the term in \(x ^ { 2 }\).
   2. Use the Maclaurin series to show that, when \(h\) is small, $$\int _ { - h } ^ { h } x ^ { 2 } \tan \left( \frac { 1 } { 4 } \pi + x \right) \mathrm { d } x \approx \frac { 2 } { 3 } h ^ { 3 } + \frac { 4 } { 5 } h ^ { 5 }$$

1

1. A curve has polar equation \(r = a ( 1 - \cos \theta )\), where \(a\) is a positive constant.
   1. Sketch the curve.
   2. Find the area of the region enclosed by the section of the curve for which \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\) and the line \(\theta = \frac { 1 } { 2 } \pi\).
2. Use a trigonometric substitution to show that \(\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { 1 } { 4 \sqrt { 3 } }\).
3. In this part of the question, \(\mathrm { f } ( x ) = \arccos ( 2 x )\).
   1. Find \(\mathrm { f } ^ { \prime } ( x )\).
   2. Use a standard series to expand \(\mathrm { f } ^ { \prime } ( x )\), and hence find the series for \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to the term in \(x ^ { 5 }\).

1

1. A curve has cartesian equation \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 3 x y ^ { 2 }\).
   1. Show that the polar equation of the curve is \(r = 3 \cos \theta \sin ^ { 2 } \theta\).
   2. Hence sketch the curve.
2. Find the exact value of \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - 3 x ^ { 2 } } } \mathrm {~d} x\).
3. 1. Write down the series for \(\ln ( 1 + x )\) and the series for \(\ln ( 1 - x )\), both as far as the term in \(x ^ { 5 }\).
   2. Hence find the first three non-zero terms in the series for \(\ln \left( \frac { 1 + x } { 1 - x } \right)\).
   3. Use the series in part (ii) to show that \(\sum _ { r = 0 } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) 4 ^ { r } } = \ln 3\).

9 \end{array} \right) + t \left( \begin{array} { c } 2
- 3
1 \end{array} \right)$$ (ii) Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect.
(iii) Find the position vector of the point \(C\) on \(l _ { 2 }\) such that \(\angle A B C = 90 ^ { \circ }\).
8. \(f ( x ) = \frac { 5 - 8 x } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } }\).
(i) Express \(\mathrm { f } ( x )\) in partial fractions.
(ii) Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
(iii) State the set of values of \(x\) for which your expansion is valid.

1 It is given that \(\mathrm { f } ( x ) = \ln ( 3 + x )\).

1. Find the exact values of \(f ( 0 )\) and \(f ^ { \prime } ( 0 )\), and show that \(f ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 9 }\).
2. Hence write down the first three terms of the Maclaurin series for \(\mathrm { f } ( x )\), given that \(- 3 < x \leqslant 3\).

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

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

1 Find the first three non-zero terms of the Maclaurin series for $$( 1 + x ) \sin x$$ simplifying the coefficients.

2

1. Given that \(\mathrm { f } ( x ) = \sin \left( 2 x + \frac { 1 } { 4 } \pi \right)\), show that \(\mathrm { f } ( x ) = \frac { 1 } { 2 } \sqrt { 2 } ( \sin 2 x + \cos 2 x )\).
2. Hence find the first four terms of the Maclaurin series for \(\mathrm { f } ( x )\). [You may use appropriate results given in the List of Formulae.]