4.08a Maclaurin series: find series for function

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.

5. $$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 x \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y ^ { 2 } = 0$$ Given that at \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\)

1. show that, at \(x = 0 , \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \frac { 3 } { 2 }\)
2. Find a series solution for \(y\) up to and including the term in \(x ^ { 3 }\)

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 \(x ^ { 3 }\).

4. Use the Taylor Series method to find the series solution, ascending up to and including the term in \(x ^ { 3 }\), of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y \frac { \mathrm {~d} y } { \mathrm {~d} x } + y ^ { 2 } = 3 x + 4$$ given that \(\frac { \mathrm { dy } } { \mathrm { dx } } = y = 1\) at \(x = 0\).
(Total 8 marks)

4. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y \frac { \mathrm {~d} y } { \mathrm {~d} x } = x , \quad y = 0 , \frac { \mathrm {~d} y } { \mathrm {~d} x } = 2 \text { at } x = 1$$ Find a series solution of the differential equation in ascending powers of ( \(x - 1\) ) up to and including the term in \(( x - 1 ) ^ { 3 }\).

1. Given that

$$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 4 } } \cos ^ { n } \theta \mathrm {~d} \theta , \quad n \geqslant 0$$

1. prove that, for \(n \geqslant 2\), $$n I _ { n } = \left( \frac { 1 } { \sqrt { 2 } } \right) ^ { n } + ( n - 1 ) I _ { n - 2 }$$
2. Hence find the exact value of \(I _ { 5 }\), showing each step of your working.

4. $$I _ { n } = \int \cosh ^ { n } x \mathrm {~d} x , \quad n \geqslant 0$$

1. Show that, for \(n \geqslant 2\) $$n I _ { n } = \sinh x \cosh ^ { n - 1 } x + ( n - 1 ) I _ { n - 2 }$$
2. Hence find the exact value of $$\int _ { 0 } ^ { \ln 2 } \cosh ^ { 5 } x \mathrm {~d} x$$

5. Given that \(y = \tan ^ { 2 } x\)

1. show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = 8 \tan x \sec ^ { 2 } x \left( p \sec ^ { 2 } x + q \right)$$ where \(p\) and \(q\) are integers to be determined.
2. Hence determine the Taylor series expansion about \(\frac { \pi } { 3 }\) of \(\tan ^ { 2 } x\) in ascending powers of \(\left( x - \frac { \pi } { 3 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 3 } \right) ^ { 3 }\), giving each coefficient in simplest form.
   
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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 \(\sum _ { r = 0 } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) 4 ^ { r } } = \ln 3\).

1

1. 1. Given that \(\mathrm { f } ( t ) = \arcsin t\), write down an expression for \(\mathrm { f } ^ { \prime } ( t )\) and show that $$\mathrm { f } ^ { \prime \prime } ( t ) = \frac { t } { \left( 1 - t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$$
   2. Show that the Maclaurin expansion of the function \(\arcsin \left( x + \frac { 1 } { 2 } \right)\) begins $$\frac { \pi } { 6 } + \frac { 2 } { \sqrt { 3 } } x$$ and find the term in \(x ^ { 2 }\).
2. Sketch the curve with polar equation \(r = \frac { \pi a } { \pi + \theta }\), where \(a > 0\), for \(0 \leqslant \theta < 2 \pi\). Find, in terms of \(a\), the area of the region bounded by the part of the curve for which \(0 \leqslant \theta \leqslant \pi\) and the lines \(\theta = 0\) and \(\theta = \pi\).
3. Find the exact value of the integral $$\int _ { 0 } ^ { \frac { 3 } { 2 } } \frac { 1 } { 9 + 4 x ^ { 2 } } \mathrm {~d} x$$

7

1. Prove that if \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), then \(z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta\).
2. Express \(\cos ^ { 6 } \theta\) in terms of cosines of multiples of \(\theta\), and hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \cos ^ { 6 } \theta \mathrm {~d} \theta$$

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.]

7 It is given that \(\mathrm { f } ( x ) = \tanh ^ { - 1 } \left( \frac { 1 - x } { 2 + x } \right)\), for \(x > - \frac { 1 } { 2 }\).

1. Show that \(\mathrm { f } ^ { \prime } ( x ) = - \frac { 1 } { 1 + 2 x }\), and find \(\mathrm { f } ^ { \prime \prime } ( x )\).
2. Show that the first three terms of the Maclaurin series for \(\mathrm { f } ( x )\) can be written as \(\ln a + b x + c x ^ { 2 }\), for constants \(a , b\) and \(c\) to be found.

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 \(x ^ { 4 }\).

5 It is given that \(y = \tan ^ { - 1 } 2 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 x \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } = 0\).
2. Find the Maclaurin series for \(y\) up to and including the term in \(x ^ { 3 }\). Show all your working.
3. The result in part (ii), together with the value \(x = \frac { 1 } { 2 }\), is used to find an estimate for \(\pi\). Show that this estimate is only correct to 1 significant figure.

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 $$\ln \left( \frac { 2 + x } { 2 - x } \right) \approx x$$

3. $$2 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \frac { \mathrm { d } y } { \mathrm {~d} x } - x y = 1$$

1. Show that $$\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = \frac { 1 } { 2 } \left( a \frac { \mathrm {~d} y } { \mathrm {~d} x } + b x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + c \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } \right)$$ where \(a , b\) and \(c\) are constants to be found. Given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = 2\)
2. find a series solution for \(y\) in ascending powers of ( \(x - 2\) ), up to and including the term in \(( x - 2 ) ^ { 4 }\). Write each term in its simplest form.
3. Use the solution to part (b) to find an approximate value for \(y\) when \(x = 2.1\), giving your answer to 3 decimal places.

4 The continuous random variable \(Y\) has a uniform (rectangular) distribution on \([ a , b ]\), where \(a\) and \(b\) are constants.

1. Show that the moment generating function \(\mathrm { M } _ { Y } ( \mathrm { t } )\) of \(Y\) is \(\frac { \left( \mathrm { e } ^ { b t } - \mathrm { e } ^ { a t } \right) } { t ( b - a ) }\).
2. Use the series expansion of \(\mathrm { e } ^ { x }\) to show that the mean and variance of \(Y\) are \(\frac { 1 } { 2 } ( a + b )\) and \(\frac { 1 } { 12 } ( b - a ) ^ { 2 }\), respectively.