4.08a Maclaurin series: find series for function

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 \(q\).

3

1. It is given that \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = ( 2 x + 1 ) \ln ( x + y )$$ and $$y ( 0 ) = 2$$ Use the improved Euler formula $$y _ { r + 1 } = y _ { r } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)$$ where \(k _ { 1 } = h \mathrm { f } \left( x _ { r } , y _ { r } \right)\) and \(k _ { 2 } = h \mathrm { f } \left( x _ { r } + h , y _ { r } + k _ { 1 } \right)\) and \(h = 0.1\), to obtain an approximation to \(y ( 0.1 )\), giving your answer to three decimal places.
2. It is given that \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 x + 1 ) \ln ( x + y )$$ and \(y = 2\) when \(x = 0\).
   1. Use implicit differentiation to find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\), giving your answer in terms of \(x\) and \(y\).
   2. Hence find the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(y ( x )\). Give your answer in an exact form.
   3. Use your answer to part (b)(ii) to obtain an approximation to \(y ( 0.1 )\), giving your answer to three decimal places.
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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.

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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}

9 A function is defined by \(y = f ( t )\) where \(f ( t ) = \ln ( 1 + a t )\) and \(a\) is a constant.

1. By considering \(\frac { d y } { d t } , \frac { d ^ { 2 } y } { d t ^ { 2 } } , \frac { d ^ { 3 } y } { d t ^ { 3 } }\) and \(\frac { d ^ { 4 } y } { d t ^ { 4 } }\), make a conjecture for a general formula for \(\frac { d ^ { n } y } { d t ^ { n } }\) in terms of \(n\) and \(a\) for any integer \(n \geqslant 1\).
2. Use induction to prove the formula conjectured in part (a).
3. In the case where \(\mathrm { f } ( t ) = \ln ( 1 + 2 t )\), find the rate at which the \(6 ^ { \text {th } }\) derivative of \(\mathrm { f } ( t )\) is varying when \(t = \frac { 3 } { 2 }\).

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 \(\pi ^ { \mathrm { e } }\).

9 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = \ln ( 1 + \sinh x )\).

1. Given that \(k\) lies in the domain of this function, explain why \(k\) must be greater than \(\ln ( \sqrt { 2 } - 1 )\).
2. 1. Find \(\mathrm { f } ^ { \prime } ( x )\).
   2. Show that \(\mathrm { f } ^ { \prime \prime } ( \mathrm { x } ) = \frac { \mathrm { a } \sinh \mathrm { x } + \mathrm { b } } { ( 1 + \sinh \mathrm { x } ) ^ { 2 } }\), where \(a\) and \(b\) are integers to be determined.
3. Hence find a quadratic approximation to \(\mathrm { f } ( x )\) for small values of \(x\).
4. Find the percentage error in this approximation when \(x = 0.1\).

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 \(\sqrt { 5 } \approx \frac { 143 } { 64 }\).

3. $$f ( x ) = \arcsin x \quad - 1 \leqslant x \leqslant 1$$

1. Determine the first two non-zero terms, in ascending powers of \(x\), of the Maclaurin series for \(\mathrm { f } ( x )\), giving each coefficient in its simplest form.
2. Substitute \(x = \frac { 1 } { 2 }\) into the answer to part (a) and hence find an approximate value for \(\pi\) Give your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers to be determined.

9. $$y = \cosh ^ { n } x \quad n \geqslant 5$$

1. 1. Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = n ^ { 2 } \cosh ^ { n } x - n ( n - 1 ) \cosh ^ { n - 2 } x$$
   2. Determine an expression for \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\)
2. Hence determine the first three non-zero terms of the Maclaurin series for \(y\), giving each coefficient in simplest form.

2. $$f ( x ) = \tanh ^ { - 1 } \left( \frac { 3 - x } { 6 + x } \right) \quad | x | < \frac { 3 } { 2 }$$

1. Show that $$f ^ { \prime } ( x ) = - \frac { 1 } { 2 x + 3 }$$
2. Hence determine \(\mathrm { f } ^ { \prime \prime } ( x )\)
3. Hence show that the Maclaurin series for \(\mathrm { f } ( x )\), up to and including the term in \(x ^ { 2 }\), is $$\ln p + q x + r x ^ { 2 }$$ where \(p , q\) and \(r\) are constants to be determined.

5. $$y = \sin x \sinh x$$

1. Show that \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = - 4 y\)
2. Hence find the first three non-zero terms of the Maclaurin series for \(y\), giving each coefficient in its simplest form.
3. Find an expression for the \(n\)th non-zero term of the Maclaurin series for \(y\).

1. Given that \(k\) is a real non-zero constant and that

$$y = x ^ { 3 } \sin k x$$ use Leibnitz's theorem to show that $$\frac { \mathrm { d } ^ { 5 } y } { \mathrm {~d} x ^ { 5 } } = \left( k ^ { 2 } x ^ { 2 } + A \right) k ^ { 3 } x \cos k x + B \left( k ^ { 2 } x ^ { 2 } + C \right) k ^ { 2 } \sin k x$$ where \(A\), \(B\) and \(C\) are integers to be determined.

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

1. Show that $$\frac { \mathrm { d } ^ { 5 } y } { \mathrm {~d} x ^ { 5 } } = a y \frac { \mathrm {~d} ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } + b \frac { \mathrm {~d} y } { \mathrm {~d} x } \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } + c \left( \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \right) ^ { 2 }$$ where \(a\), \(b\) and \(c\) are integers to be determined.
2. Hence find a series solution, in ascending powers of \(x\) as far as the term in \(x ^ { 5 }\), of the differential equation (I), given that \(y = 1\) at \(x = 0\)

1. Use l'Hospital's Rule to show that

$$\lim _ { x \rightarrow \frac { \pi } { 2 } } \frac { \left( e ^ { \sin x } - \cos ( 3 x ) - e \right) } { \tan ( 2 x ) } = - \frac { 3 } { 2 }$$

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.)

4. $$\left[ \begin{array} { l } \text { The Taylor series expansion of } \mathrm { f } ( x ) \text { about } x = a \text { is given by } \\ \mathrm { f } ( x ) = \mathrm { f } ( a ) + ( x - a ) \mathrm { f } ^ { \prime } ( a ) + \frac { ( x - a ) ^ { 2 } } { 2 ! } \mathrm { f } ^ { \prime \prime } ( a ) + \ldots + \frac { ( x - a ) ^ { r } } { r ! } \mathrm { f } ^ { ( r ) } ( a ) + \ldots \end{array} \right]$$ The curve with equation \(y = \mathrm { f } ( x )\) satisfies the differential equation $$\cos x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + \sin x = 0$$ Given that \(\left( \frac { \pi } { 4 } , 1 \right)\) is a stationary point of the curve,

1. determine the nature of this stationary point, giving a reason for your answer.
2. Show that \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \sqrt { 2 } - 2\) at this stationary point.
3. Hence determine a series solution for \(y\), in ascending powers of \(\left( x - \frac { \pi } { 4 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 4 } \right) ^ { 3 }\), giving each coefficient in simplest form.

5. $$y = \mathrm { e } ^ { 3 x } \sin x$$

1. Use Leibnitz's theorem to show that $$\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = 28 \mathrm { e } ^ { 3 x } \sin x + 96 \mathrm { e } ^ { 3 x } \cos x$$
2. Hence express \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\) in the form $$\operatorname { Re } ^ { 3 \mathrm { x } } \sin ( \mathrm { x } + \alpha )$$ where \(R\) and \(\alpha\) are constants to be determined, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)

1. Given \(k\) is a constant and that

$$y = x ^ { 3 } \mathrm { e } ^ { k x }$$ use Leibnitz theorem to show that $$\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } } = k ^ { n - 3 } \mathrm { e } ^ { k x } \left( k ^ { 3 } x ^ { 3 } + 3 n k ^ { 2 } x ^ { 2 } + 3 n ( n - 1 ) k x + n ( n - 1 ) ( n - 2 ) \right)$$

4. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 0$$

1. Show that $$\frac { \mathrm { d } ^ { 5 } y } { \mathrm {~d} x ^ { 5 } } = a x \frac { \mathrm {~d} ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } + b \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$$ where \(a\) and \(b\) are integers to be found.
2. Hence find a series solution, in ascending powers of \(x\), as far as the term in \(x ^ { 5 }\), of the differential equation (I) where \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) at \(x = 0\)

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.

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 \(2 \sin x \sin 2 x \sin 4 x = x\).