Taylor series

$$\frac{d^2 y}{dx^2} + 4y - \sin x = 0$$ Given that \(y = \frac{1}{2}\) and \(\frac{dy}{dx} = \frac{1}{8}\) at \(x = 0\), find a series expansion for \(y\) in terms of \(x\), up to and including the term in \(x^5\). [5]

$$\frac{d^2 y}{dx^2} + 4y - \sin x = 0$$ Given that \(y = \frac{1}{2}\) and \(\frac{dy}{dx} = \frac{1}{8}\) at \(x = 0\), find a series expansion for \(y\) in terms of \(x\), up to and including the term in \(x^5\). [5]

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

1. Show that $$\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = A y \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } + B \frac { \mathrm {~d} y } { \mathrm {~d} x } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$$ where \(A\) and \(B\) are integers to be determined. Given that \(y = 1\) at \(x = - 1\)
2. determine the Taylor series solution for \(y\), in ascending powers of \(( x + 1 )\) up to and including the term in \(( x + 1 ) ^ { 4 }\), giving each coefficient in simplest form.

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

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

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

$$(x^2 + 1)\frac{d^2y}{dx^2} = 2y^2 + (1 - 2x)\frac{dy}{dx}$$ (I)

1. By differentiating equation (I) with respect to \(x\), show that

Given that $$y \frac{d^3 y}{dx^3} + \left(\frac{dy}{dx}\right)^2 + 5y = 0$$

1. find \(\frac{d^3 y}{dx^3}\) in terms of \(\frac{d^2 y}{dx^2}\), \(\frac{dy}{dx}\) and \(y\). [4]

Given that \(y = 2\) and \(\frac{dy}{dx} = 2\) at \(x = 0\),

1. find a series solution for \(y\) in ascending powers of \(x\), up to and including the term in \(x^3\). [5]

Given that \(f(x) = \ln(\cos 3x)\), find \(f'(0)\) and \(f''(0)\). Hence show that the first term in the Maclaurin series for \(f(x)\) is \(ax^2\), where the value of \(a\) is to be found. [4]

4. (a) (i) Given that \(f ( x ) = \sqrt { 1 + 2 x }\), find \(f ^ { \prime } ( x )\) and \(f ^ { \prime \prime } ( x )\).
(ii) Hence, find the first three terms of the Maclaurin series for \(\sqrt { 1 + 2 x }\).
(b) Hence, using a suitable value for \(x\), show that \(\sqrt { 5 } \approx \frac { 143 } { 64 }\).
[0pt]

\(y = \sec^2 x\)

1. Show that \(\frac{d^2 y}{dx^2} = 6 \sec^4 x - 4 \sec^2 x\). [4]
2. Find a Taylor series expansion of \(\sec^2 x\) in ascending powers of \(\left(x - \frac{\pi}{4}\right)\), up to and including the term in \(\left(x - \frac{\pi}{4}\right)^3\). [6]

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.

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. $$y = \sqrt { 8 + \mathrm { e } ^ { x } } , \quad x \in \mathbb { R }$$ 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.

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 \(x ^ { 3 }\).

The function \(y\) satisfies \(\frac{d^2y}{dx^2} + x^2y = x\), and is such that \(y = 1\) and \(\frac{dy}{dx} = 1\) when \(x = 1\).

1. Using the given differential equation
   1. state the value of \(\frac{d^2y}{dx^2}\) when \(x = 1\), [1]
   2. find, by differentiation, the value of \(\frac{d^3y}{dx^3}\) when \(x = 1\). [2]
2. Hence determine the Taylor series for \(y\) about \(x = 1\) up to and including the term in \((x-1)^3\) and deduce, correct to 4 decimal places, an approximation for \(y\) when \(x = 1.1\). [3]

2 Find the Maclaurin's series for \(\ln \cosh x\) up to and including the term in \(x ^ { 4 }\).

15 The expression given in line 34 is used to calculate \(\sum _ { r = 1 } ^ { 6 } \frac { 1 } { r }\).
Show that the error in the result is less than \(1.5 \%\) of the true value.

Which of the following functions has the fourth term \(-\frac{1}{720}x^6\) in its Maclaurin series expansion? Circle your answer. [1 mark] \(\sin x\) \(\qquad\) \(\cos x\) \(\qquad\) \(e^x\) \(\qquad\) \(\ln(1 + x)\)

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.

2 It is given that \(y = 2 ^ { x }\).

1. By differentiating \(\ln y\) with respect to \(x\), show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = 2 ^ { \mathrm { x } } \ln 2\).
2. Write down \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\).
3. Hence find the first three terms in the Maclaurin's series for \(2 ^ { X }\).

It is given that \(f(x) = \tan^{-1}(1 + x)\).

1. Find \(f(0)\) and \(f'(0)\), and show that \(f''(0) = -\frac{1}{2}\). [4]
2. Hence find the Maclaurin series for \(f(x)\) up to and including the term in \(x^2\). [2]

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.]

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

4. $$f ( x ) = \sin \left( \frac { 3 } { 2 } x \right)$$

1. Find the Taylor series expansion for \(\mathrm { f } ( x )\) about \(\frac { \pi } { 3 }\) in ascending powers of \(\left( x - \frac { \pi } { 3 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 3 } \right) ^ { 4 }\)
2. Hence obtain an estimate of \(\sin \frac { 1 } { 2 }\), giving your answer to 4 decimal places.

Given that $$y = \cos x \sinh x \quad x \in \mathbb{R}$$

1. show that $$\frac{d^4y}{dx^4} = ky$$ where \(k\) is a constant to be determined. [5]
2. Hence determine the first three non-zero terms of the Maclaurin series for \(y\), giving each coefficient in simplest form. [3]

1 Find the Maclaurin's series for \(\tan \left( x + \frac { 1 } { 4 } \pi \right)\) up to and including the term in \(x ^ { 2 }\).

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 $$\lim _ { x \rightarrow 0 } \left[ \frac { 3 x \cos 2 x - \sin 3 x } { 5 x ^ { 3 } } \right]$$

Use l'Hôpital's rule to show that $$\lim_{x \to \infty} (xe^{-x}) = 0$$ Fully justify your answer. [4 marks]

The Taylor series expansion, about \(x = 1\), of the function \(y\) is $$y = 1 + \sum_{n=1}^{\infty} \frac{(-2)^{n-1}(x-1)^n}{1 \times 3 \times 5 \times \ldots \times (2n-1)}.$$ Find the value of \(\frac{\text{d}^4 y}{\text{d}x^4}\) when \(x = 1\). [3]

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)

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 \(x ^ { 3 }\).

8

1. The function f is defined as \(\mathrm { f } ( x ) = \sec x\) 8
   1. (i) Show that \(\mathrm { f } ^ { ( 4 ) } ( 0 ) = 5\) 8
   2. (ii) Hence find the first three non-zero terms of the Maclaurin series for \(\mathrm { f } ( x ) = \sec x\) 8
   3. Prove that $$\lim _ { x \rightarrow 0 } \left( \frac { \sec x - \cosh x } { x ^ { 4 } } \right) = \frac { 1 } { 6 }$$

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 \(\lim _ { x \rightarrow 0 } \left( \frac { x \tan 2 x } { \sec x - 1 } \right)\).

Which one of the expressions below is not equal to zero? Circle your answer. [1 mark] \(\lim_{x \to \infty} (x^2e^{-x})\) \quad \(\lim_{x \to 0} (x^5 \ln x)\) \quad \(\lim_{x \to \infty} \left(\frac{e^x}{x^5}\right)\) \quad \(\lim_{x \to 0^+} (x^3e^x)\)

2 It is given that \(\mathrm { f } ( x ) = \tanh ^ { - 1 } x\).

1. Show that \(\mathrm { f } ^ { \prime \prime \prime } ( x ) = \frac { 2 \left( 1 + 3 x ^ { 2 } \right) } { \left( 1 - x ^ { 2 } \right) ^ { 3 } }\).
2. Hence find the Maclaurin series for \(\mathrm { f } ( x )\), up to and including the term in \(x ^ { 3 }\).

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.