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

6

1. The function f is defined by $$\mathrm { f } ( x ) = \ln \left( 1 + \mathrm { e } ^ { x } \right)$$ Use Maclaurin's theorem to show that when \(\mathrm { f } ( x )\) is expanded in ascending powers of \(x\) :
   1. the first three terms are $$\ln 2 + \frac { 1 } { 2 } x + \frac { 1 } { 8 } x ^ { 2 }$$
   2. the coefficient of \(x ^ { 3 }\) is zero.
2. Hence write down the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln \left( \frac { 1 + \mathrm { e } ^ { x } } { 2 } \right)\).
3. Use the series expansion $$\ln ( 1 + x ) = x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } - \ldots$$ to write down the first three terms in the expansion, in ascending powers of \(x\), of \(\ln \left( 1 - \frac { x } { 2 } \right)\).
4. Use your answers to parts (b) and (c) to find $$\lim _ { x \rightarrow 0 } \left[ \frac { \ln \left( \frac { 1 + \mathrm { e } ^ { x } } { 2 } \right) + \ln \left( 1 - \frac { x } { 2 } \right) } { x - \sin x } \right]$$

15

1. Use Maclaurin's series expansion for \(\ln ( 1 + x )\) to show that the first three terms of the Maclaurin's series expansion of \(\ln ( 1 + 3 x )\) are $$3 x - \frac { 9 } { 2 } x ^ { 2 } + 9 x ^ { 3 }$$ 15
2. Julia attempts to use the series expansion found in part (a) to find an approximation for \(\ln 4\) Julia's incorrect working is shown below. $$\begin{array} { r } \text { Let } 1 + 3 x = 4 \\ 3 x = 3 \\ x = 1 \end{array}$$ $$\text { So } \begin{aligned} \ln 4 & \approx 3 \times 1 - \frac { 9 } { 2 } \times 1 ^ { 2 } + 9 \times 1 ^ { 3 } \\ & \approx 3 - 4.5 + 9 \\ & \approx 7.5 \end{aligned}$$ Explain the error in Julia's working.
   15
3. Use \(x = - \frac { 1 } { 6 }\) in the series expansion found in part (a) to find an approximation for \(\ln 4\) Fully justify your answer.

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 $$\lim _ { x \rightarrow 0 } \left( \frac { \sec x - \cosh x } { x ^ { 4 } } \right) = \frac { 1 } { 6 }$$

4 In this question you must show detailed reasoning.

1. By writing \(\sin \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\) show that $$\sin ^ { 6 } \theta = \frac { 1 } { 32 } ( 10 - 15 \cos 2 \theta + 6 \cos 4 \theta - \cos 6 \theta ) .$$
2. Hence show that \(\sin \frac { 1 } { 8 } \pi = \frac { 1 } { 2 } \sqrt [ 6 ] { 20 - 14 \sqrt { 2 } }\).
   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 \(\pi \approx 3 \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }\).

6. $$y = \ln \left( \mathrm { e } ^ { 2 x } \cos 3 x \right) \quad - \frac { 1 } { 2 } < x < \frac { 1 } { 2 }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 - 3 \tan 3 x$$
2. Determine \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\)
3. Hence determine the first 3 non-zero terms in ascending powers of \(x\) of the Maclaurin series expansion of \(\ln \left( \mathrm { e } ^ { 2 x } \cos 3 x \right)\), giving each coefficient in simplest form.
4. Use the Maclaurin series expansion for \(\ln ( 1 + x )\) to write down the first 4 non-zero terms in ascending powers of \(x\) of the Maclaurin series expansion of \(\ln ( 1 + k x )\), where \(k\) is a constant.
5. Hence determine the value of \(k\) for which $$\lim _ { x \rightarrow 0 } \left( \frac { 1 } { x ^ { 2 } } \ln \frac { \mathrm { e } ^ { 2 x } \cos 3 x } { 1 + k x } \right)$$ exists.

11

1. At all points \(( x , y )\) on the curve \(C , \frac { \mathrm {~d} y } { \mathrm {~d} x } + x y = 0\).
   1. Prove by induction that, for all integers \(n \geqslant 1\), $$\frac { \mathrm { d } ^ { n + 1 } y } { \mathrm {~d} x ^ { n + 1 } } + x \frac { \mathrm {~d} ^ { n } y } { \mathrm {~d} x ^ { n } } + n \frac { \mathrm {~d} ^ { n - 1 } y } { \mathrm {~d} x ^ { n - 1 } } = 0$$ where \(\frac { \mathrm { d } ^ { 0 } y } { \mathrm {~d} x ^ { 0 } } = y\).
   2. Given that \(y = 1\) when \(x = 0\), determine the Maclaurin expansion of \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 6 }\).
   3. Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } + x y = 0\) given that \(y = 1\) when \(x = 0\).
   4. Given that \(Z \sim \mathrm {~N} ( 0,1 )\), use your answers to parts (i) and (ii) to find an approximation, to 4 decimal places, to the probability \(\mathrm { P } ( Z \leqslant 1 )\).
      [0pt] [Note that the probability density function of the standard normal distribution is \(\mathrm { f } ( z ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - \frac { 1 } { 2 } z ^ { 2 } }\).]

2 Use the standard Maclaurin series expansions given in the List of Formulae MF20 to show that $$\frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right) \equiv \tanh ^ { - 1 } x \text { for } - 1 < x < 1$$

10 A curve has parametric equations given by $$x = - \sqrt { ( 1 - t ) ^ { 3 } } \quad y = \sqrt { ( 1 + t ) ^ { 3 } } \quad \text { for } - 1 < t < 1$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 + t } { \sqrt { 1 - t ^ { 2 } } }\).
2. Write \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a series of ascending powers of \(t\) up to and including the term in \(t ^ { 4 }\), and hence estimate the gradient of the curve when \(t = 0.5\).

6 The equation \(\sinh x + \sin x = 3 x\) has one positive root \(\alpha\).

1. Show that \(2.5 < \alpha < 3\).
2. By using the first two non-zero terms in the Maclaurin series for \(\sinh x + \sin x\), show that \(\alpha \approx \sqrt [ 4 ] { 60 }\).
3. By taking the third non-zero term in this series, find a second approximation to \(\alpha\), giving your answer correct to 4 decimal places.

5 Let $$I _ { n } = \int _ { 0 } ^ { 1 } t ^ { n } \mathrm { e } ^ { - t } \mathrm {~d} t$$ where \(n \geqslant 0\).

1. Show that, for all \(n \geqslant 1\), $$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 } .$$
2. Hence prove by induction that, for all positive integers \(n\), $$I _ { n } < n ! .$$

It is given that \(I_n = \int_{1}^{e} (\ln x)^n \mathrm{d}x\) for \(n \geqslant 0\). Show that $$I_n = (n - 1)[I_{n-2} - I_{n-1}] \text{ for } n \geqslant 2.$$ [6] Hence find, in an exact form, the mean value of \((\ln x)^3\) with respect to \(x\) over the interval \(1 \leqslant x \leqslant \mathrm{e}\). [6]

The integral \(I_n\), where \(n\) is a positive integer, is defined by $$I_n = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} x^{-n} \sin \pi x \, dx.$$

1. Show that $$n(n+1)I_{n+2} = 2^{n+1} n + \pi - \pi^2 I_n.$$ [5]
2. Find \(I_5\) in terms of \(\pi\) and \(I_1\). [2]

It is given that \(y = \sinh(x^2) + \cosh(x^2)\).

1. Use standard results from the list of formulae (MF19) to find the Maclaurin's series for \(y\) in terms of \(x\) up to and including the term in \(x^4\). [2]
2. Deduce the value of \(\frac{d^4y}{dx^4}\) when \(x = 0\). [1]
3. Use your answer to part (a) to find an approximation to \(\int_0^{\frac{1}{2}} y \, dx\), giving your answer as a rational fraction in its lowest terms. [2]

Find the first three terms in the Maclaurin's series for \(\tanh^{-1}\left(\frac{1}{2}e^t\right)\) in the form \(\frac{1}{2}\ln a + bx + cx^2\), giving the exact values of the constants \(a\), \(b\) and \(c\). [6]

\includegraphics{figure_6} The diagram shows the curve with equation \(y = e^{1-x}\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac{1}{n}\).

1. By considering the sum of the areas of these rectangles, show that \(\int_0^1 e^{1-x} \, dx < U_n\), where $$U_n = \frac{e-1}{n(1-e^{-1})}.$$ [4]
2. Use a similar method to find, in terms of \(n\), a lower bound \(L_n\) for \(\int_0^1 e^{1-x} \, dx\). [4]
3. Show that \(\lim_{n \to \infty}(U_n - L_n) = 0\). [2]
4. Use the Maclaurin's series for \(e^x\) given in the list of formulae (MF19) to find the first three terms of the series expansion of \(z(1-e^{-z})\), in ascending powers of \(z\), and deduce the value of \(\lim_{n \to \infty}(U_n)\). [3]

\(y = \sec^2 x\)

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]

\includegraphics{figure_2} The diagram shows triangle \(ABC\) in which angle \(A\) is \(60°\) and the lengths of \(AB\) and \(AC\) are \((4 + h)\) cm and \((4 - h)\) cm respectively.

1. Show that the length of \(BC\) is \(p\) cm where $$p^2 = 16 + 3h^2.$$ [2]
2. Hence show that, when \(h\) is small, \(p \approx 4 + \lambda h^2 + \mu h^4\), where \(\lambda\) and \(\mu\) are rational numbers whose values are to be determined. [4]

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

The first two non-zero terms of the Maclaurin series expansion of \(f(x)\) are \(x\) and \(-\frac{1}{2}x^3\) Which one of the following could be \(f(x)\)? Circle your answer. [1 mark] \(xe^{\frac{1}{2}x^2}\) \quad \(\frac{1}{2}\sin 2x\) \quad \(x \cos x\) \quad \((1 + x^3)^{-\frac{1}{2}}\)

Use l'Hôpital's rule to prove that $$\lim_{x \to \pi} \frac{x \sin 2x}{\cos\left(\frac{x}{2}\right)} = -4\pi$$ [5 marks]