Math topic中文

Standard series expansions

Standard series expansion of $(1+x)^n$ for any rational $n$ , and of $e^x$ , $\sin x$ , $\cos x$ and $\ln(1+x)$ .

Practice questions

1Given that $\theta$ is small, show that $\sec\theta \approx 1 + \tfrac{1}{2}\theta^{2}$ . The diagram below shows a circl…
2Express $\displaystyle\frac{2+x}{\sqrt{9-x}}$ as a series in ascending powers of $x$ , up to and including the term in $x^2$ …
3Given that the first three terms in the expansion of $\dfrac{1}{a+bx}$ are $\dfrac{1}{2}+\dfrac{1}{4}x+cx^2$ , find the v…
4Given that $\theta$ is small, show that $\sec \theta \approx 1 + \tfrac{1}{2}\theta^2$ . The diagram below shows a circle…
5It is given that $\text{e}^y = 2 + \text{e}^x$ . Show that $\frac{\text{d}^2y}{\text{d}x^2} + \left(\frac{\text{d}y}{\text{d}x}\right)^2 = \frac{\text{d}y}{\text{d}x}$ …
6It is given that $y = \sin\left(e^x - 1\right)$ . Show that $\dfrac{d^2 y}{dx^2} = \dfrac{dy}{dx} - ye^{2x}$ . Hence, find…
7It is given that $f(x)=\dfrac{1}{4+9x^2}$ . Find $\displaystyle\int f(x)\,dx$ . Find the binomial expansion for $f(x)$ , up…
8Show that $\displaystyle\int_{n}^{n+1} \ln x\, dx = (n+1)\ln(n+1) - n\ln n - 1.$ (ii) The diagram shows the graph of $y = \ln x$ …
9Section A: Pure Mathematics 1 (a) Show that $\dfrac{1}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}=\dfrac{1}{2x^{2}}\big(\sqrt{1+x^{2}}+\sqrt{1-x^{2}}\big).$ …
10In the triangle $ABC$ , $AB = 1$ , $BC = 4$ and angle $ABC = \theta$ radians. Given that $\theta$ is a sufficiently small…
11It is given that $y = \ln(2 - \text{e}^{-2x})$ . Show that $\frac{\text{d}y}{\text{d}x} = 4\text{e}^{-y} - 2$ . Hence find…
12Find the first negative coefficient in the expansion of $(4 + 3x)^{5/2}$ in a series of ascending powers of $x$ , where $|x| < \dfrac{4}{3}$ …
13In a triangle with vertices A, B and C, angle $BAC$ is a right-angle and angle $ABC=\frac{\pi}{3}-x$ . Show that $\dfrac{AB}{AC}=\dfrac{1+\sqrt{3}\tan x}{\sqrt{3}-\tan x}$ …
14Given that $\mathrm{f}(x) = 1 + \cos 2x$ , find $\mathrm{f}(0)$ , $\mathrm{f}'(0)$ , $\mathrm{f}''(0)$ , $\mathrm{f}^{(3)}(0)$ …
15Express $\dfrac{5 + x^2}{(2 + x)(1 - x)^2}$ in the form of $\dfrac{A}{(2 + x)} + \dfrac{B}{(1 - x)^2}$ , where $A$ and $B$ …
16Find the exact value of $\pi \int_{1}^{6} x^3 \text{ d}x$ . The diagram shows a sketch of the curve $y = x^{\frac{3}{2}}$ …
17A triangle $ABC$ is such that $AC = \sqrt{2}$ , $BC = 4$ and angle $ACB = \frac{1}{4}\pi + \theta$ . Given that $heta$ is…
18It is given that $\mathrm{f}(x) = \ln(1 + 2x^2)$ . A polynomial $p(x) = ax^2 + bx + c$ is used to approximate $\mathrm{f}(x)$ …
19It is given that $f(x)=\ln(a+x)$ , $x\in\mathbb{R}$ , $x>-a$ , where $a$ is a constant. Using the standard series from the…
20Use the standard series for $\ln(1+x)$ and $\ln(1-x)$ to find the first three non-zero terms in the series for $\displaystyle\ln\left(\frac{1+x}{1-x}\right).$ …
21Find the first two terms in the expansion of $\dfrac{(9 + x)^{1/2}}{1 + 2x}$ in ascending powers of $x$ .
22TMJC JC1 Promo 9758/2019/Q6 Given that $y = \ln\left(2 - e^{2x}\right)$ , where $x < \tfrac{1}{2}\ln 2$ , show that $e^y\left[\frac{d^2 y}{dx^2} + \left(\frac{dy}{dx}\right)^2\right] = -4e^{2x}.$ …
23Find the first three non-zero terms in the Maclaurin series for $\text{e}^x \sin(x + \pi)$ . It is given that the three t…
24Given that $y = \cos\{\ln(1 + x)\}$ , prove that $(1 + x)\dfrac{dy}{dx} = -\sin\{\ln(1 + x)\}$ $(1 + x)^2\dfrac{d^2y}{dx^2} + (1 + x)\dfrac{dy}{dx} + y = 0.$ …
25Given that $f(x)=1+\cos 2x$ , find $f(0)$ , $f'(0)$ , $f''(0)$ , $f^{(3)}(0)$ and $f^{(4)}(0)$ . Write down the Maclaurin ser…
26The Environment Authority wishes to build a fence surrounding a park shown by the figure below. BCEF is a square with si…
27A triangle $ABC$ is such that $AC = \sqrt{2}$ , $BC = 4$ and angle $ACB = \tfrac{1}{4}\pi + \theta$ . Given that $\theta$ …
28In the triangle $PQR$ , angle $PQR = \frac{\pi}{3}$ radians and angle $PRQ = \frac{\pi}{3} - \theta$ radians, where $\theta$ …
29It is given that $\mathrm{f}(x) = \ln(a + x)$ , $x \in \mathbb{R}$ , $x > -a$ , where $a$ is a constant. Using the standard…
30It is given that $y = \mathrm{e}^{-x}\ln(1-2x)$ . Show that $(1-2x)\frac{\mathrm{d}y}{\mathrm{d}x} = -2\mathrm{e}^{-x} - y(1-2x)$ …
31The function h is defined as follows. $h : x \mapsto \ln(x - k + 1),\ x \in \mathbb{R},\ x \ge k,$ where $k < 0$ . Find t…
32The variables $x$ and $y$ are related by $(x+y)\frac{dy}{dx} + ky = 2\quad\text{and}\quad y = 1\text{ at }x = 0,$ wher…
33Section A: Pure Mathematics 1 (a) Show that $\displaystyle \frac{1}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}=\frac{1}{2x^{2}}\left(\sqrt{1+x^{2}}+\sqrt{1-x^{2}}\right).$ …
34Expand $\dfrac{1 - x^2}{\sqrt[3]{1 + 3x}}$ in ascending powers of $x$ up to and including the term in $x^3$ . (ii) State…
35Express $\dfrac{2+x}{\sqrt{9-x}}$ as a series in ascending powers of $x$ , up to and including the term in $x^2$ . By usin…
36Given that $y = \sqrt{1+\ln(1+x)}$ , find the exact range of values of $x$ for $y$ to be well defined. Show that $2y\,\dfrac{dy}{dx} = e^{-y^2}$ …
37Express $\dfrac{2+x}{\sqrt{9-x}}$ as a series in ascending powers of $x$ , up to and including the term in $x^2$ . By usin…
38Let $x$ and $y$ be variables such that $\ln(y + 1) = 1 + \tan^{-1}x$ . Show that $(1+x^2)\frac{\text{d}^2y}{\text{d}x^2} = (1-2x)\frac{\text{d}y}{\text{d}x}$ …
39In the triangle $ABC$ , angle $BAC = \theta$ radians, angle $ACB = \frac{\pi}{6}$ radians and $AC = \sqrt{3}$ . Given that…
40It is given that $y = \ln(\cos ax - \sin ax)$ , where $a$ is a non-zero constant. Show that $\frac{d^2 y}{dx^2} + \left(\frac{dy}{dx}\right)^2 + a^2 = 0.$ …
41Use the binomial series to expand $\sqrt{\dfrac{1 + x}{1 - x}}$ as a series of ascending powers of $x$ up to and includi…
42In this question you may use expansions from the List of Formulae (MF27). Find the Maclaurin expansion of $\ln(1 + \cos 3x)$ …
43A sequence of numbers, $x_{n}$ , satisfy the relation $x_{n+1}=\tfrac{1}{3}\left(2x_{n}+\dfrac{1}{x_{n}^{2}}\right),\quad n\in\mathbb{N}^{+}\cup\{0\},\quad 0<x_{0}<1.$ …
44It is given that $y = \ln(\cos ax - \sin ax)$ , where $a$ is a non-zero constant. Show that $\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 + a^2 = 0$ …
45Find the first three non-zero terms in the Maclaurin series for $e^{x}\sin(x+\pi)$ . It is given that the three terms fou…
46It is given that $f(x) = \ln\left(1+e^{-x}\right)$ . Using standard series from the List of Formulae (MF26) or otherwise,…
47It is given that $\mathrm{e}^y = 1 + \sin 3x$ . Show that $\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 + 9 = 9\mathrm{e}^{-y}.$ …
48Given that $y = \text{e}^{\sqrt{1+2x}}$ , show that $(1+2x)\left(\frac{\text{d}y}{\text{d}x}\right)^2 = y^2 \quad \text{and} \quad (1+2x)\frac{\text{d}^2 y}{\text{d}x^2} + \frac{\text{d}y}{\text{d}x} = y.$ …
49Using standard series from the List of Formula (MF26), expand $\ln(k + 2x)^n$ in ascending powers of $x$ , as far as the…
50The coefficient of $x^3$ in the expansion of $\frac{1}{(2+ax)^5}$ is 5 times the coefficient of $x^2$ in the expansion o…
51In a triangle with vertices $A$ , $B$ and $C$ , angle $BAC$ is a right-angle and angle $ABC = \frac{\pi}{3} - x$ . Show tha…
52TMJC JC1 Promo 9758/2019/Q6 Given that $y = \ln(2 - \mathrm{e}^{2x})$ , where $x < \frac{1}{2}\ln 2$ , show that $\mathrm{e}^y \left[ \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 \right] = -4\mathrm{e}^{2x}.$ …
53The triangle $ABC$ is such that $AB = 3$ , $AC = 4$ and angle $BAC = \theta$ radians. The triangle $ABC$ is such that $AB = 3$ …
54Use the standard series for $\ln(1+x)$ and $\ln(1-x)$ to find the first three non-zero terms in the series for $\ln\left(\dfrac{1+x}{1-x}\right)$ …
55Using standard series, find the Maclaurin series expansion of $\dfrac{1}{\sqrt{1 - a^2 x^2}}$ up to and including the te…
56Find the first three non-zero terms in the Maclaurin series for $e^{x}\sin(x+\pi)$ . It is given that the three terms fou…
57It is given that $\text{f}(x) = \frac{1}{\sqrt{1 - 4x^2}}$ where $-\frac{1}{2} < x < \frac{1}{2}$ . Using standard series…
58It is given that $y = (1 + \tan^{-1} x)^2$ . Prove that $(1 + x^2)^2 \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + 2x(1 + x^2) \frac{\mathrm{d}y}{\mathrm{d}x} = 2$ …
59Given that $y = \tfrac{1}{2}\ln\left(1 + \tan x\right)$ , show that $2e^{2y}\dfrac{dy}{dx} = \sec^2 x.$ Hence write dow…
60In this question, you may use expansions from the List of Formulae (MF26). Given that $\mathrm{f}(x) = \mathrm{e}^{\sin 3x}$ …