Math topic中文

Notation

use of standard notations $f'(x)$ , $f''(x)$ , $\dfrac{dy}{dx}$ , $\dfrac{d^{2}y}{dx^{2}}\left[=\dfrac{d}{dx}\!\left(\dfrac{dy}{dx}\right)\right]$

Practice questions

1A cone of height, $h$ and radius, $r$ fits exactly inside a sphere of radius $5\text{ cm}$ as shown below. $\left(\text{Volume of cone} = \frac{1}{3}\pi r^2 h\right)$ …
2The diagram shows the glass test tube of radius $r\text{ cm}$ and height $(h+r)\text{ cm}$ with a hemisphere of radius $r\text{ cm}$ …
3Given that $y = 3e^{2x} + 2e^{-x}$ , and that $\frac{\text{d}^2 y}{\text{d}x^2} + \frac{\text{d}y}{\text{d}x} + y = Ae^{2x} + Be^{-x}$ …
4The diagram shows a prism in which the cross-section is a trapezium $ABCD$ . $AB = 4x\text{ cm}$ , $BC = DC = AD = 2x\text{ cm}$ …
5A capsule is made of a cylinder with a hemisphere at one end of the cylinder. The cylinder has a height, $h\text{ cm}$ a…
6The polynomial $f(x)$ is such that $f(x)=6x^3+ax^2-50x+b$ , where $a$ and $b$ are integers. It is given that $f(x)$ is di…
7Find the derivatives of each of the following, simplifying your answer. $y = 3\left(1 - \frac{x}{3}\right)^4$ $f(x) = (2 - 3x)(\sqrt{1 - 4x})$ …
8The figure shows the outline of a field, which consists of a semi-circular arc with radius $r\text{ m}$ and sides of a r…
9The equation of a curve is $y = \frac{\ln x}{x}$ . Find the coordinates of the stationary point of this curve. Leave your…
10It is given that $f'(x) = \sin 3x - \frac{1}{2x + 1}$ and $f(0) = \frac{2}{3}$ . Find an expression for $6f(x) + f''(x)$ .
11The equation of a curve is $y = 2x^3 + 6x^2 - 18x + 5$ . Find the coordinates of the stationary points and determine the…
12Solve the equation $6^{z}(6^{z} - 1) = 12.$ A curve has the equation $y = x e^{3x}$ . Find $\dfrac{dy}{dx}$ . Find the val…
13The diagram shows part of the curve $y = 1 - \sin x$ passing through the point $P$ . The curve touches the $x$ -axis at th…
14The figure below shows a right-angled triangular prism. The height and base of the triangular faces of the prism are $3x$ …
15A farmer just bought a piece of land to grow his crops. This piece of land consists of an equilateral triangle of sides…
16The equation of a curve is $y = \dfrac{\ln 2x}{x}$ . Find an expression for $\dfrac{d^{2}y}{dx^{2}}$ and show that $x\dfrac{d^{2}y}{dx^{2}} + 2\dfrac{dy}{dx} + \dfrac{1}{x^{2}} = 0$ …