Math topic中文

Applications differentiation

apply differentiation to gradients, tangents and normals, connected rates of change and maxima and minima problems

Practice questions

1The diagram shows a greenhouse built on a rectangular base. $140\text{ m}^2$ of polyethene is used to build the curved r…
2In the triangle $ABC$ , $AB = x\ \text{cm}$ , $AC = 3x\ \text{cm}$ and angle $BAC = \dfrac{2\pi}{3}$ . If $x$ is increasing…
3A curve has the equation $y = \frac{2x - 3}{3x + 4}$ , where $x \ne -\frac{4}{3}$ . The normal to the point $P$ on the cur…
4Find the gradient function of the curve $y = 2x^2 - \ln x + c$ . The curve touches the positive $x$ -axis at point $T$ . Fi…
5The normal to the curve $y = 2\tan 3x - 1$ at point $P\left(\frac{\pi}{3}, -1\right)$ crosses the $y$ -axis at point $A$ …
6The point $P$ lies on the curve $y = \ln\left(\frac{x+1}{x-1}\right)$ for $x > 1$ . The normal to the curve at $P$ is par…
7The radius of a circle increases at a rate of 3 cm s^{-1}. Find, in terms of $\pi$ , the rate of increase of the area whe…
8The figure shows part of the curve $y = g(x)$ . $(2, 8)$ and $(6, 1)$ are two points on the curve. Given that $\int_{2}^{6} y \text{ } dx = 32$ …
9The diagram shows part of the curve $y = 3e^{-\frac{1}{2}x} - 6$ which crosses the $x$ -axis at $P$ and the $y$ -axis at $Q$ …
10The volume of liquid in a container, $V\ \text{m}^3$ , is given by $V = 0.05\bigl[(3x+2)^3 - 8\bigr],$ where $x$ is the…
11已知函数 $f(x)={{e}^{x}}-1$ ， $g(x)=\sin x(1)$ 判断 $F(x)=f(x)-g(x)$ 在 $x\in[0,+\infty)$ 上零点的个数； (2)当 $x\in[0,\pi]$ 时， $f(x)\geqslant ag(x)(a\in R)$ …
12A motorist, travelling at a constant velocity of V m/s, passed a fixed point X and saw few vehicles ahead. He stepped on…
13A cone of height, $h$ and radius, $r$ fits exactly inside a sphere of radius $5\text{ cm}$ as shown below. $\left(\text{Volume of cone} = \frac{1}{3}\pi r^2 h\right)$ …
14The diagram shows part of the curve $y = \sqrt{3x + 4}$ . The line $AB$ is tangent to the curve at $A$ . The points $B$ an…
15Given that $y = \frac{4}{5}\left(\frac{x}{12} - 1\right)^6$ and that both $x$ and $y$ vary with time, find the value of…
16The gradient function of a curve $y = f(x)$ is given by $\frac{36}{(2x+1)^2}$ for $x > -\frac{1}{2}$ . The curve passes t…
17It is given that $y = 4x \sin 2x$ . Show that $\frac{\mathrm{d}y}{\mathrm{d}x}$ can be expressed in the form $ax \cos 2x + b \sin 2x$ …
18When the height of liquid in a container is $h$ m, the volume of liquid is $V\text{ m}^3$ where $V = 0.06 \left[ (4h + 1)^3 - 7 \right] \text{ and } h \ge 0.25$ …
19A conical vessel, whose vertical angle is $30^\circ$ as shown in the diagram, has water poured into it at such a rate th…
20The diagram shows part of the curve $y = (1 - x)(x - 4)$ and two lines $l_1$ and $l_2$ . The lines $l_1$ and $l_2$ are ta…
21Given the curve $y = \frac{1}{2}e^{2x} - \frac{1}{3}e^{-3x} + \frac{1}{3}$ and the normal to the curve at point $P\left(\ln 2, \frac{55}{24}\right)$ …
22The diagram shows the glass test tube of radius $r\text{ cm}$ and height $(h+r)\text{ cm}$ with a hemisphere of radius $r\text{ cm}$ …
23The diagram below shows the graph of $y = \frac{1}{2x+1}$ and the tangent to the curve where the curve intersects the $y$ …
24 $ABCD$ is a rectangle which fits inside a semicircle of radius 8 cm and centre $O$ . It is given that $AB = x$ cm and $BC = y$ …
25A farmer uses 160 m of fencing to enclose a plot of his land in a shape that comprises an isosceles triangle and a recta…
26The diagram shows part of the curve $y = \frac{1}{\left(\frac{3}{4}x - 5\right)^2} + 1$ . $P$ is the point on the curve w…
27Oil is dripping into an empty inverted right circular cone at the rate of 16 cm3 per second. The height, h cm, of the co…
28A particle moves along the curve $y = \frac{3(x+6)}{x+4}$ , where $x \ne -4$ , in such a way that the $y$ -coordinate of th…
29A factory manufactures miniature containers that are made up of a closed circular cylinder and a hemisphere which have t…
30The point A lies on the curve $y = x \ln x^2$ , $x > 0$ . The tangent to the curve at A is parallel to the line $y - 3x = 1$ …
31 $∀x\in(0,+∞)$ ，不等式 $xe^{x-3}-x-\ln x\geqslant a$ 恒成立，则 $a$ 的最大值为 $(\quad)$ A. $-2$ B. $0$ C. $e^{-2}-1$ D. $-\ln 3$ …
32[The volume of a sphere of radius r is $\dfrac{4}{3}\pi r^3$ , and its surface area is $4\pi r^2$ .] The diagram shows a b…
33Air is being pumped into a spherical balloon at a constant rate of 12 cm $^3$ per second. Find the rate at which the radi…
34The diagram shows part of the curve $y = \sin x$ . A is a point on the curve that lies on the line $x = \frac{\pi}{3}$ . T…
35The diagram shows part of the curve $y = 4 - e^{-2x}$ which crosses the axes at $A$ and at $B$ . Find the coordinates of…
36A cone of height 12 cm and base radius 5 cm is placed over a cylinder of radius $r$ cm and height $h$ cm. The cone is in…
37The diagram shows part of the curve $y = e^{x-1} - 2$ . The point $P(2,\ e-2)$ lies on the curve and the normal to the cu…
38The gradient function of a curve is given by $\frac{\text{d}y}{\text{d}x} = \frac{16}{x^3} - 4$ . Given that the line $2y - x - 6 = 0$ …
39[The volume of a cone of height $h$ and base radius $r$ is $\frac{1}{3}\pi r^2 h$ .] A hollow right circular cone of base…
40The diagram shows a prism in which the cross-section is a trapezium $ABCD$ . $AB = 4x\text{ cm}$ , $BC = DC = AD = 2x\text{ cm}$ …
41A miniature model in the shape of a triangular prism is shown above. A piece of wire, 36 cm in length, runs through all…
42The diagram shows part of the curve $y = \sqrt{4x + 5}$ . The normal of the curve at $P$ meets the $x$ -axis at $T$ . The $x$ …
43A curve is such that $\frac{d^2y}{dx^2} = 6e^{3x} - x$ and passes through the point $\left(2, \frac{2}{3}e^6\right)$ . Gi…
44已知函数 $g(x)= \dfrac {1}{x\cdot \sin \theta }+\ln x{在}[1,+∞)$ 上为增函数，且 $θ∈(0,π),f(x)=mx- \dfrac {m-1+2e}{x}-\ln x,m∈R$ ． (1…
45已知函数 $f(x)=x^{3}+ax^{2}+bx-a^{2}-7a$ 在 $x=1$ 处取得极大值 $10$ ，则 $\dfrac{a}{b}$ 的值为 $(\quad)$ A. $-\dfrac{2}{3}$ B. $-2$ …
46Baking powder is poured onto a flat surface at a constant rate of $2\pi\text{ cm}^3\text{s}^{-1}$ , forming a right circu…
47The graph shows part of the curve $y = -3(2x-4)^{-1/3}$ . The point $P(-2,\ 1.5)$ lies on the curve and the normal to the…
48A gardener plans to construct two flower beds, one a circle of radius $r$ m, the other a square of side $x$ m. To protec…
49The diagram shows part of the curve $y = \sqrt{2x + 4}$ , meeting the $x$ -axis at the point $A$ and the $y$ -axis at the p…
50The diagram shows part of the curve $y = 2 \sin 2x + 1$ . $P$ is the maximum point of the curve and $Q$ is the point on t…
51Point $P$ lies on the curve $y = \frac{kx^2}{\sqrt{2-x}}$ , where $k$ is a constant. The $x$ -coordinate of $P$ is $-2$ . G…
52The diagram shows part of the curve $y = x^3 + 3x^2 - 2$ . The straight line $L$ cuts the curve at $A(-3, -2)$ , the x-axi…
53A capsule is made of a cylinder with a hemisphere at one end of the cylinder. The cylinder has a height, $h\text{ cm}$ a…
54The diagram shows a glass window $ABCDEF$ , consisting of a rectangle $ABEF$ of height $3x\text{ cm}$ and width $y\text{ cm}$ …
55Solutions to this question by accurate drawing will not be accepted. $ABCD$ is a trapezium where $AD$ is parallel to…
56The diagram shows part of the curve $y = \frac{1}{4}x^2 - \frac{1}{2}\ln\left(\frac{x}{3}\right)$ and $P(3,\,2.25)$ is…
57The function is defined by $g(x) = \frac{x^2 - 4}{x^2 + 6}$ , $x > 0$ . Explain, with working, whether $g(x)$ is an increa…
58The diagram shows part of the curve $y = 27 + \frac{8}{(x-2)^3}$ which crosses the axes at points $A$ and $B$ . Explain w…
59The diagram shows part of the curve $y = 2\sin\frac{x}{2}$ , where $A$ lies on the curve. The $x$ -coordinate of $A$ is $\frac{4\pi}{3}$ …
60The diagram shows a paper weight which is made up of a solid hemisphere of radius $r$ cm, resting on top of a solid cyli…