Math topic中文

Implicit parametric

Differentiation of simple functions defined implicitly or parametrically.

Practice questions

1The curve C has equation $x + y = (x - y)^2$ . It is given that C has only one turning point. Show that $1 - \frac{dy}{dx} = \frac{2}{1 + 2x - 2y}$ …
2The curve C has equation $x + y = (x - y)^2.$ It is given that C has only one turning point. Show that $1 - \dfrac{dy}{dx} = \dfrac{2}{1 + 2x - 2y}$ …
3A curve $C_1$ has the equation $p^2x^2 - y^2 = p^2$ where $p > 1$ . Sketch $C_1$ , stating the coordinates of any points o…
4A curve $C$ has parametric equations $x = 2\cos\theta + \cos 2\theta, \quad y = 2\sin\theta - \sin 2\theta,$ where $0 < \theta < 2\pi$ …
5It is given that $y = \sqrt{1 + \tan^{-1} 2x}$ where $-\frac{1}{2} < x < \frac{1}{2}$ . Show that $y \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 + \frac{8x}{(1 + 4x^2)^2} = 0.$ …
6Find the exact solution of the inequality $\frac{2x+1}{x^2-1} \le 1.$ A curve $G$ has equation $ay^2 + bx^3 + cx = 2$ , w…
7The parametric equations of a curve C are $x = at,\; y = at^{3}$ , where $a$ is a positive constant. The point P on the c…
8The curve $C$ has equation $y = \frac{x - 2}{kx^2 + x - 2}$ , where $k > 1$ . Find the equation of the tangent at the poin…
9Show that for $x>1$ , $\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{1}{\sqrt{x^2-1}}\right)=-\dfrac{x}{(x^2-1)^{3/2}}\,.$ …
10Find an expression for $\frac{\text{d}y}{\text{d}x}$ in terms of $x$ if $y = \tan^{-1} \sqrt{x^2 - 1}$ . It is given that…
11Differentiate each of the following expressions with respect to $x$ . $\tan^{-1} \sqrt{2 + x^3}$ $\ln \left( \frac{x}{\sqrt{1 - x}} \right)$ …
12Given that $y = \tan^{-1} \dfrac{1}{5} x$ , show that $\left(25 + x^2\right) \dfrac{d^2 y}{dx^2} + 2x \dfrac{dy}{dx} = 0$ …
13It is given that $\text{e}^y = 2 + \text{e}^x$ . Show that $\frac{\text{d}^2y}{\text{d}x^2} + \left(\frac{\text{d}y}{\text{d}x}\right)^2 = \frac{\text{d}y}{\text{d}x}$ …
14It is given that $y = \sqrt{1+\tan x}$ . Show that $2y\,\dfrac{dy}{dx} = 1 + (y^{2}-1)^{2}$ and $y\,\dfrac{d^{2}y}{dx^{2}} + \left(\dfrac{dy}{dx}\right)^{2} = 2y(y^{2}-1)\left(\dfrac{dy}{dx}\right).$ …
15[The surface area and volume of a sphere are given by $4\pi r^2$ and $\dfrac{4}{3}\pi r^3$ respectively.] An aquarist wh…
16The curve C is given by the equation $\dfrac{y^2}{x^2}=\dfrac{3a-x}{a+x},\quad x \ne 0,$ where $a$ is a positive const…
17The points $(x,y)$ on a curve satisfy the equations $x = \tfrac{1}{2}t^2 - 2t$ and $\dfrac{dy}{dx} = \dfrac{1}{2t - 1}$ …
18A Dunman High alumnus plans to donate money to build two koi ponds in the middle of Zheng Xin Yuan. The ponds are in the…
19Given that $y^{\cos x} = x^{\sin 2x}$ , find the exact value of $\frac{\mathrm{d}y}{\mathrm{d}x}$ at $x = \pi$ .
20In the diagram, point $Q$ is $400\text{ m}$ vertically above point $A$ . Initially, a balloon is at point $Q$ and moves h…
21The diagram shows a shot put being projected with a velocity $v\text{ ms}^{-1}$ from the point $O$ at an angle $\theta$ …
22A curve $C$ has parametric equations $x = 3t^2, \quad y = 6t^3, \quad \text{where } t \text{ is a real parameter.}$ Sk…
23In a particular building, there is an $L$ -shaped corridor (see diagram below). Corridor 1 has a fixed width of 3 m and C…
24It is given that $x^2 y - \tan^{-1} y = \frac{3}{4}\pi$ . Find $\frac{\text{d}y}{\text{d}x}$ in terms of $x$ and $y$ . Hen…
25It is given that $y = \sqrt{1 + \tan^{-1} 2x}$ where $-\frac{1}{2} < x < \frac{1}{2}$ . Show that $y \frac{\mathrm{d}^2y}{\mathrm{d}x^2} + \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 + \frac{8x}{(1 + 4x^2)^2} = 0$ …
26The point $P$ travels along the curve $C$ with equation $y = x \sin^{-1} x$ , $-1 < x < 1$ . Let the gradient of the curve…
27A curve $C$ has equation $x^2 y + y^4 = 5$ . Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$ . Find the exa…
28Given that $y= e^{\sin^{-1}2x}$ , show that $\left(1-4x^{2}\right)\dfrac{d^{2}y}{dx^{2}}=4y+4x\dfrac{dy}{dx}$ . Given that…
29The equation of a curve is $y=\dfrac{x^2-3a^2}{x-a}$ , where $a$ is a positive constant. Find the equations of the asympt…
30A curve C is defined by the parametric equations $x = \cos\theta - \dfrac{1}{\cos\theta},\ \ y = \cos\theta + \dfrac{1}{\cos\theta},\ \ -\dfrac{\pi}{2} < \theta < \dfrac{\pi}{2}.$ …
31The equation of a curve is $x^2 - 2xy + 2y^2 = 20$ . Find the equations of the tangent and normal to the curve at the poi…
32The curve $C$ has equation $y=\ln\left(2-\lvert x\rvert\right)$ . The region $R$ is bounded by $C$ , the lines $y=\ln 2$ ,…
33The function f is defined by $f : x \mapsto e^x + \dfrac{1}{2x+2}\quad\text{for } x \in \mathbb{R},\ x \ne -1.$ A func…
34For a curve with equation $x^2 - 3xy + y^3 = 8$ , find the $x$ -coordinate of the point at which the tangent is parallel t…
35A farmer has a plot of the land, $OAB$ , with adjacent sides $OA$ and $OB$ . With reference to $O$ as the origin, $OA$ and…
36[Cambridge/2003] Sketch the curve with parametric equations $x = 3t, \quad y = \frac{3}{t}.$ The point $P$ on the…
37The variables $x$ and $y$ are related by the differential equation $\pi \dfrac{dy}{dx} + y(3 - \pi \tan x) = 0$ . Using t…
38It 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…
39In a triangle with vertices A, B and C, angle $BAC$ is a right-angle and angle $ABC=\frac{\pi}{3}-x$ . Show that $\dfrac{AB}{AC}=\dfrac{1+\sqrt{3}\tan x}{\sqrt{3}-\tan x}$ …
40Differentiate $\tan^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)$ with respect to $x$ . You should simplify your answer as a s…
41In the triangle $ABC$ , $AB = 2$ , $BC = \sqrt{3}$ , $AC = x$ and angle $ABC = \frac{\pi}{6} + \theta$ radians (see diagram…
42The parametric equations of a curve C are $x=2a\cos\left(t+\tfrac{\pi}{6}\right),\quad y=a\sin t,$ where $0\le t<2\pi$ …
43Differentiate with respect to $x$ : $e^{\frac{4}{\sqrt{x}}}$ , $\ln\left(\frac{\sin x}{2 - \sin x}\right)^2$ .
44The curve $C$ has parametric equations $x = 2t - \frac{1}{t^2}, \quad y = 2t + \frac{1}{t}, \quad t \in \mathbb{R}, t \ne 0.$ …
45The curve $C$ is defined by the parametric equations $x = t^2$ and $y = 2t + t^3$ , where $-1 \le t \le 1$ . Sketch the gr…
46Show that $\dfrac{d}{dx}(\tan x)=\sec^2 x$ Show that $\sin 2x\cot x = 2\cos^2 x$ Hence, find the exact value of $\displaystyle\int_{\tfrac{\pi}{30}}^{\tfrac{\pi}{5}} \csc(10x)\tan(5x)\,dx$ …
47A curve has parametric equations $x = 2t - \frac{1}{t}$ and $y = t + \frac{1}{t}$ where $t$ is a parameter and $t \ne 0$ …
48A curve $C$ has equation $3x^2 - 2xy + 5y^2 = 14.$ Show that $\frac{dy}{dx} = \frac{3x - y}{x - 5y}.$ Find the exact…
49It is given that $\frac{\text{d}y}{\text{d}x} = \frac{x+5}{y^2}$ and when $x=0$ , $y=5$ . Find the Maclaurin's series expa…
50A curve has parametric equations $x=2t$ , $y=e^{-t^2}$ . Sketch the curve, stating the exact coordinates of any points of…
51Projectile motion is the motion of a launched object, subject to only the constant acceleration of gravity, $g\text{ ms}^{-2}$ …
52Find $\frac{\text{d}y}{\text{d}x}$ for each of the following, simplifying your answers. $y = \sin^{-1}(e^x)$ $y = \ln\left(\frac{1+\ln x}{x^x}\right)$ …
53An astroid is a curve with parametric equations $x = a \cos^3 t, \ y = a \sin^3 t.$ Sketch an astroid with $a = 2$ a…
54By means of the substitution $x = \operatorname{cosec}\theta$ , show that $\int_{\frac{2}{\sqrt{3}}}^{2} \frac{1}{x^3 \sqrt{x^2 - 1}} \mathrm{d}x = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sin^2(\theta) \mathrm{d}\theta$ …
55The curve C is defined by the parametric equations $x = a\left(1 + \dfrac{1}{t}\right)$ and $y = a\left(t - \dfrac{1}{t^2}\right)$ …
56By using the substitution $u = x - y$ , solve the differential equation $\dfrac{dy}{dx} + 4(x - y)^2\cos^2 x = \sin^2 x.$ …
57The curve $C$ has parametric equations $x = \sin \theta + 1 \quad \text{and} \quad y = \sqrt{3} \cos \theta - 1, \text{ where } -\frac{\pi}{2} \le \theta \le \frac{\pi}{2}.$ …
58The function f is defined by $f : x \mapsto x^{3} - 1$ , $x \in \mathbb{R}$ . (i) Find an expression for $f^{-1}(x)$ . (ii)…
59[Cambridge/9709/2009] The equation of a curve is $x^3 - x^2y - y^3 = 3$ . Find $\frac{\text{d}y}{\text{d}x}$ in terms…
60The diagram below shows a region $R$ bounded by the curve $(y+5)^{2}=x-3$ and the line $y=x-10$ . Find the volume of soli…