Math topic中文

Tangents normals

Problems involving tangents and normals to curves, including cases where the curve is defined implicitly or parametrically.

Practice questions

1[Cambridge/9709/2010] The diagram shows parts of the curves $y = 9 - x^3$ and $y = \frac{8}{x^3}$ and their points o…
2A curve $C$ has parametric equations $x = 2\cos\theta + \cos 2\theta, \quad y = 2\sin\theta - \sin 2\theta,$ where $0 < \theta < 2\pi$ …
3已知函数 $f(x)=x^{2}-ax(a\neq 0)$ ， $g(x)=\ln x$ ， $f(x)$ 图象与 $x$ 轴异于原点的交点 $M$ 处的切线为 $l_{1}$ ， $g(x-1)$ 与 $x$ 轴的交点 $N$ 处的切线为…
4The parametric equations of a curve C are $x = at,\; y = at^{3}$ , where $a$ is a positive constant. The point P on the c…
5The curve $C$ has equation $y = \frac{x - 2}{kx^2 + x - 2}$ , where $k > 1$ . Find the equation of the tangent at the poin…
6A curve $C$ has equation $y = \frac{a}{x} + bx^2 + c$ , where $a$ , $b$ and $c$ are constants. It is given that $C$ passes…
7A curve $C$ is given parametrically by the equations $x = 2\cos^3 \theta$ , $y = 2\sin^3 \theta$ where $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$ …
8The 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…
9A curve $C$ has parametric equations $x = 3t^2, \quad y = 6t^3, \quad \text{where } t \text{ is a real parameter.}$ Sk…
10In the diagram, A, B and D are fixed points such that $AB = 25\text{ cm}$ , $AD = 35\text{ cm}$ and angle $BAD = 90^\circ$ …
11A 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…
12A curve C is defined by the parametric equations $x = \cos\theta - \dfrac{1}{\cos\theta},\ \ y = \cos\theta + \dfrac{1}{\cos\theta},\ \ -\dfrac{\pi}{2} < \theta < \dfrac{\pi}{2}.$ …
13The 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…
14The 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…
15For a curve with equation $x^2 - 3xy + y^3 = 8$ , find the $x$ -coordinate of the point at which the tangent is parallel t…
16[Cambridge/2003] Sketch the curve with parametric equations $x = 3t, \quad y = \frac{3}{t}.$ The point $P$ on the…
17A curve has equation $y = ax + b + \frac{c}{x^2 - 1}$ , where $a$ , $b$ and $c$ are real constants. It is given that the c…
18The equation of a curve $y = \mathrm{f}(x)$ is in the form of a cubic polynomial. The curve has a maximum point at $(-1, 0)$ …
19The parametric equations of a curve C are $x=2a\cos\left(t+\tfrac{\pi}{6}\right),\quad y=a\sin t,$ where $0\le t<2\pi$ …
20The curve C has equation $y=\dfrac{2x^{2}-a}{x+k}$ where $a,\; k>0$ . Given that $2k^{2}\ne a$ , find the range of values…
21The curve $C$ has parametric equations $x = 2t - \frac{1}{t^2}, \quad y = 2t + \frac{1}{t}, \quad t \in \mathbb{R}, t \ne 0.$ …
22The 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…
23A curve $C$ has equation $3x^2 - 2xy + 5y^2 = 14.$ Show that $\frac{dy}{dx} = \frac{3x - y}{x - 5y}.$ Find the exact…
24A curve has parametric equations $x=2t$ , $y=e^{-t^2}$ . Sketch the curve, stating the exact coordinates of any points of…
25The diagram shows the graph of $y = \text{f}'(x)$ , which passes through the origin and has turning points at $(2, 5)$ an…
26An astroid is a curve with parametric equations $x = a \cos^3 t, \ y = a \sin^3 t.$ Sketch an astroid with $a = 2$ a…
27The curve $C$ has parametric equations $x = \sin \theta + 1 \quad \text{and} \quad y = \sqrt{3} \cos \theta - 1, \text{ where } -\frac{\pi}{2} \le \theta \le \frac{\pi}{2}.$ …
28[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…
29The 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…
30A curve C is defined by the parametric equations $x=\dfrac{t}{1+t},\quad y=\dfrac{t^{2}}{1+t}$ , where $t$ takes all real…
31A curve $C$ is defined by the parametric equations $x = 2\cos\theta + 1, \quad y = \sin\theta, \quad \text{where } -\pi \le \theta \le \pi.$ …
32The parametric equations of a curve $C$ are $x = t + 2e^t, \quad y = 2t + e^{-t}, \quad t > -2.$ Sketch $C$ . Find the…
33A curve $R$ has equation $y=f(x)$ . The point $P$ with coordinates $(7,-9)$ lies on $R$ . The tangent to $R$ at point $P$ …
34It is given that $y = x^{xy}$ , where $x > 0$ , $y > 0$ . Show that $\frac{\text{d}y}{\text{d}x} = \frac{y^2 (1 + \ln x)}{1 - y \ln x}$ …
35A curve $C$ has equation $y = \frac{e^{kx}}{e^{kx} - e^{-kx}}$ , where $x \ne 0$ and $k \in \mathbb{R},\ k \ne 0$ . Show t…
36A curve $D$ has equation $y = ax + b \ln x + \frac{c}{x}, \quad x > 0,$ where $a$ , $b$ and $c$ are constants. It is gi…
37A curve $C$ is defined by the parametric equations $x = \frac{t}{1 + t}, \quad y = \frac{t^2}{1 + t},$ where $t$ takes…
38A curve $C$ has parametric equations $x = \mathrm{e}^t \cos t,\ y = \mathrm{e}^{-t} \sin t.$ Find the equation of the…
39The equation of a curve is given by $xy - 2y^{2} + 4x^{2} = 66$ . Find the exact coordinates of the points on the curve w…
40设函数 $f(x)=\ln x$ ， $g(x)=\dfrac{x}{x+1}$ ，记 $F(x)=f(x)-g(x)(1)$ 求曲线 $y=f(x)$ 在 $x=e$ 处的切线方程； (2)求函数 $F(x)$ 在 $[\dfrac{1}{e},e^{2}]$ …
41The parametric equations of a curve are $x = t + 2$ , $y = \dfrac{t}{2t - 1}$ , where $t \in \mathbb{R}$ , $t \ne \dfrac{1}{2}$ …
42The curve $C$ has equation $2x - y^2 = (x + y)^2$ . Find the equations of the tangents to $C$ which are parallel to the $x$ …
43A curve $C$ has equation $x^{3} + y^{3} - 5xy = 0$ where $x > 0$ . Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$ . Find th…
44已知函数 $f(x)=m\ln x$ ． (1)讨论函数 $F(x)=f(x)+ \dfrac {1}{x}-1$ 的单调性； (2)定义：“对于在区域 $D$ 上有定义的函数 $y=f(x)$ 和 $y=g(x)$ ，若满足 $f(x)\leqslant g(x)$ …
45A curve $C$ has parametric equations $x = t - \frac{a}{t}, \quad y = t + \frac{a}{t}, \text{ where } t \in \mathbb{R}, \ t \ne 0 \text{ and } a \text{ is a positive constant.}$ …
46A curve $C$ is defined by the parametric equations $x = t^2, \quad y = t^3.$ Prove that the equation of the tangent at…
47A curve $C$ has parametric equations $x = \mathrm{e}^{2t}, \ y = t^2.$ Find the value of $t$ at the point where the ta…
48A curve $C$ is defined by the parametric equations $\begin{aligned} x &= 4\sin t, \\ y &= b\cos t - 1, \end{aligned}$ …
49An ellipse has equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ , where $a$ and $b$ are positive real constants. Show tha…
50It is given that $y=x^{y}$ , where $x>0$ , $y>0$ . Find $\dfrac{dy}{dx}$ in terms of x and y. Hence find the equation of th…
51[Cambridge/9709/2006] The equation of a curve is $x^3 + 2y^3 = 3xy$ . Show that $\frac{\text{d}y}{\text{d}x} = \frac{y - x^2}{2y^2 - x}$ …
52The above diagram shows the curve $C$ with equation $e^{2y} = x^2 + y$ for $x>0$ . Make a copy of the diagram to include…
53A curve $C$ has parametric equations $x = 1 - 2\cos 2\theta, \quad y = \sin 2\theta - 1, \quad \text{for } 0 \le \theta \le \frac{3\pi}{8}.$ …
54The equation of a curve is given by $Ax^2 + By^2 + Cx + Dy + 13 = 0$ where $A$ , $B$ , $C$ and $D$ are constants. Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ …
55A curve $C_1$ has parametric equation $x = 3\sec^2 \theta$ , $y = 2\tan \theta$ , $0 \le \theta < \frac{\pi}{2}$ . $L_1$ an…
56Expand $\dfrac{1 - x^2}{\sqrt[3]{1 + 3x}}$ in ascending powers of $x$ up to and including the term in $x^3$ . (ii) State…
57A curve $C$ has parametric equation $x = t^2 - 1,\qquad y = 4t - t^3 + 15,$ where $t \in \mathbb{R}$ . Show that the cu…
58The equation of a curve is given by $4x^3+3x^2y=y^3-2$ . Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$ , simplifying your…
59Baby Peter is pulling a toy wagon on a flat ground which is modelled by the $xy$ -coordinate plane. Peter is initially at…
60The curve C has parametric equations $x = 2t - \dfrac{1}{t^{2}},\quad y = 2t + \dfrac{1}{t},\quad t \in \mathbb{R},\ t \ne 0. The point P on the curve has parameter$ …