Math topic中文

Derivative as gradient

derivative of $f(x)$ as the gradient of the tangent to the graph of $y = f(x)$ at a point

Practice questions

1A 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…
2The normal to the curve $y = 2\tan 3x - 1$ at point $P\left(\frac{\pi}{3}, -1\right)$ crosses the $y$ -axis at point $A$ …
3The 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…
4The figure shows part of the curve $y = g(x)$ . $(2, 8)$ and $(6, 1)$ are two points on the curve. Given that $\int_{2}^{6} y \text{ } dx = 32$ …
5The 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 $Q$ …
6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…
7The diagram shows part of the curve $y = \sqrt{3x + 4}$ . The line $AB$ is tangent to the curve at $A$ . The points $B$ an…
8The 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…
9When the height of liquid in a container is $h$ m, the volume of liquid is $V\text{ m}^3$ where $V = 0.06 \left[ (4h + 1)^3 - 7 \right] \text{ and } h \ge 0.25$ …
10曲线 $y=\sqrt{3x-2}$ 在点 $(1,f(1))$ 处的切线方程为 A. $x-2y+1=0$ B. $3x-y-2=0$ C. $3x-2y-1=0$ D. $3x+2y-5=0$
11若曲线 $y=x^{\mathrm{{-}}\frac{1}{2}}$ 在点 $(a,a^{\mathrm{{-}}\frac{1}{2}})$ 处的切线与两个坐标轴围成的三角形的面积为 $18$ ，则 $a$ 的值为___.
12The 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…
13Given the curve $y = \frac{1}{2}e^{2x} - \frac{1}{3}e^{-3x} + \frac{1}{3}$ and the normal to the curve at point $P\left(\ln 2, \frac{55}{24}\right)$ …
14The diagram below shows the graph of $y = \frac{1}{2x+1}$ and the tangent to the curve where the curve intersects the $y$ …
15The 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…
16The 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 $y - 3x = 1$ …
17The 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…
18The diagram shows part of the curve $y = 4 - e^{-2x}$ which crosses the axes at $A$ and at $B$ . Find the coordinates of…
19The 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…
20The diagram shows part of the curve $y = \sqrt{4x + 5}$ . The normal of the curve at $P$ meets the $x$ -axis at $T$ . The $x$ …
21A 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…
22The equation of a curve is $y = \frac{5x}{6 - 2x}$ . Find the $x$ -coordinates of the points at which the gradient of the…
23The 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…
24The 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…
25The 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…
26若曲线 $y=e^{-x}$ 上点 $P$ 处的切线平行于直线 $2x+y+1=0$ ，则点 $P$ 的坐标为 ___ ．
27Point $P$ lies on the curve $y = \frac{kx^2}{\sqrt{2-x}}$ , where $k$ is a constant. The $x$ -coordinate of $P$ is $-2$ . G…
28The 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…
29The function is defined by $g(x) = \frac{x^2 - 4}{x^2 + 6}$ , $x > 0$ . Explain, with working, whether $g(x)$ is an increa…
30The diagram shows part of the curve $y = 27 + \frac{8}{(x-2)^3}$ which crosses the axes at points $A$ and $B$ . Explain w…
31The diagram shows part of the curve $y = 2\sin\frac{x}{2}$ , where $A$ lies on the curve. The $x$ -coordinate of $A$ is $\frac{4\pi}{3}$ …
32设曲线 $f(x)=ax^{2}$ 在点 $(2,4a)$ 处的切线与直线 $4x-y+4=0$ 垂直，则 $a=(\quad)$ A. $2$ B. $-\dfrac{1}{16}$ C. $\dfrac{1}{2}$ D.…
33已知曲线 $y=\dfrac{|x|}{{{e}^{x}}}$ 在 $x=-1$ 处的切线和它在 $x={{x}_{0}}\left({x}_{0} > 0\right)$ 处的切线互相垂直，设 ${{x}_{0}}\in (\dfrac{m}{4},\dfrac{m+1}{4}),m\in Z$ …
34The line $2y + x + a = 0$ is a normal to the curve $y = 2x^2 - 3x + 1$ at the point $P$ . Find the coordinates of $P$ . De…
35设函数 $f(x)=ax+ \dfrac {b}{x}$ ，曲线 $y=f(x)$ 在点 $M( \sqrt {3},f( \sqrt {3}))$ 处的切线方程为 $2x-3y+2 \sqrt {3}=0$ ． $($ Ⅰ $)$ 求 $f(x)$ …
36Answer ALL the questions The function $f$ is defined, for all values of $x$ , by $f(x) = (2x - x^2)e^x$ . Find the ran…
37The diagram shows part of the curve $y = \frac{7}{6x + 1}$ intersecting the $y$ -axis at $A(0, 7)$ . The tangent to the cu…
38A curve is such that $\mathrm{f}''(x) = 6x - 10$ . The tangent to the curve $y = \mathrm{f}(x)$ at the point $(1, -2)$ pa…
39Find the set of values of $x$ for which the curve $y = -x^2 + 2x + 4$ lies below another curve $y = x^2 - 2x - 2$ and re…
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 $[\dfrac{1}{e},e^{2}]$ …
41设函数 $f(x)=x\ln\:x+2x^{3}$ ，则曲线 $y=f(x)$ 在点 $(1,2)$ 处的切线方程是___.
42已知函数 $f(x)= \dfrac {ax}{e^{x}}$ 在 $x=0$ 处的切线方程为 $y=x$ ． (1)求 $a$ 的值； (2)若对任意的 $x∈(0,2)$ ，都有 $\dfrac {1}{f(x)} > k+2x-x^{2}$ …
43The equation of a curve is $y = \frac{x}{e^{2x}}$ . Find an expression for $\frac{dy}{dx}$ . Given that $x$ is changing at…
44The diagram shows part of the graph of $y = 2\sqrt{2 + x}$ . It cuts the x-axis at $(-2,0)$ . The line $y = 2x$ meets the…
45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 $f(x)\leqslant g(x)$ …
46Given that $y = (x + 2)\sqrt{x - 1}$ , show that $\frac{\text{d}y}{\text{d}x} = \frac{kx}{2\sqrt{x - 1}}$ where $k$ is co…
47For a particular curve $\frac{\text{d}^2 y}{\text{d}x^2} = 3x - 2$ . The tangent to the curve at the point $A(4, -40)$ is…
48已知函数 $f(x)=x\sin x+a\cos x+x$ ， $a∈R$ ． $($ Ⅰ $)$ 当 $a=-1$ 时，求曲线 $y=f(x)$ 在点 $(0,f(0))$ 处的切线方程； $($ Ⅱ $)$ 当 $a=2$ 时，求 $f(x)$ …
49A curve has equation $y = x^2 - x - 12$ . Points A and B lie on the curve and have x-coordinates of $k$ and $2k$ respecti…
50曲线 $y=2x+\ln x$ 在 $x=1$ 处的切线的倾斜角为 $α$ ，则 $\cos α+\sin α$ 的值为( ) A. $\dfrac{ \sqrt{10}}{10}$ B. $\dfrac{ \sqrt{10}}{5}$ …
51The diagram shows part of the curve $y = \sqrt{3x-2}$ . The normal to the curve at $B$ meets the $x$ -axis at $C$ . Given t…
52已知函数 $f(x)=e^{x}(x^{2}-2x+a)($ 其中 $a∈R$ ， $a$ 为常数， $e$ 为自然对数的底数 $)$ ． $($ Ⅰ $)$ 讨论函数 $f(x)$ 的单调性； $($ Ⅱ $)$ 设曲线 $y=f(x)$ …
53A curve $C$ has equation $y = 1 - \frac{1}{x}$ , where $x \ne 0$ . The line $l$ is the tangent to $C$ at $x = a$ , where $a$ …
54过点 $P(2,-1)$ 作抛物线 $C$ ： $x^{2}=2y$ 的两条切线，切点分别为 $A$ ， $B$ ，则直线 $AB$ 的方程为 ___.
55The diagram shows part of the curve $y = \sqrt{2x+5}$ . A is a point on the curve and the x-coordinate of A is 2. The nor…
56The diagram shows part of the curve $y = 5 + \frac{4}{e^{2x}}$ intersecting the $y$ -axis at point $P$ . The tangent to th…
57The diagram shows part of the curve of $y = x(x-2)^2$ , which passes through the point $P(1, 1)$ . Show that the equation…
58已知函数 $f(x)=ax^{2}+x\ln x+x$ ． (1)若 $a=1$ ，求函数 $f(x)$ 在 $(1,f(1))$ 处的切线方程； (2)若 $a=-e$ ，证明：方程 $|f(x)|-\ln x= \dfrac {1}{2}x$ …
59The equation of a curve is $y = 2x^3 + 6x^2 - 18x + 5$ . Find the coordinates of the stationary points and determine the…
60过抛物线 $x^{2}=4y$ 在第一象限内的一点 $P$ 作切线，切线与两坐标轴围成的三角形的面积为 $\dfrac {1}{2}$ ，则点 $P$ 到抛物线焦点 $F$ 的距离为 $()$ A. $1$ B. $2$ C.…