Math topic中文

Related graphs

Relating the graphs of $y = \mathrm{f}'(x)$ , $y = |\mathrm{f}(x)|$ , $y = \mathrm{f}(|x|)$ and $y = \dfrac{1}{\mathrm{f}(x)}$ to the graph of $y = \mathrm{f}(x)$ .

Practice questions

15 (a) Solve the following integral. $\displaystyle\int \frac{1-2x}{\sqrt{3+2x-x^2}}\,\mathrm{d}x.$ $\displaystyle\int x\ln\left(2-x^2\right)\,\mathrm{d}x$ …
2Sketch, on the same diagram, the graphs of $y = |2x + 3|$ and $y = -2x^2 - 8x - 3$ . (You are not required to label any a…
3The diagram shows the graphs of $y = \mathrm{f}(|x|)$ and $y = \mathrm{f}'(x)$ . The graph of $y = \mathrm{f}(|x|)$ has t…
4The graph of $y = \text{f}(x)$ has a maximum turning point at $(2,3)$ and passes through the origin. The lines $x = -1$ …
5On a single diagram, sketch the graphs of $y = |2x - p|$ and $y = qx$ where the following conditions are satisfied, indi…
6Find the exact roots of the equation $\left|\dfrac{4x-5}{x-2}\right| = 3x. On the same axes, sketch the curves with equations$ …
7A function f is defined as $f(x)=\ln(5-x),\; x<5.$ Find the volume of revolution when the region bounded by the curve of…
8The figure below shows a sketch of part of the graph of $y = f(x)$ for $0 \\le x < 2a$ . The vertical asymptote $x = 2a$ …
9The functions f and g are defined by $f: x \mapsto |(x+1)(x-3)|,\quad x \in \mathbb{R},$ $g: x \mapsto x-2,\quad x \in \mathbb{R}\text{ and }x \ge -1.$ …
10The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto \sqrt{5+2x}$ , for $x \in \mathbb{R}$ , $x > -\frac{5}{2}$ …
11The diagram shows the graph of $y = f(x)$ with asymptotes $y = k$ , $x = 0$ and the graph cuts the x-axis at $(a,0)$ . On…
12A curve C has equation $y = f(x)$ , where $f(x) = \frac{2(x-1)^2}{x-3}$ . Sketch C, showing clearly the equations of the a…
13It is given that $f(x) = \begin{cases} \tfrac{1}{2}x^2, & \text{for } 1 \le x < 2, \\ \tfrac{1}{4}(-3x+14), & \text{for } 2 \le x < 4, \end{cases}$ …
14A curve $C_1$ is defined parametrically by $x = \frac{2}{t-1} \quad\text{and}\quad y = \frac{4}{t+1},\qquad t \ne \pm 1.$ …
15Find, in terms of $a$ , the roots of the equation $\dfrac{1}{(x-a)^2}=|x-a|$ . On the same axes, sketch the curves with eq…
16The figure below shows the sketch of the graph $y = f(x)$ . The graph has asymptotes at $x = 1$ , $y = 3$ and the x-axis.…
17[In this question, sketches of the given graphs are not drawn to scale] The graphs of $y = |\mathrm{f}(x)|$ and $y^2 = \mathrm{f}(x)$ …
18A curve has equation $y=f(x)$ , where $f(x)=x^{3}+ax^{2}+bx+c$ . The curve $y=f(x)$ has a maximum point at $(1,-32)$ , whil…
19The diagram shows the graphs of $y = |a - bx|$ and $y = |mx - 6|$ where $a$ , $b$ and $m$ are positive constants. The gra…
20Section A: Pure Mathematics [40 marks] Sketch, on the same axes, the graphs of $y = 3x$ and $y = |x^2 - a^2|$ where $a > 0$ …
21Given that both $a$ and $b$ are positive constants, on the same diagram, sketch the graphs of $y = \frac{2}{|x - a|} \quad \text{and} \quad y = b|x - a|.$ …
22A function $\text{f}$ is defined by $\text{f}(x) = ax^3 + bx^2 + cx + d$ . The graph of $y = \text{f}(x)$ passes through…
23A curve $R$ has equation $y=f(x)$ . The point $P$ with coordinates $(7,-9)$ lies on $R$ . The tangent to $R$ at point $P$ …
24The diagram shows the graphs of $y = f(|x|)$ and $y = f'(x)$ . The graph of $y = f(|x|)$ has turning points at $(-3,4)$ a…
25The graph of $y = f(x)$ is given below. On separate diagrams, sketch the graphs of $y = \frac{1}{f(x)}$ $y = f'(x).$ A c…
26The diagram shows the curve $y = \mathrm{f}(x)$ . The curve passes through the point $A(a,0)$ and the point $B(b,0)$ , has…
27The diagram below shows the graph of $y = \mathrm{f}(x)$ . The asymptotes of the graph are $x = 2$ and $y = 2$ . The graph…
28The curve $C$ has equation $y = \frac{x^2 + (\lambda - 1)x}{x - 1}, \quad x \ne 1,$ where $\lambda$ is a constant. Fin…
29The sketch below shows the graph of $y = \text{f}(x)$ . The curve intersects the $y$ -axis at the point $\left(0, -\frac{11}{2}\right)$ …
30The diagram shows the curve $y = f(x)$ , with turning points at $A(3, 2.5)$ and $B(0, 1)$ . The lines $y = 1$ and $x = -2$ …
31The functions f and g are defined by $f:x\mapsto |(x+1)(x-3)|,\quad x\in\mathbb{R},$ $g:x\mapsto x-2,\quad x\in\mathbb{R}\ \text{and } x\ge -1.$ …
32A curve $C_1$ is defined parametrically by $x = \frac{2}{t-1}\quad\text{and}\quad y = \frac{4}{t+1},\qquad t \ne \pm 1.$ …
33The diagram below shows the graph of $y = \mathrm{f}(x)$ . The curve cuts the axes at $A(0, 1.5)$ and $B(-3, 0)$ . The asy…
34The diagrams below show the graphs of $y = |\mathrm{f}(x)|$ and $y = \mathrm{f}'(x)$ . On separate diagrams, sketch the g…
35The diagram shows the graphs of $y = |a - bx|$ and $y = |mx - 6|$ where $a$ , $b$ and $m$ are positive constants. The gra…
36Show that $y=\ln\left(\dfrac{e^{2}}{3x}\right)$ can be written in the form $y=a+b\ln(cx)$ , where $a$ , $b$ and $c$ are in…
37The graphs of $y = \mathrm{f}'(x)$ and $y = |\mathrm{f}(x)|$ are shown below. State the nature of all turning point(s) o…
38The graph of $y = |\mathrm{f}(x)|$ has a turning point at $(0,2)$ and passes through the points $(-4,0)$ and $(4,0)$ . Th…
39The curve $C$ has equation $y=\ln\left(2-\lvert x\rvert\right)$ . The region $R$ is bounded by $C$ , the lines $y=\ln 2$ ,…
40The graph of $y = \text{f}(x)$ is given above. Sketch, on separate diagrams, the graphs of Your sketches should show cle…
41The graphs of $y=f'(x)$ and $y=|f(x)|$ are shown below. State the nature of all turning point(s) of the graph of $y=f(x)$ …
42Describe fully a sequence of transformations which transforms the curve $y^2=x$ onto the curve $\dfrac{y^2}{4}=1-x$ . A c…
43On the same axes, sketch the curves with equations $y = \left| \frac{ax + 1}{1 - ax} \right|$ and $y = \frac{1}{a}x + 1$ …
44A curve $G$ with equation $y = \mathrm{f}(3x-2)+a$ , where $a$ is a positive real constant, is transformed onto the curve…
45A curve $C$ has equation $\text{f}(x) = \frac{a}{x^2} + bx + c$ , where $a$ , $b$ and $c$ are constants. It is given that…
46Sketch the graphs of $y = \dfrac{1}{f(x)}$ , $y = f'(x)$ , $y = |f(x)|$ , $y = f(|x|)$ .
47A function f is defined as $f(x)=\ln(5-x)$ , $x<5$ . Find the volume of revolution when the region bounded by the curve of…
48The function f is defined by $f(x)=ax^3+bx^2+cx+d$ , where $a,b,c$ and $d$ are real numbers. Given that $1-2i$ and $-1$ a…
49A function f is defined as $f(x) = \ln(5-x)$ , $x < 5$ . Find the volume of revolution when the region bounded by the curv…
50The graph of $y = \mathrm{f}(x)$ , where $\mathrm{f}(x) = ax^2 + bx + c$ , passes through the point $(10, -28)$ . When the…
51Without using a calculator, solve the inequality $\dfrac{2x^2 - x}{x^2 + 3x - 4} > 1$ . On the same axes, sketch the grap…
52Describe a sequence of two transformations that maps the graph of $y = \ln\left(\frac{x^2}{x+1}\right)$ onto the graph o…
53The diagram below shows the graph of $y=f(x)$ for $x<\frac{3}{2}$ . The graph has stationary points at $(-\frac{\pi}{4}, -2)$ …
54The curve $C$ has equation $y = 2 + \frac{3a}{x - a}$ , where $a$ is a positive constant. Describe fully a sequence of tr…
55Find the exact roots of the equation $|x^2 - 7x + 3| = 13 - x$ . On the same axes, sketch the curves with equations $y = |x^2 - 7x + 3|$ …