Math topic中文

Relating graphs fprime f

Relating the graph of $y = \mathrm{f}'(x)$ to the graph of $y = \mathrm{f}(x)$ .

Practice questions

1A 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…
2The diagram shows the graphs of $y = \mathrm{f}(|x|)$ and $y = \mathrm{f}'(x)$ . The graph of $y = \mathrm{f}(|x|)$ has t…
3The diagram shows the curve $y = \text{f}(1-x)$ . The curve has a maximum point at $(-1, 5)$ and two asymptotes $x = -2$ …
4The graph of $y = \text{f}(x)$ has a maximum turning point at $(2,3)$ and passes through the origin. The lines $x = -1$ …
5The diagram shows the graph of $y = f(x)$ . The curve has a turning point at $(0,a)$ and the lines $y = \pm kx$ are asymp…
6The 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…
7The diagram below shows the graph of $y = \text{f}(x)$ . The curve passes through the $x$ -axis at $(-2, 0)$ and $(1, 0)$ ,…
8The diagram shows the graph of $y = \text{f}'(x)$ , which passes through the origin and has turning points at $(2, 5)$ an…
9The 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…
10The 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…
11The curve $C$ has equation $y = \frac{x^2 + (\lambda - 1)x}{x - 1}, \quad x \ne 1,$ where $\lambda$ is a constant. Fin…
12The diagram shows the curve with equation $y = \mathrm{f}(x)$ . The curve crosses the $x$ -axis at $x = 1$ and $x = 2.5$ ,…
13It is given that $\mathrm{f}(x) = 2x^6 - 4x^4 - 6x^2 - 7$ . The diagram shows the curve with equation $y = \mathrm{f}(x)$ …
14The sketch below shows the graph of $y = \text{f}(x)$ . The curve intersects the $y$ -axis at the point $\left(0, -\frac{11}{2}\right)$ …
15The diagram below shows the graph of curve $C_1$ , with equation $y = \mathrm{f}(x)$ , defined for $x \in \mathbb{R} \setminus \{4\}$ …
16The diagram below shows the curve $y = \text{f}'(x)$ . It has turning points at $(0, 6)$ , $(5, 0)$ and $(7, 7)$ and inter…
17The 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…
18The diagrams below show the graphs of $y = |\mathrm{f}(x)|$ and $y = \mathrm{f}'(x)$ . On separate diagrams, sketch the g…
19Show 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…
20The diagram below shows the graph of y = f(x). The curve passes through the x-axis at (-2, 0) and (1, 0), and has a maxi…
21The graphs of $y = \mathrm{f}'(x)$ and $y = |\mathrm{f}(x)|$ are shown below. State the nature of all turning point(s) o…
22The curve $C$ has equation $y=\dfrac{x^2+(\lambda-1)x}{x-1},\quad x\ne 1,$ where $\lambda$ is a constant. Find the equ…
23The graph of $y = \text{f}(x)$ is given above. Sketch, on separate diagrams, the graphs of Your sketches should show cle…
24The graphs of $y=f'(x)$ and $y=|f(x)|$ are shown below. State the nature of all turning point(s) of the graph of $y=f(x)$ …
25A curve $C$ has parametric equations $x = (a^2 - t^2)^{\frac{1}{2}}, \quad y = \ln t, \quad \text{for } 0 < t \le a \text{ and } a > 1.$ …
26A curve $C$ has parametric equations $x = 3t^3,\ y = \frac{3}{1+t^2},\quad t \in \mathbb{R}.$ Find $\frac{\text{d}y}{\text{d}x}$ …
27Sketch the graphs of $y = \dfrac{1}{f(x)}$ , $y = f'(x)$ , $y = |f(x)|$ , $y = f(|x|)$ .
28The function f is given by $f : x \mapsto a + \dfrac{3a}{(x+3)(x-1)}$ , where $a>0$ . Using calculus, find the x-coordinat…
29The diagram below shows the graph of $y=f(x)$ for $x<\frac{3}{2}$ . The graph has stationary points at $(-\frac{\pi}{4}, -2)$ …
30The curve $C$ has equation $y = \dfrac{x^2 + (\lambda - 1)x}{x - 1}, \quad x \ne 1,$ where $\lambda$ is a constant. Fi…