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Date	May 2018	Marks available	2	Reference code	18M.2.sl.TZ2.3
Level	SL only	Paper	2	Time zone	TZ2
Command term	Find	Question number	3	Adapted from	N/A

Question

Let $f\left( x \right) = \,\,{\text{sin}}\,\left( {{e^x}} \right)$ for 0 ≤ $x$ ≤ 1.5. The following diagram shows the graph of $f$ .

Find the x-intercept of the graph of $f$ .

[2]

The region enclosed by the graph of $f$ , the y-axis and the x-axis is rotated 360° about the x-axis.

Find the volume of the solid formed.

[3]

Markscheme

valid approach (M1)
eg $f\left( x \right) = 0,\,\,\,\,{e^x} = 180$ or 0…

1.14472

$x = {\text{ln}}\,\pi$ (exact), 1.14 A1 N2

[2 marks]

attempt to substitute either their limits or the function into formula involving ${f^2}$ . (M1)

eg ${\int_0^{1.14} {{f^2},\,\,\pi \int {\left( {{\text{sin}}\,\left( {{e^x}} \right)} \right)} } ^2}dx,\,\,0.795135$

2.49799

volume = 2.50 A2 N3

[3 marks]

Examiners report

[N/A]

Syllabus sections

Topic 2 - Functions and equations » 2.2 » Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes, symmetry, and consideration of domain and range.

Show 94 related questions

18M.2.sl.TZ2.7c: The line y = k, where $k \in \mathbb{R}$ intersects the graph...
18M.2.sl.TZ2.7b: Write down the equation of the horizontal asymptote to the graph of f.
18M.2.sl.TZ2.7a: The line x = 3 is a vertical asymptote to the graph of f. Find the value of c.
18M.2.sl.TZ2.3b: The region enclosed by the graph of $f$ , the y-axis and the x-axis is rotated 360° about...
18M.2.sl.TZ1.4c: Find the area of the region enclosed by the graphs of f and g.
18M.2.sl.TZ1.4b: On the grid above, sketch the graph of g for −2 ≤ x ≤ 4.
18M.2.sl.TZ1.4a: Write down the coordinates of the vertex of the graph of g.
17N.2.sl.TZ0.2c: On the following grid, sketch the graph of $f$ .
17N.2.sl.TZ0.2b: The graph of $f$ has a maximum at the point A. Write down the coordinates of A.
17N.2.sl.TZ0.2a: Find the $x$ -intercept of the graph of $f$ .
17M.2.sl.TZ2.8d: Let $R$ be the region enclosed by the graph of $f$ , the $x$ -axis, the line $x = b$ ...
17M.2.sl.TZ2.8c.i: Find the coordinates of B.
17M.2.sl.TZ2.8b.ii: Write down the rate of change of $f$ at A.
17M.2.sl.TZ2.8b.i: Write down the coordinates of A.
17M.2.sl.TZ2.8a: Find the value of $p$ .
17M.1.sl.TZ2.10d: Given that the area of triangle ABC is $p$ times the area of $R$ , find the value of $p$ .
17M.1.sl.TZ2.10c: Find the area of triangle ABC, giving your answer in terms of $k$ .
17M.1.sl.TZ2.10b: Show that the $x$ -coordinate of B is $- \frac{k}{2}$ .
17M.1.sl.TZ2.10a.ii: Find the gradient of $L$ .
17M.1.sl.TZ2.10a.i: Write down $f'(x)$ .
17M.1.sl.TZ1.9c: The line $y = kx - 5$ is a tangent to the curve of $f$ . Find the values of $k$ .
17M.1.sl.TZ1.9b: Find the value of $a$ .
17M.1.sl.TZ1.9a: Find the value of $p$ .
16M.2.sl.TZ2.9e: There is a value of $x$ , for $1 < x < 4$ , for which the graphs of $f$ and $g$ ...
16M.2.sl.TZ2.9d: Given that $g'(1) = - e$ , find the value of $a$ .
16M.2.sl.TZ2.9c: Write down the value of $b$ .
16M.2.sl.TZ2.9b: Find $f'(x)$ .
16M.2.sl.TZ2.9a: Write down the equation of the horizontal asymptote of the graph of $f$ .
16M.1.sl.TZ2.1c: Find the $y$ -intercept.
16M.1.sl.TZ2.1b: Write down the value of $a$ and of $b$ .
16M.1.sl.TZ2.1a: Write down the value of $h$ and of $k$ .
16M.2.sl.TZ1.9e: Find the maximum speed of P.
16M.2.sl.TZ1.9d: Find the acceleration of P after 3 seconds.
16M.2.sl.TZ1.9c: Write down the number of times P changes direction.
16M.2.sl.TZ1.9b: Find when P is first at rest.
16M.2.sl.TZ1.9a: Find the displacement of P from O after 5 seconds.
16M.1.sl.TZ1.5a: Show that the two zeros are 3 and $- 6$ .
16N.2.sl.TZ0.2b: (i) sketch the graph of $f$ , clearly indicating the point A; (ii) sketch the...
12N.2.sl.TZ0.3a: Write down the x-coordinate of P.
12N.2.sl.TZ0.3b: Write down the gradient of the curve at P.
12N.2.sl.TZ0.3c: Find the equation of the normal to the curve at P, giving your equation in the form...
12N.2.sl.TZ0.5a(i) and (ii): Write down the value of (i) $a$ ; (ii) $c$ .
12N.2.sl.TZ0.5b: Find the value of b .
12N.2.sl.TZ0.5c: Find the x-coordinate of R.
12M.1.sl.TZ2.10a: Use the quotient rule to show that...
12M.1.sl.TZ2.10b: Hence find the coordinates of B.
12M.1.sl.TZ2.10c: Given that the line $y = k$ does not meet the graph of f , find the possible values of k .
12M.2.sl.TZ2.9a: Show that $8a + 4b + c = 9$ .
12M.2.sl.TZ2.9b: The graph of f has a local minimum at $(1{\text{, }}4)$ . Find two other equations in a ,...
12M.2.sl.TZ2.9c: Find the value of a , of b and of c .
08N.2.sl.TZ0.4b: The graph of f intersects the x-axis when $x = a$ , $a \ne 0$ . Write down the value of a.
08M.2.sl.TZ2.9b: Write down the equation of the horizontal asymptote.
10M.1.sl.TZ2.10a(i) and (ii): Solve for $0 \le x < 2\pi$ (i) $6 + 6\sin x = 6$ ; (ii) $6 + 6\sin x = 0$ .
10M.1.sl.TZ2.10b: Write down the exact value of the x-intercept of f , for $0 \le x < 2\pi$ .
10M.1.sl.TZ2.10c: The area of the shaded region is k . Find the value of k , giving your answer in terms of...
10M.1.sl.TZ2.10d: Let $g(x) = 6 + 6\sin \left( {x - \frac{\pi }{2}} \right)$ . The graph of f is transformed...
10M.1.sl.TZ2.10e: Let $g(x) = 6 + 6\sin \left( {x - \frac{\pi }{2}} \right)$ . The graph of f is transformed...
09N.1.sl.TZ0.9a: (i) Find the coordinates of A. (ii) Show that $f'(x) = 0$ at A.
09N.1.sl.TZ0.9d: Write down the range of $f$ .
09N.1.sl.TZ0.10b: Point A is the x-intercept of L . Find the x-coordinate of A.
09N.2.sl.TZ0.7: The fencing used for side AB costs $\$ 11$ per metre. The fencing for the other three sides...
09M.2.sl.TZ2.10b: Write down (i) the amplitude; (ii) the period; (iii) the x-intercept that lies...
10N.2.sl.TZ0.7a: There are two points of inflexion on the graph of f . Write down the x-coordinates of these...
10N.2.sl.TZ0.7b: Let $g(x) = f''(x)$ . Explain why the graph of g has no points of inflexion.
10N.2.sl.TZ0.8a: Find the value of a and of b .
10N.2.sl.TZ0.8b: The graph of f has a maximum value when $x = c$ . Find the value of c .
10N.2.sl.TZ0.8c: The region under the graph of f from $x = 0$ to $x = c$ is rotated ${360^ \circ }$ ...
10N.2.sl.TZ0.8d: Let R be the region enclosed by the curve, the x-axis and the line $x = c$ , between...
10M.2.sl.TZ2.6a: Write down the x-coordinate of A.
10M.2.sl.TZ2.6c: Find $\int_p^q {f(x){\rm{d}}x}$ . Explain why this is not the area of the shaded region.
10M.2.sl.TZ2.6b(i) and (ii): Find the value of (i) p ; (ii) q .
10M.2.sl.TZ2.10b: Let $g(x) = {x^3}\ln (4 - {x^2})$ , for $- 2 < x < 2$ . Show that...
10M.2.sl.TZ2.10a(i) and (ii): Let P and Q be points on the curve of f where the tangent to the graph of f is parallel to...
10M.2.sl.TZ2.10c: Let $g(x) = {x^3}\ln (4 - {x^2})$ , for $- 2 < x < 2$ . Sketch the graph of $g'$ .
10M.2.sl.TZ2.10d: Let $g(x) = {x^3}\ln (4 - {x^2})$ , for $- 2 < x < 2$ . Consider $g'(x) = w$ ....
SPNone.2.sl.TZ0.2a: Find the x-intercepts of the graph of f .
SPNone.2.sl.TZ0.9b(i), (ii) and (iii): (i) Sketch the graph of h for $- 4 \le x \le 4$ and $- 5 \le y \le 8$ , including...
11N.2.sl.TZ0.10b(i) and (ii): (i) Write down the x-coordinate of the maximum point on the graph of f . (ii) Write...
11N.2.sl.TZ0.10a: Sketch the graph of f .
11N.2.sl.TZ0.10c: Show that $f'(x) = \frac{{20 - 6x}}{{{{\rm{e}}^{0.3x}}}}$ .
11N.2.sl.TZ0.10d: Find the interval where the rate of change of f is increasing.
11M.1.sl.TZ1.7a: Find the value of k .
11M.1.sl.TZ1.7b: The line $y = p$ intersects the graph of f . Find all possible values of p .
13M.2.sl.TZ1.5b.i: Find the total distance travelled by the particle in the first five seconds.
14M.2.sl.TZ1.9b: Find the values of $x$ where the function is decreasing.
14M.2.sl.TZ2.2a: Find the $x$ -coordinate of ${\text{A}}$ and of ${\text{B}}$ .
13N.1.sl.TZ0.8c(i): Find the $y$ -intercept of the graph of $h$ .
15M.2.sl.TZ1.7a: Find the coordinates of the local minimum point.
15M.2.sl.TZ1.10a: The graph of $f$ has a local maximum point when $x = p$ . State the value of $p$ , and...
15M.2.sl.TZ2.7: Let $f(x) = k{x^2} + kx$ and $g(x) = x - 0.8$ . The graphs of $f$ and $g$ intersect at...
15M.2.sl.TZ1.4a: For the graph of $f$ (i) find the $x$ -intercept; (ii) write down the equation...
15M.2.sl.TZ1.7b: The graph of $f$ is translated to the graph of $g$ by the vector...
15N.2.sl.TZ0.3a: Find the equation of the vertical asymptote to the graph of $f$ .
15N.2.sl.TZ0.3b: Find the $x$ -intercept of the graph of $f$ .

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