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Date	May 2017	Marks available	1	Reference code	17M.2.SL.TZ2.S_8
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Write down	Question number	S_8	Adapted from	N/A

Question

Let $f(x) = - 0.5{x^4} + 3{x^2} + 2x$ . The following diagram shows part of the graph of $f$ .

M17/5/MATME/SP2/ENG/TZ2/08

There are $x$ -intercepts at $x = 0$ and at $x = p$ . There is a maximum at A where $x = a$ , and a point of inflexion at B where $x = b$ .

Find the value of $p$ .

[2]

a.

Write down the coordinates of A.

[2]

b.i.

Write down the rate of change of $f$ at A.

[1]

b.ii.

Find the coordinates of B.

[4]

c.i.

Find the the rate of change of $f$ at B.

[3]

c.ii.

Let $R$ be the region enclosed by the graph of $f$ , the $x$ -axis, the line $x = b$ and the line $x = a$ . The region $R$ is rotated 360° about the $x$ -axis. Find the volume of the solid formed.

[3]

d.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

evidence of valid approach (M1)

eg $\,\,\,\,\,$ $f(x) = 0,{\text{ }}y = 0$

2.73205

$p = 2.73$ A1 N2

[2 marks]

a.

1.87938, 8.11721

$(1.88,{\text{ }}8.12)$ A2 N2

[2 marks]

b.i.

rate of change is 0 (do not accept decimals) A1 N1

[1 marks]

b.ii.

METHOD 1 (using GDC)

valid approach M1

eg $\,\,\,\,\,$ $f’’ = 0$ , max/min on $f’,{\text{ }}x = - 1$

sketch of either $f^{'}$ or $f^{″}$ , with max/min or root (respectively) (A1)

$x = 1$ A1 N1

Substituting their $x$ value into $f$ (M1)

eg $\,\,\,\,\,$ $f(1)$

$y = 4.5$ A1 N1

METHOD 2 (analytical)

$f’’ = - 6{x^2} + 6$ A1

setting $f’’ = 0$ (M1)

$x = 1$ A1 N1

substituting their $x$ value into $f$ (M1)

eg $\,\,\,\,\,$ $f(1)$

$y = 4.5$ A1 N1

[4 marks]

c.i.

recognizing rate of change is $f^{'}$ (M1)

eg $\,\,\,\,\,$ $y’,{\text{ }}f’(1)$

rate of change is 6 A1 N2

[3 marks]

c.ii.

attempt to substitute either limits or the function into formula (M1)

involving ${f^2}$ (accept absence of $\pi$ and/or ${\text{d}}x$ )

eg $\,\,\,\,\,$ $\pi \int {{{( - 0.5{x^4} + 3{x^2} + 2x)}^2}{\text{d}}x,{\text{ }}\int_1^{1.88} {{f^2}} }$

128.890

${\text{volume}} = 129$ A2 N3

[3 marks]

d.

Examiners report

[N/A]

a.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

c.i.

[N/A]

c.ii.

[N/A]

d.

Syllabus sections

Topic 4—Statistics and probability » SL 4.1—Concepts, reliability and sampling techniques

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18M.1.SL.TZ1.S_2b:
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18N.2.AHL.TZ0.H_10e:
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17M.2.SL.TZ2.S_8c.i:
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17M.1.SL.TZ1.S_9b:
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16N.2.AHL.TZ0.H_11d:
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EXN.1.SL.TZ0.1b:
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16N.2.SL.TZ0.S_10b:
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17M.2.SL.TZ2.S_8d:
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17M.2.SL.TZ2.S_8a:
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SPM.1.SL.TZ0.3c:
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17N.1.SL.TZ0.S_5b:
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19M.2.SL.TZ1.S_9a:
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19M.2.SL.TZ1.S_3b:
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21M.2.SL.TZ1.1a:
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18N.2.SL.TZ0.S_10b.i:
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16N.2.SL.TZ0.S_10c:
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21M.3.AHL.TZ1.2b:
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21M.2.SL.TZ2.1a:
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16N.2.AHL.TZ0.H_11c:
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21M.3.AHL.TZ2.1b:
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EXN.1.SL.TZ0.1a:
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17M.2.SL.TZ2.S_8c.ii:
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18M.1.SL.TZ1.S_7:
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Given that g(3) = 7 , g′ (3) = 4 and f ′ (7) = −5 , find the gradient of the normal to the curve of h at x = 3.

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