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Date May 2017 Marks available 2 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 + 2 x . 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 ,   y = 0

2.73205

p = 2.73     A1     N2

[2 marks]

a.

1.87938, 8.11721

( 1.88 ,   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 ,   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 ,   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 π and/or d x )

eg π ( 0.5 x 4 + 3 x 2 + 2 x ) 2 d x ,   1 1.88 f 2

128.890

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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Topic 5 —Calculus » SL 5.8—Testing for max and min, optimisation. Points of inflexion
Topic 4—Statistics and probability
Topic 5 —Calculus

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