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Date November 2018 Marks available 4 Reference code 18N.2.AHL.TZ0.H_9
Level Additional Higher Level Paper Paper 2 Time zone Time zone 0
Command term Find Question number H_9 Adapted from N/A

Question

The function  f is defined by f ( x ) = 2 ln x + 1 x 3 , 0 <  x < 3.

Draw a set of axes showing  x and  y  values between −3 and 3. On these axes

Find f ( x ) .

[4]
a.

Hence, or otherwise, find the coordinates of the point of inflexion on the graph of  y = f ( x ) .

[4]
b.

sketch the graph of y = f ( x ) , showing clearly any axis intercepts and giving the equations of any asymptotes.

[4]
c.i.

sketch the graph of y = f 1 ( x ) , showing clearly any axis intercepts and giving the equations of any asymptotes.

[4]
c.ii.

Hence, or otherwise, solve the inequality f ( x ) > f 1 ( x ) .

[3]
d.

Markscheme

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

METHOD 1

f ( x ) = 2 ( x 3 ) x ( 2 ln x + 1 ) ( x 3 ) 2 ( = 2 ( x 3 ) x ( 2 ln x + 1 ) x ( x 3 ) 2 )       (M1)A1A1A1

Note: Award M1 for attempt at quotient rule, A1A1 for numerator and A1 for denominator.

 

METHOD 2

f ( x ) = ( 2 ln x + 1 ) ( x 3 ) 1       (A1)

f ( x ) = ( 2 x ) ( x 3 ) 1 ( 2 ln x + 1 ) ( x 3 ) 2 ( = 2 ( x 3 ) x ( 2 ln x + 1 ) x ( x 3 ) 2 )       (M1)A1A1

Note: Award M1 for attempt at product rule, A1 for first term, A1 for second term.

 

[4 marks]

a.

finding turning point of  y = f ( x ) or finding root of y = f ( x )        (M1)

x = 0.899        A1

y = f ( 0.899048 ) = 0.375       (M1)A1

(0.899, −0.375)

Note: Do not accept x = 0.9 . Accept y-coordinates rounding to −0.37 or −0.375 but not −0.38.
 

[4 marks]

b.

smooth curve over the correct domain which does not cross the y-axis

and is concave down for x  > 1       A1

x -intercept at 0.607       A1

equations of asymptotes given as x  = 0 and x  = 3 (the latter must be drawn)       A1A1
 

[4 marks]

c.i.

attempt to reflect graph of f in y  = x        (M1)

smooth curve over the correct domain which does not cross the x -axis and is concave down for y  > 1       A1

y -intercept at 0.607       A1

equations of asymptotes given as y  = 0 and y  = 3 (the latter must be drawn)       A1

Note: For FT from (i) to (ii) award max M1A0A1A0.


[4 marks]

c.ii.

solve  f ( x ) = f 1 ( x ) or  f ( x ) = x to get x  = 0.372        (M1)A1

0 <  x < 0.372      A1

Note: Do not award FT marks.


[3 marks]

d.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.i.
[N/A]
c.ii.
[N/A]
d.

Syllabus sections

Topic 5—Calculus » AHL 5.9—Differentiating standard functions and derivative rules
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Topic 5—Calculus

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