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Date May 2019 Marks available 2 Reference code 19M.1.AHL.TZ1.H_8
Level Additional Higher Level Paper Paper 1 Time zone Time zone 1
Command term Find Question number H_8 Adapted from N/A

Question

The graph of y = f ( x ) , 0 ≤ x  ≤ 5 is shown in the following diagram. The curve intercepts the x -axis at (1, 0) and (4, 0) and has a local minimum at (3, −1).

The shaded area enclosed by the curve y = f ( x ) , the x -axis and the y -axis is 0.5. Given that f ( 0 ) = 3 ,

The area enclosed by the curve y = f ( x ) and the x -axis between x = 1 and x = 4 is 2.5 .

Write down the x -coordinate of the point of inflexion on the graph of  y = f ( x ) .

[1]
a.

find the value of  f ( 1 ) .

[3]
b.

find the value of  f ( 4 ) .

[2]
c.

Sketch the curve y = f ( x ) , 0 ≤ x ≤ 5 indicating clearly the coordinates of the maximum and minimum points and any intercepts with the coordinate axes.

[3]
d.

Markscheme

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

3     A1

[1 mark]

a.

attempt to use definite integral of  f ( x )         (M1)

0 1 f ( x ) d x = 0.5

f ( 1 ) f ( 0 ) = 0.5         (A1)

f ( 1 ) = 0.5 + 3

= 3.5      A1

[3 marks]

b.

1 4 f ( x ) d x = 2.5        (A1)

Note: (A1) is for −2.5.

f ( 4 ) f ( 1 ) = 2.5

f ( 4 ) = 3.5 2.5

= 1      A1

[2 marks]

c.

    A1A1A1

A1 for correct shape over approximately the correct domain
A1 for maximum and minimum (coordinates or horizontal lines from 3.5 and 1 are required),
A1 for y -intercept at 3

[3 marks]

d.

Examiners report

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

Syllabus sections

Topic 2—Functions » SL 2.2—Functions, notation domain, range and inverse as reflection
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Topic 5—Calculus

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