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Date May 2017 Marks available 1 Reference code 17M.1.SL.TZ1.S_6
Level Standard Level Paper Paper 1 Time zone Time zone 1
Command term Write down Question number S_6 Adapted from N/A

Question

The following diagram shows the graph of f , the derivative of f .

M17/5/MATME/SP1/ENG/TZ1/06

The graph of f has a local minimum at A, a local maximum at B and passes through ( 4 ,   2 ) .

The point P ( 4 ,   3 ) lies on the graph of the function, f .

Write down the gradient of the curve of f at P.

[1]
a.i.

Find the equation of the normal to the curve of f at P.

[3]
a.ii.

Determine the concavity of the graph of f when 4 < x < 5 and justify your answer.

[2]
b.

Markscheme

2     A1     N1

[1 mark]

a.i.

gradient of normal = 1 2     (A1)

attempt to substitute their normal gradient and coordinates of P (in any order)     (M1)

eg y 4 = 1 2 ( x 3 ) ,   3 = 1 2 ( 4 ) + b ,   b = 1

y 3 = 1 2 ( x 4 ) ,   y = 1 2 x + 1 ,   x 2 y + 2 = 0     A1     N3

[3 marks]

a.ii.

correct answer and valid reasoning     A2     N2

answer:     eg     graph of f is concave up, concavity is positive (between 4 < x < 5 )

reason:     eg     slope of f is positive, f is increasing, f > 0 ,

sign chart (must clearly be for f and show A and B)

M17/5/MATME/SP1/ENG/TZ1/06.b/M

 

Note:     The reason given must refer to a specific function/graph. Referring to “the graph” or “it” is not sufficient.

 

[2 marks]

b.

Examiners report

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

Syllabus sections

Topic 4—Statistics and probability » SL 4.1—Concepts, reliability and sampling techniques
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Topic 4—Statistics and probability
Topic 5—Calculus

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