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Date May 2017 Marks available 1 Reference code 17M.1.sl.TZ1.6
Level SL only Paper 1 Time zone TZ1
Command term Write down Question number 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 \((4,{\text{ }} - 2)\).

The point \({\text{P}}(4,{\text{ }}3)\) 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 \( = \frac{1}{2}\)     (A1)

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

eg\(\,\,\,\,\,\)\(y - 4 = \frac{1}{2}(x - 3),{\text{ }}3 = \frac{1}{2}(4) + b,{\text{ }}b = 1\)

\(y - 3 = \frac{1}{2}(x - 4),{\text{ }}y = \frac{1}{2}x + 1,{\text{ }}x - 2y + 2 = 0\)     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 6 - Calculus » 6.3 » Graphical behaviour of functions, including the relationship between the graphs of \(f\) , \({f'}\) and \({f''}\) .

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