17M.1.SL.TZ1.S_6

pestleMathematicsAISLPaper 117M· sl-4-1-concepts-reliability-and-sampling-techniquessource ↗

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]

Markscheme / solution

$- 2$ 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]

Examiners’ report

[N/A]

a.i.

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a.ii.

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