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Date	November 2021	Marks available	3	Reference code	21N.1.SL.TZ0.5
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Hence and Find	Question number	5	Adapted from	N/A

Question

The function $f$ is defined for all $x \in ℝ$ . The line with equation $y = 6 x - 1$ is the tangent to the graph of $f$ at $x = 4$ .

The function $g$ is defined for all $x \in ℝ$ where $g (x) = x^{2} - 3 x$ and $h (x) = f (g (x))$ .

Write down the value of $f' (4)$ .

[1]

a.

Find $f (4)$ .

[1]

b.

Find $h (4)$ .

[2]

c.

Hence find the equation of the tangent to the graph of $h$ at $x = 4$ .

[3]

d.

Markscheme

$f' (4) = 6$ A1

[1 mark]

a.

$f (4) = 6 \times 4 - 1 = 23$ A1

[1 mark]

b.

$h (4) = f (g (4))$ (M1)

$h (4) = f (4^{2} - 3 \times 4) = f (4)$

$h (4) = 23$ A1

[2 marks]

c.

attempt to use chain rule to find $h'$ (M1)

$f' (g (x)) \times g' (x)$ OR $(x^{2} - 3 x)' \times f' (x^{2} - 3 x)$

$h' (4) = (2 \times 4 - 3) f' (4^{2} - 3 \times 4)$ A1

$= 30$

$y - 23 = 30 (x - 4)$ OR $y = 30 x - 97$ A1

[3 marks]

d.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

[N/A]

d.

Syllabus sections

Topic 2—Functions » SL 2.1—Equations of straight lines, parallel and perpendicular

22M.1.SL.TZ1.1a:
Find the equation of $L$ .
22M.2.SL.TZ2.3b:
Assuming Kaia’s annual salary can be approximately modelled by the equation $S = a x + b$ , show that Kaia had a higher salary than Gemma in the year 2021, according to the model.
21N.1.SL.TZ0.5b:
Find $f (4)$ .
18N.2.SL.TZ0.T_4a:
Sketch the graph of y = f (x), for −4 ≤ x ≤ 3 and −50 ≤ y ≤ 100.
18N.2.SL.TZ0.T_4b.iii:
Use your graphic display calculator to find the equation of the tangent to the graph of y = f (x) at the point (–2, 38.75).

Give your answer in the form y = mx + c.
18N.2.SL.TZ0.T_4c:
Sketch the graph of the function g (x) = 10x + 40 on the same axes.
21N.1.SL.TZ0.5a:
Write down the value of $f' (4)$ .
22M.1.SL.TZ1.1b:
The line $L$ passes through the point $(k, 2)$ .

Find the value of $k$ .
22M.1.SL.TZ2.7c:
Find the equation of $L$ .
17M.1.SL.TZ2.T_4c:
Find the $x$ -coordinate of point A.
21N.1.SL.TZ0.5c:
Find $h (4)$ .
17M.1.SL.TZ2.T_4b:
Find the equation of $L$ in the form $y = mx + c$ .

Topic 5 —Calculus » SL 5.4—Tangents and normal

Topic 2—Functions » SL 2.5—Composite functions, identity, finding inverse

Topic 5 —Calculus » SL 5.6—Differentiating polynomials n E Q. Chain, product and quotient rules

Topic 2—Functions

Topic 5 —Calculus