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Date	May 2022	Marks available	2	Reference code	22M.2.AHL.TZ1.6
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Show that and Hence	Question number	6	Adapted from	N/A

Question

Consider the curve $y = \sqrt{x}$ .

The shape of a piece of metal can be modelled by the region bounded by the functions $f$ , $g$ , the $x$ -axis and the line segment $[AB]$ , as shown in the following diagram. The units on the $x$ and $y$ axes are measured in metres.

The piecewise function $f$ is defined by

$f (x) = {\begin{matrix} \sqrt{x} & 0 \leq x \leq 0.16 \\ 1.25 x + 0.2 & 0.16 < x \leq 0.5 \end{matrix}$

The graph of $g$ is obtained from the graph of $f$ by:

a stretch scale factor of $\frac{1}{2}$ in the $x$ direction,
followed by a stretch scale factor $\frac{1}{2}$ in the $y$ direction,
followed by a translation of $0.2$ units to the right.

Point $A$ lies on the graph of $f$ and has coordinates $(0.5, 0.825)$ . Point $B$ is the image of $A$ under the given transformations and has coordinates $(p, q)$ .

The piecewise function $g$ is given by

$g (x) = {\begin{matrix} h (x) & 0.2 \leq x \leq a \\ 1.25 x + b & a < x \leq p \end{matrix}$

The area enclosed by $y = g (x)$ , the $x$ -axis and the line $x = p$ is $0.0627292 m^{2}$ correct to six significant figures.

Find $\frac{d y}{d x}$ .

[2]

a.i.

Hence show that the equation of the tangent to the curve at the point $(0.16, 0.4)$ is $y = 1.25 x + 0.2$ .

[2]

a.ii.

Find the value of $p$ and the value of $q$ .

[2]

Find an expression for $h (x)$ .

[2]

c.i.

Find the value of $a$ .

[1]

c.ii.

Find the value of $b$ .

[2]

c.iii.

Find the area enclosed by $y = f (x)$ , the $x$ -axis and the line $x = 0.5$ .

[3]

d.i.

Find the area of the shaded region on the diagram.

[4]

d.ii.

Markscheme

$y = x^{\frac{1}{2}}$ (M1)

$\frac{d y}{d x} = \frac{1}{2} x^{- \frac{1}{2}}$ A1

[2 marks]

a.i.

gradient at $x = 0.16$ is $\frac{1}{2} \times \frac{1}{\sqrt{0.16}}$ M1

$= 1.25$

EITHER

$y - 0.4 = 1.25 (x - 0.16)$ M1

$0.4 = 1.25 (0.16) + b$ M1

Note: Do not allow working backwards from the given answer.

THEN

hence $y = 1.25 x + 0.2$ AG

[2 marks]

a.ii.

$p = 0.45, q = 0.4125$ (or $0.413$ ) (accept " $(0.45, 0.4125)$ ") A1A1

[2 marks]

$(h (x) =) \frac{1}{2} \sqrt{2 (x - 0.2)}$ A2

Note: Award A1 if only two correct transformations are seen.

[2 marks]

c.i.

$(a =) 0.28$ A1

[1 mark]

c.ii.

EITHER

Correct substitution of their part (b) (or $(0.28, 0.2)$ ) into the given expression (M1)

$\frac{1}{2} (1.25 \times 2 (x - 0.2) + 0.2)$ (M1)

Note: Award M1 for transforming the equivalent expression for $f$ correctly.

THEN

$(b =) - 0.15$ A1

[2 marks]

c.iii.

recognizing need to add two integrals (M1)

$\int_{0}^{0.16} \sqrt{x} d x + \int_{0.16}^{0.5} (1.25 x + 0.2) d x$ (A1)

Note: The second integral could be replaced by the formula for the area of a trapezoid $\frac{1}{2} \times 0.34 (0.4 + 0.825)$ .

$0.251 m^{2} (0.250916 \dots)$ A1

[3 marks]

d.i.

EITHER

area of a trapezoid $\frac{1}{2} \times 0.05 (0.4125 + 0.825) = 0.0309375$ (M1)(A1)

$\int_{0.45}^{0.5} (8.25 x - 3.3) d x = 0.0309375$ (M1)(A1)

Note: If the rounded answer of $0.413$ from part (b) is used, the integral is $\int_{0.45}^{0.5} (8.24 x - 3.295) d x = 0.03095$ which would be awarded (M1)(A1).

THEN

shaded area $= 0.250916 \dots - 0.0627292 - 0.0309375$ (M1)

Note: Award (M1) for the subtraction of both $0.0627292 \dots$ and their area for the trapezoid from their answer to (a)(i).

$= 0.157 m^{2} (0.15725)$ A1

[4 marks]

d.ii.

Examiners report

The differentiation using the power rule was well done. In part (ii) some candidates felt it was sufficient to refer to the equation being the same as the one generated by their calculator. Generally, for ‘show that’ questions an algebraic derivation is expected.

a.i.

[N/A]

a.ii.

The candidates were successful at applying transformations to points but very few were able to apply these transformations to derive the correct function h. In most cases it was due to not appreciating the effect the horizontal transformations have on x.

c.i.

[N/A]

c.ii.

[N/A]

c.iii.

Part (i) was frequently done well using the inbuilt functionality of the GDC. Part (ii) was less structured, and candidates needed to create a clear diagram so they could easily see which areas needed to be subtracted. Most of those who were successful used the formula for the trapezoid for the area they needed to find, though others were also successful through finding the equation of the line AB.

d.i.

[N/A]

d.ii.

Syllabus sections

Topic 5—Calculus » SL 5.4—Tangents and normal

Show 112 related questions

18N.2.SL.TZ0.T_4a:
Sketch the graph of y = f (x), for −4 ≤ x ≤ 3 and −50 ≤ y ≤ 100.
18M.2.SL.TZ1.T_4e:
Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.
Clearly indicate the minimum point P and the x-intercepts on your graph.
19M.1.AHL.TZ1.H_5:
A camera at point C is 3 m from the edge of a straight section of road as shown in the following diagram. The camera detects a car travelling along the road at $t$ = 0. It then rotates, always pointing at the car, until the car passes O, the point on the edge of the road closest to the camera.

A car travels along the road at a speed of 24 ms⁻¹. Let the position of the car be X and let OĈX = θ.

Find $\frac{{{\text{d}}\theta }}{{{\text{d}}t}}$ , the rate of rotation of the camera, in radians per second, at the instant the car passes the point O .
19N.2.SL.TZ0.T_4c:
Use the symmetry of the graph to show that the second solution is $x = 4$ .
17N.2.AHL.TZ0.H_10d:
This region is now rotated through $2\pi$ radians about the $x$ -axis. Find the volume of revolution.
17M.2.SL.TZ1.T_6d.i:
Find $g’( - 1)$ .
EXN.1.SL.TZ0.7b:
Show that the normal to the curve at the point where $x = 1$ is $2 y - x + 3 = 0$ .
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.
22M.1.SL.TZ2.11a:
Find $f' (x)$ .
22M.1.SL.TZ2.11b:
Use your answer to part (a) to find the gradient of $L$ .
22M.1.SL.TZ1.9b:
Find the equation of the normal to the curve $y = f (x)$ at $(1, 2)$ in the form $a x + b y + d = 0$ , where $a, b, d \in ℤ$ .
17M.1.SL.TZ2.T_13a:
Write down the value of $f(1)$ .
17M.2.SL.TZ2.S_8b.i:
Write down the coordinates of A.
18N.2.SL.TZ0.T_4b.i:
Use your graphic display calculator to find the zero of f (x).
19M.2.SL.TZ2.T_5f:
Find the equation of the tangent line to the graph of $y = f\left( x \right)$ at $x = 2$ . Give the equation in the form $ax + by + d = 0$ where, $a$ , $b$ , and $d \in \mathbb{Z}$ .
18M.2.SL.TZ1.T_4a:
Find the value of k.
17N.1.SL.TZ0.T_14b:
Find the point on the graph of $f$ at which the gradient of the tangent is equal to 6.
17M.2.SL.TZ2.T_6f:
Write down the number of possible solutions to the equation $f(x) = 5$ .
18M.1.SL.TZ2.S_10b:
Show that the graph of g has a gradient of 6 at P.
18M.2.SL.TZ2.T_6f:
Given that y = 2x³ − 9x² + 12x + 2 = k has three solutions, find the possible values of k.
17N.1.SL.TZ0.S_5a:
Find $(g \circ f)(x)$ .
18M.1.SL.TZ1.T_5a:
Write down the coordinates of C, the midpoint of line segment AB.
17M.2.SL.TZ2.S_8b.ii:
Write down the rate of change of $f$ at A.
18M.2.SL.TZ2.T_6a:
Sketch the curve for −1 < x < 3 and −2 < y < 12.
17M.1.SL.TZ2.T_13c:
Draw the line $N$ on the diagram above.
17M.2.SL.TZ2.T_6d.i:
Write down the $x$ -coordinates of these two points;
18M.2.SL.TZ2.T_6b:
A teacher asks her students to make some observations about the curve.

Three students responded.
Nadia said “The x-intercept of the curve is between −1 and zero”.
Rick said “The curve is decreasing when x < 1 ”.
Paula said “The gradient of the curve is less than zero between x = 1 and x = 2 ”.

State the name of the student who made an incorrect observation.
17M.2.SL.TZ2.T_6a:
Write down the $y$ -intercept of the graph.
18M.2.AHL.TZ2.H_11b.i:
Find the coordinates of P and Q.
17N.1.SL.TZ0.T_14a:
Write down the derivative of $f$ .
17M.2.SL.TZ1.T_6d.ii:
Hence justify that $g$ is decreasing at $x = - 1$ .
17M.2.SL.TZ2.T_6c.ii:
Find $f(2)$ .
18M.1.SL.TZ2.S_10a.ii:
Find $f\left( 2 \right)$ .
19N.1.SL.TZ0.T_14a:
Write down the value of $k$ .
17M.2.SL.TZ2.S_8c.i:
Find the coordinates of B.
18M.2.SL.TZ1.T_4b:
Using your value of k , find f ′(x).
18M.1.SL.TZ2.T_14c:
Find the x-coordinate of the point at which the normal to the graph of f has gradient ${ - \frac{1}{8}}$ .
18N.1.SL.TZ0.T_11b:
Find the gradient of this tangent at point P.
16N.1.AHL.TZ0.H_9a:
Find an expression for $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ in terms of $x$ and $y$ .
16N.2.SL.TZ0.S_10a:
(i) Find the value of $c$ .

(ii) Show that $b = \frac{\pi }{6}$ .

(iii) Find the value of $a$ .
18N.1.SL.TZ0.T_11a:
Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ .
19N.2.SL.TZ0.T_4b:
Write down the equation for the axis of symmetry of the graph.
19N.2.SL.TZ0.T_4d:
Write down the $x$ -intercepts of the graph.
18N.2.SL.TZ0.T_4b.ii:
Use your graphic display calculator to find the coordinates of the local minimum point.
19N.2.SL.TZ0.T_4f.ii:
Draw the tangent $T$ on your graph.
17M.2.SL.TZ2.T_6c.i:
Show that $a = 8$ .
18M.1.SL.TZ2.T_14a:
Find f'(x)
16N.1.AHL.TZ0.H_9b:
Find the equations of the tangents to this curve at the points where the curve intersects the line $x = 1$ .
18M.2.AHL.TZ2.H_11c:
Find the coordinates of the three points on C, nearest the origin, where the tangent is parallel to the line $y = - x$ .
18M.1.SL.TZ1.T_5c:
Find the equation of the line DC. Write your answer in the form ax + by + d = 0 where a , b and d are integers.
17N.2.AHL.TZ0.H_10a.i:
Show that the $x$ -coordinate of the minimum point on the curve $y = f(x)$ satisfies the equation $\tan x = 2x$ .
19M.2.SL.TZ1.T_6a:
Show that $k = - 6$ .
17M.2.SL.TZ1.T_6b.ii:
Find the equation of the tangent to the graph of $y = g(x)$ at $x = 2$ . Give your answer in the form $y = mx + c$ .
18M.1.SL.TZ2.S_10c:
Let L₂ be the tangent to the graph of g at P. L₁ intersects L₂ at the point Q.

Find the y-coordinate of Q.
18M.2.AHL.TZ2.H_11b.ii:
Given that the gradients of the tangents to C at P and Q are m₁ and m₂ respectively, show that m₁ × m₂ = 1.
19N.2.SL.TZ0.T_4e:
On graph paper, draw the graph of $y = f (x)$ for $- 10 \leq x \leq 4$ and $- 14 \leq y \leq 14$ . Use a scale of $1 cm$ to represent $1$ unit on the $x$ -axis and $1 cm$ to represent $2$ units on the $y$ -axis.
17M.2.SL.TZ2.T_6d.ii:
Write down the intervals where the gradient of the graph of $y = f(x)$ is positive.
18N.2.SL.TZ0.S_10c:
When $t$ = 0, the volume of water in the container is 2.3 m³. It is known that the container is never completely full of water during the 4 hour period.

Find the minimum volume of empty space in the container during the 4 hour period.
18M.1.SL.TZ2.S_10a.i:
Write down $f'\left( 2 \right)$ .
19N.1.SL.TZ0.T_14c:
At the point where $x = 2$ , the gradient of the tangent to the curve is $0.5$ .

Find the value of $a$ .
18M.2.AHL.TZ2.H_11a:
Show that $\frac{{{\text{d}}y}}{{{\text{d}}x}} = - \left( {\frac{{1 + y\,{\text{sin}}\left( {xy} \right)}}{{1 + x\,{\text{sin}}\left( {xy} \right)}}} \right)$ .
17N.2.AHL.TZ0.H_10c:
Find the coordinates of the point on the graph of $f$ where the normal to the graph is parallel to the line $y = - x$ .
17N.2.AHL.TZ0.H_10b:
Sketch the graph of $y = f(x)$ showing clearly the minimum point and any asymptotic behaviour.
17N.2.AHL.TZ0.H_10a.ii:
Determine the values of $x$ for which $f(x)$ is a decreasing function.
19N.2.SL.TZ0.T_4f.i:
Write down the equation of $T$ .
20N.1.SL.TZ0.S_10a.ii:
Show that the equation of $L_{1}$ is $k x + p^{2} y - 2 p k = 0$ .
18M.2.AHL.TZ1.H_9a:
Show that there are exactly two points on the curve where the gradient is zero.
20N.1.SL.TZ0.T_13b:
Write down the gradient of this tangent.
17N.1.SL.TZ0.T_2c.ii:
Write down, in the form $y = mx + c$ , the equation of ${L_2}$ .
17M.2.SL.TZ2.T_6g:
The equation $f(x) = m$ , where $m \in \mathbb{R}$ , has four solutions. Find the possible values of $m$ .
16N.2.SL.TZ0.S_10b:
(i) Write down the value of $k$ .

(ii) Find $g(x)$ .
17M.2.SL.TZ2.S_8d:
Let $R$ be the region enclosed by the graph of $f$ , the $x$ -axis, the line $x = b$ and the line $x = a$ . The region $R$ is rotated 360° about the $x$ -axis. Find the volume of the solid formed.
18N.1.SL.TZ0.T_11c:
Find the equation of this tangent. Give your answer in the form y = mx + c.
17M.2.SL.TZ2.S_8a:
Find the value of $p$ .
18M.2.SL.TZ2.T_6d:
Find $\frac{{{\text{dy}}}}{{{\text{dx}}}}$ .
18N.2.SL.TZ0.S_10a:
Find the volume of the container.
17M.2.SL.TZ2.T_6b:
Find $f'(x)$ .
17M.2.AHL.TZ1.H_2a:
Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ in terms of $x$ and $y$ .
17M.2.SL.TZ1.T_6b.i:
Show that $k = 6$ .
17M.2.SL.TZ1.T_6a:
Find $g'(x)$ .
18M.2.SL.TZ1.T_4c:
Use your answer to part (b) to show that the minimum value of f(x) is −22 .
18M.2.AHL.TZ1.H_9d:
The shaded region is rotated by 2 $\pi$ about the $y$ -axis. Find the volume of the solid formed.
18N.2.SL.TZ0.S_10b.ii:
During the interval $p$ < $t$ < $q$ , he volume of water in the container increases by $k$ m³. Find the value of $k$ .
19M.2.SL.TZ2.T_5d:
Find $f'\left( x \right)$ .
17M.2.SL.TZ2.T_6e:
Write down the range of $f(x)$ .
17M.1.SL.TZ2.T_13b:
Find the equation of $N$ . Give your answer in the form $ax + by + d = 0$ where $a$ , $b$ , $d \in \mathbb{Z}$ .
20N.1.SL.TZ0.S_10a.i:
Find $f' (p)$ in terms of $k$ and $p$ .
20N.1.SL.TZ0.T_13c:
Find the value of $k$ .
19M.2.SL.TZ1.T_6b:
Find the coordinates of the local minimum.
18M.1.SL.TZ1.T_5b:
Find the gradient of the line DC.
19N.1.SL.TZ0.T_14b:
Find $\frac{d y}{d x}$ .
16N.1.SL.TZ0.T_14b:
Find the coordinates of P.
18M.1.SL.TZ2.T_14b:
Find the gradient of the graph of f at $x = - \frac{1}{2}$ .
17N.1.SL.TZ0.S_5b:
Given that $\mathop {\lim }\limits_{x \to + \infty } (g \circ f)(x) = - 3$ , find the value of $b$ .
17M.2.SL.TZ1.T_6e:
Find the $y$ -coordinate of the local minimum.
17M.2.AHL.TZ1.H_2b:
Determine the equation of the tangent to $C$ at the point $\left( {\frac{2}{{\text{e}}},{\text{ e}}} \right)$
19M.2.SL.TZ1.T_6d:
Determine the equation of the normal at $x = - 2$ in the form $y = mx + c$ .
19N.2.SL.TZ0.T_4a:
Find the value of $c$ .
17M.2.SL.TZ1.T_6c:
Use your answer to part (a) and the value of $k$ , to find the $x$ -coordinates of the stationary points of the graph of $y = g(x)$ .
19N.2.SL.TZ0.T_4g:
Given $f (a) = 5.5$ and $f' (a) = - 6$ , state whether the function, $f$ , is increasing or decreasing at $x = a$ . Give a reason for your answer.
18N.2.SL.TZ0.S_10b.i:
Find the value of $p$ and of $q$ .
16N.2.SL.TZ0.S_10c:
(i) Find $w$ .

(ii) Hence or otherwise, find the maximum positive rate of change of $g$ .
18M.2.AHL.TZ1.H_9c:
The normal at P cuts the curve again at the point Q. Find the $x$ -coordinate of Q.
20N.1.SL.TZ0.S_10c:
The graph of $f$ is translated by $(\begin{matrix} 4 \\ 3 \end{matrix})$ to give the graph of $g$ .
In the following diagram:
- point $Q$ lies on the graph of $g$
- points $C$ , $D$ and $E$ lie on the vertical asymptote of $g$
- points $D$ and $F$ lie on the horizontal asymptote of $g$
- point $G$ lies on the $x$ -axis such that $FG$ is parallel to $DC$ .
Line $L_{2}$ is the tangent to the graph of $g$ at $Q$ , and passes through $E$ and $F$ .

Given that triangle $EDF$ and rectangle $CDFG$ have equal areas, find the gradient of $L_{2}$ in terms of $p$ .
19M.2.SL.TZ1.T_6c:
Write down the interval where the gradient of the graph of $f\left( x \right)$ is negative.
20N.1.SL.TZ0.T_13a:
Write down $f' (x)$ .
20N.1.SL.TZ0.S_10b:
Find the area of triangle $AOB$ in terms of $k$ .
18M.2.AHL.TZ1.H_9b:
Find the equation of the normal to the curve at the point P.
16N.1.SL.TZ0.T_14a:
Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ .
EXN.1.SL.TZ0.7a:
Find an expression for $\frac{d y}{d x}$ .
17M.2.SL.TZ2.S_8c.ii:
Find the the rate of change of $f$ at B.
19M.2.SL.TZ2.T_5e:
Find the gradient of the graph of $y = f\left( x \right)$ at $x = 2$ .

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