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Date	May 2021	Marks available	4	Reference code	21M.2.SL.TZ1.9
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	9	Adapted from	N/A

Question

Consider the function $f$ defined by $f (x) = 90 e^{- 0.5 x}$ for $x \in ℝ^{+}$ .

The graph of $f$ and the line $y = x$ intersect at point $P$ .

The line $L$ has a gradient of $- 1$ and is a tangent to the graph of $f$ at the point $Q$ .

The shaded region $A$ is enclosed by the graph of $f$ and the lines $y = x$ and $L$ .

Find the $x$ -coordinate of $P$ .

[2]

a.

Find the exact coordinates of $Q$ .

[4]

b.

Show that the equation of $L$ is $y = - x + 2 \ln 45 + 2$ .

[2]

c.

Find the $x$ -coordinate of the point where $L$ intersects the line $y = x$ .

[1]

d.i.

Hence, find the area of $A$ .

[4]

d.ii.

The line $L$ is tangent to the graphs of both $f$ and the inverse function $f^{- 1}$ .

Find the shaded area enclosed by the graphs of $f$ and $f^{- 1}$ and the line $L$ .

[2]

e.

Markscheme

Attempt to find the point of intersection of the graph of $f$ and the line $y = x$ (M1)

$x = 5.56619 \dots$

$= 5.57$ A1

[2 marks]

a.

$f' (x) = - 45 e^{- 0.5 x}$ A1

attempt to set the gradient of $f$ equal to $- 1$ (M1)

$- 45 e^{- 0.5 x} = -1$

$Q$ has coordinates $(2 \ln 45, 2)$ (accept ( $- 2 \ln \frac{1}{45}, 2$ ) A1A1

Note: Award A1 for each value, even if the answer is not given as a coordinate pair.

Do not accept $\frac{\ln \frac{1}{45}}{- 0.5}$ or $\frac{\ln 45}{0.5}$ as a final value for $x$ . Do not accept $2.0$ or $2.00$ as a final value for $y$ .

[4 marks]

b.

attempt to substitute coordinates of $Q$ (in any order) into an appropriate equation (M1)

$y - 2 = - (x - 2 \ln 45)$ OR $2 = - 2 \ln 45 + c$ A1

equation of $L$ is $y = - x + 2 \ln 45 + 2$ AG

[2 marks]

c.

$x = \ln 45 + 1 (= 4.81)$ A1

[1 mark]

d.i.

appropriate method to find the sum of two areas using integrals of the difference of two functions (M1)

Note: Allow absence of incorrect limits.

$\int_{4.806 \dots}^{5.566 \dots} (x - (- x + 2 \ln 45 + 2)) d x + \int_{5.566 \dots}^{7.613 \dots} (90 e^{- 0.5 x} - (- x + 2 \ln 45 + 2)) d x$ (A1)(A1)

Note: Award A1 for one correct integral expression including correct limits and integrand.
Award A1 for a second correct integral expression including correct limits and integrand.

$= 1.52196 \dots$

$= 1.52$ A1

[4 marks]

d.ii.

by symmetry $2 \times 1.52 \dots$ (M1)

$= 3.04$ A1

Note: Accept any answer that rounds to $3.0$ (but do not accept $3$ ).

[2 marks]

e.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

[N/A]

d.i.

[N/A]

d.ii.

[N/A]

e.

Syllabus sections

Topic 5 —Calculus » SL 5.4—Tangents and normal

Show 95 related questions

22M.3.AHL.TZ1.2c:
Sketch the curve $y = f (x)$ and the tangent to the curve at point $A$ , clearly showing where the tangent crosses the $x$ -axis.
19M.1.SL.TZ2.S_9d:
The following diagram shows the graph of $f$ for 0 ≤ $x$ ≤ 3. Line $M$ is a tangent to the graph of $f$ at point P.

Given that $M$ is parallel to $L$ , find the $x$ -coordinate of P.
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.
19M.2.SL.TZ2.T_5b:
Write down the $y$ -intercept of the graph of $y = f\left( x \right)$ .
SPM.1.SL.TZ0.8b:
Find the value of $a$ and the value of $b$ .
19M.2.SL.TZ1.S_9b:
Find u.
19M.2.SL.TZ1.S_3b:
The graph of $f$ has a horizontal tangent line at $x = 0$ and at $x = a$ . Find $a$ .
17N.1.AHL.TZ0.H_11c:
Hence or otherwise, find an expression for the derivative of ${f_n}(x)$ with respect to $x$ .
17M.2.SL.TZ2.T_6f:
Write down the number of possible solutions to the equation $f(x) = 5$ .
19M.2.SL.TZ1.S_9d.iii:
Hence or otherwise, find the obtuse angle formed by the tangent line to $f$ at $x = 8$ and the tangent line to $f$ at $x = 2$ .
SPM.1.SL.TZ0.8c.i:
Sketch the graph of $y = f'\left( x \right)$ .
19M.1.AHL.TZ2.H_6b:
Hence find the equation of the normal to $C$ at the point (1, 1).
19N.1.SL.TZ0.S_10c:
The line ${L_2}$ is tangent to the graph of $g$ at ${\text{A}}$ and has equation $y = \left( {{\text{ln}}\,p} \right)x + q + 1$ .

The line ${L_2}$ passes through the point $\left( { - 2{\text{, }} - 2} \right)$ .

The gradient of the normal to $g$ at ${\text{A}}$ is $\frac{1}{{{\text{ln}}\left( {\frac{1}{3}} \right)}}$ .

Find the equation of ${L_1}$ in terms of $x$ .
21N.1.SL.TZ0.5a:
Write down the value of $f' (4)$ .
17N.1.AHL.TZ0.H_11b:
By using mathematical induction, prove that

${f_n}(x) = \frac{{\sin {2^{n + 1}}x}}{{{2^n}\sin 2x}},{\text{ }}x \ne \frac{{m\pi }}{2}$ where $m \in \mathbb{Z}$ .
19N.2.AHL.TZ0.H_11a.i:
Using implicit differentiation, find an expression for $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ .
19N.1.SL.TZ0.S_10a:
Write down the coordinates of ${\text{B}}$ .
19M.2.SL.TZ1.S_9d.i:
Find $\left( {f \circ f} \right)\left( x \right)$ .
SPM.1.SL.TZ0.8a:
Find $f'\left( x \right)$ .
19N.2.SL.TZ0.S_3b:
Let ${\text{C}}$ be a point on the graph of $h$ . The tangent to the graph of $h$ at ${\text{C}}$ is parallel to the graph of $f$ .

Find the $x$ -coordinate of ${\text{C}}$ .
19M.2.SL.TZ2.T_5c:
Sketch the graph of $y = f\left( x \right)$ for −3 ≤ $x$ ≤ 3 and −4 ≤ $y$ ≤ 12.
19N.2.SL.TZ0.S_3a:
Find $h\left( x \right)$ .
18N.1.SL.TZ0.S_10b.i:
Find $f'\left( x \right)$ .
16N.1.SL.TZ0.T_14b:
Find the coordinates of P.
19N.2.AHL.TZ0.H_11a.ii:
Find the equation of the tangent to the curve at the point $\left( {\frac{1}{4}{\text{, }}\frac{{5\pi }}{6}} \right)$ .
19M.1.SL.TZ2.S_9a:
Find the value of ${\text{tan}}\,\theta$ .
18M.2.AHL.TZ1.H_9a:
Show that there are exactly two points on the curve where the gradient is zero.
18M.2.AHL.TZ1.H_9c:
The normal at P cuts the curve again at the point Q. Find the $x$ -coordinate of Q.
19M.2.AHL.TZ1.H_1:
Let $l$ be the tangent to the curve $y = x{{\text{e}}^{2x}}$ at the point (1, ${{\text{e}}^2}$ ).

Find the coordinates of the point where $l$ meets the $x$ -axis.
19M.1.SL.TZ2.S_9b:
Line $L$ passes through the origin and has a gradient of ${\text{tan}}\,\theta$ . Find the equation of $L$ .
22M.3.AHL.TZ1.2b:
Show that the line $y = x - 1$ is tangent to the curve $y = f (x)$ at the point $A (4, 3)$ .
22M.3.AHL.TZ1.2d.ii:
Hence, or otherwise, prove that the tangent to the curve $y = g (x)$ at the point $A (a, g (a))$ intersects the $x$ -axis at the point $R (r, 0)$ .
17M.2.AHL.TZ1.H_2a:
Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ in terms of $x$ and $y$ .
22M.1.SL.TZ2.7c:
Find the equation of $L$ .
SPM.1.SL.TZ0.8c.ii:
Hence explain why the graph of $f$ has a local maximum point at $x = a$ .
20N.1.SL.TZ0.T_13b:
Write down the gradient of this tangent.
SPM.1.SL.TZ0.8d.i:
Find $f''\left( b \right)$ .
19M.1.AHL.TZ2.H_6a:
At the point (1, 1) , show that $\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{2 + \pi }}{{2 - \pi }}$ .
21M.1.SL.TZ1.5c:
Hence, show that $k = e + \frac{e^{2}}{4}$ .
21M.1.SL.TZ1.5a:
Find $f' (x)$ .
21M.1.SL.TZ1.5b:
Show that $h = \frac{e + 6}{2}$ .
19M.2.SL.TZ1.S_9d.ii:
Hence, write down ${f^{ - 1}}\left( x \right)$ .
20N.1.AHL.TZ0.H_11a:
Show that $\frac{d y}{d x} = \frac{y \cos (x y)}{2 y - x \cos (x y)}$ .
20N.1.SL.TZ0.S_10b:
Find the area of triangle $AOB$ in terms of $k$ .
18N.2.AHL.TZ0.H_5:
Differentiate from first principles the function $f\left( x \right) = 3{x^3} - x$ .
19M.2.SL.TZ1.S_9a:
Find the gradient of $L$ .
18M.2.AHL.TZ1.H_9b:
Find the equation of the normal to the curve at the point P.
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 $p$ .
20N.1.AHL.TZ0.H_11b:
Prove that, when $\frac{d y}{d x} = 0, y = \pm 1$ .
20N.1.AHL.TZ0.H_2:
Find the equation of the tangent to the curve $y = e^{2 x} - 3 x$ at the point where $x = 0$ .
19N.1.SL.TZ0.S_10b:
Given that $f'\left( a \right) = \frac{1}{{{\text{ln}}\,p}}$ , find the equation of ${L_1}$ in terms of $x$ , $p$ and $q$ .
20N.1.SL.TZ0.S_10a.ii:
Show that the equation of $L_{1}$ is $k x + p^{2} y - 2 p k = 0$ .
EXN.2.AHL.TZ0.6b:
The tangent to $C$ at the point Ρ is parallel to the $y$ -axis.

Find the $x$ -coordinate of Ρ.
21M.1.SL.TZ2.5b:
Given that the gradient of $L$ is $\frac{1}{3}$ , find the $x$ -coordinate of $B$ .
SPM.1.SL.TZ0.8e:
The normal to the graph of $f$ at $x = a$ and the tangent to the graph of $f$ at $x = b$ intersect at the point ( $p$ , $q$ ) .

Find the value of $p$ and the value of $q$ .
20N.1.SL.TZ0.S_10a.i:
Find $f' (p)$ in terms of $k$ and $p$ .
17M.1.SL.TZ2.T_13b:
Find the equation of $N$ . Give your answer in the form $ax + by + d = 0$ where $a$ , $b$ , $d \in \mathbb{Z}$ .
21N.2.AHL.TZ0.8b:
Hence find the equation of the tangent to $C$ at the point where $x = 1$ .
22M.1.SL.TZ1.5:
Consider the curve with equation $y = (2 x - 1) e^{k x}$ , where $x \in ℝ$ and $k \in ℚ$ .

The tangent to the curve at the point where $x = 1$ is parallel to the line $y = 5 e^{k} x$ .

Find the value of $k$ .
22M.3.AHL.TZ2.1f.i:
Find the equation of the tangent to $C$ at $P$ .
17M.2.SL.TZ2.T_6d.i:
Write down the $x$ -coordinates of these two points;
17M.1.SL.TZ2.T_13c:
Draw the line $N$ on the diagram above.
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.
17M.1.SL.TZ2.T_13a:
Write down the value of $f(1)$ .
17M.1.SL.TZ1.S_9c:
The line $y = kx - 5$ is a tangent to the curve of $f$ . Find the values of $k$ .
17N.1.AHL.TZ0.H_11a:
Determine whether ${f_n}$ is an odd or even function, justifying your answer.
21N.1.SL.TZ0.5c:
Find $h (4)$ .
SPM.1.SL.TZ0.8d.ii:
Hence, use your answer to part (d)(i) to show that the graph of $f$ has a local minimum point at $x = b$ .
19M.2.SL.TZ1.S_9c:
Find the acute angle between $y = x$ and $L$ .
17M.2.AHL.TZ1.H_2b:
Determine the equation of the tangent to $C$ at the point $\left( {\frac{2}{{\text{e}}},{\text{ e}}} \right)$
20N.1.AHL.TZ0.H_11c:
Hence find the coordinates of all points on $C$ , for $0 < x < 4 π$ , where $\frac{d y}{d x} = 0$ .
16N.1.SL.TZ0.T_14a:
Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ .
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.
17N.1.AHL.TZ0.H_11d:
Show that, for $n > 1$ , the equation of the tangent to the curve $y = {f_n}(x)$ at $x = \frac{\pi }{4}$ is $4x - 2y - \pi = 0$ .
21M.2.SL.TZ1.9c:
Show that the equation of $L$ is $y = - x + 2 \ln 45 + 2$ .
EXN.2.AHL.TZ0.6a:
Show that $\frac{d y}{d x} = \frac{3 x^{2}}{2 e^{2 y} - 1}$ .
18M.1.SL.TZ1.T_5a:
Write down the coordinates of C, the midpoint of line segment AB.
18N.1.SL.TZ0.T_11c:
Find the equation of this tangent. Give your answer in the form y = mx + c.
20N.1.SL.TZ0.T_13c:
Find the value of $k$ .
18M.1.SL.TZ1.T_5b:
Find the gradient of the line DC.
17M.2.SL.TZ2.T_6a:
Write down the $y$ -intercept of the graph.
17M.2.SL.TZ2.T_6c.i:
Show that $a = 8$ .
17M.2.SL.TZ2.T_6b:
Find $f'(x)$ .
20N.1.SL.TZ0.T_13a:
Write down $f' (x)$ .
19M.2.SL.TZ2.T_5h:
Determine the range of $f\left( x \right)$ for $p$ ≤ $x$ ≤ $q$ .
17M.2.SL.TZ2.T_6g:
The equation $f(x) = m$ , where $m \in \mathbb{R}$ , has four solutions. Find the possible values of $m$ .
21N.1.SL.TZ0.5d:
Hence find the equation of the tangent to the graph of $h$ at $x = 4$ .
21N.2.AHL.TZ0.8a:
Show that $\frac{d y}{d x} + (x \frac{d y}{d x} + y) (1 + \ln (x y)) = 1$ .
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.
17M.2.SL.TZ2.T_6c.ii:
Find $f(2)$ .
17M.2.SL.TZ2.T_6d.ii:
Write down the intervals where the gradient of the graph of $y = f(x)$ is positive.
17M.2.SL.TZ2.T_6e:
Write down the range of $f(x)$ .

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