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Date	May 2022	Marks available	5	Reference code	22M.1.SL.TZ1.5
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 1
Command term	Find	Question number	5	Adapted from	N/A

Question

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 $k$ .

Markscheme

evidence of using product rule (M1)

$\frac{d y}{d x} = (2 x - 1) \times (k e^{k x}) + 2 \times e^{k x} (= e^{k x} (2 k x - k + 2))$ A1

correct working for one of (seen anywhere) A1

$\frac{d y}{d x}$ at $x = 1 \Rightarrow k e^{k} + 2 e^{k}$

slope of tangent is $5 e^{k}$

their $\frac{d y}{d x}$ at $x = 1$ equals the slope of $y = 5 e^{k} x (= 5 e^{k})$ (seen anywhere) (M1)

$k e^{k} + 2 e^{k} = 5 e^{k}$

$k = 3$ A1

[5 marks]

Examiners report

The product rule was well recognised and used with 𝑥=1 properly substituted into this expression. Although the majority of the candidates tried equating the derivative to the slope of the tangent line, the slope of the tangent line was not correctly identified; many candidates incorrectly substituted 𝑥=1 into the tangent equation, thus finding the y-coordinate instead of the slope.

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 ${f^{ - 1}}\left( x \right)$ .
21M.2.SL.TZ1.9b:
Find the exact coordinates of $Q$ .
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.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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