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Date	May 2022	Marks available	3	Reference code	22M.3.AHL.TZ2.1
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 2
Command term	Show that	Question number	1	Adapted from	N/A

Question

This question asks you to explore properties of a family of curves of the type $y^{2} = x^{3} + a x + b$ for various values of $a$ and $b$ , where $a, b \in ℕ$ .

On the same set of axes, sketch the following curves for $- 2 \leq x \leq 2$ and $- 2 \leq y \leq 2$ , clearly indicating any points of intersection with the coordinate axes.

Now, consider curves of the form $y^{2} = x^{3} + b$ , for $x \geq - \sqrt[3]{b}$ , where $b \in ℤ^{+}$ .

Next, consider the curve $y^{2} = x^{3} + x, x \geq 0$ .

The curve $y^{2} = x^{3} + x$ has two points of inflexion. Due to the symmetry of the curve these points have the same $x$ -coordinate.

$P (x, y)$ is defined to be a rational point on a curve if $x$ and $y$ are rational numbers.

The tangent to the curve $y^{2} = x^{3} + a x + b$ at a rational point $P$ intersects the curve at another rational point $Q$ .

Let $C$ be the curve $y^{2} = x^{3} + 2$ , for $x \geq - \sqrt[3]{2}$ . The rational point $P (- 1, - 1)$ lies on $C$ .

$y^{2} = x^{3}, x \geq 0$

[2]

a.i.

$y^{2} = x^{3} + 1, x \geq - 1$

[2]

a.ii.

Write down the coordinates of the two points of inflexion on the curve $y^{2} = x^{3} + 1$ .

[1]

b.i.

By considering each curve from part (a), identify two key features that would distinguish one curve from the other.

[1]

b.ii.

By varying the value of $b$ , suggest two key features common to these curves.

[2]

Show that $\frac{d y}{d x} = \pm \frac{3 x^{2} + 1}{2 \sqrt{x^{3} + x}}$ , for $x > 0$ .

[3]

d.i.

Hence deduce that the curve $y^{2} = x^{3} + x$ has no local minimum or maximum points.

[1]

d.ii.

Find the value of this $x$ -coordinate, giving your answer in the form $x = \sqrt{\frac{p \sqrt{3} + q}{r}}$ , where $p, q, r \in ℤ$ .

[7]

Find the equation of the tangent to $C$ at $P$ .

[2]

f.i.

Hence, find the coordinates of the rational point $Q$ where this tangent intersects $C$ , expressing each coordinate as a fraction.

[2]

f.ii.

The point $S (- 1, 1)$ also lies on $C$ . The line $[QS]$ intersects $C$ at a further point. Determine the coordinates of this point.

[5]

Markscheme

approximately symmetric about the $x$ -axis graph of $y^{2} = x^{3}$ A1

including cusp/sharp point at $(0, 0)$ A1

[2 marks]

Note: Final A1 can be awarded if intersections are in approximate correct place with respect to the axes shown. Award A1A1A1A0 if graphs ‘merge’ or ‘cross’ or are discontinuous at $x$ -axis but are otherwise correct. Award A1A0A0A0 if only one correct branch of both curves are seen.

Note: If they sketch graphs on separate axes, award a maximum of 2 marks for the ‘best’ response seen. This is likely to be A1A1A0A0.

a.i.

approximately symmetric about the $x$ -axis graph of $y^{2} = x^{3} + 1$ with approximately correct gradient at axes intercepts A1
some indication of position of intersections at $x = - 1$ , $y = \pm 1$ A1

[2 marks]

Note: If they sketch graphs on separate axes, award a maximum of 2 marks for the ‘best’ response seen. This is likely to be A1A1A0A0.

a.ii.

$(0, 1)$ and $(0, - 1)$ A1

[1 mark]

b.i.

Any two from:

$y^{2} = x^{3}$ has a cusp/sharp point, (the other does not)

graphs have different domains

$y^{2} = x^{3} + 1$ has points of inflexion, (the other does not)

graphs have different $x$ -axis intercepts (one goes through the origin, and the other does not)

graphs have different $y$ -axis intercepts A1

Note: Follow through from their sketch in part (a)(i). In accordance with marking rules, mark their first two responses and ignore any subsequent.

[1 mark]

b.ii.

Any two from:

as , $x \to \infty, y \to \pm \infty$

as $x \to \infty, y^{2} = x^{3} + b$ is approximated by $y^{2} = x^{3}$ (or similar)

they have $x$ intercepts at $x = - \sqrt[3]{b}$

they have $y$ intercepts at $y = (\pm) \sqrt{b}$

they all have the same range

$y = 0$ (or $x$ -axis) is a line of symmetry

they all have the same line of symmetry $(y = 0)$

they have one $x$ -axis intercept

they have two $y$ -axis intercepts

they have two points of inflexion

at $x$ -axis intercepts, curve is vertical/infinite gradient

there is no cusp/sharp point at $x$ -axis intercepts A1A1

Note: The last example is the only valid answer for things “not” present. Do not credit an answer of “they are all symmetrical” without some reference to the line of symmetry.

Note: Do not allow same/ similar shape or equivalent.

Note: In accordance with marking rules, mark their first two responses and ignore any subsequent.

[2 marks]

METHOD 1

attempt to differentiate implicitly M1

$2 y \frac{d y}{d x} = 3 x^{2} + 1$ A1

$\frac{d y}{d x} = \frac{3 x^{2} + 1}{2 y}$ OR $(\pm) 2 \sqrt{x^{3} + x} \frac{d y}{d x} = 3 x^{2} + 1$ A1

$\frac{d y}{d x} = \pm \frac{3 x^{2} + 1}{2 \sqrt{x^{3} + x}}$ AG

METHOD 2

attempt to use chain rule $y = (\pm) \sqrt{x^{3} + x}$ M1

$\frac{d y}{d x} = (\pm) \frac{1}{2} {(x^{3} + x)}^{- \frac{1}{2}} (3 x^{2} + 1)$ A1A1

Note: Award A1 for $(\pm) \frac{1}{2} {(x^{3} + x)}^{- \frac{1}{2}}$ , A1 for $(3 x^{2} + 1)$

$\frac{d y}{d x} = \pm \frac{3 x^{2} + 1}{2 \sqrt{x^{3} + x}}$ AG

[3 marks]

d.i.

EITHER

local minima/maxima occur when $\frac{d y}{d x} = 0$

$1 + 3 x^{2} = 0$ has no (real) solutions (or equivalent) R1

$(x^{2} \geq 0 \Rightarrow) 3 x^{2} + 1 > 0$ , so $\frac{d y}{d x} \neq 0$ R1

THEN

so, no local minima/maxima exist AG

[1 mark]

d.ii.

EITHER

attempt to use quotient rule to find $\frac{d^{2} y}{d x^{2}}$ M1

$\frac{d^{2} y}{d x^{2}} = (\pm) \frac{12 x \sqrt{x + x^{3}} - (1 + 3 x^{2}) {(x + x^{3})}^{- \frac{1}{2}} (1 + 3 x^{2})}{4 (x + x^{3})}$ A1A1

Note: Award A1 for correct $12 x \sqrt{x + x^{3}}$ and correct denominator, A1 for correct $- (1 + 3 x^{2}) {(x + x^{3})}^{- \frac{1}{2}} (1 + 3 x^{2})$ .

Note: Future A marks may be awarded if the denominator is missing or incorrect.

stating or using $\frac{d^{2} y}{d x^{2}} = 0$ (may be seen anywhere) (M1)

$12 x \sqrt{x + x^{3}} = (1 + 3 x^{2}) {(x + x^{3})}^{- \frac{1}{2}} (1 + 3 x^{2})$

attempt to use product rule to find $\frac{d^{2} y}{d x^{2}}$ M1

$\frac{d^{2} y}{d x^{2}} = \frac{1}{2} (3 x^{2} + 1) (- \frac{1}{2}) (3 x^{2} + 1) {(x^{3} + x)}^{- \frac{3}{2}} + 3 x {(x^{3} + x)}^{- \frac{1}{2}}$ A1A1

Note: Award A1 for correct first term, A1 for correct second term.

setting $\frac{d^{2} y}{d x^{2}} = 0$ (M1)

attempts implicit differentiation on $2 y \frac{d y}{d x} = 3 x^{2} + 1$ M1

$2 {(\frac{d y}{d x})}^{2} + 2 y \frac{d^{2} y}{d x^{2}} = 6 x$ A1

recognizes that $\frac{d^{2} y}{d x^{2}} = 0$ (M1)

$\frac{d y}{d x} = \pm \sqrt{3 x}$

$(\pm) \frac{3 x^{2} + 1}{2 \sqrt{x^{3} + x}} = (\pm) \sqrt{3 x}$ (A1)

THEN

$12 x (x + x^{3}) = {(1 + 3 x^{2})}^{2}$

$12 x^{2} + 12 x^{4} = 9 x^{4} + 6 x^{2} + 1$

$3 x^{4} + 6 x^{2} - 1 = 0$ A1

attempt to use quadratic formula or equivalent (M1)

$x^{2} = \frac{- 6 \pm \sqrt{48}}{6}$

$(x > 0 \Rightarrow) x = \sqrt{\frac{2 \sqrt{3} - 3}{3}} (p = 2, q = - 3, r = 3)$ A1

Note: Accept any integer multiple of $p, q$ and $r$ (e.g. $4, - 6$ and $6$ ).

[7 marks]

attempt to find tangent line through $(- 1, - 1)$ (M1)

$y + 1 = - \frac{3}{2} (x + 1)$ OR $y = - 1.5 x - 2.5$ A1

[2 marks]

f.i.

attempt to solve simultaneously with $y^{2} = x^{3} + 2$ (M1)

Note: The M1 mark can be awarded for an unsupported correct answer in an incorrect format (e.g. $(4.25, - 8.875)$ ).

obtain $(\frac{17}{4}, - \frac{71}{8})$ A1

[2 marks]

f.ii.

attempt to find equation of $[QS]$ (M1)

$\frac{y - 1}{x + 1} = - \frac{79}{42} (= - 1.88095 \dots)$ (A1)

solve simultaneously with $y^{2} = x^{3} + 2$ (M1)

$x = 0.28798 \dots (= \frac{127}{441})$ A1

$y = - 1.4226 \dots (= \frac{13175}{9261})$ A1

$(0.228, - 1.42)$

attempt to find vector equation of $[QS]$ (M1)

$(\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} - 1 \\ 1 \end{matrix}) + λ (\begin{matrix} \frac{21}{4} \\ - \frac{79}{8} \end{matrix})$ (A1)

$x = - 1 + \frac{21}{4} λ$

$y = 1 - \frac{79}{8} λ$

attempt to solve ${(1 - \frac{79}{8} λ)}^{2} = {(- 1 + \frac{21}{4} λ)}^{3} + 2$ (M1)

$λ = 0.2453 \dots$

$x = 0.28798 \dots (= \frac{127}{441})$ A1

$y = - 1.4226 \dots (= \frac{13175}{9261})$ A1

$(0.228, - 1.42)$

[5 marks]

Examiners report

This was a relatively straightforward start, though it was disappointing to see so many candidates sketch their graphs on two separate axes, despite the question stating they should be sketched on the same axes.

a.i.

[N/A]

a.ii.

Of those candidates producing clear sketches, the vast majority were able to recognise the points of inflexion and write down their coordinates. A small number embarked on a mostly fruitless algebraic approach rather than use their graphs as intended. The distinguishing features between curves tended to focus on points of intersection with the axes, which was accepted. Only a small number offered ideas such as $y \to \infty$ on both curves. A number of (incorrect) suggestions were seen, stating that both curves tended towards a linear asymptote.

b.i.

[N/A]

b.ii.

A majority of candidates' suggestions related to the number of intersection points with the coordinate axes, while the idea of the x-axis acting as a line of symmetry was also often seen.

The required differentiation was straightforward for the majority of candidates.

d.i.

[N/A]

d.ii.

The majority employed the quotient rule here, often doing so successfully to find a correct expression for $\frac{d^{2} y}{d x^{2}}$ . Despite realising that $\frac{d^{2} y}{d x^{2}} = 0$ , the resulting algebra to find the required solution proved a step too far for most. A number of slips were seen in candidates' working, though better candidates were able to answer the question confidently.

Mistakes proved to be increasingly common by this stage of the paper. Various equations of lines were suggested, with the incorrect $y = 1.5 x + 2.5$ appearing more than once. Only the better candidates were able to tackle the final part of the question with any success; it was pleasing to see a number of clear algebraic (only) approaches, though this was not necessary to obtain full marks.

f.i.

[N/A]

f.ii.

Significant work on this question part was rarely seen, and it may have been the case that many candidates chose to spend their remaining time on the second question, especially if they felt they were making little progress with part f. Having said that, correct final answers were seen from better candidates, though these were few and far between.

Syllabus sections

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

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18M.2.SL.TZ1.S_10e:
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17M.2.SL.TZ1.S_10b.i:
Find $\int_{0.111}^{3.31} {\left( {h(x) - x} \right){\text{d}}x}$ .
21N.1.SL.TZ0.5a:
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19N.1.AHL.TZ0.H_10a.i:
Find $f'\left( x \right)$ .
17M.2.AHL.TZ1.H_12b:
Sketch the graph of $y = f(x)$ showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.
16N.1.SL.TZ0.S_10a:
(i) Find the first four derivatives of $f(x)$ .

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17N.1.SL.TZ0.S_7:
Consider $f(x) = \log k(6x - 3{x^2})$ , for $0 < x < 2$ , where $k > 0$ .

The equation $f(x) = 2$ has exactly one solution. Find the value of $k$ .
18N.1.SL.TZ0.S_10b.i:
Find $f'\left( x \right)$ .
SPM.2.SL.TZ0.6b:
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Show that $C = 20\pi {r^2} + \frac{{320\pi }}{r}$ .
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local minimum points.
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Find the maximum speed of P.
18N.1.SL.TZ0.S_10c:
The graph of $f$ has a local minimum at the point Q. The line L passes through Q.

Find the value of $a$ .
19M.2.SL.TZ1.S_4c:
Hence find the values of $x$ for which the graph of $f$ is concave-down.
19M.2.SL.TZ2.S_5a:
Find the population of fish at $t$ = 10.
SPM.3.AHL.TZ0.2c:
On a new set of axes, sketch the graphs of $y = {f_2}(x)$ and $y = {f_4}(x)$ for −1 ≤ $x$ ≤ 1.
19N.1.AHL.TZ0.H_10a.ii:
Show that, if $f'\left( x \right) = 0$ , then $x = 2 - \sqrt 3$ .
22M.3.AHL.TZ1.2d.i:
Show that $g' (x) = 2 (x - r) (x - a) + x^{2} - 2 a x + a^{2} + b^{2}$ .
22M.1.AHL.TZ1.12c.ii:
Hence, deduce that $g^{(4)} (x) = 2 (g''' (x) - g'' (x))$ .
22M.1.AHL.TZ2.6b:
The range of $f$ is $a \leq y \leq b$ , where $a, b \in ℝ$ .

Find the value of $a$ and the value of $b$ .
22M.2.SL.TZ1.5c:
Find the total distance travelled by the particle.
22M.2.SL.TZ2.6b:
Find the times when the particle’s acceleration is $- 1.9 {m s}^{- 2}$ .
22M.2.SL.TZ2.6c:
Find the particle’s acceleration when its speed is at its greatest.
22M.2.SL.TZ2.8c:
For $0 \leq t \leq 9$ , find the total amount of time when the rate of growth of Plant $B$ was greater than the rate of growth of Plant $A$ .
22M.2.AHL.TZ2.7b:
Using l’Hôpital’s rule, show algebraically that the value of the limit is $- \frac{1}{4}$ .
20N.2.SL.TZ0.S_10a:
Show that $f'' (x) = \frac{24 - 6 x^{2}}{{(x^{2} + 4)}^{2}}$ .
21M.1.SL.TZ1.5c:
Hence, show that $k = e + \frac{e^{2}}{4}$ .
21M.1.SL.TZ1.8a:
Show that $\frac{d y}{d x} = \frac{1 - 4 \ln x}{x^{5}}$ .
21M.1.SL.TZ1.5a:
Find $f' (x)$ .
21M.1.SL.TZ1.5b:
Show that $h = \frac{e + 6}{2}$ .
21M.1.SL.TZ1.8b:
The graph of $f$ has a horizontal tangent at point $P$ . Find the coordinates of $P$ .
17M.2.AHL.TZ1.H_8b:
Calculate $\frac{{{\text{d}}\theta }}{{{\text{d}}t}}$ when $\theta = \frac{\pi }{3}$ .
21M.2.SL.TZ1.9b:
Find the exact coordinates of $Q$ .
SPM.3.AHL.TZ0.2g:
Use an appropriate trigonometric identity to show that ${f_{n + 1}}(x) = {\text{cos}}\left( {n\,{\text{arccos}}\,x} \right){\text{cos}}\left( {{\text{arccos}}\,x} \right) - {\text{sin}}\left( {n\,{\text{arccos}}\,x} \right){\text{sin}}\left( {{\text{arccos}}\,x} \right)$ .
SPM.3.AHL.TZ0.2e:
Solve the equation ${f_n}^\prime (x) = 0$ and hence show that the stationary points on the graph of $y = {f_n}(x)$ occur at $x = {\text{cos}}\frac{{k\pi }}{n}$ where $k \in {\mathbb{Z}^ + }$ and 0 < $k$ < $n$ .
SPM.3.AHL.TZ0.2f:
Use an appropriate trigonometric identity to show that ${f_2}(x) = 2{x^2} - 1$ .
17M.2.AHL.TZ1.H_12d:
Explain why the inverse function ${f^{ - 1}}$ does not exist.
17M.2.AHL.TZ1.H_12g.i:
Hence, show that there are no solutions to $g'(x) = 0$ ;
17M.2.AHL.TZ1.H_12a:
Find the largest possible domain $D$ for $f$ to be a function.
20N.2.SL.TZ0.S_10b:
Find the least value of $n$ .
SPM.3.AHL.TZ0.2a:
On the same set of axes, sketch the graphs of $y = {f_1}(x)$ and $y = {f_3}(x)$ for −1 ≤ $x$ ≤ 1.
18M.2.SL.TZ1.S_10b.ii:
For the graph of $f$ , write down the period.
21M.3.AHL.TZ2.1c:
Show that ${f_{n}}^{'} (x) = n x^{n - 1} (a - 2 x) {(a - x)}^{n - 1}$ .
18M.1.SL.TZ1.S_7:
Consider f(x), g(x) and h(x), for x∈ $\mathbb{R}$ where h(x) = $\left( {f \circ g} \right)$ (x).

Given that g(3) = 7 , g′ (3) = 4 and f ′ (7) = −5 , find the gradient of the normal to the curve of h at x = 3.
SPM.1.SL.TZ0.9d:
The region R is enclosed by the graph of $f$ , the x-axis, and the vertical lines through the maximum point P and the point of inflexion Q.

Given that the area of R is 3, find the value of $k$ .
19M.2.SL.TZ2.S_5b:
Find the rate at which the population of fish is increasing at $t$ = 10.
SPM.1.SL.TZ0.9b:
Find the x-coordinate of P.
18M.2.SL.TZ2.S_9c:
Write down the number of times that the acceleration of P is 0 m s⁻² .
17M.2.AHL.TZ1.H_12f:
Find $g'(x)$ .
20N.2.AHL.TZ0.H_8b:
At the point on the curve where $x = 4$ , it is given that $\frac{d y}{d t} = - 0.1 {m s}^{- 1}$

Find the value of $\frac{d x}{d t}$ at this exact same instant.
18M.1.SL.TZ2.S_9a:
Express h in terms of r.
EXN.1.AHL.TZ0.11d:
Show that $m = - \frac{6 x}{{(x^{2} + 2)}^{2}}$ .
17M.2.AHL.TZ1.H_12e:
Find the inverse function ${g^{ - 1}}$ and state its domain.
SPM.1.SL.TZ0.9c:
Show that the x-coordinate of Q is $\frac{{\text{1}}}{5}{{\text{e}}^{\frac{3}{2}}}$ .
21M.1.SL.TZ2.5b:
Given that the gradient of $L$ is $\frac{1}{3}$ , find the $x$ -coordinate of $B$ .
20N.2.SL.TZ0.S_10c:
Find $\int \frac{6 x}{x^{2} + 4} d x$ .
17M.2.SL.TZ1.S_10a.i:
Write down the value of $q$ ;
21N.3.AHL.TZ0.1b:
Show that ${(f (t))}^{2} + {(g (t))}^{2} = f (2 t)$ .
21N.3.AHL.TZ0.1c.i:
$f (i u)$ .
21N.3.AHL.TZ0.1g:
The hyperbola with equation $x^{2} - y^{2} = 1$ can be rotated to coincide with the curve defined by $x y = k, k \in ℝ$ .

Find the possible values of $k$ .
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.1.AHL.TZ2.11a:
Sketch the curve $y = f (x)$ , clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.
22M.2.AHL.TZ2.7a:
Show that a finite limit only exists for $k = \frac{π}{4}$ .
22M.3.AHL.TZ2.1f.i:
Find the equation of the tangent to $C$ at $P$ .
21N.3.AHL.TZ0.1a:
Verify that $u = f (t)$ satisfies the differential equation $\frac{d^{2} u}{d t^{2}} = u$ .
21N.3.AHL.TZ0.1e:
Show that ${(f (t))}^{2} - {(g (t))}^{2} = {(f (i u))}^{2} - {(g (i u))}^{2}$ .
21N.3.AHL.TZ0.2b.ii:
By substituting $Y = \frac{d y}{d t}$ , show that $Y = B e^{2 t}$ where $B$ is a constant.
21N.3.AHL.TZ0.2c.iii:
Let the two values found in part (c)(ii) be $λ_{1}$ and $λ_{2}$ .

Verify that $y = F e^{λ_{1} t} + G e^{λ_{2} t}$ is a solution to the differential equation in (c)(i),where $G$ is a constant.
17M.2.AHL.TZ1.H_8a:
Find an expression for the volume of water $V{\text{ }}({{\text{m}}^3})$ in the trough in terms of $\theta$ .
18N.1.SL.TZ0.S_10b.ii:
Hence, find the equation of L in terms of $a$ .
21N.1.SL.TZ0.5c:
Find $h (4)$ .
18M.1.SL.TZ2.S_9c:
Given that there is a minimum value for C, find this minimum value in terms of $\pi$ .
16N.2.AHL.TZ0.H_6:
An earth satellite moves in a path that can be described by the curve $72.5{x^2} + 71.5{y^2} = 1$ where $x = x(t)$ and $y = y(t)$ are in thousands of kilometres and $t$ is time in seconds.

Given that $\frac{{{\text{d}}x}}{{{\text{d}}t}} = 7.75 \times {10^{ - 5}}$ when $x = 3.2 \times {10^{ - 3}}$ , find the possible values of $\frac{{{\text{d}}y}}{{{\text{d}}t}}$ .

Give your answers in standard form.
18M.2.SL.TZ1.S_10c:
Hence, write $f\left( x \right)$ in the form $p\,\,{\text{cos}}\,\left( {x + r} \right)$ .
SPM.3.AHL.TZ0.2d.i:
local maximum points;
21M.2.AHL.TZ1.11b:
Find an expression for $f' (x)$ .
SPM.1.SL.TZ0.9a:
Show that $f'\left( x \right) = \frac{{1 - {\text{ln}}\,5x}}{{k{x^2}}}$ .
18M.1.AHL.TZ1.H_7b:
Find $\int_0^1 {{\text{arccos}}\left( {\frac{x}{2}} \right){\text{d}}x}$ .
17M.2.AHL.TZ1.H_12g.ii:
Hence, show that there are no solutions to $({g^{ - 1}})'(x) = 0$ .
21M.2.AHL.TZ2.12c.i:
Show that $f' (x) = \frac{2 x}{\sqrt{x^{2}} (x^{2} + 1)}$ for $x \in ℝ, x \neq 0$ .
21M.3.AHL.TZ2.1d:
State the three solutions to the equation ${f_{n}}^{'} (x) = 0$ .
SPM.3.AHL.TZ0.2h.ii:
Hence express ${f_3}(x)$ as a cubic polynomial.
20N.2.SL.TZ0.S_10d:
Let $R$ be the region enclosed by the graph of $f$ , the $x$ -axis and the lines $x = 1$ and $x = 3$ . The area of $R$ is $19.6$ , correct to three significant figures.

Find $f (x)$ .
SPM.3.AHL.TZ0.2h.i:
Hence show that ${f_{n + 1}}(x) + {f_{n - 1}}(x) = 2x{f_n}\left( x \right)$ , $n \in {\mathbb{Z}^ + }$ .
EXN.1.SL.TZ0.9e:
Find the rate of change of the ball’s height above the ground when $t = \frac{1}{3}$ . Give your answer in the form $p π \sqrt{q} {ms}^{- 1}$ where $p \in ℚ$ and $q \in ℤ^{+}$ .
21M.2.SL.TZ1.9c:
Show that the equation of $L$ is $y = - x + 2 \ln 45 + 2$ .
20N.2.AHL.TZ0.H_8a:
Find an expression for $\frac{d y}{d x}$ .
18M.1.AHL.TZ2.H_6b:
Hence, or otherwise, find $\int\limits_0^\pi {{{\text{e}}^{ - x}}\,{\text{sin}}\,x\,{\text{d}}x}$ .
19M.2.SL.TZ1.S_4b:
Find the $x$ -coordinates of the points of inflexion of the graph of $f$ .
SPM.3.AHL.TZ0.2b.i:
local maximum points;
21M.3.AHL.TZ2.1e:
Show that the point $(\frac{a}{2}, f_{n} (\frac{a}{2}))$ on the graph of $y = f_{n} (x)$ is always above the horizontal axis.
17M.2.SL.TZ1.S_10a.iii:
Write down the value of $k$ .
17M.2.SL.TZ1.S_10c:
Let $d$ be the vertical distance from a point on the graph of $h$ to the line $y = x$ . There is a point ${\text{P}}(a,{\text{ }}b)$ on the graph of $h$ where $d$ is a maximum.

Find the coordinates of P, where $0.111 < a < 3.31$ .
17M.2.SL.TZ1.S_10a.ii:
Write down the value of $h$ ;
17M.2.SL.TZ1.S_10b.ii:
Hence, find the area of the region enclosed by the graphs of $h$ and ${h^{ - 1}}$ .
21N.1.SL.TZ0.5d:
Hence find the equation of the tangent to the graph of $h$ at $x = 4$ .
21N.3.AHL.TZ0.1c.ii:
$g (i u)$ .
21N.3.AHL.TZ0.1d:
Hence find, and simplify, an expression for ${(f (i u))}^{2} + {(g (i u))}^{2}$ .
21N.3.AHL.TZ0.2a.i:
By solving the differential equation $\frac{d y}{d t} = y$ , show that $y = A e^{t}$ where $A$ is a constant.
21N.3.AHL.TZ0.2a.ii:
Show that $\frac{d x}{d t} - x = - A e^{t}$ .
21N.3.AHL.TZ0.2a.iii:
Solve the differential equation in part (a)(ii) to find $x$ as a function of $t$ .
21N.3.AHL.TZ0.2b.i:
By differentiating $\frac{d y}{d t} = - x + y$ with respect to $t$ , show that $\frac{d^{2} y}{d t^{2}} = 2 \frac{d y}{d t}$ .
21N.3.AHL.TZ0.2b.iii:
Hence find $y$ as a function of $t$ .
21N.3.AHL.TZ0.2b.iv:
Hence show that $x = - \frac{B}{2} e^{2 t} + C$ , where $C$ is a constant.
21N.3.AHL.TZ0.2c.i:
Show that $\frac{d^{2} y}{d t^{2}} - 2 \frac{d y}{d t} - 3 y = 0$ .
21N.3.AHL.TZ0.2c.ii:
Find the two values for $λ$ that satisfy $\frac{d^{2} y}{d t^{2}} - 2 \frac{d y}{d t} - 3 y = 0$ .

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