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Date	November 2020	Marks available	3	Reference code	20N.2.SL.TZ0.T_4
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	T_4	Adapted from	N/A

Question

Hyungmin designs a concrete bird bath. The bird bath is supported by a pedestal. This is shown in the diagram.

The interior of the bird bath is in the shape of a cone with radius $r$ , height $h$ and a constant slant height of $50 cm$ .

Let $V$ be the volume of the bird bath.

Hyungmin wants the bird bath to have maximum volume.

Write down an equation in $r$ and $h$ that shows this information.

[1]

Show that $V = \frac{2500 π h}{3} - \frac{π h^{3}}{3}$ .

[1]

Find $\frac{d V}{d h}$ .

[2]

Using your answer to part (c), find the value of $h$ for which $V$ is a maximum.

[2]

Find the maximum volume of the bird bath.

[2]

To prevent leaks, a sealant is applied to the interior surface of the bird bath.

Find the surface area to be covered by the sealant, given that the bird bath has maximum volume.

[3]

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$h^{2} + r^{2} = 50^{2}$ (or equivalent) (A1)

Note: Accept equivalent expressions such as $r = \sqrt{2500 - h^{2}}$ or $h = \sqrt{2500 - r^{2}}$ . Award (A0) for a final answer of $\pm \sqrt{2500 - h^{2}}$ or $\pm \sqrt{2500 - r^{2}}$ , or any further incorrect working.

[1 mark]

$\frac{1}{3} \times π \times (2500 - h^{2}) \times h$ OR $\frac{1}{3} \times π \times {(\sqrt{2500 - h^{2}})}^{2} \times h$ (M1)

Note: Award (M1) for correct substitution in the volume of cone formula.

$V = \frac{2500 π h}{3} - \frac{π h^{3}}{3}$ (AG)

Note: The final line must be seen, with no incorrect working, for the (M1) to be awarded.

[1 mark]

$(\frac{d V}{d h} =) \frac{2500 π}{3} - π h^{2}$ (A1)(A1)

Note: Award (A1) for $\frac{2500 π}{3}$ , (A1) for $- π h^{2}$ . Award at most (A1)(A0) if extra terms are seen. Award (A0) for the term $- \frac{3 π h^{2}}{3}$ .

[2 marks]

$0 = \frac{2500 π}{3} - π h^{2}$ (M1)

Note: Award (M1) for equating their derivative to zero. Follow through from part (c).

sketch of $\frac{d V}{d h}$ (M1)

Note: Award (M1) for a labelled sketch of $\frac{d V}{d h}$ with the curve/axes correctly labelled or the $x$ -intercept explicitly indicated.

$(h =) 28.9 (cm) (\sqrt{\frac{2500}{3}}, \frac{50}{\sqrt{3}}, \frac{50 \sqrt{3}}{3}, 28.8675 \dots)$ (A1)(ft)

Note: An unsupported $28.9 cm$ is awarded no marks. Graphing the function $V (h)$ is not an acceptable method and (M0)(A0) should be awarded. Follow through from part (c). Given the restraints of the question, $h \geq 50$ is not possible.

[2 marks]

$(V =) \frac{2500 \times π \times 28.8675 \dots}{3} - \frac{π {(28.8675 \dots)}^{3}}{3}$ (M1)

$\frac{1}{3} π {(40.828 \dots)}^{2} \times 28.8675 \dots$ (M1)

Note: Award (M1) for substituting their $28.8675 \dots$ in the volume formula.

$(V =) 50400 ({cm}^{3}) (50383.3 \dots)$ (A1)(ft)(G2)

Note: Follow through from part (d).

[2 marks]

$(S =) π \times \sqrt{2500 - {(28.8675 \dots)}^{2}} \times 50$ (A1)(ft)(M1)

Note: Award (A1) for their correct radius seen $(40.8248 \dots, \sqrt{2500 - {(28.8675 \dots)}^{2}})$ .
Award (M1) for correctly substituted curved surface area formula for a cone.

$(S =) 6410 ({cm}^{2}) (6412.74 \dots)$ (A1)(ft)(G2)

Note: Follow through from parts (a) and (d).

[3 marks]

Examiners report

[N/A]

Syllabus sections

Topic 5 —Calculus » SL 5.8—Testing for max and min, optimisation. Points of inflexion

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18M.2.SL.TZ2.S_9a:
Find the initial velocity of P.
18M.2.SL.TZ2.S_9d:
Find the acceleration of P when it changes direction.
22M.1.SL.TZ1.7e.ii:
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22M.1.SL.TZ1.7e.i:
Write down the value of $a$ .
19M.2.AHL.TZ2.H_9c:
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18N.2.AHL.TZ0.H_9c.i:
sketch the graph of $y = f\left( x \right)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.
17M.1.AHL.TZ2.H_9a.i:
Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.

$y = f(x)$ ;
18M.1.AHL.TZ1.H_9c:
Sketch the graph of $y = f\left( x \right)$ showing clearly the position of the points A and B.
19M.2.AHL.TZ2.H_9b:
Sketch the graph of $y = {x^4} - 6{x^3} - 2{x^2} + 58x - 51$ , stating clearly the coordinates of any maximum and minimum points and intersections with axes.
18M.2.SL.TZ1.S_10a:
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18M.2.SL.TZ1.S_10b.i:
For the graph of $f$ , write down the amplitude.
19M.2.SL.TZ1.S_4a:
Sketch the graph of ${f''}$ on the grid below:
18M.2.SL.TZ1.T_6c:
Find the height of the cylinder, h , of the new trash can, in terms of r.
19M.2.SL.TZ2.S_5c:
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18M.1.SL.TZ2.T_13a:
Find the cost of producing 70 shirts.
18M.1.SL.TZ2.T_13c:
Find the number of shirts produced when the cost of production is lowest.
18M.2.SL.TZ1.T_6b:
Find the total volume of the trash can.
18N.2.SL.TZ0.T_6a:
Calculate the area of cloth, in cm², needed to make Haruka’s bag.
17M.1.AHL.TZ2.H_9a.ii:
Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.

$y = \frac{1}{{f(x)}}$ ;
19N.2.SL.TZ0.S_8d:
Let $R$ be the region enclosed by the graph of $f$ , the $x$ -axis and the lines $x = p$ and $x = b$ . The region $R$ is rotated 360º about the $x$ -axis. Find the volume of the solid formed.
18M.2.SL.TZ1.T_6e:
Using your graphic display calculator, find the value of r which maximizes the value of V.
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 $b$ .
19N.2.SL.TZ0.S_8c.ii:
Find the rate of change of $f$ at ${\text{B}}$ .
17M.2.SL.TZ2.T_6f:
Write down the number of possible solutions to the equation $f(x) = 5$ .
17M.2.SL.TZ2.S_8b.i:
Write down the coordinates of A.
17M.2.SL.TZ2.S_8c.i:
Find the coordinates of B.
18M.2.SL.TZ2.S_9e:
Find the total distance travelled by P.
SPM.1.SL.TZ0.8c.i:
Sketch the graph of $y = f'\left( x \right)$ .
17N.2.AHL.TZ0.H_11a.i:
Determine an expression for $f’(x)$ in terms of $x$ .
16N.1.AHL.TZ0.H_11h:
Find the value $\kappa$ for $x = \frac{\pi }{2}$ and comment on its meaning with respect to the shape of the graph.
18M.2.SL.TZ1.S_10e:
Find the first time when the ball’s speed is changing at a rate of 2 cm s⁻².
17M.2.SL.TZ1.S_10b.i:
Find $\int_{0.111}^{3.31} {\left( {h(x) - x} \right){\text{d}}x}$ .
19N.1.SL.TZ0.S_8a:
Write down an expression for $y$ in terms of $x$ .
19N.2.SL.TZ0.S_8b.ii:
Find the equation of the tangent to the graph of $f$ at ${\text{A}}$ .
18N.1.AHL.TZ0.H_10b:
Hence, show that $\int {{{\text{e}}^x}\,{\text{cos}}{\,^2}x{\text{d}}x = } \frac{{{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{{10}}{\text{cos}}\,2x + \frac{{{{\text{e}}^x}}}{2} + c$ , $c \in \mathbb{R}$ .
21N.1.SL.TZ0.9c.ii:
Find the values of $x$ where the graph of $f$ has points of inflexion. Justify your answer.
19N.1.SL.TZ0.S_8d.i:
Find the value of $x$ for which $V$ is a maximum.
19N.2.SL.TZ0.S_8a:
Find the value of $p$ .
16N.1.AHL.TZ0.H_11c:
Show that the function $f$ has a local maximum value when $x = \frac{{3\pi }}{4}$ .
17M.1.AHL.TZ2.H_9b:
Find $\int {f(x)\cos x{\text{d}}x}$ .
SPM.1.SL.TZ0.8a:
Find $f'\left( x \right)$ .
18N.2.AHL.TZ0.H_9c.ii:
sketch the graph of $y = {f^{ - 1}}\left( x \right)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.
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 $k$ .
17M.2.SL.TZ2.S_8b.ii:
Write down the rate of change of $f$ at A.
19M.2.SL.TZ2.T_5c:
Sketch the graph of $y = f\left( x \right)$ for −3 ≤ $x$ ≤ 3 and −4 ≤ $y$ ≤ 12.
17N.1.SL.TZ0.T_15a:
Write down how many kilograms of cheese Maria sells in one week if the price of a kilogram of cheese is 8 EUR.
18N.1.SL.TZ0.S_10b.i:
Find $f'\left( x \right)$ .
18M.2.SL.TZ1.S_10d:
Find the maximum speed of the ball.
18M.1.SL.TZ2.S_9b:
Show that $C = 20\pi {r^2} + \frac{{320\pi }}{r}$ .
17M.1.AHL.TZ2.H_9a.iii:
Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.

$y = \left| {\frac{1}{{f(x)}}} \right|$ .
18M.2.SL.TZ2.S_9b:
Find the maximum speed of P.
18M.2.SL.TZ1.S_7a:
Given that x_k_{+ 1} = x_k + a, find a.
17M.2.SL.TZ2.S_8a:
Find the value 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 $a$ .
18M.1.AHL.TZ1.H_9b.ii:
The coordinates of B can be expressed in the form B $\left( {{2^a},\,b \times {2^{ - 3a}}} \right)$ where a, b $\in \mathbb{Q}$ . Find the value of a and the value of b.
17N.2.AHL.TZ0.H_10b:
Sketch the graph of $y = f(x)$ showing clearly the minimum point and any asymptotic behaviour.
19N.1.SL.TZ0.S_8b:
Find an expression for $V$ in terms of $x$ .
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.
19N.2.SL.TZ0.S_8b.i:
Write down the coordinates of ${\text{A}}$ .
18N.1.AHL.TZ0.H_10a:
Use integration by parts to show that $\int {{{\text{e}}^x}\,{\text{cos}}\,2x{\text{d}}x = } \frac{{2{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{5}{\text{cos}}\,2x + c$ , $c \in \mathbb{R}$ .
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.
22M.3.AHL.TZ1.2g.ii:
Hence describe numerically the horizontal position of point $P$ relative to the horizontal positions of the points $R$ and $A$ .
22M.3.AHL.TZ2.1d.ii:
Hence deduce that the curve $y^{2} = x^{3} + x$ has no local minimum or maximum points.
EXN.1.SL.TZ0.7c:
Find an expression for $A$ in terms of $x$ .
EXN.1.SL.TZ0.7e:
Determine the maximum area of rectangle $ORST$ .
22M.1.SL.TZ2.7e:
Find the values of $x$ for which the graph of $g$ is concave-up.
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 $b$ .
SPM.1.SL.TZ0.8c.ii:
Hence explain why the graph of $f$ has a local maximum point at $x = a$ .
22M.2.AHL.TZ1.12c.iii:
Using the graph of $y = \frac{8 x + x^{4}}{4 - x^{3}}$ , suggest a reason why the approximation given by Euler’s method in part (a) is not a good estimate to the actual value of $y$ at $x = 1.5$ .
22M.2.AHL.TZ2.11e:
Let $D (t)$ represent the distance between airplane $A$ and airplane $B$ for $0 \leq t \leq 2.5$ .

Find the minimum value of $D (t)$ .
22M.2.AHL.TZ2.12d:
Hence determine the maximum value of $\frac{d P}{d t}$ in terms of $k$ and $N$ .
17N.2.AHL.TZ0.H_10a.ii:
Determine the values of $x$ for which $f(x)$ is a decreasing function.
20N.2.SL.TZ0.S_10a:
Show that $f'' (x) = \frac{24 - 6 x^{2}}{{(x^{2} + 4)}^{2}}$ .
EXN.2.AHL.TZ0.12c:
The curve has a point of inflexion at $(x_{1}, y_{1})$ where $e^{- \frac{π}{2}} < x_{1} < e^{\frac{π}{2}}$ . Determine the coordinates of this point of inflexion.
SPM.1.SL.TZ0.8d.i:
Find $f''\left( b \right)$ .
17N.2.AHL.TZ0.H_11b.iii:
Hence show that $f(x) = 0$ can be expressed as ${u^3} - 7{u^2} + 15u - 9 = 0$ .
17M.1.AHL.TZ2.H_9c:
By finding $g'(x)$ explain why $g$ is an increasing function.
18N.2.AHL.TZ0.H_9d:
Hence, or otherwise, solve the inequality $f\left( x \right) > {f^{ - 1}}\left( x \right)$ .
16N.1.AHL.TZ0.H_11f:
Find the area of the region enclosed by the graph of $f$ and the $x$ -axis.

The curvature at any point $(x,{\text{ }}y)$ on a graph is defined as $\kappa = \frac{{\left| {\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}}} \right|}}{{{{\left( {1 + {{\left( {\frac{{{\text{d}}y}}{{{\text{d}}x}}} \right)}^2}} \right)}^{\frac{3}{2}}}}}$ .
21M.2.AHL.TZ1.11c:
Find the $x$ -coordinate of the point of inflexion.
17M.2.SL.TZ2.S_8c.ii:
Find the the rate of change of $f$ at B.
20N.2.SL.TZ0.S_10b:
Find the least value of $n$ .
20N.2.SL.TZ0.T_4a:
Write down an equation in $r$ and $h$ that shows this information.
18M.2.SL.TZ1.S_10b.ii:
For the graph of $f$ , write down the period.
19M.1.AHL.TZ2.H_8a:
Show that the volume of the cone may be expressed by $V = \frac{\pi }{3}\left( {2R{h^2} - {h^3}} \right)$ .
17N.1.SL.TZ0.T_11b:
Write down the value of $c$ .
21M.3.AHL.TZ1.1c.i:
a point of inflexion with zero gradient.
21M.3.AHL.TZ1.1c.iii:
no points where the gradient is equal to zero.
21M.3.AHL.TZ1.1c.ii:
one local maximum point and one local minimum point.
21M.3.AHL.TZ1.1e.i:
exactly one $x$ -axis intercept.
19N.2.SL.TZ0.S_8c.i:
Find the coordinates of ${\text{B}}$ .
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 $k$ .
16N.1.AHL.TZ0.H_11e:
Sketch the graph of $f$ , clearly indicating the position of the local maximum point, the point of inflexion and the axes intercepts.
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 $y = - x$ .
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.
18N.1.AHL.TZ0.H_10d:
Find the area enclosed by the curve and the $x$ -axis between B and D, as shaded on the diagram.
18M.2.SL.TZ2.S_9c:
Write down the number of times that the acceleration of P is 0 m s⁻² .
EXN.1.SL.TZ0.7b:
Find the dimensions of rectangle $ORST$ that has maximum perimeter and determine the value of the maximum perimeter.
18M.1.SL.TZ2.S_9a:
Express h in terms of r.
19N.1.SL.TZ0.S_8e:
Find the maximum volume.
19N.1.SL.TZ0.S_8d.ii:
Justify your answer.
19N.1.SL.TZ0.S_8c:
Find $\frac{{{\text{d}}V}}{{{\text{d}}x}}$ .
19M.1.SL.TZ2.T_15c:
Hence, find the number of vases that will maximize the profit.
EXN.1.SL.TZ0.7a:
Show that $P = - 2 x^{2} + 2 x + 8$ .
21M.3.AHL.TZ2.1f:
Hence, or otherwise, show that ${f_{n}}^{'} (\frac{a}{4}) > 0$ , for $n \in ℤ^{+}$ .
17N.1.SL.TZ0.S_6:
Let $f(x) = 15 - {x^2}$ , for $x \in \mathbb{R}$ . The following diagram shows part of the graph of $f$ and the rectangle OABC, where A is on the negative $x$ -axis, B is on the graph of $f$ , and C is on the $y$ -axis.

Find the $x$ -coordinate of A that gives the maximum area of OABC.
21M.3.AHL.TZ2.1g.ii:
a point of inflexion with zero gradient for odd values of $n$ , where $n > 1$ and $a \in ℝ^{+}$ .
21M.3.AHL.TZ2.1h:
Consider the graph of $y = x^{n} {(a - x)}^{n} - k$ , where $n \in ℤ^{+}$ , $a \in ℝ^{+}$ and $k \in ℝ$ .

State the conditions on $n$ and $k$ such that the equation $x^{n} {(a - x)}^{n} = k$ has four solutions for $x$ .
SPM.1.SL.TZ0.9c:
Show that the x-coordinate of Q is $\frac{{\text{1}}}{5}{{\text{e}}^{\frac{3}{2}}}$ .
21M.3.AHL.TZ1.1d.i:
the $y$ -coordinate of the local maximum point is $2 c^{\frac{3}{2}} + 2$ .
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.2.SL.TZ0.S_10c:
Find $\int \frac{6 x}{x^{2} + 4} d x$ .
19M.1.AHL.TZ2.H_8b:
Given that there is one inscribed cone having a maximum volume, show that the volume of this cone is $\frac{{32\pi {R^3}}}{{81}}$ .
EXN.1.SL.TZ0.7d:
Find the dimensions of rectangle $ORST$ that has maximum area.
17M.2.SL.TZ1.S_10a.i:
Write down the value of $q$ ;
21N.1.SL.TZ0.9a:
Find all the values of $x$ where the graph of $f$ is increasing. Justify your answer.
21N.1.SL.TZ0.9c.i:
Find the value of $x$ where the graph of $f$ has a local minimum. Justify your answer.
21N.1.SL.TZ0.9d:
The total area of the region enclosed by the graph of $f'$ , the derivative of $f$ , and the $x$ -axis is $20$ .

Given that $f (p) + f (t) = 4$ , find the value of $f (0)$ .
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.3.AHL.TZ2.1b.i:
Write down the coordinates of the two points of inflexion on the curve $y^{2} = x^{3} + 1$ .
22M.3.AHL.TZ2.1e:
Find the value of this $x$ -coordinate, giving your answer in the form $x = \sqrt{\frac{p \sqrt{3} + q}{r}}$ , where $p, q, r \in ℤ$ .
21N.1.SL.TZ0.9b:
Find the value of $x$ where the graph of $f$ has a local maximum.
19M.1.SL.TZ1.T_15a:
Write down an equation for the area, $A$ , of the curved surface in terms of $r$ .
17N.1.SL.TZ0.T_11a:
Find the equation of the axis of symmetry of the graph of $y = f(x)$ .
19M.1.SL.TZ2.T_15a:
Find the value of $P$ if no vases are sold.
17M.2.SL.TZ2.T_6d.i:
Write down the $x$ -coordinates of these two points;
17M.2.SL.TZ2.T_5c:
Find the value of $x$ that maximizes the area of the plot.
16N.2.SL.TZ0.T_6b:
Express this volume in ${\text{c}}{{\text{m}}^3}$ .
16N.2.SL.TZ0.T_6c:
Write down, in terms of $r$ and $h$ , an equation for the volume of this water container.
16N.2.SL.TZ0.T_6e:
Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$ .
16N.2.SL.TZ0.T_6f:
Using your answer to part (e), find the value of $r$ which minimizes $A$ .
18M.2.SL.TZ1.T_6a:
Write down the height of the cylinder.
18M.1.SL.TZ2.T_13b:
Find the value of s.
16N.1.AHL.TZ0.H_11g:
Find the value of the curvature of the graph of $f$ at the local maximum point.
18N.1.SL.TZ0.S_10b.ii:
Hence, find the equation of L in terms of $a$ .
18M.1.SL.TZ2.S_9c:
Given that there is a minimum value for C, find this minimum value in terms of $\pi$ .
17N.2.AHL.TZ0.H_11a.ii:
Sketch a graph of $y = f’(x)$ for $0 \leqslant x < \frac{\pi }{2}$ .
18M.2.SL.TZ1.S_10c:
Hence, write $f\left( x \right)$ in the form $p\,\,{\text{cos}}\,\left( {x + r} \right)$ .
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 $x = b$ .
17N.1.SL.TZ0.T_15d:
Find the price, $p$ , that will give Maria the highest weekly profit.
EXN.1.AHL.TZ0.11e:
Show that the maximum value of $m$ is $\frac{27}{32} \sqrt{\frac{2}{3}}$ .
20N.2.SL.TZ0.T_4d:
Using your answer to part (c), find the value of $h$ for which $V$ is a maximum.
17N.2.AHL.TZ0.H_11c:
Solve the equation $f(x) = 0$ , giving your answers in the form $\arctan k$ where $k \in \mathbb{Z}$ .
SPM.1.SL.TZ0.9a:
Show that $f'\left( x \right) = \frac{{1 - {\text{ln}}\,5x}}{{k{x^2}}}$ .
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 $\tan x = 2x$ .
17N.1.SL.TZ0.T_15b:
Find how much Maria earns in one week, from selling cheese, if the price of a kilogram of cheese is 8 EUR.
21M.3.AHL.TZ1.1e.iii:
exactly three $x$ -axis intercepts.
17N.2.AHL.TZ0.H_11a.iii:
Find the $x$ -coordinate(s) of the point(s) of inflexion of the graph of $y = f(x)$ , labelling these clearly on the graph of $y = f’(x)$ .
17N.2.AHL.TZ0.H_10d:
This region is now rotated through $2\pi$ radians about the $x$ -axis. Find the volume of revolution.
18N.1.AHL.TZ0.H_10c:
Find the $x$ -coordinates of A and of C , giving your answers in the form $a + {\text{arctan}}\,b$ , where $a$ , $b \in \mathbb{R}$ .
17N.2.AHL.TZ0.H_11b.i:
Express $\sin x$ in terms of $\mu$ .
16N.1.AHL.TZ0.H_11d:
Find the $x$ -coordinate of the point of inflexion of the graph of $f$ .
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 $f (x)$ .
18N.2.AHL.TZ0.H_9b:
Hence, or otherwise, find the coordinates of the point of inflexion on the graph of $y = f\left( x \right)$ .
17N.2.AHL.TZ0.H_11b.ii:
Express $\sin 2x$ in terms of $u$ .
EXN.2.AHL.TZ0.12d:
Use the differential equation $\frac{d y}{d x} = \frac{y^{2} + 3 x y + 2 x^{2}}{x^{2}}$ to show that the points of zero gradient on the curve lie on two straight lines of the form $y = m x$ where the values of $m$ are to be determined.
19M.2.SL.TZ1.S_4b:
Find the $x$ -coordinates of the points of inflexion of the graph of $f$ .
21M.2.AHL.TZ1.12e:
Find the value of $t$ when the rate of change of the population is at its maximum.
21M.3.AHL.TZ1.1d.ii:
the $y$ -coordinate of the local minimum point is $- 2 c^{\frac{3}{2}} + 2$ .
21M.1.SL.TZ1.8c:
Given that $f'' (x) = \frac{20 \ln x - 9}{x^{6}}$ , show that $P$ is a local maximum point.
21M.3.AHL.TZ1.1e.ii:
exactly two $x$ -axis intercepts.
21M.3.AHL.TZ2.1g.i:
a local minimum point for even values of $n$ , where $n > 1$ and $a \in ℝ^{+}$ .
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 ${h^{ - 1}}$ .
18N.2.SL.TZ0.T_6g:
Use your answer to part (f) to show that the width of Nanako’s bag is 12 cm.
17N.1.SL.TZ0.T_11c:
Find the value of $a$ and of $b$ .
19M.1.SL.TZ1.T_15c:
Find the value of $r$ when the area of the curved surface is maximized.
18N.2.SL.TZ0.T_6d:
Write down and simplify an expression in x and y for the area of cloth, A, used to make Nanako’s bag.
19M.1.SL.TZ2.T_15b:
Differentiate $P\left( x \right)$ .
17N.1.SL.TZ0.T_15c:
Write down an expression for $W$ in terms of $p$ .
20N.2.SL.TZ0.T_4b:
Show that $V = \frac{2500 π h}{3} - \frac{π h^{3}}{3}$ .
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.2.SL.TZ0.T_4e:
Find the maximum volume of the bird bath.
17M.2.SL.TZ2.T_5b:
Write down the area of the plot in terms of $x$ .
19M.2.SL.TZ2.T_5h:
Determine the range of $f\left( x \right)$ for $p$ ≤ $x$ ≤ $q$ .
17M.2.SL.TZ2.T_5a:
Show that the width of the plot, in metres, is given by $100 - x$ .
17M.2.SL.TZ2.T_6g:
The equation $f(x) = m$ , where $m \in \mathbb{R}$ , has four solutions. Find the possible values of $m$ .
20N.2.SL.TZ0.T_4c:
Find $\frac{d V}{d h}$ .
16N.2.SL.TZ0.T_6d:
Show that $A = \pi {r^2} + \frac{{1\,000\,000}}{r}$ .
16N.2.SL.TZ0.T_6g:
Find the value of this minimum area.
19M.1.SL.TZ1.T_15b:
Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$ .
18N.2.SL.TZ0.T_6e:
Use your answers to parts (c) and (d) to show that

$A = 3{x^2} + \frac{{10368}}{x}$ .
18M.2.SL.TZ1.T_6d:
Show that the volume, V cm³ , of the new trash can is given by

$V = 110\pi {r^3}$ .
16N.2.SL.TZ0.T_6a:
Write down a formula for $A$ , the surface area to be coated.
16N.2.SL.TZ0.T_6h:
Find the least number of cans of water-resistant material that will coat the area in part (g).
18M.2.SL.TZ1.T_6f:
The designer claims that the new trash can has a capacity that is at least 40% greater than the capacity of the original trash can.

State whether the designer’s claim is correct. Justify your answer.
17M.2.SL.TZ2.T_5d:
Show that Violeta earns 5000 BGN from selling the flowers grown on the plot.
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 $f(x)$ .
18N.2.SL.TZ0.T_6b:
Calculate the volume, in cm³, of the bag.
18N.2.SL.TZ0.T_6c:
Use this value to write down, and simplify, the equation in x and y for the volume of Nanako’s bag.
18N.2.SL.TZ0.T_6f:
Find $\frac{{{\text{d}}A}}{{{\text{d}}x}}$ .

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