Applications of Differentiation (Edexcel IGCSE Further Maths): Exam Questions

Two numbers $x$ and $y$ are such that $2x+y=13$

The sum of the squares of $2x$ and $y$ is $S$.

Show that $S=8x^{2}--52x+169$

Using calculus, find the value of $x$ for which $S$ is a minimum, justifying that this value of $x$ gives a minimum value for $S$.

Using calculus, find the minimum value of $S$.

Figure 1 shows the curve $M$ with equation $y=x^{3}--13x--12$

The point $P$, with $x$ coordinate −2, lies on $M$ and line$l_{1}$ is the tangent to $M$ at the point $P$.

Find an equation for $l_{1}$

The point $Q$ lies on $M$ and the line $l_{2}$ is the tangent to $M$ at the point $Q$.

Given that $l_{1}$ and $l_{2}$ are parallel,

find an equation for $l_{2}$

The normal to $M$ at $P$ meets $l_{2}$ at the point $R$.

Find the coordinates of $R$.

Find the exact length of the line $PR$.

The tangent and normal at $P$and the tangent and normal at $Q$ form a rectangle.

Find the exact area of this rectangle.

Figure 2 shows a right circular cone with a base radius of $x$ cm. The slant height of the cone is $l$ cm and the height of the cone is $h$ cm. The vertex of the cone is $B$ and the points $A$ and $C$, on the base of the cone, are such that $AC$ is a diameter of the base.

The cone is increasing in size in such a way that the size of the angle $ABC$ is constant at $60^{\circ}$ and the total surface area of the cone is increasing at a constant rate of 10 cm² /s.

Find the exact rate of increase of the volume of the cone when $x=6$

$f(x)=2x^{2}+4x+9$

Given that $f(x)$ can be written in the form $A ( x + B )^{2 } + C$ , where $A,B$ and $C$ are integers,

find the value of $A$, the value of $B$and the value of $C$

$f(x)=2x^{2}+4x+9$

Given that $f(x)$ can be written in the form $A ( x + B )^{2} + C$ , where $A,B$ and $C$ are integers,

(i) Hence, or otherwise, find the value of $x$ for which $\frac{1}{ \mathrm{f} ( x )}$is a maximum

(ii) find the maximum value of $\frac{1}{ \mathrm{f} ( x )}$

The surface area of a sphere with radius $r$ cm is increasing at a constant rate of $50π$ cm² /s

Find, in cm³ , the exact volume of the sphere at the instant when the rate of increase of $r$ is $\frac{5}{12}$ cm/s

Figure 1 shows a sketch of part of the curve $C$ with equation

$y=\frac{x^{2}}{4}-3\sqrt{x}+8$

The point $P$ lies on $C$ and has coordinates $(4,a)$

Show that $a=6$

The line $L$is the normal to $C$ at the point $P$

Show that an equation of $L$ is $5y+4x-46=0$

The finite region $R$ is bounded by the curve $C$, the line $L$, the $x$-axis and the line with equation $x=1$

Use calculus to find the exact area of $R$

Figure 4 shows a solid right triangular prism $ABCDEF$

The cross section of the prism is an isosceles triangle.

• $\text{angle} D E C = \text{angle} A F B = 90 ^{\circ}$
• $AB=DC=x$ cm
• $AD=BC=FE=y$cm
• $AF=BF=DE=CE$

The triangular faces of the prism are vertical and the edges $AD,BC$ and $FE$are horizontal. The volume of the prism is $3.6$ cm³ The total external surface area of the prism is $S$ cm²

Show that $S$ satisfies the equation

$S = \frac{x^{2}}{2} + \frac{72 ( \sqrt{2} + 1 )}{5 x}$

Given that $x$ can vary, use calculus, to find to 3 significant figures, the value of $x$ for which $S$ is a minimum.

Justify that this value of $x$ gives a minimum value of $S$

Hence find, to 2 significant figures, the minimum value of $S$

Figure 2 shows a shape $ABCDEOFG$

$ABCO$ is a quarter circle with radius $r$ cm $CDEO$ and $AOFG$are congruent rectangles of length $r$ cm and width $x$cm

The total area of the shape is 100cm² The perimeter of the shape is $P$ cm

Show that $P=\frac{200}{r}+2r$

Use calculus to find the value of $r$ for which $P$ is a minimum, justifying that this value of $r$ gives a minimum value of $P$

Find the minimum value of $P$

$\mathrm{f} ( x ) = 3 - 4 x - 9 x^{2}$

Given that $\mathrm{f} ( x )$can be expressed in the form $A - B ( x + C )^{2}$ where $A$, $B$ and $C$ are positive constants

find the value of $A$, the value of $B$and the value of $C$

$\mathrm{f} ( x ) = 3 - 4 x - 9 x^{2}$

Given that $\mathrm{f} ( x )$can be expressed in the form $A - B ( x + C )^{2}$ where $A$, $B$ and $C$ are positive constants

Hence write down the maximum value of $\mathrm{f} ( x )$

The equation $\mathrm{f} ( x ) = 0$ has roots $α$ and $β$

Without solving the equation $\mathrm{f} ( x ) = 0$, form a quadratic equation, with integer coefficients, that has roots $\frac{3α}{β}$ and $\frac{3β}{α}$

Show that $(x+y)^{3}=x^{3}+y^{3}+3xy(x+y)$

$g ( x ) = 3 x^{2} + q x + r$ where $q$and $r$ are constants

The equation $g(x)=0$ has roots $α^{2}-β$ and $β^{2}-α$ where $α$ and $β$are the roots of the equation $\mathrm{f} ( x ) = 0$

Using your answer to part (d), find in simplified exact form, the value of $q$ and the value of $r$

The height of liquid in a vessel $P$ is $h$ The volume, $V$, of the liquid in $P$ is given by $V=6h^{3}$ Liquid is leaking from $P$ at a constant rate of $36\mathrm{cm}^{3}/s$

Find the exact rate of change, in cm/s, of $h$ when $V=384\mathrm{cm}^{3}$

A curve $C$ has equation

$y=\frac{5x-2}{3x+2}x\neq\frac{2}{3}$

Find the coordinates of the point where $C$ intersects the

(i) $x$-axis

(ii) $y$-axis

A curve $C$ has equation

$y=\frac{5x-2}{3x+2}x\neq\frac{2}{3}$

Write down an equation of the asymptote to $C$ that is

(i) parallel to the $x$-axis

(ii) parallel to the $y$-axis

Sketch $C$ on the opposite page. Show and label the asymptotes and the coordinates of the points where $C$ crosses the coordinate axes.

Point $A$ lies on $C$ such that the gradient of $C$ at $A$is parallel to the line with equation $4y-x-7$

The normal to $C$ at $A$intersects the $x$-axis at point $D$and the $y$-axis at point $E$ Given that the $x$ coordinate of $A$ is positive,

find, in its simplified form, the exact length of line $DE$

A solid cuboid has width $x$ cm, length 4$x$ cm and height $h$cm. The volume of the cuboid is 75cm³ and the surface area of the cuboid is $S\mathrm{cm}^{2}$

Show that $S=8x^{2}+\frac{375}{2x}$

Given that $x$can vary, using calculus,

(i) find to 3 significant figures, the value of $x$ for which $S$ is a minimum,

(ii) justify that this value of $x$ gives a minimum value of $S$

Find, to 3 significant figures, the minimum value of $S$

The equation of a curve is $y=\sqrt{\frac{e^{4x}}{2x-3}}$

When $x$ is increased to $(x+δx)$, $y$ increases to $(y+δy)$ where $δx$ and $δy$ are small.

Show that $δy\approx\frac{e^{2x}(4x-7)}{(2x-3)^{\frac{3}{2}}}δx$

The equation of a curve is $y=\sqrt{\frac{e^{4x}}{2x-3}}$

When $x$ is increased to $(x+δx)$, $y$ increases to $(y+δy)$ where $δx$ and $δy$ are small.

Given that $x$= 2.5

find an estimate, to 2 significant figures, of the value of $δy$ when the value of $x$ increases by 0.2%

Figure 3 shows a right triangular prism $ABCDEF$. A cross section $ABC$ of the prism is a triangle in which $AB=AC=r\mathrm{cm}$ and angle C A B equals pi over italic 3radians.

In the prism

$AE=BF=CD=5\text{cm}ED=EF=r\text{cm}$ and angle D E F equals pi over italic 3 radians.

Show that the volume of the prism is $\frac{5\sqrt{3}}{4}r^{2}\text{cm}^{3}$

Figure 3 shows a right triangular prism $ABCDEF$. A cross section $ABC$ of the prism is a triangle in which $AB=AC=r\mathrm{cm}$ and angle C A B equals pi over italic 3radians.

The volume of the prism is increasing in such a way that the size of angle C A B and the size of angle D E F remain constant and the length of $AE$, the length of $BF$ and the length of $CD$ remain constant. The lengths of $AB,AC,ED$ and $EF$ are each increasing at a constant rate of 0.2cm / s

Find the exact rate of increase, in cm³ / s, of the volume of the prism when the area of the rectangular face $BCDF$ is 60 cm²

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

Express $\mathrm{g} ( x )$ in the form $p ( x + q )^{2}$ where $p$, $q$ and $r$ are rational numbers to be found.

Find

(i) the minimum value of $\mathrm{g} ( x )$

(ii) the value of $x$at which this minimum occurs.

$h(x)=2x^{6}+\frac{1}{2}x^{3}-3$

Hence, or otherwise, write down

(i) the minimum value of $\mathrm{h} ( x )$

(ii) the value of $x$ at which this minimum occurs.

Figure 1 shows part of the curve $S$ with equation $y=px^{2}+qx+r$ where $p$, $q$ and $r$ are constants.

The points $A$, $B$ and $P$ with coordinates (– 2, 0), (6, 0) and (4, – 6) respectively lie on $S$

Show that an equation of $S$ is $y=\frac{x^{2}}{2}-2x-6$

Figure 1 shows part of the curve $S$ with equation $y=px^{2}+qx+r$ where $p$, $q$ and $r$ are constants.

The line $l$ is the normal to $S$ at the point $P$

Show that an equation of $l$ is $2y+x+8=0$

Figure 1 shows part of the curve $S$ with equation $y=px^{2}+qx+r$ where $p$, $q$ and $r$ are constants.

The finite region shown shaded in Figure 1 is bounded by $S$ and $l$

Use algebraic integration to find the exact area of the shaded region.

The volume of oil in a container is $V$ cm³ when the height of the oil is $h$ cm. Oil is pouring into the container at a constant rate of 12 cm³ /s. Given that $V=3h^{3}$

find the exact rate, in cm/s, at which the height of the oil is increasing when $V=1536\mathrm{cm}^{3}$

Two numbers $x$ and $y$ are such that $3x-y=4$

$S=5x^{3}+y^{2}$

Show that $S=5x^{3}+9x^{2}-24x+16$

Given that $x$ can vary,

use calculus to find the value of $x$for which $S$ is a minimum, justifying that this value of $x$ gives a minimum value of $S$

Find the minimum value of $S$

Figure 1 shows a badge, shown shaded, made from two identical rectangles, $ABCD$ and $DEFG$, and a sector $DCG$ of a circle with centre $D$.

Each rectangle measures $x$ cm by $y$ cm. The radius of the sector is $x$ cm and the angle $CDG$ is 0.5 radians.

The area of the badge is 50 cm² The perimeter of the badge is $P$ cm.

Show that

$P=2x+\frac{100}{x}$

Given that $x$ can vary,

use calculus, to find the exact value of $x$ for which $P$ is a minimum. Justify that this value of $x$ gives a minimum value for $P$

Find the minimum value of $P$ Give your answer in the form $k\sqrt{2}$, where $k$ is an integer to be found.

The equation of a curve is $y=x^{3}\mathrm{sin}x$

Find an equation of the tangent to the curve at the point on the curve where $x=\frac{1}{2}π$

Give your answer in the form $y=mx+c$

A cube has edges of length $x$ cm.

The total surface area, $A\mathrm{cm}^{2}$ , of the cube is increasing at a constant rate of 0.45cm² /s

Find the rate of increase, in cm³ /s, of the volume of the cube at the instant when the total surface area of the cube is 384cm²

Figure 3 shows part of the curve $C$ with equation $y=4-e^{2x}$ The curve $C$ crosses the $y$-axis at the point $A$and the $x$-axis at the point $B$.

(i) Write down the $y$ coordinate of point $A$.

(ii) Show that the $x$ coordinate of $B$ is $x=\mathrm{ln}2$

The line $l$ is the normal to $C$ at the point $B$.

Find an equation for $l$ , giving your answer in the form $y=mx+c$

The finite region $R$ is bounded by $C$, $l$ and the $y$-axis.

Using calculus, find the area of $R$. Give your answer to one decimal place.