Further Pure Math (App of Differentation)

 1. (2011/june/paper01/q9)

A curve has equation

$$y=\dfrac{2 x^{2}-6}{3 x-6} \quad x \neq 2$$

(a) Write down an equation of the asymptote to the curve which is parallel to the $y$-axis.

(b) Find the coordinates of the stationary points on the curve. (7)

The curve crosses the $y$-axis at the point $A$.

(c) Find an equation of the normal to the curve at $A .$

The normal at $A$ meets the curve again at $B$.

(d) Find the $x$-coordinate of $B$.


2. (2011/june/Paper02/q7) 


A rectangular box has length $3 x \mathrm{~cm}$, width $x \mathrm{~cm}$ and height $h \mathrm{~cm}$, as shown in Figure 2 . The top of the box, $A B C D$, is open. The volume of the box is $30 \mathrm{~cm}^{3}$ and the total external surface area of the box is $S \mathrm{~cm}^{2}$.

(a) Show that $S=3 x^{2}+\dfrac{80}{x}$

Given that $x$ can vary,

(b) find, to 3 significant figures, the minimum value of $S$.

(c) Verify that your answer to part (b) does give the minimum value for $S$.


3. (2012/jan/paper01/q11) 


The centre of the circle $C$, with equation $x^{2}+y^{2}-10 y=0$, has coordinates $(0,5)$. The circle passes through the origin $O$. The region bounded by the circle, the positive $y$-axis and the line $y=h$, where $h<5$, is shown shaded in Figure $3 .$ The shaded region is rotated through $2 \pi$ radians about the $y$-axis.

(a) Show that the volume of the solid formed is $\dfrac{1}{3} \pi h^{2}(15-h)$.

The point $A$ with coordinates $(5,5)$ lies on $C$. A bowl is formed by rotating the $\operatorname{arc} O A$ through $2 \pi$ radians about the $y$-axis, as shown in Figure 4 . Water is poured into the bowl at a constant rate of $6 \mathrm{~cm}^{3} / \mathrm{s}$. The volume of water in the bowl is $V \mathrm{~cm}^{3}$ when the depth of water above $O$ is $h \mathrm{~cm}$.

(b) Use the formula given in part (a) to find an expression for $\dfrac{\mathrm{d} V}{\mathrm{~d} h}$ in terms of $h$.

(c) Find, to 3 significant figures, the rate at which $h$ is changing when the water above $O$ is $1.5 \mathrm{~cm}$ deep.

The area of the surface of the water is $W \mathrm{~cm}^{2}$ when the depth of water above $O$ is $h \mathrm{~cm}$.

(d) Show that, for $0<h<5$, the rate of change of the depth of water above $O$ is $\dfrac{k}{W}$, stating the value of $k$ (3)


4. (2012/jan/paper01/q7)

The curve $C$ with equation $y=\dfrac{2 x-3}{x-3}, x \neq 3$, crosses the $x$-axis at the point $A$ and the $y$-axis at the point $B$

(a) Find the coordinates of $A$ and the coordinates of $B .$ (2)

(b) Write down an equation of the asymptote to $C$ which is

(i) parallel to the $y$-axis,

(ii) parallel to the $x$-axis. (2)

(c) Sketch C showing clearly the asymptotes and the coordinates of the points $A$ and $B$.

(d) Find an equation of the normal to $C$ at the point $B$.

The normal to $C$ at the point $B$ crosses the curve again at the point $D$.

(e) Find the $x$-coordinate of $D$.


5. (2012/jan/ paper02/q6) 


A container in the shape of a right circular cone of height $10 \mathrm{~cm}$ is fixed with its axis of symmetry vertical. The vertical angle of the container is $60^{\circ}$, as shown in Figure $3 .$ Water is dripping out of the container at a constant rate of $2 \mathrm{~cm}^{3} / \mathrm{s}$. At time $t=0$ the container is full of water. At time $t$ seconds the depth of water remaining is $h \mathrm{~cm}$.

(a) Show that $h=\left[1000-\dfrac{18 t}{\pi}\right]^{\dfrac{1}{3}}$

(b) Find, in $\mathrm{cm}^{2} / \mathrm{s}$, to 3 significant figures, the rate of change of the area of the surface of the water when $t=15$ $(6)$


6. (2012/jan / рареr02/q9)

The curve $C$, with equation $y=\mathrm{f}(x)$, passes through the point with coordinates $(0,4)$.

Given that $\mathrm{f}^{\prime}(x)=x^{3}-3 x^{2}-x+3$

(a) find $\mathrm{f}(x)$ (3)

(b) Show that $C$ has a minimum point at $x=-1$ and a minimum point at $x=3$ (6)

(c) (i) Find the coordinates of the maximum point on $C$.

(ii) Show that the point found in (i) is a maximum point.

(d) State the ranges of values of $x$ for which $\mathrm{f}^{\prime}(x)>0$ (2)


7. (2012/june/paper01/q6) 


The points $P$ and $Q$ lie on the circumference of a circle with centre $O$ and radius $r \mathrm{~cm}$. Angle $P O Q=\theta$ radians. The segment shaded in Figure 1 has area $A \mathrm{~cm}^{2}$.

(a) Show that $A=\dfrac{1}{2} r^{2}(\theta-\sin \theta)$,

When angle $P O Q$ is increased to $(\theta+\delta \theta)$ radians, where $\delta \theta$ is small, the area of the shaded segment is increased to $(A+\delta A) \mathrm{cm}^{2}$, where $\delta A$ is small.

(b) Show that $\delta A \approx \dfrac{1}{2} r^{2}(1-\cos \theta) \delta \theta$

For a circle of radius $4 \mathrm{~cm}$, the area of the shaded segment is increased by $0.05 \mathrm{~cm}^{2}$ when angle $P O Q$ increases by $0.02$ radians.

(c) Find, to 1 decimal place, an estimate for $\theta$


8. (2012/june/paper01/q9)

The point $P$ with coordinates $(4,4)$ lies on the curve $C$ with equation $y=\dfrac{1}{4} x^{2}$

(a) Find an equation of

(i) the tangent to $C$ at $P$,

(ii) the normal to $C$ at $P$.

The point $Q$ lies on the curve $C$. The normal to $C$ at $Q$ and the normal to $C$ at $P$ intersect at the point $R$. The line $R Q$ is perpendicular to the line $R P$.

(b) Find the coordinates of $Q$.

(c) Find the $x$-coordinate of $R$. (4)

The tangent to $C$ at $P$ and the tangent to $C$ at $Q$ intersect at the point $S$.

(d) Show that the line $R S$ is parallel to the $y$-axis.


9. (2012/june/paper02/q8)

The curve $C$ has equation $y=4 x+8+\dfrac{25}{x-2}, x \neq 2$

(a) Find the coordinates of the stationary points on $C$. (6)

(b) Determine the nature of each of these stationary points. $(3)$


10. (2013/jan/Paper01/q11) 

$$f(x)=x^3+px^2+qx+6,p,q\in Z$$

Given that $\mathrm{f}(x)=(x-1)(x-3)(x+r)$

(a) find the value of $r$.

Hence, or otherwise,

(b) find the value of $p$ and the value of $q$.


Figure 2 shows the curve $C$ with equation $y=\mathrm{f}(x)$ which crosses the $x$-axis at the points with coordinates $(3,0)$ and $(1,0)$ and at the point $A$. The point $B$ on $C$ has $x$-coordinate 2

(c) Find an equation of the tangent to $C$ at $B$.

(d) Show that the tangent at $B$ passes through $A$.

(e) Use calculus to find the area of the finite region bounded by $C$ and the tangent at $B$.


11. $(2013 / \mathrm{jan} /$ paper02/q6)

A solid paperweight in the shape of a cuboid has volume $15 \mathrm{~cm}^{3} .$ The paperweight has a rectangular base of length $5 x \mathrm{~cm}$ and width $x \mathrm{~cm}$ and a height of $h \mathrm{~cm} .$ The total surface area of the paperweight is $A \mathrm{~cm}^{2} .$

(a) Show that $A=10 x^{2}+\dfrac{36}{x}$ (3)

(b) Find, to 3 significant figures, the value of $x$ for which $A$ is a minimum, justifying that this value of $x$ gives a minimum value of $A$

(c) Find, to 3 significant figures, the minimum value of $A .$


12. (2013/june/paper01/q10)

The curve $C$ has equation $y=x^{4}-4 x^{3}-2 x^{2}+13 x+5$ and the line $l_{1}$ is the tangent to $C$ at the point $R(1,13)$

(a) Find an equation for $l_{1}$

The points $P$ and $Q$ lie on $C .$ The $x$-coordinates of $P$ and $Q$ are $p$ and $q$ respectively, where $p<q .$ The tangent to $C$ at $P$ is parallel to $l_{1}$ and the tangent to $C$ at $Q$ is parallel to $I_{1}$

(b) Find the coordinates of $P$ and the coordinates of $Q .$

The line $l_{2}$ passes through $P$ and $Q$

(c) Find an equation for $l_{2}$

(d) Show that $l_{2}$ is a tangent to $C$ at $P$ and a tangent to $C$ at $Q .$

The normal to $C$ at $R(1,13)$ intersects $l_{2}$ at the point $S .$

(e) Find the exact length of $R S$.

(f) Find the area of the triangle $P Q R$.


13. (2013/june/paper01/q5)

The volume of liquid in a container is $V \mathrm{~cm}^{3}$ when the depth of the liquid is $h \mathrm{~cm}$. Liquid is added to the container at a rate of $36 \mathrm{~cm}^{3} / \mathrm{s}$. Given that $V=4 h^{3}$, find the rate at which the depth of the liquid is increasing when $V=500$


14. (2013/june/paper02/q5)


Figure 2 shows a shape $B C D E F$ of area $A \mathrm{~cm}^{2}$. In the shape, $B C D F$ is a rectangle and $D E F$ is a semicircle with $F D$ as diameter.

$B F=C D=y \mathrm{~cm}$ and $B C=F D=2 x \mathrm{~cm} .$ The perimeter of the shape $B C D E F$ is $30 \mathrm{~cm}$.

(a) Find an expression for $y$ in terms of $x$.

(b) Show that $A=30 x-2 x^{2}-\dfrac{1}{2} \pi x^{2}$

(c) Find, to 2 significant figures, the maximum value of $A$, justifying that the value you have found is a maximum.


15. (2014/jan/paper01/q11)

The curve $C$ has equation $5 y=4\left(x^{2}+1\right)$. The coordinates of the point $P$ on the curve are $(p, 8), p>0$

The line $l$ with equation $5 y-24 x+q=0$ is the tangent to $C$ at $P$.

(a) (i) Show that $p=3$

(ii) Find the value of $q$ (4)

(b) Find an equation, with integer coefficients, for the normal to $C$ at $P$.

(c) Find the exact value of the area of the triangle formed by the tangent to $C$ at $P$, the normal to $C$ at $P$ and the $x$-axis.

The finite region bounded by $C$, the tangent to $C$ at $P$, the $x$-axis and the $y$-axis is rotated through $360^{\circ}$ about the $x$-axis.

(d) Find, to 2 significant figures, the volume of the solid generated.


16. (2014/jan/paper02/q2)

The volume of a right circular cone is increasing at a constant rate of $12 \mathrm{~cm}^{3} / \mathrm{s}$. The radius of the base of the cone is always half the height of the cone. Find, in $\mathrm{cm} / \mathrm{s}$, the exact value of the rate of increase of the height of the cone when the height is $4 \mathrm{~cm}$.


17. (2014/jan/paper02/q6) 


A rectangular sheet of card measures $80 \mathrm{~cm}$ by $40 \mathrm{~cm}$. A square of side $x \mathrm{~cm}$ is cut away from each corner of the card as shown in Figure 2. The card is then folded along the dotted lines to form an open box.

The volume of the box is $V \mathrm{~cm}^{3}$.

(a) Show that $V=3200 x-240 x^{2}+4 x^{3}$

(b) Find, to 3 significant figures, the value of $x$ for which $V$ is a maximum, justifying that this value of $x$ gives a maximum value of $V$.

(c) Find, to 3 significant figures, the maximum value of $V$.


18. $(2014 /$ june $/$ paper01/q3)

Given that $2 x y-3 y=\mathrm{e}^{2 x}$

(a) show that $\dfrac{\mathrm{d} y}{\mathrm{~d} x}=\dfrac{4 \mathrm{e}^{2 x}(x-2)}{(2 x-3)^{2}}$

(b) find the value of $\dfrac{\mathrm{d} y}{\mathrm{~d} x}$ when $x=0$

(c) find an equation, with integer coefficients, of the tangent to the curve with equation $2 x y-3 y=\mathrm{e}^{2 x}$ at the point on the curve where $x=0$


19. (2014/june/paper01/q5)

The volume of a right circular cone is increasing at the rate of $72 \mathrm{~cm}^{3} / \mathrm{s}$. The height of the cone is always four times the radius of the base of the cone. Find the rate of increase of the radius of the base, in $\mathrm{cm} / \mathrm{s}$ to 3 significant figures, when the height of the cone is $12 \mathrm{~cm}$.


20. $(2015 / \mathrm{jan} /$ paper02/q10) 


Figure 3 shows the curve $C$ with equation $y=9 x^{3}-18 x^{2}-8 x+24$

The curve cuts the $y$-axis at the point $P$ with coordinates $(0,24)$.

The point $Q$ lies on $C$ and the line $P Q$ is the tangent to $C$ at $P .$

(a) Find an equation of $P Q$

(b) Find the coordinates of $Q$ $(5)$

The point $R$ lies on $C$ and $S$ is the point such that $P Q R S$ is a parallelogram.

Given that $R S$ is the tangent to $C$ at $R$,

(c) find the coordinates of $R$

(d) find the coordinates of $S .$

(e) Show that $S$ lies on $C$. (2)


21. $(2015 / \mathrm{jan} /$ paper02/q2)

A solid right circular cylinder has height $h \mathrm{~cm}$ and base radius $r \mathrm{~cm} .$ The total surface area of the cylinder is $S \mathrm{~cm}^{2}$ and the volume of the cylinder is $V \mathrm{~cm}^{3}$

(a) Show that $S=\dfrac{2 V}{r}+2 \pi r^{2}$

Given that $V=1600$

(b) find, to 3 significant figures, the minimum value of $S .$

Verify that the value you have found is a minimum.


22. (2015/june/paper01/q10)

A solid right circular cylinder has base radius $r \mathrm{~cm}$ and height $h \mathrm{~cm}$. The volume of the cylinder is $50 \mathrm{~cm}^{3}$ and the total surface area is $A \mathrm{~cm}^{2}$.

(a) Show that $A=2 \pi r^{2}+\dfrac{100}{r}$

(b) Use calculus to find, to 4 significant figures, the value of $r$ for which $A$ is a minimum.

(c) Use calculus to verify that the value of $r$ found in part (b) does give a minimum value of $A$

(d) Find, to the nearest whole number, the minimum value of $A$. $(2)$


23. (2016/jan/paper01/q3)

The volume, $V \mathrm{~cm}^{3}$, of a sphere of radius $r \mathrm{~cm}$ is increasing at the rate of $60 \mathrm{~cm}^{3} / \mathrm{s}$.

Find the rate of increase of the radius, in $\mathrm{cm} / \mathrm{s}$ correct to 2 significant figures, when the volume is $36000 \pi \mathrm{cm}^{3}$.


24. (2016/jan/paper02/q7) 


Figure 1 shows the curve with equation $y=\dfrac{x^{2}-2}{2 x-3} \quad$ where $x \neq \dfrac{3}{2}$

(a) Write down an equation of the asymptote to the curve which is parallel to the $y$-axis. (1)

(b) Find $\dfrac{\mathrm{d} y}{\mathrm{~d} x}$ (3)

(c) Find the coordinates of the stationary points on the curve.


25. (2016/june/paper02/q10) 


A conical container is fixed with its axis of symmetry vertical. Oil is dripping into the container at a constant rate of $0.4 \mathrm{~cm}^{3} / \mathrm{s}$. At time $t$ seconds after the oil starts to drip into the container, the depth of the oil is $h \mathrm{~cm}$. The vertical angle of the container is $60^{\circ}$, as shown in Figure 1

When $t=0$ the container is empty.

(a) Show that $h^{3}=\dfrac{18 t}{5 \pi}$

Given that the area of the top surface of the oil is $A \mathrm{~cm}^{2}$

(b) show that $\dfrac{\mathrm{d} A}{\mathrm{~d} t}=\dfrac{4}{5 h}$

(c) Find, in $\mathrm{cm}^{2} / \mathrm{s}$ to 3 significant figures, the rate of change of the area of the top surface of the oil when $t=10$


26. (2016/june/paper02/q5)

A solid cuboid has volume $772 \mathrm{~cm}^{3}$

The cuboid has width $x \mathrm{~cm}$, length $4 x \mathrm{~cm}$ and height $h \mathrm{~cm}$.

The total surface area of the cuboid is $A \mathrm{~cm}^{2}$.

(a) Show that $A=8 x^{2}+\dfrac{1930}{x}$

(b) Find, to 3 significant figures, the value of $x$ for which $A$ is a minimum, justifying that this value of $x$ gives a minimum value of $A$.

(c) Find, to 3 significant figures, the minimum value of $A$.


27. (2017/jan/paper02/q3)

The radius of a circular pool of oil is increasing at a constant rate of $0.5 \mathrm{~cm} / \mathrm{s}$.

Find, in $\mathrm{cm}^{2} / \mathrm{s}$ to 3 significant figures, the rate at which the area of the pool is increasing when the radius of the pool is $200 \mathrm{~cm}$. (5)


28. (2017/jan/paper02/q5)

Given that $y=3 x \sqrt{2 x-1} \quad x>\dfrac{1}{2}$

(a) show that $\dfrac{\mathrm{d} y}{\mathrm{~d} x}=\dfrac{3(3 x-1)}{\sqrt{2 x-1}}$

The straight line $l$ is the normal to the curve with equation $y=3 x \sqrt{2 x-1}$ at the point on the curve where $x=1$

(b) Find an equation, with integer coefficients, for $l$. (6)


29. (2017/june/paper01/q2)

Sand is poured onto horizontal ground at a rate of $50 \mathrm{~cm}^{3} / \mathrm{s} .$ The sand forms a right circular cone with its base on the ground. The volume of the cone increases in such a way that the radius of the base is always three times the height of the cone. Find the rate of change, in $\mathrm{cm} / \mathrm{s}$ to 3 significant figures, of the radius of the cone when the radius is $10 \mathrm{~cm}$.


30. $(2017 / \mathrm{june} / \mathrm{paper0} 2 / \mathrm{q} 7)$

A solid cuboid has width $x \mathrm{~cm}$, length $5 x \mathrm{~cm}$ and height $h \mathrm{~cm}$. The total surface area of the block is $480 \mathrm{~cm}^{2} .$ The volume of the block is $V \mathrm{~cm}^{3}$.

(a) Show that $V=200 x-\dfrac{25}{6} x^{3}$

(b) Find the maximum value of $V$.


31. $(2018 / \mathrm{jan} /$ paper01/q3)

The volume of a right circular cone is increasing at a constant rate of $27 \mathrm{~cm}^{3} / \mathrm{s}$. The radius of the base of the cone is always $1.5$ times the height of the cone.

Calculate the rate of change of the height of the cone, in $\mathrm{cm} / \mathrm{s}$ to 3 significant figures, when the height of the cone is $4 \mathrm{~cm}$. (6)


32. (2018/jan/paper02/q8) 


A solid right circular cylinder has radius $r \mathrm{~cm}$ and height $h \mathrm{~cm}$, as shown in Figure $4 .$ The cylinder has a volume of $355 \mathrm{~cm}^{3}$ and a total surface area of $S \mathrm{~cm}^{2}$

(a) Show that $S=2 \pi r^{2}+\dfrac{710}{r}$ (4)

Given that $r$ can vary,

(b) using calculus find, to 3 significant figures, the minimum value of $S .$

(c) Verify that your answer to part (b) does give the minimum value of $S$. $(2)$


33. (2018/june/paper02/q3)

The volume of liquid in a container is $V \mathrm{~cm}^{3}$ when the depth of the liquid is $h \mathrm{~cm}$. Liquid is leaking from the container at a rate of $24 \mathrm{~cm}^{3} / \mathrm{s}$

Given that $V=5 h^{3}$, find the rate, in $\mathrm{cm} / \mathrm{s}$, at which the depth of the liquid is decreasing when $V=800$ Give your answer to 2 significant figures. (7)


34. (2018/june/paper02/q6)


Figure 1 shows a rectangular box with length $5 x \mathrm{~cm}$, width $2 x \mathrm{~cm}$ and height $h \mathrm{~cm}$. The box has a base but no top. The volume of the box is $1000 \mathrm{~cm}^{3}$ and the total external surface area of the box is $S \mathrm{~cm}^{2}$

(a) Show that $S=10 x^{2}+\dfrac{1400}{x}$ (4)

Given that $x$ can vary,

(b) find, to 3 significant figures, the minimum value of $S .$ (5)

(c) Verify that your answer to part (b) does give the minimum value of $S .$


35. (2019/june/paper01/q11)

The curve $C$ has equation $3 y=x^{2}+2$

The point $P$ lies on $C$ and has $x$ coordinate 4

The line $k$ is the tangent to $C$ at $P$.

(a) Find an equation for $k$, giving your answer in the form $a y=b x+c$ where $a, b$ and $c$ are integers.

The line $l$ is the normal to $C$ at $P$

(b) Find an equation for $l$, giving your answer in the form $d y=e x+f$ where $d, e$ and $f$ are integers. (2)

(c) Find the area of the triangle bounded by the line $k$, the line $l$ and the $x$-axis. (3)

The finite region bounded by $C$, the line $l$, the $x$-axis and the $y$-axis is rotated through $360^{\circ}$ about the $x$-axis.

(d) Use algebraic integration to find, to the nearest whole number, the volume of the solid generated.


36. (2019/june/рaper02/q2)

Oil is leaking from a pipe and forms a circular pool on a horizontal surface. The area of the surface of the pool is increasing at a constant rate of $8 \mathrm{~cm}^{2} / \mathrm{s}$. Find, in $\mathrm{cm} / \mathrm{s}$ to 3 significant figures, the rate at which the radius of the pool is increasing when the area of the pool is $50 \mathrm{~cm}^{2}$ (6)


37. (2019/june/рареr02/q9)

The curve $C$, with equation $y=\mathrm{f}(x)$, passes through the point with coordinates $\left(-2,-\dfrac{28}{3}\right)$ Given that $f^{\prime}(x)=x^{3}-x^{2}-4 x+4$

(a) show that $C$ passes through the origin. (4)

(b) (i) Show that $C$ has a minimum point at $x=2$ and a maximum point at $x=1$

(ii) Find the exact value of the $y$ coordinate at each of these points. $(7)$

The curve has another turning point at $A$.

(c) (i) Find the coordinates of $A$.

(ii) Determine the nature of this turning point.


38. (2019/juneR/paper01/q4)

$$\begin{aligned}&f(x)=e^{3 x} \sqrt{1+2 x} \\&f^{\prime}(x)=\dfrac{2 e^{3 x}(2+3 x)}{\sqrt{1+2 x}}\end{aligned}$$

(a) Show that

(b) Find an equation of the normal to the curve with equation $y=\mathrm{f}(x)$ at the point on the curve where $x=0$

Give your answer in the form $a x+b y+c=0$ where $a, b$ and $c$ are integers.


39. (2019/juneR/paper01/q5)

A circle has radius $3 \mathrm{rcm}$ and area $A \mathrm{~cm}^{2}$

Given that the value of $r$ increases by $0.05 \%$

use calculus to find an estimate for the percentage increase in the value of $A .$


40. (2019/juneR/paper01/q9)


Figure 3 shows a solid cuboid $A B C D E F G H$

$$ A B=x \mathrm{~cm} \quad B C=3 x \mathrm{~cm} \quad A H=h \mathrm{~cm}$$

The volume of the cuboid is $540 \mathrm{~cm}^{3}$

The total surface area of the cuboid is $S \mathrm{~cm}^{2}$

(a) Show that $S=6 x^{2}+\dfrac{1440}{x}$ (4)

Given that $x$ can vary,

(b) 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 $S$.

(c) Find, to 3 significant figures, the minimum value of $S$. (1)


41. (2019/juneR/paper02/q3)


Figure 2 shows a solid right circular cylindrical metal rod.

The diameter of the rod is $x \mathrm{~cm}$ and the length of the rod is $10 x \mathrm{~cm}$.

The rod is being heated so that the length of the rod is increasing at a rate of $0.005 \mathrm{~cm} / \mathrm{s}$.

Find the rate of increase, in $\mathrm{cm}^{3} / \mathrm{s}$ to 2 significant figures, of the volume of the rod when $x=3$



Answer


1.9(a) $x=2$ (b) $(3,4),\left(1, \dfrac{4}{3}\right)$ (c) $y=1-2 x$ (d) $x=\dfrac{15}{8}$

2.7(a) Show (b) $S_{\min }=50.6$ (c) $\dfrac{d^{2} s}{d x^{2}}>0$

3.11(a) Show (b) $\dfrac{d V}{d h}=\pi\left(10 h-h^{2}\right)$ (c) $0.150$ (d) Show

4.7(a) $\quad A=\left(1 \dfrac{1}{2}, 0\right), B=(0,1)$ (b) (i) $x=3$ (ii) $y=2$ (c) Graph (d) $y=3 x+1$ (e) $x=\dfrac{10}{3}$

5.6(a) Show (b) $\dfrac{d A}{d t}=-0.412$

6.9(a) $f(x)=\dfrac{1}{4} x^{4}-x^{3}-\dfrac{1}{2} x^{2}+3 x+4$ (b) Show (c)(i) $\left(1,5 \dfrac{3}{4}\right)$ (ii) $f^{\prime \prime}(x)<0$ (d) $-1<x<1, \quad x>3$

7.6(a) Show (b) Show (c) $\theta=0.8$

8.9(a) (i) $y=2 x-4$ (ii) $y=-\dfrac{1}{2} x+6$ (b) $Q\left(-1, \dfrac{1}{4}\right)$ (c) $x=1 \dfrac{1}{2}$ (d) Show

9.8(a) $\left(\dfrac{1}{2}, 4\right), \max$ (b) $\left(4 \dfrac{1}{2}, 36\right) \mathrm{min}$

10.11 (a) $r=2$ (b) $p=-2, q=-5$ (c) $y=-x-2$ (d) Show (e) $21 \dfrac{1}{3}$

11.6 (a) show (b) $x=1.22$ (c) $A_{\min }=44.4$

12.10 (a) $y=x+12$ (b) $P(-1,-5), Q(3,-1)$ (c) $y=x-4$ (d) Show (e) $R S=8 \sqrt{2}(f) 32$

13.5) $\dfrac{d h}{d t}=0.12$

14.5 (a) $y=15-x-\dfrac{1}{2} \pi x$ (b) Show (c) 63

15.11(a) (i) Show (ii) $q=32$ (b) $24 y+5 x=207$ (c) $A=160 \dfrac{4}{15}$(d) 28

16.2) $\dfrac{d V}{d t}=\dfrac{3}{\pi}$

17.6) (a) Show (b) $x=8.45$ (c) $V_{\max }=12300$

18.3(a) Show (b) $\dfrac{d y}{d x}=\dfrac{-8}{9}$ (c) $9 y+8 x+3=0$

19.5 $\dfrac{d r}{d t}=0.637$

20.10(a) $y=-8 x+24$ (b) $Q(2,8)$ (c) $R\left(\dfrac{4}{3}, \dfrac{8}{3}\right)$ (d) $S\left(-\dfrac{2}{3}, 18 \dfrac{2}{3}\right)$ (e) Show

21.2(a) Show (b) $S_{\min }=757$

22.10(a) Show (b) $1.996$ (c) $\dfrac{d^{2} A}{dr^2}>0$ (d) $A_{\min }=75$

23.3) $\dfrac{d r}{d t}=0.0053 \mathrm{~cm} / \mathrm{s}$

24.7(a) $x=\dfrac{3}{2}$ (b) $\dfrac{d y}{d x}=\dfrac{2 x^{2}-6 x+4}{(2 x-3)^{2}}$ (c) $(1,1),(2,2)$

25.10 (a) Show (b) Show (c) $0.355$

26.5(a) show (b) $x=4.94$ (c) $A_{\min }=586$

27.3 $\dfrac{dA}{dt}=628$

28.5) (a) Show (b) $6 y+x-19=0$

29.2) $\dfrac{d r}{d t}=0.477 \mathrm{~cm} / \mathrm{s}$

30.7) (a) Show (b) $v=533 \dfrac{1}{3}$

31.3 $\dfrac{dh}{dt}=0.239$

32.8(a) Show (b) $S_{\min}=278$ (c) Verify

33.3 $\dfrac{dh}{dt}=0.054$

34.6(a) Show (b) $S_{\min}=510$ (c) $\dfrac{d^2S}{dx^2}>0$

35.(11)(a) $3 y=8 x-14$ (b) $8 y=-3 x+60$ (c) $54 \dfrac{3}{4}$ (d) 710

36.2) $0.139 \mathrm{~cm} / \mathrm{s}$

37.9) (a) Show $(b)(i)$ show (ii) $\left(1,1 \dfrac{11}{12}\right),\left(2,1 \dfrac{1}{3}\right)($ c $)\left(-2, \dfrac{-28}{3}\right), \min$

38.4(a) Show (b) $x+4y-4=0$

39.5 $0.1\%$

40.9(a) Show (b) $x=4.93$ (c) $S_{\min}=438$

41.3 $\dfrac{dV}{dt}\approx 0.11$ (cm$^3$/s)












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