Consider the function f(x) =ura;exr *rkl5 6;n- ;npqno-r.z : 7ja y-geag $x^3$ - 6$x^2$ +5x +18 , Part of the graph of f is shown below.

1.Find f'(x) = ax$^2 - bx + c ; a=   , b=   , c=  .

2.There are two points at which the gradient of the graph of f is 20. Find the x-coordinates of these points.
two points are ( -   ,   ) and (  ,   ).

  Let f(x)=$e^{3x}$. The line L is the tangent to the curve of f at (0,1).
Find the eqation of L in the form y= mx +c.
m=  ; c=  .

  The equation of the cuw ra0:: .eferfrve C is y= $\frac{3}{4}x^4$ + $\frac{1}{3}x^3$ - 2.

1.Find $\frac{dy}{dx}$ = a$x^3$+b$x^2$, a=  , b=  .

Point A lies on the curve C. The gradient of the tangent to the curve C at point A is equal to 4.

2.Calculate the coordinates of A( a , -$\frac{b}{c}$ ) , a=  , b=  , c=  .

  The daily cost of production of a company producing lithium batteriesl: 9na9 zh1gm**0barz0ss ivl is modelled by the cl l9ah:bz 90gvss*n*za0mir1 ubic function
C(X)= 900+ 18X - 0.01X$^2$ - 0.0002X$^3$ , 0≤X≤150

where x is the number of lithium batteries produced and C the cost in USD.

1.Write down the daily cost to the company if no lithium batteries are produced.
we get is   .
The marginal cost of production is the cost of producing one additional unit. This can be approximated by the gradient of the cost function.

2.Find an expression for the marginal cost, C′(x), of producing x lithium batteries.
C′(x)=a-bx-cx$x^2$ ; a=  , b=  , c=  .
3.Find the marginal cost of producing
3.1.
50 lithium batteries;C′(50)=  .
3.2.
100 lithium batteries.C′(100)=  .

  The daily cost of production of a company 8n(wxe :cbul .producing lithium batteries is modelled by the cubic functixu(:ew bc.l8 non
C(X)= 5000+ 65X - 0.1X$^2$ - 0.0005X$^3$ , 0≤X≤150

where x is the number of car tires produced and C the cost in USD.

1.Write down the daily cost to the company if no car tires are produced.
we get is   .
The marginal cost of production is the cost of producing one additional unit. This can be approximated by the gradient of the cost function.

2.Find an expression for the marginal cost, C′(x), of producing x car tires.
C′(x)=a+bx-cx$x^2$ ; a=  , b=  , c=  .
3.Find the marginal cost of producing
3.1.
100 lithium batteries;C′(100)=  .
3.2.
200 lithium batteries.C′(200)=  .

  The equation of a curd / )-jl(haeqjve is y=-x$^3$ +4x$^2$ +x -4.A section of the curve is shown on the diagram below, with the three x-intercepts labelled.

1.Find $\frac{dy}{dx}$.
$\frac{dy}{dx}$=-ax$^2$ +bx +c ; a=  , b=   ,c=   .
2.Write down the coordinates of the local maximum.
the local maximum is located at (x, y) ; x(≈)=   , y(≈)=  .
3.Write down an integral representing the area of the shaded region.
its area is determined by A=$\int_{1}^{4}(-x^3+ax^2+x-b)dx$, a=  , b=  .
4.Find the area of the shaded region.
A=$\frac{a}{b}$, a=   , b=   .

  The equation of a curve is y=6c v+b(tk at(,f:8k sikp 2xae-x$^2$ +8x -12.A section of the curve is shown on the diagram below, with the two x-intercepts labelled.

1.Find $\frac{dy}{dx}$.
$\frac{dy}{dx}$=-ax+b ; a=  , b=  .
2.Write down the coordinates of the local maximum.
the local maximum is located at (x, y) ; x=   , y=  .
3.Write down an integral representing the area of the shaded region.
its area is determined by A=$\int_{2}^{6}(-x^2+ax-b)dx$, a=  , b=  .
4.Find the area of the shaded region.
A=$\frac{a}{b}$, a=   , b=   .

  Consider the curve y=0 vq8vcv1rvx *k:po, e$\frac{2x}{1+x}$ , x∈R, x≠−1.

1.Find $\frac{dy}{dx}$.
$\frac{dy}{dx}$=$\frac{a}{(1+x)^2}$; a=  .

2.Determine the equation of the normal to the curve at the point P(−2,4).
hence, we get y=-$\frac{1}{2}$x+b ; b=  .

  Consider the curve y .iilvs eh.pc )y(o8nu-;4ttc=$\frac{12x}{1+4x}$ , x∈R, x≠−1.

1.Find $\frac{dy}{dx}$.
$\frac{dy}{dx}$=$\frac{a}{(1+4x)^2}$; a=  .

2.Determine the equation of the normal to the curve at the point P(−1,4).
hence, we get y=-$\frac{3}{4}$x+$\frac{a}{b}$ ;a=  , b=  .

  The equation of the curvefbm.i /7b+yr;ou jid6 C is y=$\frac{}{}$x$^5$- $\frac{2}{3}$x$x^3$+6x. A section of the curve C is shown on the diagram below.

1.Find $\frac{dy}{dx}$ = $\frac{a}{b}$x$^4$ - cx$^2$ + d.a=  , b=   , c=   , d=  .

There are two points, A and B, in the domain of the curve shown at which the gradient of the tangent line to the curve C is equal to −3.

2.Calculate the x-coordinates of points A and B.Hence the x-coordinates of points A and B are -a and b .
a=  , b=  .

  Consider the graph op jvf,df 40v( s ym+(dqxo)5inf the function f(x)= -x$^3$ +3x$^2$ +1.

1.Label the local minimum as A on the graph.

2.Label the local maximum as B on the graph.

3.Write down the interval where f′(x)>0.a , b=  .

4.Draw the tangent to the curve at x=−1 on the graph. we get the tangent passes through the coordinate (-  ,   ) and is parallel to the graph at this point.

5.Write down the equation of the tangent at x=−1.
Hence, the equation of the tangent at x=−1 is y=-ax-b, a=  , b=  .

  Consider the functiod 5.nmx 2d0hqwn f(X)=2x$^4$ + 1.

1.Find f′(x)=ax$^3$, a=  .

2.Find the gradient of the graph of f at x=−1.
f′(-1)=-  .
3.Find the x-coordinate of the point at which the normal to the graph of f has a gradient of −8.
x=$\frac{a}{b}$, a=  , b=  .

  A toy racing car is drigr 5w32v) gzvhg-deo1ving over a hill on a path that can be modelled by part of the fun5w)1 r-g3 odzvg 2evhgction
f(x)=-$\frac{1}{4}x^2+5$.

1.Find the gradient of the hill at the point where x=2.
f'(2)=-  .
2.Find the x value of the point where the gradient perpendicular to the hill is equal to $\frac{3}{2}$.
the number is $\frac{a}{b}$ ,a=  , b=  .

  The number of bacteria in a culture is gikubt1ukb dc +a-q 3a;j333u8bzo v k8dven by the function n(t)=310e$^{kt}$ , where t is the time in minutes after the start of the culture and k is a constant. It is known that the number of bacteria doubles every 10 minutes.

1.Find the exact value of k.
k=$\frac{lna}{b}$, a=  , b=  .
2.Find the rate of change of the number of bacteria, n ′(t), when t=11. Round your answer to the nearest integer.
Therefore, at the instant when t=11, the number of bacteria in the culture is increasing at a rate of    bacteria per minute
​

  Consider the function f(x)=ln(2x−1). Let point A be the poin n(0w2scl( v1l4rtwqo t on the curve where x=lw 2c4o(rqnv l(tw01s3.

1.Find the gradient of the curve at A.
f'(3)=  .
2.The normal to the curve at A cuts the x-axis at P. find the coordinates of P.
The coordinates of P are (  ,  )

  The equation of a cuq 5 dunl05/kmgrve is y=1-$\frac{9}{x}$.
1.Find the equation of the tangent to the curve at x=3.
y=ax-b;a=  , b=  .
2.Find the coordinates of the points on the curve where the gradient is $\frac{1}{4}$.
the points are (a,-$\frac{b}{2}) and (-6,$\frac{c}{d}$).
a=  , b=  , c=  , d=  .

  The export fee per tonne of apples, E (in thousand of dolln( ax*hob/bfs2s n5/ lars), when x tonnes of apples are exportes2 *bbon hs/n la/5x(fd is given by

E(x)=5x+$\frac{147}{x}$,x>0.
1.Find an expression for E′(x)=a-$\frac{b}{x^c}$. a=  ,b=  ,c=  .

2.Find the value of x for which the export fee per tonne is a minimum.

3.Find the value of the minimum export fee.

Let f(x)=g(x)h(x), where g(3)=6, h(3)=2, g6t yoktl-3y)kwiia3r d, 6o- i′(3)=4 and h′(3)=1.

Find the equation of the normal to the graph of f at x=3.

The equation of the normal is y=-$\frac{1}{a}$+$\frac{b}{c}$ ; a=__, b=__, c=__.

  Let f(x)= $x^2e^x$ and g(x) = 4x - $x^2$.

1.Find f'(x) = xe$^x$(x+  ).

2.Find the x-coordinate where the tangents of f(x) and g(x) are parrallel.
x=  .

  Let f(x) = $\frac{5x}{e^{3x}}$ , for 0≤x≤10.
1.Sketch the graph of f.

2.1.Write down the x-coordinate of the maximum point on the graph of f.
x=  .
2.2.Write down the interval where f is increasing.
a≤x≤b; a=  , b=  .
3.Find f′(x) using the quotient rule.
f′(x)=$\frac{ae^{bx}-15xe^{bx}}{e^{cx}}$; a=  ,b=  ,c=  .
4.Find the interval where the rate of change of f is increasing.

  The following diagram shows the graph of f′, the derivative xv) axmqr0e v k/ie3)1of f.

The graph of f′ has a local maximum at A, a local minimum at B and passes through P(2,−3).
1.The point Q(2,6) lies on the graph of the function f.

1.1Write down the gradient of the curve of f at Q is -  .

1.2.Find the equation of the normal to the curve of f at Q, the equation of the normal is given by y=$\frac{a}{b}$x+$\frac{c}{d}$; a=  , b=  , c=  , d=  .

2.Determine the concavity of the graph of f when 2

  The values of the functions f and g and their derivatives foz; fn3xt )dgfq+o*;kur x=3 and x=7 are shown in the follow;f*t gzx k 3d;uo)f+nqing table.

Let h(x)=f(x)g(x).

1.Find h(3)=-  .

2.Find the equation of the normal to h when x=7.
y=-$\frac{a}{b}$x+$\frac{c}{d}$ ; a=  ,b=  , c=  , d=  .

  A cannon-ball is fired from the top of a tower. The height, .w xuvyk +fk5*9 mljo7h, in metres, of the cannon-ball above th +k mxu*w9ov.lj7yf5ke ground is modelled by the function
h(t)=-2t$^2$ + 20t +8, t≥0,
​
where t is the time, in seconds, since the moment the cannon-ball was fired.

1.Write down the height of the tower.
h(0)=  m.
2.Calculate the height of the cannon-ball 5 seconds after it was fired.
h(5)=  m.
The cannon-ball hits its target on the ground n seconds after it was fired.

3.Find the value of n.

4.Find h′(t).
h'(t)=-at+b ; a=-  , b=  .
5.Calculate the maximum height reached by the cannon-ball, and write down the corresponding time t.
the maximum height is    m and it is reached after    seconds
​
6.Determine the total time the cannon ball was above the height of the tower.
t=  seconds

  Consider the function ojb*6t)n dwt7f(x)=$\frac{1}{2x}$ + $\frac{x^4}{8}$ - 1.625 , x≠=0.
1.Find f′(x)=-$\frac{1}{ax^2}$+$\frac{x^b}{c}$.
a=  , b=  , c=  .
2.Calculate f(1)=-  .

3.Find f'(1)=  .

4.Sketch the graph of f(x), for −3≤x≤3 and −3≤y≤7.

5.Draw the tangent line, A, to the graph of y=f(x) at the point x=1.

6.Find the equation of the tangent line A.

7.The graph of y=f(x) and the tangent line A have a second intersection at point K. Use your graphic display calculator to find the x-coordinate of K.
x$_k$=-  .

  A food company produces ice creams in the shai4,mh gi -k:.udtd a.e:sh 5vepe of a cone with a hemisphere on top. Each ice cream consists of a cone base with height h and radius r, and a hemisphere on top of the cone's base, also w g e:d4i -.ah:te, s.mhduki5vith a radius of r. The total surface area of the ice cream cone is in cm$^2$ and is given by the formula
A=2$\pi$$r^2$+$\frac{60\pi}{r}$,
where r is the radius of the cone, in cm.

The ice cream designers of the company have been instructed to minimize the surface area of the cone in order to reduce the melting rate of ice cream.
1.Find $\frac{dA}{dr}$=a$\pi$r - $\frac{b\pi}{r^2}$.
a=  , b=  .
2.Calculate the value of r that minimizes the total surface area of the ice cream cone.
r=  cm.

  A cement silo made of steel consists of (ceto55egmu ( a cylinder with height h and radius r and a cone at the bottom eemo5tu( 5cg (of the cylinder's base. The total surface area of the cement silo is given by
A=$\frac{48\pi}{r}$ + 3$\pi r^2$,
where r is the radius of the cylinder, in metres.

The silo is designed such that the total cost for the steel used is minimized. This is the same as minimizing the total surface area of the silo.

1.Find $\frac{dA}{dr}$=-$\frac{a\pi}{r^2}$ + b$\pi r$.
a=  ,b=  .
2.Calculate the value of r that minimizes the total surface area of the silo.
r=  m

  Mustafa is an ice sculptor. In an ice, pa0ioayjov9p( v fyc1z2b ,( and snow festival, he is about to build an ice tent in the shape of a cylinder with a cone of the same radius at the top. The total surface ,2 vo(pcvfi,byp9 1a(a zy j0oarea of the tent is in $m^2$ and given by
A=$\frac{18\pi}{r}$ + $\pi r^2$ - 2
where r is the radius of the cylinder, in metres.

In order for his work to last as long as possible, Mustafa aims to reduce the evaporation rate and hence minimize the total surface area of the ice tent.
1.Find $\frac{dA}{dr}$=-$\frac{a\pi}{r^2}$ + b$\pi r$;
a=  , b=  .
2.Determine the value of r that minimizes the total surface area of the ice tent.
r≈  .08m.

  A curve y=f(x) passes througqo*afypm2 3j,- ,yto ih point P(2,7) and has a gradient of f '(x)=4x−3.

1.Find the gradient of the curve at point P.
f'(2)=  .
2.Find the equation of the tangent to the curve at point P.
y=ax-b , a=  , b=  .
3.Determine the equation of the curve.
y=ax$^2$-bx+c ; a=  , b=  , c=  .

  A particle P moves so that its velocity,0,sp5m5xrckzj/i -*:yc hnv z in m s$^{−1}$ , at time t seconds can be described by the function
v(t)=2cost, where t≥0. The kinetic energy of the particle, in joules (J), is given by the function E(v)=3v$^2$ .

1.Find an expression for E as a function of time.
E(t)=a cos$^2$ t; a=  .
2.Hence, find E'(t)=-a cost sint; a=  .E'(t)=  

3.Hence or otherwise, find the first time at which the kinetic energy is changing at a rate of 4 J s$^{−1}$.
t=  s

  Tom uses a compass to draw a circle. The ar*)t*pld w 5xowm of the compass with the needle is 7 cm long, and the arm with the pencil is 8 cm long. The angle p)5wtxdw*o*l between the arms is θ as shown in the diagram below.

1.Find an expression for the radius of the circle to be drawn in terms of θ.
r=$\sqrt{a-b cosθ}$, a=  , b=  .
2.Hence, find the rate of change in r with respect to θ when θ=30$^∘$ .
the rate of change is   cm per radian
​

  The concentration in mg of a ceu1 o9vox(rs; xrtain medicine t hours after administration can be modeled by C(t) = 100s x 19vo(;xoru0te$^{-1.25t}$ , t≥0.
1.Find the rate of change of the concentration with respect to time.
C'(t) = 1000e$^{-1.25t}(a-b1.25t)$ , a=  , b=  .
2.Find the value of t when the rate of change of C is zero.
t=  .
3.Determine if the time found in part (b) is a local minima or maxima for C. Justify your answer.

  The following diagram sh; n)2*nslfk8u (bl20nje:kasi5 yj qc ows the graph of the function
f(x)=ax$^3$+bx$^2$+cx+d , for -2≤x≤2.



1.State whether the function is increasing or decreasing at x=−1,0 and 1.
x=-1: the function is  ----待选择----decreasingincreasing 
x=0: the function is  ----待选择----decreasingincreasing 
x=1: the function is  ----待选择----decreasingincreasing 
2.Write down the value of d.
d=-  .
The values of a and b are such that f(x) = -x$^3$ + $\frac{x^2}{2}$+cx+d.

3.Point K(−1,−1) lies on the graph of y=f(x). Find the value of c.
c=$\frac{a}{b}$;a=  ,b=  .
4.Use your graphic display calculator to find the coordinates of the local maximum, M.
f(x)=-x$^3$+$\frac{x^2}{a}$+$\frac{b}{c}$x-d; a=  ,b=  ,c=  ,d=  .
5.Find f'(x).
f'(x)=-ax$^2$+x+$\frac{3}{2}$; a=  .
6.1.Calculate f′(0)=$\frac{a}{b}$;a=  ,b=  .

6.2.Find the equation of the tangent to the graph at the point (0,−1).

6.3.Write down the gradient of the normal to the graph at x=0.

7.Use your graphic display calculator to find the x coordinates where f(x)=−x, for the given domain.

  The function f is in the ft *xxh:n-ygc-k1io w+dw ehhj o-07w; ollowing form, where p is a constant.
f(x)=x$^3$+px$^2$-9x-2
A part of the graph of the function y=f(x) is shown in the diagram below.

1.Write down the y-intercept of the graph.
Thus the y-intercept is
(0,−  ).
2.Find f′(x)=$ax^2+2px-9$, a=  .

The tangent line L to the graph y=f(x) at x=1 is parallel to the x-axis.

3.1Find:the value of p=  .;

3.2.the equation of line L.
y=-  .
4.Determine the coordinates of another point on the graph y=f(x), where the tangent line to the graph is parallel to the x-axis.

5.Write down the interval where the gradient of the graph of y=f(x) is negative.

6.Determine all the possible values of m, for which the equation f(x)=m has three solutions. Write down your answer using interval notation.m is (-  ,  )

  Jack makes an open container in the shape of a c15r.thcxm / yh75fsh z+uo0tzuboid with square base, as shown in the followinhy /5chs 0t+mhzuf1t o 5r.x7zg diagram.


The container has base length x m and height y m. The volume is 32 $m^3$.
Let A(x) be the outside surface area of the container.
1. Show that A(x)=$\frac{128}{x}+x^{2}$ .
2. Find $A^{\prime}(x)$=ax-$\frac{128}{x^2}$,a=  .
3. Given that the outside surface area is a minimum, find the base length of the container. x=  m.
4. Jack coats the outside of the container with waterproof resin. A can of resin covers a surface area of 5$ \mathrm{~m}^{2}$ and costs $\$$ 15 . Find the total cost of the cans needed to coat the container.

  Consider the functionj;f(xbap *8+skhvi t 8 $f(x)=2 x^{2}-b x+c$

1. Find $f^{\prime}(x)$ = ax-b.a=  .

The equation of the tangent line to the graph of y=f(x) at x=-1 is y=2 x-7 .

2. Calculate the value of b =-  .

3. Calculate the value of c =-  .

  The cost to manufacture badminton rackets at a company1a;cx:o u rd/ 64t jtchqft,*m in Japan can be modelled by the cost fttcr /mx,6 *4chotja;d1 :qfuunction

$C(x)=8 x^{3}-24 x^{2}+28x$

where x is in hundreds of rackets and C(x) is in hundreds of Japanese Yen (JPY).
1. Find $C^{\prime}(x)$ = a$x^2$ - bx + c;a=  ,b=  ,c=  .

The marginal cost of production is the cost of producing one additional unit. This can be approximated by the gradient of the cost function.
2. Find the marginal cost when 100 rackets are produced and interpret its meaning in this context.
$C^{\prime}(1)$=JPY  .

The revenue from selling the rackets is given by the function R(x)=26 x where x is in hundreds of rackets and R(x) is in hundreds of JPY.
3. Given that Profit = Revenue - Cost, determine a function for the profit, P(x) , in hundreds of JPY from selling x hundreds of badminton rackets.
P(x)=-a$x^3$+24x$^b$-cx ; a=  ,b=  ,c=  .
4. Find $P^{\prime}(x)$= -ax$^2$+bx-c; a=  ,b=  ,c=  ..

5. Determine the intervals where P(x) is increasing and decreasing.

The derivative $P^{\prime}(x)$ gives the marginal profit. The production will reach its optimal level when the marginal profit is zero and P(x) is positive.
6. Find the optimal production level and the expected profit at this level.

  A small population of rabbits in mqs8rqe- bmr5os odi1 (9 /ug(a forest is observed. After t weeks the populatio(qorg(ubmse8 q5-o9 d sim /1rn is modelled by

$P(t)=\frac{15000}{1+50 e^{-0.6 t}}, \text { where } 0 \leq t \leq 30$.

1. Find $P^{\prime}(t)$=$\frac{ae^{-0.6t}}{\left(1+50 e^{-0.6 t}\right)^{2}}$. a=  .
2. Find the rate at which the population is increasing after 10 weeks.$p^'(10)$≈  .
3. Determine the time(s) at which the population is increasing at 1860 rabbits per week. Round your answer(s) to the nearest integer.
4. During which week does the rate at which the population is increasing reach its maximum.

  A ladder that is 6 m long rests against a wall with its feet on 6 1he2nmvl6+ga i / mtj k:t7;f2fh ah:reew4thorizontal ground x m from the wall, as igrane 6fl;kwhh 6i /:vtt4+h21 7m2ajetmfe :llustrated.


1.Show that the height the ladder reaches up the wall, y, can be expressed by the equation,
$y=\sqrt{36-x^{2}}$
where x is the horizontal distance in metres from the wall to the ladder.

The bottom of the ladder is then pulled away from the wall at a constant speed of 3 m s$^{-1}$.

2.Calculate the speed the top of the ladder is moving down the wall at the instant when the bottom of the ladder is 4m away from the wall.
$\frac{dy}{dx}$≈-  .

  Let $f(x)=x^{3}+3 x^{2}-9 x+k$. Part of the graph of f is shown below. The graph of f has a local maximum at A, a local minimum at B and a point of inflection at C.

1.1. Find $f^{\prime}(x)$ =a$x^2$+bx-c;a=  ,b=  ,c=  .
1.2. Find $f^{\prime \prime}(x)$ =ax+b;a=  ,b=  .
2. Find the x -coordinate of the point of inflection at C.
x=-  .
Given that f(-1)=14 .
3.1. Find f(0) =   .
3.2.Hence, find the coordinates of the local maximum A(x,y) and justify your answer.
A(-  ,  )
4.Write down in order from least to greatest f$^{\prime \prime}$(B),f$^{\prime}$(B),f(B).

  Consider the function u /qhi cp23k6r$f(x)=-\frac{1}{3} x+\frac{a}{2 x^{2}}$ , where a is a constant and $x \neq 0 $.
1. Find $f^{\prime}(x)$ = -$\frac{1}{a_1}$-$\frac{a}{x^3}$;$a_1$=  .

The function f(x) has a local maximum at x=3 .
2. Show that a=-9 .
3. Find the y -coordinate of the local maximum of the function.f(x)=-$\frac{1}{3}$x-$\frac{b}{2x^2}$;b=  .
4. Sketch the graph of f(x) , for $-6 \leq x \leq 8$ and $ -6 \leq y \leq 2 $.
5. State the values of x for which f(x) is increasing,   .
6 . Find the x -intercept of the graph of the function f(x) .
7. Calculate $f^{\prime}(1)$ =$\frac{a}{b}$;a=  ,b=  .
8. Find the equation of the normal to the graph of y=f(x) at x=1 .

  The number of fish in a lakxs9:, qa61;z)h hr4mre+teak6uoccxe after t weeks is modelled by
$F(t)=\frac{4500}{1+4 e^{-0.5 t}}, t \geq 0$
1.Find the initial fish population,The number is   .

2.Find the percentage increase of the population after 1 week, the population increase is   .

3.Find the limit of the population size.that the population size stabilizes at    fish, this size is known as the carrying capacity.

4.Find F′(t), and determine if the population is increasing or decreasing over time.

5.Find the rate at which the population is increasing after 8 weeks.F'(8)≈  .

6.Sketch the graph of F(t).

  Consider the function defined byf7u gxfabh 6.4*5 oxkrxzm-tu /+9sbx

$f(x)=\frac{-4 x^{2}+12}{x^{3}}, \quad x \neq 0$

1. Find an expression for $f^{\prime}(x)$ , the derivative of f(x) .
$f^{\prime}(x)$=$\frac{a(x^2-b)}{x^c}$;a=  ,b=  ,c=  .
2. Find the equation of the tangent to the curve at the point x=1 .
at x=1 as follows y=-ax+b;a=  ,b=  .
3. Find the x -coordinates of the points on the curve where the gradient is zero.
$x_1$=-  ,$x_2$=  .
4. Determine the intervals on which f(x) is increasing.
f is increasing on the intervals (−∞,−  ) and (  ,∞)
​

  A trough, containing water, of length 8 m c(/8qg er,pmf 5pzow3has a uniform cross-section in the shape of a trapezcgfr /,mpe85p( o3zq woid, with side lengths shown below.



When the water is h \mathrm{~m} deep, the volume of water in the trough is given by

$V(h)=16 h+\frac{8 h^{2}}{\sqrt{3}}$


Water leaks from the bottom of the trough at a constant rate of $3.1 \mathrm{~m}^{3}$ per minute. Find the rate at which the height of the water level h is falling at the instant when the water has $50 \mathrm{~cm}$ of depth.

the height of the water level is falling at a rate of    cm/min
​

  Points A and B lie on a circle with centre O and radiz8sn foef,+ ;9 yff ;jyrd*mq0us $6 \mathrm{~cm}$ . Suppose the angle $\theta=A \widehat{O} B$ is increasing at a rate of 0.5 radians per second.

Find the rate at which the area of the shaded region is increasing with respect to time, at the instant $\theta=2 $ radians.
the shaded area is increasing at a rate of    cm$^2$/sec
​

  Consider the functiom4 ;:pu0;;rao d xrv gkbdv.g.n $f(x)=\frac{2}{x^{2}}-ax$ , where a is a constant and $x \neq 0$.
1. Find $f^{\prime}(x)$ .

The function f(x) has a local minimum at x=-2 .${f'}^x$=-a-$\frac{a_1}{x^{b_1}}$;$a_1$=  ,$b_1$=  .
2. Show that $a=\frac{1}{2}$.
3 . Find the y -coordinate of the local minimum of the function.
4. Sketch the graph of f(x) , for $-5 \leq x \leq 6$ and $-3 \leq y \leq 6$ .
5. State the values of x for which f(x) is increasing.
6 . Find the x -intercept of the graph of the function f(x) .
7. Calculate $f^{\prime}(2)$=-  .
8. Find the equation of the normal to the graph of y=f(x) at x=2 .
y=x-$\frac{a}{b}$,a=  ,b=  .

  The Burj Khalifa, located in Dubai, is faw-o. 4zn)a0 x0kkqm the tallest building in the world. It has a height of 830 m and has a square base thakkqnzoam0- ) wa0x4f.t covers a floor area of 556 m×556 m. A tourism shop located near the building sells souvenirs of the tower, which sit inside glass pyramids, as illustrated by the diagram below. The souvenir tower is an accurate scale replica of the actual tower.

The scaled model of Burj Khalifa has a base area of 20 cm×20 cm. The height and base area dimensions of the glass pyramid are 10% larger than the model.
1.1.Find the height of the souvenir tower, in cm, correct to the nearest mm.

1.2.Find the volume of the glass pyramid, rounding your answer correct to the nearest cubic centimetre.
V≈  cm$^3$
The shop owner aims to maximise profits from selling the souvenirs. The more the owner orders from the manufacturer, the cheaper the souvenirs are to buy. However, if too many are ordered, profits may decrease due to surplus stock unsold.

The number of souvenirs ordered from previous years and the resulting profits are shown in the following table.

The shop owner decides to fit a cubic model of the form

$P(x)=a x^{3}+b x^{2}+c x+d$

to model the profit, P , for x thousand souvenirs ordered.
2. Explain why d=0 .
3. Construct three linear equations to solve for a, b and c , and hence write down the completed function P(x) .
P(x)=-$\frac{a}{b}$x$^3$+$\frac{c}{b}$+$\frac{x}{y}$;a=  ,b=  ,c=  ,x=  ,y=  .
4. Find $ P^{\prime}(x) $.
5. Find, to the nearest hundred souvenirs, the optimal number of souvenirs the owner should buy to maximise profit, and the resulting profit from this number.

  Olivia designs a logo for a mountain campi*d7y /q0 t0f s; 5xxssz5yyvhing club. The logo is in the shape of a right-angled triangle, ABC, which represents a mountain. it v/qx fh7*yxszy;5ssyd 050A rectangular section, ADEF, is inscribed inside the triangle to create a view of two smaller mountains. The lengths of BD, DE, EF and FC are p cm, 4 cm, 6 cm and q cm respectively.

The total area of the logo is $A \mathrm{~cm}^{2}$ .
1. 1. Find A in terms of p and q , giving your answer in the form A=a p+b q+c ;A=ap+bq+c;a=  ,b=  ,c=  .
2. Show that A=$\frac{48}{q}+3q+24$ .
2. Find $\frac{\mathrm{d} A}{\mathrm{~d} q}$=-$\frac{a}{q^b}$+c；a=  ,b=  ,c=   .

Olivia wishes to find the value of q that will minimize the area of the club logo.
3. 1. Write down an equation Olivia could solve to find this value of q .
2. Hence, or otherwise, find this value of q =  .

  The cross-sectional view of a two-lane road tunno+gbdc, qxybd fwo gx01p,ka 8k;bwy26v 4,;gel system is shown on the axes below. The left and right lane tunnels are separated by a 2 metre thick concrete wall. The right-hand tunnel passes through the points A, B, C and D and its height, in metres, above the base of theoc xx;g v0bg ,b;k28ayo1 , 6w+pfkb,dwyd4gq tunnel, is modelled by
$ f(x)=-0.04 x^{3}+0.41 x^{2}, 4 \leq x \leq 10$ , relative to an origin O .

Point A has coordinates (4,4) and point $ \mathrm{D} $ has coordinates (10,1) .
1. Find the height of the right-hand tunnel when:
1. x=6 ;f(6)=  .
2. x=8 .f(8)=  .

The left-hand tunnel can be modelled by a function g(x) , found by reflecting f(x) in the line x=3 .
2. Find the equation of g(x) .
3. 1. Find $g^{\prime}(x)$ .
3.2. Hence find the maximum height of the left-hand tunnel.The maximum height of the left-hand tunnel is approximately   m.
4. 1. Write down an integral which can be used to find the cross-sectional area of the left-hand tunnel.
2. Hence find the combined cross-sectional area of both tunnels.
The combined cross-sectional area of both tunnels is approximately   m$^2$.

  A function, f , is giveoc3u1zm/ /dur n by

$f(x)=2 \cdot 3^{-x+1}+x-3 $.

1. Calculate f(1) =  .
2. Use your graphic display calculator to solve f(x)=0 . x=  or x=  .
3. Sketch the graph of y=f(x) for $0 \leq x \leq 5$ and $-1 \leq y \leq 4$ . Clearly show the coordinates of the x and y intercepts. Use a scale of $1 \mathrm{~cm}$ to represent 1 unit on both the horizontal and vertical axes.

The function f is the derivative of the function g . It is known that g(0)=2 .
4. 1. Calculate $g^{\prime}(0)$ =  .
2. Find the equation of the normal of the graph of y=g(x) at the point (0,2) . Give your answer in the form a y+b x+c=0 .
we get ax+by-c=0; a=  ,b=  ,c=  .

  Sandy is on a holiday in Hawaii and takes a parasailing ridp:d (i uwrw 1itc xd4;p-1ea1ne on a beach. She is towed behind a motor boat and attached to the parasail. The vertical height of the parasail over the first part of the ride canin 4(ci1 du:wtex1 1a-p ;pwrd be modelled by the equation

$h(x)=\frac{2 x^{3}}{45}-\frac{17 x^{2}}{15}+8 x$

for $0 \leq x \leq 15$ , where $x \mathrm{~m}$ is the horizontal distance from the start, and $h \mathrm{~m} $ is the vertical height.
1. Find $h^{\prime}(x)$ =$\frac{a}{b}$$x^2$-$\frac{c}{b}$+d;a=  ,b=  ,c=  ,d=   .
2. Solve $h^{\prime}(x)$=0 x=   or x=  .
3. Using your answer to part (b), find the coordinates of the turning points of h(x) .
4. Sketch the graph of h(x) for $0 \leq x \leq 15$ , labelling the turning points.
5. If the parasail reaches a height level of $12 \mathrm{~m}$ , Sandy might suffer from vertigo. Find the x value for which this first happens.
6. When the gradient is negatively steeper than $-\frac{3}{2}$ , Sandy screams. Find the x value for which this first happens.
x=  .

  The diagram below shows the tr61o wvvo p10vk s5cr/savel path of the Orange Star cargo ship from an aer1v65kcs o w1ov/v0rspial view.

The travel path of the Orange Star can be modelled by the quadratic equation
$y=-\frac{x^{2}}{12}+\frac{35 x}{2}, x \geq 0, y \geq 0$

where x and y represent the distance, in kilometres, due east and north from a port located at O(0,0) , respectively. The ship stops at a port located at \mathrm{P}(180,450) for refuelling before continuing its journey.
There is another port located at a point $\mathrm{C}(120,900) $.
1. Determine whether the Orange Star cargo ship passes through port $\mathrm{C}$ . Give a reason for your answer.
2. 1. Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$=-$\frac{x}{a}$+$\frac{b}{c}$;a=  .b=  ,c=   .
2. Hence determine whether port C is the point furthest north on the travel path between the ports O and P . Give a reason for your answer.
3. 1. Find the midpoint of the line segment O P ; M=_(  ,  )
2. Find the gradient of the line segment O P ;M$_{op}$=_(  ,  )

Another cargo ship, The Black Pearl, departs from a new point B(0,100) and travels along a straight path that is parallel to the line segment O P .
4. Find the equation of the Black Pearl's travel path. Express your answer in the form a x+b y+d=0 , where a, b and d $\in \mathbb{Z}$ .
5. Find the coordinates of the point where the Black Pearl's travel path crosses the Orange Star's travel path for a second time.

  A steel pail is made in the shape of a cylinder with an intibphpw/ w);7sernal height h \mathrpi wh)w; 7sp/bm{~cm} and internal base radius r \mathrm{~cm} .

The steel pail has an open top. The inner surfaces of the pail are to be coated with a protective resin.
1. Write down a formula for A , the internal surface area to be coated.

The volume of the steel pail is $10000 \mathrm{~cm}^{3}$ .
A=a$\pi rh$ + $\pi r^b$ ; a=  ,b=  .
2. Write down, in terms of r and h , an equation for the volume of this steel pail.
3. Show that $A=\pi r^{2}+\frac{20000}{r}$ .

The steel pail is designed so that the area to be coated is minimised.
4. Find $\frac{\mathrm{d} A}{\mathrm{~d} r}$ =-$\frac{a}{r^2}$+b$\pi r$; a=  ,b=  .
5. Using your answer to part (d), find the value of r which minimizes A .
r≈   cm
6. Hence, find the value of this minimum area, correct to the nearest $\mathrm{cm}^{2}$ .
A≈  cm$^2$
One can of protective resin coats a surface area of $350 \mathrm{~cm}^{2}$ .

7. Find the minimum number of cans of protective resin required to coat the area found in part (f).
n≈   cans

  Charlotte decides to build a storage box w8zw23p1;f t,nwndc kfith an open top from a rectangular piece of cardboard, 45 cm by 24 cm. She removes squares with side length x cm from each corner, as shown in the ;fpwz knftc w1d83n,2following diagram.

After the corner squares are removed, the remainder of the cardboard is folded up to form the storage box as shown in the following diagram.

1. Write down, in terms of x , the length and the width of the storage box.
l=a-bx;a=  ,b=  .
w=c-dx;c=  ,d=  .
2. 1. State whether x can have a value of 12 . Give a reason for your answer.
2. Write down the interval for the possible values of x .
3. Show that the volume, $V \mathrm{~cm}^{3}$ , of this storage box is given by

$V=4 x^{3}-138 x^{2}+1080 x$ .

4. Find $\frac{\mathrm{d} V}{\mathrm{~d} x}$ = $ax^3-bx^2+cx$ ; a=  ,b=  c=  .
5. Using your answer from part (d), find the value of x that maximizes the volume of the storage box.
x=  or x=  .
6. Calculate the maximum volume of the storage box.V=  $cm^3$
7. Sketch the graph of V for the possible values of x found in part (b)(ii), and $0 \leq V \leq 2500$ . Label the maximum point.

  Jeremy is making an open-top rectangular box as 5t*: wof k9bnu (-vtfgc*fz ;t47 uuqapart of a science project. He makes the box from a rectangular piece of cardboard, 30 cm x 18 cm. To construc4 wt;n fv(uk9u: foqau75bg** cft- tzt the box, Jeremy cuts off squares of side length x cm from each corner, as shown in the following diagram.

After removing the squares, Jeremy folds up the edges to form the box as shown.

1. Write down, in terms of x , expressions for the length and width of the box.
l=a-bx ;a=  ,b=  .
w=c-dx; c=  ,d=  .
2. 1. State whether x can have a value of 10 . Give a reason for your answer.
2. Write down the interval for the possible values of x .
3. Show that the volume, $V \mathrm{~cm}^{3}$ , of the box is given by

$V=4 x^{3}-96 x^{2}+540 x$ .

4. Find $\frac{\mathrm{d} V}{\mathrm{~d} x}$=ax^2-bx+c;a=  ,b=  ,c=   .
5. Using your answer from part (d), find the value of x that maximizes the volume of the box.
x≈  cm.
6. Calculate the maximum volume of the box.V≈  $cm^3$.
7. Sketch the graph of V , for the possible values of x found in part (b)(ii), and $0 \leq V \leq 1000$ . Label the maximum point.

  The diagram below shows the cross-sectional area of a mound of beach sand creaounf bo0d9) *wted after a dfwou)bo9n *0high tide.

The curve of the cross section can be modelled by the following equation

$y=\frac{x^{2}(90-x)}{1800}$

where y represents the vertical height of the mound in \mathrm{cm} and x denotes the horizontal width in \mathrm{cm} , from the start of the mound.
1. At a horizontal width of x=30 , determine
1. The vertical height of the mound at this point;y=  cm.
2. The gradient of the mound curve at this point.y'=$\frac{a}{b}$;a=  ,b=  .
2. 1. Find the value of x which corresponds to the maximum the vertical height of the mound.
x=  .
2. Hence, find the maximum vertical height of the mound.A=  cm$^2$.
3. Calculate the cross-sectional area of the mound, rounding your answer to one decimal place.

A child uses a toy shovel to remove the top of sand mound, as illustrated by the line segment MN below. Point M has coordinates at (30,30) .

4. Determine the coordinates of point N .
N(  ,  )
The cross-sectional area removed by the child can be expressed by the following integral

$\int_{p}^{q} \frac{x^{2}(90-x)}{1800} \mathrm{~d} x-R$

5. Determine the values of p, q and R .

R=  .

  Consider the function deff6lt4;-( v2 oaha w9m ueh.xznined by

$f(x)=\frac{8 x^{2}}{3 x^{3}+x}, x \quad \neq 0$

1. Find an expression for $f^{\prime}(x)$ , the derivative of f(x) .
f'(x)=$\frac{-ax^4+bx^2}{(3x^3+cx)^2}$;a=  ,b=  ,c=  .
2. Find the equation of the tangent to the curve at the point x=1 .
y=-ax+b;a=  ,b=  .
3 . Find the x -coordinates of the points on the curve where the gradient is zero.x=$\pm \frac{1}{\sqrt{a}}$;a=  .
4. Determine the intervals on which f(x) is increasing.

  The volume V of a cube increases at a ratnxjany e442 )ze(cc/ qz,x0yre of $4 \mathrm{~cm}^{3} \mathrm{sec}^{-1} $. Find the rate of change of the surface area of the cube, with respect to time t , at the instant when the volume of the cube is $8 \mathrm{~cm}^{3}$ .
the surface area is increasing at a rate of    $\mathrm{~cm}^{2} / \mathrm{sec}$

  Let $f(x)=\sqrt{ } 2 x+1$ for $ x \geq-0.5 $.
1. Find
1. f(12)=  ;
2. $f^{\prime}(12)=\frac{a}{b} $;a=  ,b=  .

Consider another function g(x) . Let $\mathrm{P}$ be a point on the graph g . The x -coordinate of $\mathrm{P}$ is 12 . The equation of the tangent to the graph at $\mathrm{P}$ is y=x+3 .
2. Write down $g^{\prime}(12) $=  .
3. Find g(12)=   .
4. Let $ h(x)=f(x) \times g(x)$ . Find the equation of the tangent of h at the point where x=12 .

  Engineers at a laboratory are designing a new type of gas storage containe+ ztsnrjjzl6c/ ;myampv1-j;x+2zw 6r. The design consists of a cone with radius and vertical height r, on top of a cylinder with length ℓ, where r and ℓ are measured in meters. A diagram of the containerz6 zlt1-z 6 w;sjmnc+/a + x;y2jjpmvr is shown below.

1. Find an expression for the volume, V , of the container, in terms of r, $\ell and \pi$ .
V=$\frac{{\pi}r^a}{b}$+ℓ$\pi$$r^2$;a= (数值) ,b= (数值) .
2. Find an expression for the surface area of the container, A , in terms of r, $\ell$ and $\pi$ .
A=$({\sqrt{2}}+a)$$\pi$$r^2$+b$\pi$rℓ;a= (数值) ,b= (数值) .
3. Given the design constraint $\ell=\frac{10-2 \pi r^{2}}{\pi r} $, show that V=$10 r-\frac{5 \pi r^{3}}{3}$ .
V=ar-$\frac{b\pi r^3}{3}$;a= (数值) ,b= (数值) .
4. Find $ \frac{\mathrm{d} V}{\mathrm{~d} r}$ .
$\frac{dV}{dr}$=a-b$\pi$$r^2$;a= (数值) ,b= (数值) .
The engineers aim to maximise the volume of the container for the given design constraints.
5. Using your answer to part (d), show that V is a maximum when $r=\sqrt{\frac{2}{\pi}} $
r=$\sqrt{\frac{a}{\pi}} \mathrm{m}$;a= (数值) .
6. Find the length of the cylinder, $ \ell $, for which V is a maximum.
7. Calculate the maximum volume, V , of the container.

  A function, f , is given).5smx eee ntn*hhi+ 7 by

f(x)=$\frac{1}{3} \cdot 2^{-2 x}+\frac{1}{2} x-1$ .

1. Calculate f(-2)=$frac{a}{b}$ ; a=  ,b=   .
2. Use your graphic display calculator to solve f(x)=0 .
3. Sketch the graph of y=f(x) for $-2 \leq x \leq 5$ and $-1 \leq y \leq 4$ . Clearly label the coordinates of the x and y intercepts.

The function f is the derivative of the function g . It is known that g(0)=1 .
4. 1. Calculate $g^{\prime}(0)$=-$\frac{a}{b}$ ; a=  ,b=   . .
2. Find the equation of the tangent of the graph of y=g(x) at the point (0,1) . Give your answer in the form y=m x+c ;m=-$\frac{a}{b}$;a=  ,b=  ,c=  

  The owner of a bakery has found that the profit obtained from sellinfyg q8+ rne df/+mdi2s*w;wwpn/9ef7 g x cakes is given by the fufn;79p/+f 8g+i ynww erde/d mwfq2 *snction

P(x)=$\frac{x}{20}\left(600-\frac{x^{2}}{2 k^{2}}\right)$

where k is a positive constant and $x \geq 0$ .
1. Find an expression for $P^{\prime}(x)$ in terms of k and x ；P’(X)=c-$\frac{ax^2}{bk^2}$;a=  ,b=  ,c=  .
2. Hence, find the maximum value of P in terms of k,The maximum value of P is   k.

The owner knows that the bakery makes a profit of $\$ 873$ when they sell 30 cakes.
3. Find the value of k=  .
4. Determine how many cakes the bakery should sell to maximize their profit,The bakery should sell    cakes to maximise their profit.
5 . Sketch the graph of P , labelling the maximum point and x -intercepts.
6. Determine the maximum number of cakes the bakery can sell before they start losing money, the maximum number of cakes they can sell before they start losing money is   .

  A water tank is made from glass in the shape of a cylinder with a h-r4v )vwt5vakp+ aiwa6s ,-j, ir(esbemisphere on its bottom base. The internaa 6ri ,+,eiv5v-)wkws st(ja-p4b r val height of the cylinder is
h cm and the internal radius is r cm. The tank has no top.

1. Write down a formula, in terms of r and h , for A , the total surface area of the tank.
A=a$\pi$rh+b$\pi$r^2;a=  ,b=  .
The volume of the tank is 120 litres.
2. Express this volume in $\mathrm{cm}^{3}$;120L=   cm$^3$ .
3. Write down an equation, in terms of r and h , for the volume of this tank.
V=$\pi$$r^2$h + $\frac{a}{b}$$\pi$$r^3$;a=  ,b=  .
4. Show that $A=\frac{2}{3} \pi r^{2}+\frac{240000}{r}$, A=$\frac{240000}{r}$-$\frac{a}{b}$$\pi$$r^2$;a=  ,b=  .
5. Find $\frac{\mathrm{d} A}{\mathrm{~d} r}$=-$\frac{a}{r^2}$+$\frac{b}{c}$$\pi$$r$,a=  ,b=  ,c=  .
6. Find the value of r which minimizes the surface area, A .
7. Hence, find the minimum surface area, A , correct to the nearest $\mathrm{cm}^{2}$ .

The glass surface of these water tanks is to be covered by a special dirt protection spray. Each bottle of the spray can be used to cover 6400 $\mathrm{~cm}^{2}$ of glass surface.
8. Using your answer from part ($\mathrm{g}$) , calculate the number of tanks the company can coat using 40 bottles this dirt protection spray,The number of tanks that the company can coat is    tanks

  A greenhouse is made in the shape of a half cylinder. It is consj jsyj2k:,g+ t2cyksov 5xd9zn5 9(o btructed from a galvanized steel frame with a multiwall polycarbonate sheeting. The steel frame consists of a rectangular base, four semicircular arches and three further support rods, as showt29+y:sk95j 5odkv(cj,nx sbojg 2z y n in the following diagram.

The semicircular arches have radius r and the support rods each have length l .
Let S be the total length of steel used in the frame of the greenhouse.
1. Write down an expression for S in terms of r, l and $\pi$ .
S= (数值) l+ (数值) r+ (数值) $\pi$r
The volume of the greenhouse is 37.5 $\mathrm{~m}^{3}$ .
2. Write down an equation for the volume of the greenhouse in terms of r, l and $\pi$ .

3. Show that S=$\frac{375}{\pi r^{2}}+4 r(1+\pi)$.
S=$\frac{375}{{\pi}r^2}$+ar(b+$\pi$);a= (数值) ,b= (数值) .
4. Find $\frac{\mathrm{d} S}{\mathrm{~d} r}$=-$\frac{a}{{\pi}r^3}$+4(b+$\pi$);a= (数值) ,b= (数值) .

The greenhouse is designed so that the length of the steel used in the frame is minimized.
5. Show that the value of r for which S is a minimum is 2.43 $\mathrm{~m}$ , correct to 3 significant figures
6. Calculate the value of l for which S is a minimum.
7. Calculate the minimum value of S ≈    m.

  A manufacturer makes chemicaf)vowvd-n w)rvw5m tg, ek:l ofk91,+l transport vessels in the form of a cylinder with a hemispherical front. The vessel has a total le okv9k) m wfo:f1wwgvn er-lv,,d+t)5 ngth of 4 m. The base radius of both the cylinder and the hemispherical front is 1m.

1. Write down the length of the cylinder.
$l_c$=   m.
2. Find the total volume of the vessel.
$V_t$≈    m$^3$.
A truck company is looking to optimize and improve the dimensions of the vessel. The new vessel will also be in the form of a cylinder with a hemispherical front. It will have a length of l $\mathrm{~m}$ and a base radius of r $\mathrm{~m}$ .

There is a design constraint such that l+4 r=8 $\mathrm{~m}$ . The manufacturer wants to maximize the volume of the vessel.
3. Write down the length of the new vessel, l , in terms of r ; l=a-br , a=  ,b=  .
4. Show that the volume, V $\mathrm{~m}^{3}$ , of the new vessel is given by

V=8 $\pi r^{2}-\frac{13 \pi r^{3}}{3}$

5. Using your graphic display calculator, find the value of r which maximizes the value of V . The manufacturer claims that the new vessel has a capacity 20 $\%$ greater than the capacity of the original vessel.
6. State whether the manufacturer's claim is correct. Justify your answer.

  A conical tank is being filled wit/ v*h,wli2/icif8ujj d5ly3k j;s( juh water. The dimensions of the tank are shown in the follow ,w8u;u5li/lh i(jif3j jd*2cv/yskj ing diagram.

1.Show that an expression for the volume, V, in the tank in terms of the water level height, h, is

$V=\frac{\pi h^{3}}{27}$

2. The tank is being filled at a rate of 1 $\mathrm{~m}^{3} \mathrm{~min}^{-1}$ . Find the rate of change of the water height h at the instant when the water level is 2 $\mathrm{~m}$ high.
the water level height is rising at a rate of    m/min(Omit to three decimal places)
​

  A company that manufactures and sells cardboard boxes hasl,nbl x.s mbt a/:j1+dlc yp75l 0sy,n a box with an open-top design. Thbd0s,yca+1s 5 lnj:p t, ll7/nl .ybxmis box is constructed from a rectangular cardboard sheet with a length of 2 meters and a width of 1.2 meters, as illustrated in the diagram below. The box is formed by cutting squares of equal side length (x meters) from each corner and folding up the sides.

1. Show that the volume of the box can be described by the function $V(x)=4 x^{3}-6.4 x^{2}+2.4 x$ .
2. 1. Find $V^{\prime}(x)$= a$x^3$ - bx + c; a=  ,b=  ,c=  .
2. Hence or otherwise, find the value for x that maximises the volume of the box;x=  m
3. Hence, find the maximum volume of the box.
3. Sketch the graph of V(x) on the axes below for the domain 0
4. 1. Write down an integral representing the area between the graph of V(x) and the x -axis;
2. Hence, find this area between the graph of V(x) and the x -axis.

Let A(x) be a function representing the outside surface area of the box.
5. Determine the function A(x) =-a$x^2$ + b ;a=  ,b=   .
6. Given that the volume of the box is maximised, find the outside surface area of the box.

  For a homework assignment, Tiffany makes a cylindeg*( 8,8tkpincapd e2 ar out of paper with a volume of exactly 200g8n cta k82*e,api(dp $\mathrm{~cm}^{3}$ . To make the curved surface of the cylinder, Tiffany cuts a rectangular shape from a standard A4 sheet of paper (A4 dimensions: 21 $\mathrm{~cm}$ width, 29.7 $\mathrm{~cm}$ length). The circumference of the cylinder base (the length of the rectangle cut) is denoted as x $\mathrm{~cm}$ and the height of the cylinder (the width of the rectangle cut) is denoted as y $\mathrm{~cm}$ .

1. Using the formula for the volume of a cylinder, find an expression of the width of the rectangular cut y , in terms of x .
y=$\frac{a{\pi}}{x^2}$;a=  .
2. State whether x can have a value of 2 $\mathrm{~cm}$ . Give a reason for your answer.
y=   cm
3. Show that the curved surface area of the cylinder can be expressed, in terms of x , as A=$\frac{800 \pi}{x} \mathrm{~cm}^{2}$ ,A=$\frac{a{\pi}}{x}$ $cm^2$ , a=  .
4. Find $\frac{\mathrm{d} A}{\mathrm{~d} x}$ .
5. Calculate the value of x that minimises the curved surface area of the cylinder. Also find the corresponding value of y when this area is mini-mised.
6. Calculate the minimum curved surface area of the cylinder.
A≈   cm$^2$

  All lengths in this mdwu+ludd5/l:qe /d* question are in metres.
Consider the function f(x)=$\sqrt{\frac{16-4 x^{2}}{7}}$ , for $-2 \leq x \leq 2$ . In the following diagram, the shaded region is enclosed by the graph of f and the x -axis.

A rainwater collection tank can be modelled by revolving this region by $360^{\circ}$ about the x -axis.
1. Find the volume of the tank.
V=  m$^3$.

Rainwater in the tank is used for drinking, cooking, bathing and other needs during the week.
The volume of rainwater in the tank is given by the function g(t) , for $0 \leq t \leq 7$ , where t is measured in days and g(t) is measured in $\mathrm{m}^{3}$ . The rate of change of the volume of rainwater in the tank is given by $g^{\prime}(t)=1.5- 4 \cos \left(0.12 t^{2}\right)$ .
2. The volume of rainwater in the tank is increasing only when a1. Find the value of a =   days and the value of b =   days.

2. During the interval a $m^3$.

When t=0 , the volume of rainwater in the tank is 8.2 $\mathrm{~m}^{3}$ . It is known that the tank is never completely full of rainwater during the 7 day period.
3. Find the minimum volume of empty space in the tank during the 7 day period.
$V_{min}$=   m$^3$

  The table below shows the average number of words of a child's vocabulap;go wacp6rzp.4)lk;pk: 1az h*d )yf5v l,eyry as age increases on a monthly basis;,g5l)py aa vkf. rce)pyowh416z *;dz:pp kl . The first row shows the child's age in months (t), and the second row shows the average number of words, W, in hundreds.

1. Generate a scatter diagram for this data on the axes below and describe the trend.


2. Linearise the data and determine an appropriate linear model for the number of words, W , in terms of the months, t .
W=  log(t) -   .
3. Using your model, predict the approximate number of words in the vocabulary of a child who has just turned 6 years old.

The rate of change for the number of words a child learns per month can be estimated by $W^{\prime}(t)$.

4. Estimate the month at which the child learns 100 words per month.
t≈   month

  Consider the function defined by 0zcpkpsp5s 2 7 $f(x)=(1-x) \sqrt{2 x-x^{2}}$ where $0 \leq x \leq 2$ .
1. Show that $f(1-x)=-f(1+x)$ , for $-1 \leq x \leq 1$ .
2. Find $f^{\prime}(x)$ = $\frac{ax^{2}-bx+c}{\sqrt{d x-x^{2}}}$; a=   , b=  , c=   , d=   .
3. Find the x -coordinates of any local minimum or maximum points.
4. Find the range of f is [-   ,   ].
5. Sketch the graph of y=f(x) , indicating clearly the coordinates of the x -intercepts and any local maximum or minimum points.
6. Find the area of the region enclosed by the graph of y=f(x) on the x -axis, for $0\leq x \leq1$.
A=$\frac{a}{b}$; a=   , b=  .

  In a triangle $\mathrm{ABC}$, $\mathrm{B} \hat{\mathrm{AC}}=60^{\circ}$, $\mathrm{AB}=(1-x) \mathrm{cm}$, $\mathrm{AC}=(x+3)^{2} \mathrm{~cm}$,-31. Show that the area, A $\mathrm{~cm}^{2}$ , of the triangle is given by

$A=\frac{\sqrt{3}}{4}\left(9-3 x-5 x^{2}-x^{3}\right)$.

2. 1. Calculate $\frac{\mathrm{d} A}{\mathrm{~d} x}$=$-\frac{\sqrt{a}}{b}\left[3+10 x+3 x^{2}\right]$ ; a=  ,b=  .
2. Verify that $\frac{\mathrm{d} A}{\mathrm{~d} x}=0$ when $x=-\frac{1}{3}$,$\frac{dA}{dx}$=  .
3. 1. Find $\frac{\mathrm{d}^{2} A}{\mathrm{~d} x^{2}}$ and hence verify that x=-$\frac{1}{3}$ gives the maximum area of triangle A B C .
2. Calculate the maximum area of triangle A B C .
3. Find the length of [B C] when the area of triangle A B C is a maximum.
[BC]≈   cm(Omit to two decimal places)[/BC][/B C]