Consider the function f(x) i yf, s2zlb; 4.pj-yol,t0wca= $x^3$ - 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 curve Cbh)k)cbs3 7ds8/,7qf qgv qdj 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 i 3dz3fl/y7 0bv2 d;mgi qauy4a company producing lithium batteries is modelled by the cubic function vy30dy2 zq4; igd37uibafml /
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 producing lv-4:ia 7n r82a+zeb8 tj apwiiithium batteries is modelled by the cubic functiozr8ji7 wi-4 eataabni :+2p8 vn
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 curve isv).y/ xl8rh;iqe.- + (nidn rqo-v mmx 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= wjwh xh) mgb85jw(5v/-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=f.6snj*oz6: iy.3p2 xe 9;e xiczh eep$\frac{2x}{1+x}$ , 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=yz8ig dixovv.c 1;f.9)hm( c rlcm.p)$\frac{12x}{1+4x}$ , 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 curve C isnt/ p )/0gbfam 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 of th 7nyc -+h*zgrhe 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 function f(X8cudost (uqa5aui fr 17gjjfy+2+-1p )=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 driving over a hill on a path that can be modelled by part ,04 .h7d tocpp osdl6ky bj:3wof 3cp d0o 47t6k pysol,h:.bjdwthe function
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 i : qhlh locodv;,n/3bhjxzcl8+-v. 8 ls given 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 h8rdc:7mt5ac, 3e0e;r aslnqf(x)=ln(2x−1). Let point A be the point on the curve where x=3e tsl5a0emhrr8,7 c;:qacdn3.

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 curh *ws6 h+igv;oh/x*exve 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 dollars), when x tonnes os071om 9mx:z+zlrr p9kl. oybf apples are exported is givex 7bsmloy:oz 1z0l.+ k9r9mr pn 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)=gre8:ih 6iw,gea)g4a2, g′(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 thc4jc5pm/e 1 hye9wm x3e graph of f′, the derivative of 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 for x=3eog.e :m-oa7r and x=7 are sho-a7mo: greo e.wn in the following 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 hkmx-g zr5pfw qom 7;7-eight, h, in metres, of the cannon-ball above the ground is model gfzm7-pw roqm;5k-7xled 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 f *yt3*1el6e: ocabjzm(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 creamsl v w vycgnu1)3a.cmrb 6v 7k)9m.uer. in the shape of a cone with a hemisphere on top. Each ice cream consists of a cone 1 evycukr9r)m73 lnvg u .mv)b6cw.a.base with height h and radius r, and a hemisphere on top of the cone's base, also with 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 a cylq* ,lg4,q hveninder with height h and radius r and a cone at the bottom of the cylinder's base. The total surface area of the cement silo is given hq g,enqv4,l*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 and snow fer7dh rji4fy02i 7g* kuwjf) yuv8o4e6stival, 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 area ourkg yihu04yd4 8 wf7fjo)jr72*e6ivf 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 through point P(2,z-e,37*nc ez y(v:byjigo d4a7) 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 velocit/fytwb+op : i5y, 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 arm of tblq j (;6+deydhe compass with the needle is 7 cm long, and th6l+jey;qd b(d e arm with the pencil is 8 cm long. The angle 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 iniw5kemlnwsu (m55/h/yh 4 6dy mg of a certain medicine t hours after administration can be modeled by C(t mwen/ikys655w (/mh45h d yul) = 1000te$^{-1.25t}$ , 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 shows the graph of n/df k9,.gpwl 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 followiw0:epl(hf p j4ng 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 cuboid with square base, asd.mimes 7qn))/: as wy shown in the following diagram. saei.) :wmnymd s)/7q


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 function 1 yse+huea11xfvy/m) pn g*ycyg-c:1$f(x)=2 x^{2}-b x+c$

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* 7h1cm gn x5whu-xzo2yh* +he at a company in Japan can be modelled by the cost funct 5 *cx*u1hn2x y7o-hge+hhmwz ion

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

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 a forest is observed. After t weeks the popfym g8q47 4sgy2yq n8dulatiodgq yy2f4nymqg78 s84 n 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 agai mmulghc94c(80,hlvd nst a wall with its feet on horizontal ground x m from tcu(m4hl8mc9d h lgv0 ,he wall, as illustrated.


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 l83+hytnhm n9;ad-ze $f(x)=-\frac{1}{3} x+\frac{a}{2 x^{2}}$ , 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 lake after t weeks is modelrf ,x9 ;:uhv1mol/y + 5nshrvnled 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 function28n-rky lx41pzlt0rb, e-ucc defined by

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

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 dys +i;vcrtd/l9o8s 3 c h4tyzk5 q(m)has a uniform cross-section in the shape of a tivt89 zc5(tyho / dr+qd3scl4m)sky;rapezoid, 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 ri e hlh) 8,dbu)id-co,adius $6 \mathrm{~cm}$ . 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 function lgcl ttrd e1d0+hro lmap*cl8i (8.49 $f(x)=\frac{2}{x^{2}}-ax$ , 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 the tql7+.sga27c y8a sdt6,bopwr xz 5u*ballest building in the world. It has a height of 830 m and has a square base that 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 scaleao2g8l*tx,+ 6 uq r5sysbcd7w7pb .a z 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 of(pre b4os x;4f8xx8mountain camping club. The logo is in the shape of a right-angled triangle, ABC, which represents a mountain. A rectangular section, rf8p4 (8 ex4xxb;osfoADEF, 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 tunnel system is show0i;osw tboqx8ke. ;bk92 .xby n on the axes below. The left and right lane tunnel;xx .e.b9si tbyw;8boko2q0 ks 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 the 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 gloe( 1lx :v:nt:v v+qmd6vf *yiven 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 Hawaiik7x0f 4i0(xfpp9wiuj xq 1ss1 and takes a parasailing ride on a beach. She is towed behind a motor boat and attached to the parasail. The vertical height of the parasail ovk700x(4qffpijw pxsus1 9 1ix er the first part of the ride can 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 travej1*:5hiiq m xul path of the Orange Star cargo ship from an aerial view. ux5ji:h*1 i mq

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 internal hjlz4hw x m*4l4eight h \math w 4xllz44*hjmrm{~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 storagj(x7 4hl7vrr he box with an open top from a rectangular piece of cardboard, 45 cm by 24 cm. She removes squares with sij47 7hv(xrh rlde length x cm from each corner, as shown in the following 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 rectan7 8llkqyq-v4 wgular box as part of a science project. He makes the box from a rectangular piece of cardboard, 30 cm x 18 cm. To construct the box, Jeremy cuts off squares of side length x cm from each y kwvl-lq78 4qcorner, 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 beachhf-r*ll -n8lq sand created afte lhf --ll8*nrqr a high 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 def-o3gg73n t4p ;fb m15ds9 lqdfgmue)yined 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 rao hp, . rgiwzvuviz3,k8d5vm402btr3te 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 cur(kz*t cpdnja,i;ue /;6+k iontainer. The design consists of a cone with radius and vertical height i+/e ,*prtz;d;i(ukcuj n ka6r, on top of a cylinder with length ℓ, where r and ℓ are measured in meters. A diagram of the container 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 girs18lg6f glf( ven 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 has3bp:n;awk)*hc 5rlk w found that the profit obtained from selling x cakes is given by the functwh)cwbka*:5 ;3r pnlkion

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 cylindeml d6uppk6(hkq g j,y;o y:(m-r with a hemisphere on its b6dk,j upmgq(:pk6 ho -yy ;(mlottom base. The internal 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. Ig s)y2ox8/e: bqy7 hjet is constructed from a galvanized steel frame with a multiwall polycarbonate sheeting. The steel frame consists of a rectangular base, four semicircula:ee8 2)/sy xhgoqj7byr arches and three further support rods, as shown 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 ≈  (数值)  m.

  A manufacturer makes chemical transport vessels in the form of a cylinderno*rv ccx4kn-z b 90r a:z30xo with a hemispherical front. The vessel has a total length of 4 m. The kn9n40 b ar0ozo c*:3vxcxz-rbase 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 with water. The dimc guw6f+iw(,k ys; (p-e vpz;wensions of the tank are shown in the following k (spfwp,vz+w6(weu y gc;;-i 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 boxeso m/ugt envwx5/e- 9+/ueuv(u has a box with an open-top design. This box is co(meeuguout59 v ex/-/v wu /+nnstructed 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 cylz en9)l4bkum8/6g0v te0ya kkkn 0; zqinder out of paper with a volume of exactly)9k0lkz6beeg n8;0m 4y0 nz/autkqk v 200 $\mathrm{~cm}^{3}$ . 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 question are in m scz6 na3 bs( btiy;(ntrb19-cetres.
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 chil;d6gjt 0jm2 ko0 rubp+d's vocabulary as age increases on a monthly basis. The first row shows the child's age in months (t), andu;mgjrjp 2tk+0odb6 0 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 functio vwzr (- hmo6a8h7i (ae//ibpen defined by $f(x)=(1-x) \sqrt{2 x-x^{2}}$ 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]