$\text { The following diagram shows the graph of } f(x)=\frac{4 x}{x^{2}+1} \text {, for } 0 \leq x \leq 6 \text {, and the line } x=6 \text {. }$


Let R be the region enclosed by the graph of f , the x -axis and the line x=6 . Find the area of R .   

$\text { A particle } P \text { starts from a point } O \text { and moves along a horizontal straight line. Its velocity } v \mathrm{~ms}^{-1} \text { after } t \text { seconds is given by }$


$\text { The following diagram shows the graph of } v$


1. Find the initial velocity of particle P .
2. Find the acceleration of the particle in the first second.
3. How many times does the particle change direction in the first 8 seconds. Explain your answer.
4. Find the total distance travelled by the particle in the first 8 seconds.

  $\text { Let } f^{\prime}(x)=\frac{x}{\sqrt{x^{2}+1}} \text {. Given that } f(0)=7 \text {, find } f(x) \text {. }$  (代数式) 

  $\text { Let } f^{\prime}(x)=\frac{12 x}{\left(x^{2}-1\right)^{4}} \text {. Given that } f(0)=3 \text {, find } f(x)$  (代数式) 

  A particle moves in a straight line and its 58cp83-jh dw-uogfcemoi7h( velocity, $ v \mathrm{~ms}^{-1}$ , at time t seconds, is given by $ v(t)=\left(t^{2}-2\right)^{2} $, for $ 0 \leq t \leq 2$ .
1. Find the initial velocity of the particle.   
2. Find the value of t for which the particle is at rest.   
3. Find the total distance travelled by the particle in the first 2 seconds.   
4. Show that the acceleration of the particle is given by $a(t)=4 t^{3}-8 t $.  (代数式) 
5. Find the values of t for which the velocity is positive and the acceleration is negative. a $\lt$t$lt$ b a =    b =   

  $\text { Show that } \int_{1}^{3} x^{2} \ln x \mathrm{~d} x=9 \ln 3-\frac{26}{9} \text {. }$   

  Let f(x)=$\frac{1}{\sqrt{3 x-6}}$ , for x>2 .
1. Find $\int(f(x))^{2} \mathrm{~d} x$ .  (代数式) 

Part of the graph of f is shown in the following diagram.

The shaded region R is enclosed by the graph of f , the x -axis, and the lines x=3 and x=11 .
2. Find the exact volume of the solid formed when R is rotated $ 360^{\circ} $ about the x -axis.   

A cannonball is fired from mx7 nlvoi;i /wp n38x2the top of a tower. The rate of change of the height, h , of t lpx2/m3 ;7ovninixw 8he cannonball above the ground is modelled by

$h^{\prime}(t)=-4 t+20, \quad t \geq 0$,

where h is in metres and t is the time, in seconds, since the moment the cannonball was fired.
1. Determine the time t at which the cannonball reached its maximum height. __

After one second, the cannonball is 26 metres above the ground.
2. a. Find an expression for h(t) , the height of the cannonball above the ground at time t . __
b. Hence, determine the maximum height reached by the cannonball. __
3. Write down the height of the tower.
4. Calculate the height of the cannonball 4 seconds after it was fired. __

The cannonball hits its target on the ground n seconds after it was fired.
5. Find the value of n . __
6. Determine the total time the cannonball was above the height of the tower. __

A second cannonball is fired from exactly halfway up the tower, with the same projectile motion as the first cannonball.
7. Given that both cannonballs land at the same time, determine the length of time between the first cannonball and the second cannonball being fired. __

$\text { Let } f(x)=x^{2}-3 x+2 \text {, for } x \in \mathbb{R} \text {. The following diagram shows part of the graph of } f \text {. }$



The graph of f crosses the x -axis at the point $\mathrm{P}(1,0)$ and at the point $\mathrm{Q}(2,0)$ .
1. Show that $ f^{\prime}(1)=-1$ .
The line L is the normal to the graph of f at P .
2. Find the equation of L in the form y=m x+c .

The line L intersects the graph of f at another point R , as shown in the following diagram.



3 . Find the x -coordinate of R .
4. Find the area of the region enclosed by the graph of f and the line L .

  The values of the functions k e*sp/i78+z6.y aehqi nkq1ff and g and their derivatives for x=2,x=3 and x=6 areae7qiyh k8e s+ zqk*p1n/fi6. shown in the following table.


1. Evaluate $\int_{2}^{3} g^{\prime \prime}(x) \mathrm{d} x$ .   
2. Let $ k(x)=\frac{f(x)}{g(x)}$ . Find $k^{\prime}(2)$   
3. Let h(x)=f(g(x)) . Find the equation of the tangent to h at x=6 .  (代数式) 

A particle P moves along a straight line so that :qc ms)m+1mcda90lio its velocity, $v \mathrm{~ms}^{-1}$ , after t seconds, is given by $v=\sin 3 t-2 \cos t-2 $, for $0 \leq t \leq 6$ . The initial displacement of P from a fixed point O is 5 metres.
1. Find the displacement of P from O after 6 seconds.

The following sketch shows the graph of v .

2. Find when the particle is first at rest.
3. Write down the number of times the particle changes direction.
4. Find the acceleration of P after 2 seconds.
5. Find the maximum speed of P .

  A function f is defined babpm *j/ 41hxhy $f(x)=\frac{3 x+2}{4 x+1}$,$ x \in \mathbb{R}$, $x \neq-\frac{1}{4} $
1. Find an expression for $f^{-1}(x)$ .  (代数式) 
2. Given that f(x) can be written in the form $f(x)=A+\frac{B}{4 x+1}$ , find the values of the constants A and B . A =    B =   
3. Hence write down $\int \frac{3 x+2}{4 x+1} \mathrm{~d} x$ .  (代数式) 

  $\text { Using the substitution } u=\sqrt{x}-1 \text {, find the value of } \int_{1}^{4} \frac{2 \sqrt{\sqrt{x}-1}}{\sqrt{x}} \mathrm{~d} x$   

  $\text { Let } g(x)=\frac{8-2 x}{\sqrt{9+8 x-x^{2}}} \text {. The following diagram shows part of the graph of } g \text {. }$


$\text { The region } R \text { is enclosed by the graph of } g \text {, the } x \text {-axis, and the } y \text {-axis. Find the area of } R \text {. }$   

  John drops a stone from the top qm;kgo 0l(m7 uof a cliff which is h metres above sea level. The stone strikes the water surface aft(mgk qmoul 07;er 9 seconds. The velocity of the falling stone, $v \mathrm{~m} \mathrm{~s}^{-1}$, t seconds after John releases it, can be modelled by the function



1. Find the velocity of the stone when t=12 , giving your answer to the nearest $ \mathrm{m} \mathrm{s}^{-1}$ . ≈   
2. Calculate the value of h , giving your answer to the nearest metre.≈   

The velocity of the stone when it reaches the bottom of sea is 10 $\mathrm{~m} \mathrm{~s}^{-1}$ .
3. Determine the depth of sea near the cliff, giving your answer to the nearest metre.≈   

1. Show that $\cos \theta=\sin \left(\frac{\pi}{2}-\theta\right)$ for $0\lt \theta \lt \frac{\pi}{2}$ .
2. Find $\int_{\sin \theta}^{\cos \theta} \frac{1}{\sqrt{1-x^{2}}} \mathrm{~d} x $ where $0 \lt \theta \lt \frac{\pi}{2}$ .

  The graph of y=f(x) is obtained froe2 y(kevz,8u(:tj9ctylk1 j q m the graph of $y=\ln x $ by a horizontal translation 3 units in the negative x direction and a vertical translation $\ln 6$ units in the positive y direction
1. Find f(x) .  (代数式) 
2. The region bounded by the graph of y=f(x) and the two axes is rotated through $2 \pi$ radians about the y -axis. Find the volume generated.   

  $\text { Let } f(x)=x(x-3)^{2} \text {, for } 0 \leq x \leq 4 \text {. The following graph shows } f \text {. }$





Let R be the region enclosed by the x -axis and the curve of f .
1. Find the area of R .   
2. Find the volume of the solid formed when R is rotated $360^{\circ}$ about the x -axis.   
3. The diagram below shows part of the graph of the quadric function g(x)=x(a-x) . The graph of g crosses the x -axis when x=a .   



The area enclosed by the graph of g is equal to the area of R . Find the value of a .

  Note: In this question, distancezchnsqb (b8eti5 )6vhy:yz; f/ j; eg, is in metres and time is in seconds.
A particle P moves in a straight line for six seconds. Its acceleration during this period is given by $a(t)=-2 t^{2}+13 t-15$ , for $0 \leq t \leq 6$ .
1. Write down the values of t when the particle's acceleration is zero.      
2. Hence or otherwise, find all possible values of t for which the velocity of P is increasing.$a \lt t \lt b $ a =    b =   

The particle has an initial velocity of $7 \mathrm{~ms}^{-1}$ .
3. Find an expression for the velocity of P at time t .  (代数式) 
4. Find the total distance travelled by P when its velocity is decreasing.   

  $\text { A particle moves in a straight line such that its velocity, } v \mathrm{~m} \mathrm{~s}^{-1} \text {, at time } t \text { seconds, is given by }$



1. Find the value of t , for t>0 , when the particle is instantaneously at rest.   

The particle returns to its initial position at t=T .
2. Find the value of T . Give your answer correct to three significant figures. ≈   

  A function f satisfies the conditions f(0rf * ledmabc)mt9b88/q5cyhgs:*evf)3c k3 y )=ln 2, f(1)=0 and its second derivative is* *b9r8em83 ts)maed hcc:)5lfkgfq cy/b 3yv $f^{\prime \prime}(x)=40 x^{\frac{2}{3}}+(x+1)^{-2}$,$ x \geq 0$ . Find f(x) .  (代数式) 

Consider the function f(x) defined by f(x)=12-12 sin x , kse3wiz40wcx9dweqsz 8, 6* wfor $0 \leq x \leq 3 \pi$ . The following diagram shows the graph of y=f(x) .



The graph of f touches the x -axis at points A and B , as shown. The shaded region is enclosed by the graph of y=f(x) and the x -axis, between the points A and B .
1. Find the x -coordinate of A and B .
2. Show that the area of the shaded region is 24 $\pi$ .

The following right cone has a total surface area of 24 $\pi$ , equal to the shaded area in the previous diagram.



The cone has a base radius of 3 , height h , and slant height l .
3 . Find the value of l .
4. Hence, find the volume of the cone.

A particle P moves alopr ,q +6g u .tjrlhea8/3bnz5hng a straight line so that its velocity, v, $\mathrm{~ms}^{-1}$ , after t seconds, is given by v=2 $\sin t-\cos 5 t+0.1$ , for $ 0 \leq t \leq 4 $. The initial displacement of P from a fixed point O is 2 metres.



1. Find the displacement of P from O after 4 seconds.
2. Find the second time for t , when the particle is at rest.
3. Write down the number of times P changes direction.
4. Write down the number of times P is neither accelerating or decelerating.
5. Find the maximum distance of P from O during the time $0 \leq t \leq 4$ and justify your answer.

  $\text { Using integration by parts find } \int x^{2} \cos x \mathrm{~d} x \text {. }$  (代数式) 

  $\text { Using integration by parts, find } \int x^{2} \sin x \mathrm{~d} x \text {. }$  (代数式) 

$\text { Consider a function with domain } a



From the graph above p, 0 and s are x -intercepts of $f^{\prime}$ , and there is a local minimum at x=q and a local maximum at x=r .
1. Find all the values of x where the graph of f is increasing. Justify your answer.
2. Find the value of x where the graph of f has a local minimum. Justify your answer.
3 . Find the value of x where the graph of f has a local maximum. Justify your answer.
4. Find the values of x where the graph of f has points of inflexion. Justify your answer.

The total area of the region enclosed by the graph of $ f^{\prime}$ and the x -axis between x=p and x=s is 25 .
5. Given that f(p)+f(s)=13 , find the value of f(0) .

  Let f(x)=ln x and kyu* 2nh)ig;t4 dzqi* $g(x)=2+3 \ln (x-1)$ , for x>1 .
The graph of g can be obtained from the graph of f by two transforms:
a vertical stretch of scale factor q
a translation of $\binom{h}{k}$
1. Write down the value of
1. q ;   
2. h ;   
3. k .   

Let $h(x)=g(x) \cdot \cos (0.1 x) $, for 1\lt x \lt8 . The following diagram shows the graph of h and the line y=x .

2. a. Find $\int_{2.02}^{5.57}(h(x)-x) \mathrm{d} x$ .   
b. Hence, find the area of the region enclosed by the graphs of h and $h^{-1}$ .   
3. Let d be the vertical distance from a point on the graph of h to the line y=x . There is a point Q(x, y) on the graph of h where d is a maximum. Find the coordinates of Q , where $2.02\lt x \lt 5.57$ . x =    y =   

  The following table shoud vosp(2 1gz:b4 l;i qtkd7.qws the probability distribution of a discrete random variable Z, in tezbd.lu 7qtp 24o 1:(kvdgiq;srms of an angle θ.


1. Show that $\cos \theta=\frac{3}{4}$.   
2. Find $\tan \theta $, given that $\tan \theta>0$ .   

Let $f(x)=\frac{1}{\cos x}$ , for $0\lt x\lt \frac{\pi}{2}$ .
The graph of y=f(x) between $x=\theta$ and $x=\frac{\pi}{4}$ is rotated $360^{\circ}$ about the x -axis.
3. Find the volume of the solid formed.v   

  Consider the ellipse defined by the equatiombva) 7wm- o)yn $x^{2}+3 y^{2}=12$ .
1. Find the equation of the normal to the ellipse at the point P(3,1) . y =  (代数式) 
2. Find the volume of the solid formed when the region bounded by the ellipse, the x -axis for $x \geq 0$ and the y -axis for $y \geq 0$ is rotated through $2 \pi$ about the y -axis.   

  Consider the ellipse defined pj*fojr:ho f an4r, m :a4zwx,q(,m 0eby the equation $x^{2}+3 y^{2}=12$ .
1. Find the equation of the normal to the ellipse at the point P(3,1) . y =  (代数式) 
2. Find the volume of the solid formed when the region bounded by the ellipse, the x -axis for $x \geq 0$ and the y -axis for $y \geq 0$ is rotated through $2 \pi$ about the y -axis.   

1. Show that 3 $\log _{a^{3}} x=\log _{a}$ x where a, x $\in \mathbb{R}^{+} $.

It is given that $ \log _{2} y+\log _{8} 4 x^{2}+\log _{8} 2 x=0$ .
2. Express y in terms of x . Give your answer in the form $y=b x^{c}$ where b, c are constants.

The region R , is bounded by the graph of the function found in part (b), the x -axis, and the lines x=1 and x=k where k>1 . The area of R is $\frac{3}{2} $.
3. Find the value of k .

  Consider the functio d s7u)vm :c4;pmtsnp:n f defined by $ f(x)=2 \ln (24-1.5 x) $ for x<16 .
The line $L_{1}$: y=x intersects the graph of f at point P .
The line $ L_{2}$ is perpendicular to $ L_{1}$ and tangent to the graph of f at point Q .

1. Find the x -coordinate of point P , to three significant figures. ≈   
2. a. Find the exact coordinates of point Q . x =    y =   
b. Show that the equation of $L_{2} $ is $ y=-x+2 \ln 3+14$ . y =  (代数式) 

The shaded region A , as shown in the previous diagram, is enclosed by the graph of f , the line $L_{1}$ and the line $ L_{2}$
3.a. Find the exact x -coordinate of the point where $ L_{2}$ intersects $ L_{1}$ .   
b. Hence, find the area of A , to two decimal places.≈   

The line $L_{2}$ is also tangent to the graph of the inverse function $f^{-1}$ .

$\text { 4. Find the shaded area enclosed by the graphs of } f, f^{-1} \text { and the line } L_{2} \text {. }$   

Let $f(x)=2 e^{\frac{t}{5}}$ and g(x)=m x , where $m \geq 0 $, and $-6 \leq x \leq 6$ . Let R be the region enclosed by the y -axis, the graph of f , and the graph of g .
1. Let m=2 .
a. Sketch the graphs of f and g on the same axes.
b. Find the area of R .
2. Consider all values of m such that the graphs of f and g intersect. Find the value of m that gives the greatest value for the area of R .

  A continuous random variable X has a probability d8 cr 7/igb uvlg7ak g3:7ds.poensity function f given by


$\text { Find } \mathrm{P}(0 \leq X \leq 2) \text {. }$   

$\text { Find } \int \arccos x \mathrm{~d} x$ __

$\text { Using the substitution } u=e^{x}-4 \text {, find } \int \frac{e^{x}}{e^{2 x}-8 e^{x}+25} \mathrm{~d} x \text {. }$

  Consider the functiony l3hm6*,objb : sq.ymeer,df , i(yl(s f, g , defined for $x \in \mathbb{R}$ , given by $f(x)=e^{2 x} \sin x$ and $g(x)=e^{2 x} \cos x$ .
1. Find
a.f ′(x):  (代数式) 
b.g ′(x).  (代数式) 

$\text { 2. Hence, or otherwise, find } \int_{0}^{\pi} e^{2 x} \cos x \mathrm{~d} x \text {. }$   

  All lengths in this question are i(d47zolaxr b0n 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.   

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 $a\lt t \lt b$ .
a. Find the value of a and the value of b . a =    b =   
b. During the interval $a\lt t \lt b$ , the volume of rainwater in the tank increases by $d \mathrm{~m}^{3}$ . Find the value of d .   

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.   

$\text { The following diagram shows the curve } \frac{x^{2}}{400}+\frac{(y+5)^{2}}{225}=1 \text {, where } 0 \leq y \leq h \text {. }$



The curve from point C to point P is rotated $360^{\circ}$ about the y -axis to form a lamp shade. The rectangle ABCD , of height $(10-h) \mathrm{cm}$ , is rotated $360^{\circ}$ about the y -axis to form a solid ceiling fixture.
The lamp shade is assumed to have a negligible thickness. Given that the interior volume of the lamp shade is to be 6000 $\mathrm{~cm}^{3}$ , determine the height of the ceiling fixture, length A D in the diagram.

1. Using the substitution .l4e fm q)lofly. 9p/k$x=\cot \theta$ , show that $\int_{0}^{1} \frac{1}{\left(x^{2}+1\right)^{2}} \mathrm{~d} x=\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \sin ^{2} \theta \mathrm{d} \theta$ .
2. Hence find the value of $\int_{0}^{1} \frac{1}{\left(x^{2}+1\right)^{2}} \mathrm{~d} x$ .

  $\text { Using the substitution } u=1-\sqrt{2 x} \text {, find } \int \frac{\sqrt{2 x}}{1-\sqrt{2 x}} \mathrm{~d} x \text {. }$  (代数式) 

Let $ f(x)=\frac{\ln \left(8 x^{3}\right)}{k x}$ where x>0, $k \in \mathbb{R}^{+}$.
1. Show that $f^{\prime}(x)=\frac{3-\ln \left(8 x^{3}\right)}{k x^{2}}$ .

The graph of f has exactly one maximum point A .
2. Find the x -coordinate of A .

The second derivative of f is given by $ f^{\prime \prime}(x)=\frac{2 \ln \left(8 x^{3}\right)-9}{k x^{3}}$ . The graph of f has exactly one point of inflexion B .
3. Show that the x -coordinate of B is $ \frac{e^{3 / 2}}{2}$ .

The region R is enclosed by the graph of f , the x -axis, and the vertical lines through the maximum point A and the point of inflexion B .

$\text { 4. Given that the area of } R \text { is } 5 \text {, find the value of } k \text {. }$

$\text { Using the substitution } x^{3}=3 \sec \theta \text {, show that } \int \frac{\mathrm{d} x}{x \sqrt{x^{6}-9}}=\frac{1}{9} \arccos \left(\frac{3}{x^{3}}\right)+C \text {. }$

  $\text { A particle moves in a straight line such that at time } t \text { seconds }(t \geq 0) \text {, its velocity is given by } v=18 t^{3} e^{-3 t^{2}} \text {. Find the exact distance travelled by the particle in the first two seconds. }$   

By using the substitutio,kyc fva jy39-n $s=\tan x$ , find $ \int \frac{\mathrm{d} x}{2+\cos 2 x}$ .
Express your answer in the form $A \arctan (B \tan x)+C $, where A, B are constants to be determined.

Consider the functionq-o*lzcq 4 5elkq,j4i $f(x)=\sqrt{\frac{10}{x^{2}}-1}$ , where $1 \leq x \leq \sqrt{10} $.
1. Sketch the curve y=f(x) , indicating the coordinates of the endpoints.
2. 1. Show that $f^{-1}(x)=\sqrt{\frac{10}{x^{2}+1}}$.
2. State the domain and range of $ f^{-1}$ .

The curve y=f(x) is rotated through $2 \pi$ about the y -axis to form a solid of revolution that is used to model a vase.
3. 1. Show that the volume $ V \mathrm{~cm}^{3}$ , of liquid in the vase when it is filled to a height of h centimetres is given by $V=10 \pi \arctan (h)$ .
2. Hence, determine the volume of the vase.

At t=0 , the vase is filled to its maximum volume with water. Water is then removed from the vase at a constant rate of $4 \mathrm{~cm}^{3} \mathrm{~s}^{-1}$ .
4. Find the time it takes to completely empty the vase.
5. Find the rate of change of the height of the water when half of the water has been emptied from the vase.

Consider the functions f and g defi:0iunngvk z /u:bo .89b 21ov bfuy1yxned on the domain $0\lt x\lt 2 \pi$ by

$f(x)=4 \cos 2 x \text { and } g(x)=2-8 \cos x $.

The following diagram shows the graphs of y=f(x) and y=g(x) .

1. Find the x -coordinates of the points of intersection of the two graphs.
2. Find the exact area of the shaded region, giving your answer in the form $a \pi+b \sqrt{3}$ , where a, b $\in \mathbb{Q}$ .

At the points P and Q on the diagram, the gradients of the two graphs are equal.
3. Determine the y -coordinate of P on the graph of g .

Consider the functions ,s v6)b/ibpf9z dnywf-q-p u 2 $f(x)=\sin x,-\frac{\pi}{2} \leq x \leq \frac{\pi}{2} $ and $ g(x)=\frac{2 x \sqrt{1-x^{2}}}{1-2 x^{2}}$, $x \in \mathbb{R}$, $x \neq \pm \frac{1}{\sqrt{2}}$
1. Find an expression for $(g \circ f)(x) $, stating its domain.
2. Hence show that $(g \circ f)(x)=\tan 2 x$ .
3. Letting $y=(g \circ f)(x)$ , find an exact value for $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at x=\frac{\pi}{3}$ .
4. Show that the area bounded by the graph of $ y=(g \circ f)(x)$ , the x -axis and the lines x=0 and $x=\frac{\pi}{3}$ is $\frac{1}{2} \ln 2 $.

Consider the function0a.hlb 3d38 l2)*aoyrm dnj*t aanf o( $f(x)=\frac{\sqrt{x}}{2 \cos x}$, $\frac{\pi}{2}\lt x\lt \frac{3 \pi}{2}$ .
1. 1. Show that the x -coordinate of the maximum point on the curve y=f(x) satisfies the equation $1+2 x \tan x=0$ .
2. Determine the values of x for which f(x) is an increasing function.
2. Sketch the graph of y=f(x) , showing clearly the maximum point and any asymptotic behaviour.
3. Find the coordinates of the point on the curve y=f(x) where the normal to the curve is perpendicular to the line y=x . Give your answers correct to two decimal places.

Consider the region bounded by the curve y=f(x) , the x -axis and the lines

$x=\frac{3 \pi}{4}, x=\frac{4 \pi}{3} \text {. }$

4. The region is now rotated through $ 2 \pi $ radians about the x -axis. Find the volume of revolution, giving your answer correct to two decimal places.

Consider the function definedukx k1 cb8if,4r.m12 p)mnsnd 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)$ .
3. Hence find the x -coordinates of any local minimum or maximum points.
4. Find the range of f .
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 \leq 1 $.
7. Show that $\int_{0}^{2}|f(x)| d x>\left|\int_{0}^{2} f(x) d x\right|$ .

$\text { The following graph shows the relation } x=5 \sin \left(\frac{\pi y}{30}\right)+10,0 \leq y \leq 60 \text {. }$



The curve is rotated $360^{\circ}$ about the y -axis to form a volume of revolution.
1. Calculate the value of the volume generated.

A vase with this shape is made with a solid base of diameter 20 cm . The vase is filled with water from a faucet at a constant rate of 150 $\mathrm{~cm}^{3} \mathrm{sec}^{-1}$ . At time $t \mathrm{sec}$ , the water depth is h $\mathrm{cm}, 0 \leq h \leq 60 $ and the volume of water in the vase is V $\mathrm{~cm}^{3} $.
2. a. Given that $ \frac{\mathrm{d} V}{\mathrm{~d} h}=\pi\left[5 \sin \left(\frac{\pi h}{30}\right)+10\right]^{2}$ , find an expression for $\frac{\mathrm{d} h}{\mathrm{~d} t}$.
b. Find the value of $ \frac{\mathrm{d} h}{\mathrm{~d} t}$ when h=45 \mathrm{~cm} .
3. a. Find $\frac{\mathrm{d}^{2} h}{\mathrm{~d} t^{2}}$ .
b. Find the values of h for which $ \frac{\mathrm{d}^{2} h}{\mathrm{~d} t^{2}}=0$.
c. By making specific reference to the shape of the vase, interpret $\frac{\mathrm{d} h}{\mathrm{~d} t} $ at the values of h found in part (c) (ii).

Consider the functiony9mbkx kr,l ;fbl1 q-9 $f(x)=\frac{a e^{-x}}{b-a e^{-x}}$ where $a\lt 0, b\lt 0$ .
1. Show that $f^{\prime}(x)=\frac{-a b e^{-x}}{\left(b-a e^{-x}\right)^{2}}$ .
2. Explain why $f^{\prime \prime}(x)$ is never zero.
3. Find the equation of:
a. the vertical asymptote of f ;
b. the horizontal asymptote of f .
4. Draw a sign diagram for $f^{\prime}(x)$ .
5. If a=3 and b=1 ,
a. sketch the graph of f labelling all asymptotes;
b. find the area of the region enclosed by f , the x and y axes and the line $x=\ln 2$ .

1. Use L'Hôpital's rule to fina m6pr-l vcjjgvso;i0q( z3 w37n/rr4d $\lim _{x \rightarrow \infty} x^{3} e^{-x} $.
2. Show that the proper integral $\int_{0}^{\infty} x^{3} e^{-x} \mathrm{~d} x$ converges, and state its value.

Let $ y=e^{-\frac{x}{2}} \cos \left(\frac{x}{2}\right) $
1. Find an expression for $\frac{\mathrm{d} y}{\mathrm{~d} x} $.
2. Show that $\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=\frac{1}{2} e^{-\frac{x}{2}} \sin \left(\frac{x}{2}\right)$ .

Consider the function f defined by $f(x)=e^{-\frac{x}{2}} \cos \left(\frac{x}{2}\right),-\pi \leq x \leq \pi $.
3. Show that the function f has a local maximum value when $ x=-\frac{\pi}{2}$ .
4. Find the x -coordinate of the point of inflexion of the graph of y=f(x) .
5. Sketch the graph of y=f(x) , clearly indicating the positions of the local maximum point, the point of inflexion and the intercepts with the axes.
6. Find the area of the region enclosed by the graph of y=f(x) and the x -axis.

The curvature at any point (x, y) on a graph is defined as $ \kappa=\frac{\left|\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\right|}{\left[1+\left[\frac{\mathrm{d} y}{\mathrm{~d} x}\right]^{2}\right]^{\frac{3}{2}}}$ .
7. Find the value of the curvature of the graph of y=f(x) at the local maximum point.
8 . Find the value of $\kappa$ for x=0 and comment on its meaning with respect to the shape of the graph.

$\text { The following diagram shows the graph of } y=x(\ln x)^{2}, x>0 \text {. }$



1. Given that the curve passes through the point (p, 0) , state the value of p .

The region R is enclosed by the curve ( $p \leq x \leq e $ ), the x -axis and the line x=e .
2. Integrate by parts twice to find the area of the region R .

Let $I_{n}=\int_{1}^{e} x^{2}(\ln x)^{n} \mathrm{~d} x, n \in \mathbb{N}$
3. a. Find the value of I_{0} .
b. Show that $I_{n}=\frac{1}{3}\left(e^{3}-n I_{n-1}\right)$.
c. Hence find the values of $ I_{1}$,$ I_{2}$ and $I_{3}$ .

The region R is rotated through $2 \pi$ radians about the x -axis.
4. Find the volume of the solid formed.