Note: In this question, distance is in metres and t22*ze g -zczhbime is in seconds.
A tennis ball is thrown in the air. Its height h above the ground after time t is given by

h(t)=-5 $t^{2}+20 t+4$,$ \text { for } 0 \leq t \leq 4 $.

1. Find $h^{\prime}(t)$ .  (代数式) 
2. Find the maximum height attained by the ball.   

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

  $\text { Let } f(x)=2 x^{2}-5 x+7 \text {. The line } L \text { intersects } f \text { at } \mathrm{P}(3,10) \text { and is perpendicular to the tangent to the curve of } f \text { at } \mathrm{P} \text {. Find the equation of } L \text { in the form } y=m x+c \text {. }$  (代数式) 

  $\text { Find the equation of the tangent to the curve } y=e^{2 x-4}-x \text { at the point where } x=2 \text {. }$ (代数式) 

  Let $f(x)=e^{-3 x}$ .
1. Write down $f^{\prime}(x), f^{\prime \prime}(x) $ and $f^{\prime \prime \prime}(x)$ .$f^{\prime}(x) $  (代数式) 
$f^{\prime \prime}(x) $  (代数式) 
$f^{\prime \prime \prime}(x) $  (代数式) 
2. Find an expression for $ f^{(n)}(x)$ .$f^{(n)}(x)$  (代数式) 

Let f(x)=0.5 $x^{3}-3.2 x $. There is a local maximum at point A and a local minimum at point B .
1. 1. Find the coordinates of point A .
2. Find the coordinates of point B .
2. Find the coordinates of the point of inflection of f .
3. Write down the values of x for which the graph of f has a negative rate of change.

Let $f^{\prime}(x)=x^{3}-9$ $x^{2}+24 x+3 $.
1. There are two points of inflection on the graph of f . Write down the x -coordinates of these points.
2. Let $h(x)=f^{\prime \prime}(x)$ . Explain why the graph of h has no points of inflection.

  Consider $f(x)=x^{3}-\frac{p}{x}$, $x \neq 0$ , where p is a constant.
1. Find $f^{\prime}(x)$ .  (代数式) 
2. There is a minimum value of f(x) when x=1 . Find the value of p .   

  A particle moves in a st2 )o gq5p4)znl+wnl cnraight line with velocity $v(t)=2 t-0.3 t^{3}+2$ , for $t \geq 0$ , where v is in $\mathrm{ms}^{-1}$ and t in seconds.
1. Find the acceleration of the particle after 2.2 seconds.   
2. a. Find the time when the acceleration is zero.   
b. Find the velocity when the acceleration is zero.   

The diagram shows part of the graph of y=f ′(x). The x-intercepts j2g p+3 m(pqx ix:rqo.wq7i4gare at points A and C. There is a minimum at point B and a max mgp w7po4: qx ijqr+q32(gi.ximum at point D.


1. 1. Write down the value of $f^{\prime}(x)$ at A .
2. Hence, show that A corresponds to a maximum on the graph of f .
2. Which of the points A, B, C, D corresponds to a minimum on the graph of f .
3. Which of the points A, B, C, D corresponds to a point of inflection of f . Justify your answer.

The following diagram show part 3 zm0cfml t)1fof the graph of y=f(x).



The graph has a local maximum at A , where x=-2 , and a local minimum at B , where x=8 .
1. On the graph above, sketch the graph of $y=f^{\prime}(x)$ .
2. Write down in order from least to greatest: $f(-2), f^{\prime}(8)$, $f^{\prime \prime}(-2) $.

  $\text { Consider the function } f(x)=\frac{2 x}{\cos x} \text {. Find } f^{\prime}(\pi) \text {. }$   

  The following diagram shows the grak et cul )m5ylggw3/syy;7;ot3eske-zo:x(4 ph of f ′, the derivative of f.



The points $\mathrm{M}(-\sqrt{2},-60)$ and $\mathrm{N}(\sqrt{2},-60)$ lie on the graph of $f^{\prime}$ . The point $\mathrm{P}(\sqrt{2}, 40)$ lies on the graph of the function f .
1. Write down the gradient of the curve of f at P .   
2. Find the equation of the tangent to the curve of f at P .  (代数式) 

Let $f(x)=k x^{3}$ .
1. Show that the point $\mathrm{P}(2,8 k)$ lies on the curve of f .

At P , the normal to the curve is parallel to $y=\frac{1}{6} x $.
2. Find the value of k .

  Let $f(x)=4 x \sin (2 x)$ .
1. Find $f^{\prime}(x)$ .  (代数式) 
2. Find the gradient of the curve when $ x=\frac{\pi}{2}$ .   

  Let $f(x)=6 x-\ln x $, for x>0 .
1. Find $f^{\prime}(x)$ .  (代数式) 
2. Find $f^{\prime \prime}(x)$ .  (代数式) 
3. Solve $f^{\prime}(x)=f^{\prime \prime}(x)$ .   

  Consider the curve $ y=\frac{2 x}{1+x}$,$ x \in \mathbb{R}$,$ x \neq-1$.
1. Find $\frac{\mathrm{d} y}{\mathrm{~d} x} $.  (代数式) 
2. Determine the equation of the normal to the curve at the point $ \mathrm{P}(-2,4) $.  (代数式) 

  Let f(x)=p x^{3}-q x . At x=0 , the gradient of the curve of f is 2 . Given that $ f^{-1}(12)=2 $, find the value of p and q . p =    q =   

  A function f(x) has derivatfoi1rh1ad.7g g hi 6k0ive $f^{\prime}(x)=6 x^{2}-24 x$ . The graph of f has an x -intercept at x=1 .
1. Find f(x) .  (代数式) 
2. The graph of f has a point of inflexion at x=k . Find k .   
3. Find the values of x for which the graph of f is concave-up. x>  

  Let $ f(x)=x^{2} e^{x}$ and $g(x)=4 x-x^{2}$ .
1. Find $f^{\prime}(x)$ .  (代数式) 
2. Find the x -coordinate where the tangents of f(x) and g(x) are parallel.   

  A farmer wants to build a rectangular enclosure for his chickens. The area of t6or c qk)fo)e ;oqu((the enclosure me6ru();) (fktqoooqcust be 350 $\mathrm{~m}^{2}$ . The fencing used for the side A B costs \$ 13 per metre. The fencing for the other three sides costs $ 4 per metre. The farmer wants the cost of the enclosure to be a minimum. Find the minimum cost.   

  The function f is ddgh6z 84vdi1i mflz60 efined for all $x \in \mathbb{R}$ . The tangent to the graph of f at x=3 has equation y=4 x-2 .
1. Write down the value of $ f^{\prime}(3)$ .   
2. Find f(3) .

The function g is defined for all $ x \in \mathbb{R}$ where $g(x)=11-2 x^{2}$ and $ h(x)=f(g(x)) $.   
3. Find h(2) .   
4. Hence, find the equation of the tangent to the graph of h at x=2 .  (代数式) 

$\text { A function } f \text { is defined by } f(x)=-x^{5}+e^{-x}+2, x \in \mathbb{R} \text {. By considering } f^{\prime}(x) \text {, determine whether } f \text { is a one-to-one or many-to-one function. }$

  Let f(x)=g(x) h(x) , wib jkpzw ua ,6:c 7t ymq6ey.l6c(sz,9here g(3)=6, h(3)=2, $g^{\prime}(3)=4 $ and $h^{\prime}(3)=1$ . Find the equation of the normal to the graph of f at x=3 .  (代数式) 

  Consider the curve zohd-q,; dejg10kf.d $y=\frac{9}{5-x}+\frac{1}{x-1}$ .
Find the x -coordinates of the points on the curve where the gradient is zero.      

  Consider the function f(fwqit6ucz b.m :)c- 5dx)=ln (2 x-1) . Let point A be the point on the curve whb)f- ct6cz5mqw .d:iuere x=3 .
1. Write down the gradient of the curve at A .   
2. The normal to the curve at A cuts the x -axis at P . find the coordinates of P . (a,b) a=   b=  

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

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

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.

  Let $f(x)=\frac{g(x)}{h(x)}+15$ , where g(5)=30, h(5)=10, $g^{\prime}(5)=20 $ and $h^{\prime}(5)=5$.
Find the equation of the normal to the graph of f at x=5 .  (代数式) 

  Consider the function f u(zm(ty,8geo8jpx b8defined by $ f(x)=\ln \left(4 x^{2}-9\right) $ for $x>\frac{3}{2}$ . The following diagram shows part of the graph of f which crosses the x -axis at point A , with coordinates (a, 0) . The line L is the tangent to the graph of f at the point B .



1. Find the exact value of a .   
2. Given that the gradient of L is $\frac{1}{2}$ , find the x -coordinate of B .   

The following diagram shows the graph of f ′ , the deri 4 e)c8mlt/itovative of f.


The graph of $f^{\prime}$ has a local maximum at A , a local minimum at B and passes through $\mathrm{P}(2,-3)$ .
1. The point $ \mathrm{Q}(2,6)$ lies on the graph of the function f .
1. Write down the gradient of the curve of f at Q .
2. Find the equation of the normal to the curve of f at Q .
2. Determine the concavity of the graph of f when 2

A particle moves in a straight linegov5cahw3s-n b1- h 71)xo igf and its 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 $t^{3}-8 t$ .
5. Find the values of t for which the velocity is positive and the acceleration is negative.

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



1. 1. Find $f^{\prime}(x)$ .
2. Find $f^{\prime \prime}(x)$ .
2. Find the x -coordinate of the point of inflection at C .

Given that f(-1)=14 .
3. 1. Find f(0) .
2. Hence, find the coordinates of the local maximum $\mathrm{A}(x, y)$ and justify your answer.
4. Write down in order from least to greatest $f^{\prime \prime}(\mathrm{B})$, $f^{\prime}(\mathrm{B}), f(\mathrm{~B})$ .v

  The values of the functions f and g and their dq (6;j)s d uz9e1 baqwr-u.jqlerivatives for x=3 and x=7 are shown i l(q j.)ds-qeq;ujbu16aw r9zn 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 .  (代数式) 

Let $ f(x)=\frac{\ln x}{x}$ , for x>0 .
1. Find $f^{\prime}(x)$ .
2. The graph of f has a maximum at point P . Find the coordinates of P .

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



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 .  (代数式) 

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 . y =   

  Jack makes an open container in the shape of a cuboid with square base, as qg4+0wo6wcy( skoylef (+ w d*shown in the following diq yd(w0+o4ywk f+ cle(*s6wogagram.


The container has base length $x \mathrm{~m}$ and height $y \mathrm{~m}$ . The volume is $32 \mathrm{~m}^{3}$ .
Let A(x) be the outside surface area of the container.
1. Show that $A(x)=\frac{128}{x}+x^{2}$.  (代数式) 
2. Find $A^{\prime}(x)$ .  (代数式) 
3. Given that the outside surface area is a minimum, find the base length of the container.   
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.   

  Let $ f(x)=24-2 x^{2}$ , for x $\in \mathbb{R}$ . The following diagram shows part of the graph of f and the rectangle OPQR , where R is on the positive x -axis, P is on the y -axis and Q is on the graph of f .




Find the coordinates of point R(x,y) that gives the maximum area of OPQR.x =    y =   

  Consider the functionr3l tp 8w2 q-;nmc2kbg)ffm-u $ f(x)=\frac{4 x^{2}+p x}{x+1}$ , where $x \neq-1$ and $p \in \mathbb{R}$ .
Find the value of p for which the graph of f has exactly one point with a gradient of zero.   

  Let $ f(x)=\left(x^{2}+a\right)^{5} $.
In the expansion of the derivative, $f^{\prime}(x)$ , the coefficient of the term in $ x^{5} $ is 960 . Find the possible values of a .±   

  $\text { A closed cylindrical can with radius } r \mathrm{~cm} \text { and height } h \mathrm{~cm} \text { has a volume of } 24 \pi \mathrm{cm}^{3} \text {. }$


1. Express h in terms of r .  (代数式) 

The material for the base and top of the can costs 15 cents per $\mathrm{cm}^{2}$ and the material for the curved side costs 10 cents per $\mathrm{cm}^{2}$ . The total cost of the material, in cents, is C .
2. Show that C=30 $\pi r^{2}+\frac{480 \pi}{r}$ .  (代数式) 
3. Given that there is a minimum value for C , find this minimum value in terms of $\pi$ .   

  Let f(x)=$\sin (3 x)$ and g(x)=$\ln (2 x+1)$ .
1. a. Find $ f^{\prime}(x)$ ;  (代数式) 
b. Find $g^{\prime}(x)$ .  (代数式) 
2. Let $h(x)=f(x) \times g(x)$ . Find $h^{\prime}\left(\frac{\pi}{2}\right)$ .   

  The values of the functions f and g and their derivatives for x=2, x=3 and x=6 a;bj/h0dd(gsmxw4wm f k **h .ire shown in the following table. f*j(h xsddh ;4/kmw.mi0*bw g


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) 

  The displacement, s , in metres, of a particle t seconds after it p5hg vbmkvh0u6 u(i,)iasses through the origin is given by the expression ig v kmb hi(065uv)h,u$s=\ln \left(3+t-2 e^{-t}\right)$,$ t \geq 0$ .
1. Find an expression for the velocity, v , of the particle at time t .  (代数式) 
2. Find an expression for the acceleration, a , of the particle at time t .  (代数式) 
3. Find the acceleration of the particle at time t=0 .   

A particle P moves along a straight line so that its velocity, lcm1jza w 4p)/*.m dka$v \mathrm{~ms}^{-1}$ , 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 .

  Consider the curve $ y=(k x-1) \ln (2 x)$ where $ k \in \mathbb{R}$ and x>0 .
The tangent to the curve at x=2 is perpendicular to the line $y=\frac{2}{5+4 \ln 4} x$ .
Find the value of k .   

  $\text { Consider } f(x), g(x) \text { and } h(x)=(f \circ g)(x) \text {, for } x \in \mathbb{R} \text {. Given that } g(5)=8, g^{\prime}(5)=2 \text { and } f^{\prime}(8)=-4 \text {, find the gradient of the normal to the curve } y=h(x) \text { at } x=5 \text {. }$   

  

The graph of f has a minimum at $\mathrm{P}(4,8)$ and a maximum at $\mathrm{Q}(12,16) $.
1. a. Find the value of c .   
b. Show that $k=\frac{\pi}{8} $.   
c. Find the value of a .   

The graph of g is obtained from the graph of f by a translation of $\binom{d}{0}$ .
The minimum point on the graph of g has coordinates (6.5,8) .
2. a. Write down the value of d .   
b. Find g(x) .  (代数式) 

The graph of g changes from concave-up to concave-down when $x=\nu$ .
3. a. Find $\nu$ .   
b. Hence, or otherwise, find the maximum positive rate of change of g .≈   

  The displacement, s , in metres, of a particzwwhd+bb5ro/ adu 7-30/z u hu6yy-dkle t seconds after it passes through the origin is given by a+ohz/ku7r/d dh36yy- - zubd5w buw 0the expression $s=\ln \left(1+t e^{-t}\right)$,$ t \geq 0$ .
1. Find an expression for the velocity, v , of the particle at time t .  (代数式) 
2. Find an expression for the acceleration, a , of the particle at time t .  (代数式) 
3. Find the acceleration of the particle at time t=0 .   

  $\text { Use L'Hôpital's rule to determine the value of } \lim _{x \rightarrow 0} \frac{x^{2} e^{2 x}}{1-\cos x}$   

  Use L'Hôpital's ruleddceimnm el79fy .p1t h+1z.f e)c0(tv9ua1 m to find

$\lim _{x \rightarrow 0} \frac{\arctan (3 x)}{\tan (4 x)}$

  

  Note: In this question,.qfo9 -tkm1yh distance 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.   

  $\text { The diagram below shows the graph of } f(x)=a \sin (k(x-d))+c \text {, for } 2 \leq x \leq 14 \text {. }$



The graph of f has a maximum at $\mathrm{P}(5,15)$ and a minimum at $\mathrm{Q}(11,-5)$ .
1. Write down the value of:
a. a ;   
b. c .   
2. a. Show that $k=\frac{\pi}{6}$ .   
b. Find the smallest possible value of d , given d>0 .   
3. Find $f^{\prime}(x)$ .  (代数式) 
4. At a point R , the gradient is $-\frac{5 \pi}{3}$ . Find the x -coordinate of R .   

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

Consider another function g(x) . Let P be a point on the graph g . The x -coordinate of P is 12 . The equation of the tangent to the graph at 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 .  (代数式) 

  Let $f(x)=\sin x-\sqrt{3} \cos x, 0 \leq x \leq 2 \pi$ .
The following diagram shows the graph of f .

The curve crosses the x -axis at A and C and has a maximum at point B .
1. Find the exact coordinates of A and of C . A (a,b) a =    b =    C (c,d) c =    d =   
2. Find $f^{\prime}(x) $.  (代数式) 
3. Find the coordinates of B . B (a,b) a =    b =   

Let $f(x)=\frac{2 x+1}{x-3}$,$ x \in \mathbb{R}$,$ x \neq 3 $
1. a. Find the equation of the vertical and horizontal asymptote of f .
b. Write down the coordinates of the point Q at which the asymptotes intersects.
c. Find the x and y intercepts of f .
2. Show that $f^{\prime}(x)=-\frac{7}{(x-3)^{2}}$ .
3. Hence, find the equation of the tangent at point $\mathrm{P}(4, y) $.
4. The point S also lies on the graph of f . The tangent to S is parallel to the tangent at point P . Find the coordinates of S .
5. Show that Q is the midpoint of [PS].

  Let $f(x)=16-x^{2}$ , for $x \in \mathbb{R}$ .
1. Find the x -intercepts of the graph of f . x = $\pm $   

The following diagram shows part of the graph of f .

Rectangle ABCD is drawn with $\mathrm{A} \& \mathrm{~B}$ on the x -axis and $\mathrm{C} \& \mathrm{D} $ on the graph of f . Let $\mathrm{OA}=a$ .
2. Show that the area of ABCD is $32 a-2 $a^{3}$$ .  (代数式)  = 0
3. Hence find the value of a>0 such that the area of ABCD is a maximum.   

Let $g(x)=(x-4)^{2}+k$ , for $x \in \mathbb{R}$ , where k is a constant.
4. Show that when the graphs of f and g intersect, 2 x^{2}-8 x+k=0 .  (代数式) 
5. Given that the graphs of f and g intersect only once, find the value of k .   

  In a triangle $\mathrm{ABC}, \mathrm{AB}=2 \mathrm{~cm}$, $\mathrm{CBA}=\frac{\pi}{4}$ and $\mathrm{BA} \mathrm{A}= x $ .
1. Show that $\mathrm{AC}=\frac{2}{\cos x+\sin x}$ .  (代数式) 
2. Given that AC has a minimum value, find the value of x for which this occurs.   

  The curve C is defi(g vt -:9cc36hv kf0kj qxada)ned by the equation $x^{2} y+\ln (x y)=1$, x $\lt$0, y $\lt$0 .
1. Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ in terms of x and y .  (代数式) 
2. Determine the equation of the tangent to C at the point $\mathrm{P}(1,1)$.  (代数式) 

  The function f is defined by iwp)tn a/ni35 g 2, q6gnpn.tn




where a and b are real constants.
Given that both f and its derivative are continuous at t=4 , find the value of a and the value of b . a =    b =   

  Use L'Hôpital's rule to determine theiih,9u)5bgu jr w2pf 8 value of$\lim _{x \rightarrow 0} \frac{2 \sin ^{2}(x)}{\ln \left(1+x^{2}\right)}$   

$\text { Consider a function with domain } a\lt x \lt b \text {. The following diagram shows the graph of } f^{\prime} \text {, the derivative of } f \text {. }$



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) .

In a triangle $\mathrm{ABC}$, $\mathrm{BÂC}=60^{\circ}$, $\mathrm{AB}=(1-x) \mathrm{cm}$, $\mathrm{AC}=(x+3)^{2} \mathrm{~cm}$,$-3\lt x \lt 1$
1. 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. a. Calculate $\frac{\mathrm{d} A}{\mathrm{~d} x}$ .
b. Verify that $\frac{\mathrm{d} A}{\mathrm{~d} x}=0 $ when $x=-\frac{1}{3}$ .
c. 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 [BC] when the area of triangle A B C is a maximum.

[/BC]

  $\text { Let } f(x)=x^{3}+2 x^{2}+k x-3 \text {. Part of the graph of } f \text { is shown in the following diagram. }$



The graph of f crosses the y -axis at the point Q . The line L is tangent to the graph of f at Q .
1. Find the coordinates of Q . y =   
2. a. Find $f^{\prime}(x)$ . f'(x) =  (代数式) 
b. Hence find the equation of L in terms of k . y =  (代数式) 

The graph of f has a local maximum at the point P . The line L passes through P .
3 . Find the value of k .   

Consider $f(x)=x \ln (x)-x$ , for x>0
1. Find f(1) .
2. Solve f(x)=0 .

The graph of f has a local minimum at point $\mathrm{P}(x, y)$ .
3. Find the coordinates of point P and explain why it is a local minimum.
4. Find the set of x values for which f is increasing.
5. Hence, sketch the graph of f , for x>0 .

The displacement, in centimeters, of a particle from an origin, O , at time t3lkk kxa ab2ywbk410 - seconds, is givenk baklxakw-k 3b1y24 0 by $s(t)=t \sin 2 t-7 \sin t \cos t, 0 \leq t \leq 3$ .
1. Find the maximum distance of the particle from O .
2. Find the acceleration of the particle at the instant it changes direction for the the second time.

Let $f(x)=\frac{1}{3} x^{3}+2 x^{2}-5 x+10$ .
1. Find $f^{\prime}(x)$ .

The graph of f has horizontal tangents at the points where x=a and x=b, a$\lt$b .
2. Find the value of a and the value of b .
3. a. Sketch the graph of $y=f^{\prime}(x)$ .
b. Hence explain why the graph of f has a local maximum point at x=a .
4. a. Find $f^{\prime \prime}(b)$ .
b. Hence, use your answer to part (d) (i) to show that the graph of f has a local minimum point at x=b .

The tangent to the graph of f at x=a and the normal to the graph of f at x=b intersect at the point (p, q) .
5. Find the value of p and the value of q .

Let $f(x)=(x+1) e^{-2 x}, x \in \mathbb{R}$
1. Find $f^{\prime}(x)$ .
2. Prove by induction that $\frac{\mathrm{d}^{n} f}{\mathrm{~d} x^{n}}=\left[n(-2)^{n-1}+(-2)^{n}(x+1)\right] e^{-2 x}$ for all $ n \in \mathbb{Z}^{+}$ .

  Consider the ellipse defined by the msc+;:j3bqu x equation $x^{2}+3 y^{2}=12$ .
1. Find the equation of the normal to the ellipse at the point $\mathrm{P}(3,1)$ .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.   

Air is being pumped into a spherical balloon so that its vo( j63hl1ggorw lume is increasing at a conwh gr631(oj glstant rate of $15 \mathrm{~cm}^{3} \mathrm{~min}^{-1}$ . Find the rate at which the surface area of the balloon is increasing when its radius hits 10 cm .
The surface area S and the volume V of a sphere of radius r are given by $S=4 \pi r^{2}$ and $V=\frac{4}{3} \pi r^{3}$ .

Consider a function f .fxhu9 ,xl): bqw)gt0 i The line $L_{1}$ with equation y=2 x-1 is a tangent to the graph of f when x=3 .
1. a. Write down $f^{\prime}(3)$ .
b. Find f(3) .

Let $g(x)=f\left(x^{2}-1\right)$ and P be the point on the graph of g where x=2 .
2. Show that the graph of g has a gradient of 8 at P .

Let $L_{2}$ be the tangent to the graph of g at P . The line $ L_{1}$ intersects $ L_{2}$ at the point Q .
3. Find the y -coordinate of Q .

Let $y=\left(x^{2}+x\right)^{\frac{s}{2}}$ , for $x \geq 0$ .
1. Find $ \frac{\mathrm{d} y}{\mathrm{~d} x}$ .
2. Hence find $\int(2 x+1) \sqrt{x^{2}+x} \mathrm{~d} x$.

Consider the functions $ f(x)=\frac{1}{4} \sqrt{x^{2}+x}$ and $g(x)=5-\frac{x}{2} \sqrt{x^{2}+x}$ , for $x \geq 0$ . The graphs of f and g are shown in the following diagram.

The shaded region S is enclosed by the graph of f , the graph of g , the y -axis and the line x=2 .
3. Write down an expression for the area of S .
4. Hence find the exact area of S .

  Consider the function f defined byz -b/+/a;kxq r0 )udldryd52 ub1owzj $f(x)=2 \ln (24-1.5 x)$ for x $\lt $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 . (a,b) a =    b =   
b. Show that the equation of $ L_{2}$ is $y=-x+2 \ln 3+14$ .  (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 $f(x)=2 e^{\frac{x}{5}}$ 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 .
1. Sketch the graphs of f and g on the same axes.
2. 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 biased coin is tossed 10 times. Letsmm351 0zux gpg1ht9 t X be the number of tails obtained.
The probability of obtaining a tail in any one throw is p>0 .
1. Find, in terms of p , an expression for $\mathrm{P}(X=8)$ .  (代数式) 
2. a. Determine the value of p for which $\mathrm{P}(X=8)$ is a maximum.   
b. For this value of p , determine the expected number of tails.   

  Consider the curve defined by the equi8j5 v zcllf4(ggle+*ation $x^{5}+y^{5}=5 x^{2} y$ .
1. Show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{2 x y-x^{4}}{y^{4}-x^{2}}$ .  (代数式) 

The normal to this curve is parallel to the y -axis at the point where x=h, h>0 .
2. Find the value of h , giving your answer correct to two decimal places. ≈   

A ladder of length 13 m rests on horizontal ground and leans against a7fz36 bmzx gpw he5mqs(kuq)1kw 26l6 vertical wall. The bottom of the ladder is pulled away from thekzh7 pezq5(gqm) 366s2lkx6 1uwwfbm wall at a constant speed of $1.2 \mathrm{~ms}^{-1}$ . Calculate the speed of descent of the top of the ladder when the bottom is 5 m away from the wall.

Farmer Thomas wants to build a sheep farming field in the shape of a re( b/4o nsrfyphxu/ 6*nctangle with semicircles of radius r on two sides, as shown on the diagram. He has decided to use in total 350 metres of woodensu*4 (by6hfp n/xo r/n fencing.


1. 1. Find an expression for the area of the farming field in terms of r .
2. Find the width of the farming field when the area is a maximum.
2. Show that in this case the length of the rectangle is equal to zero and the farming field is the circle of radius 175 $\pi^{-1}$ metres.

  A curve has equation :v xs ,)zb5nxfe) eet: $2 y^{2} e^{x+1}-5 x^{2}=3$ .
1. Find an expression for $\frac{\mathrm{d} y}{\mathrm{~d} x}$ in terms of x and y .  (代数式) 
2. Find the equations of the tangents to this curve when x=-1 . y = $\pm$    m = $\pm$   

  $\text { A bird is } 4 \mathrm{~km} \text { East and } 6 \mathrm{~km} \text { North of its nest. It is flying East at a rate of } 14 \mathrm{~km} \mathrm{~h}^{-1} \text { and North at a rate of } 18 \mathrm{~km} \mathrm{~h}^{-1} \text {. Calculate the rate that its distance from the nest is changing. }$≈   

  Consider the functions 2w1 uk gkqyhhu/n; .m* 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. $f^{\prime}(x)$ ;  (代数式) 
b. $g^{\prime}(x)$ .  (代数式) 
2. Hence, or otherwise, find $\int_{0}^{\pi} e^{2 x} \cos x \mathrm{~d}$ x .  (数值) 

Consider the curves /lil xu72ycg1x+lq 5 t$C_{1}$ and $C_{2}$ defined as follows

$\begin{array}{l}
C_{1}: \quad 3 y^{2}+2 x^{2}=5, y>0 \\
C_{2}: \quad y^{2}-5 x^{3}=0, y>0
\end{array}$

1. Using implicit differentiation, or otherwise, find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ for each curve in terms of x and y .

Let $\mathrm{P}(a, b)$ be the unique point where the curves $C_{1}$ and $C_{2}$ intersect.
2. Show that the tangent to $C_{1}$ at P is perpendicular to the tangent to $C_{2}$ at P .

  $\text { Consider the function } f(x)=\sqrt{\frac{16-4 x^{2}}{7}} \text {, for }-2 \leq x \leq 2 \text {. In the following diagram, the shaded region is enclosed by the graph of } f \text { and the } x \text {-axis. }$



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 aa. Find the value of a and the value of b . a =    b =   
b. During the interval $a\lt t \ltb$ , 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.   

Let the Maclaurin ser 9e( )muu1xolxies for $\cot x$ be

$\cot x=\frac{a_{1}}{x}+a_{2} x+a_{3} x^{3}+\cdots$

where $a_{1}, a_{2}$ and $ a_{3}$ are non zero constants.
1. Find the series for $\csc ^{2}$ x , in terms of $a_{1}, a_{2}$ and $a_{3}$ , up to and including the $x^{2}$ term
a. by differentiating the above series for $\cot x$ ;
b. by using the relationship $\csc ^{2} x=1+\cot ^{2} x$ .
2. Hence, by comparing your two series, determine the values of $a_{1}, a_{2}$ and $ a_{3}$ .

The function f is defined by j0h1 im8f d9hm$f(x)=e^{-x} \sin x-x+x^{2}$ .
By finding a suitable number of derivatives of f , determine the first non-zero term in its Maclaurin series.

A physicist is studying the motion of t/eyh 89y /fxp7czmfil9-9z ybwo separate particles moving in a straight line. She measures the displacement of each particle from a fixed origin over the course of 10 seconds The physicist found that the displacement of 9xic pz9b/ m l/ ef-y9fh8zyy7particle A, $s_{A} \mathrm{~cm}$ , at time t seconds can be modelled by the function $s_{A}(t)=7 t+9$ , where $ 0 \leq t \leq 10$ .
The physicist found that the displacement of particle B,$s_{B} \mathrm{~cm}$ , at time t seconds can be modelled by the function $s_{B}(t)=\cos (3 t+5)+8 t+4 $.
1. Use the physicist's models to find the initial displacement of
1. Particle A ;
2. Particle B correct to three significant figures.
2. Find the values of t when $s_{A}(t)=s_{B}(t)$ .
3. For t>6 , prove that particle B was always further away from the fixed origin than particle A .
4. For $ 0 \leq t \leq 10$ , find the total amount of time that the velocity of particle A was greater than the velocity of particle B .

A curve is given by t6lsfl2-9ygeeci20 u n he equation $y=\cos (2 \pi \sin x)$ .
Find the coordinates of all the points on the curve for which $\frac{\mathrm{d} y}{\mathrm{~d} x}=0,0 \leq x \leq \pi$ .

A curve is given by thed v4t9 dpk)q:m equation $y=\cos (2 \pi \cos x)$ .
Find the coordinates of all the points on the curve for which $\frac{\mathrm{d} y}{\mathrm{~d} x}=0,0 \leq x \leq \pi$ .

The function f is defined by 6l3sj 7y18reqyy/g sn$f(x)=e^{2 x}-4 e^{x}+2$ , for $x \in \mathbb{R}, x \leq a$ , where $a \in \mathbb{R}$ . Part of the graph of y=f(x) is shown in the following diagram.



1. Find the largest value of a such that f has an inverse function.
2. For this value of a , find an expression for $ f^{-1}(x)$ , stating its domain.

The function h is k 3x ,f3v2/8 flstzd+w3ijdxzdefined by $h(x)=\cos (\pi \cos x),-\pi \leq x \leq \pi$ .
1. Determine whether h is even, odd or neither even nor odd.
2. Show that $h^{\prime \prime}(0)=0$ .
3. Jack states that, because $ h^{\prime \prime}(x)=0$ , the graph of h has an inflexion at the point $\mathrm{P}(0,-1)$ . Explain briefly whether Jack's statement is correct or not.

  Find the x -coordinates of all the points on o0pv3dcd1ux2w8m e( mthe curve $y=2 x^{4}-\frac{34}{3} x^{3}+\frac{41}{2} x^{2}-10 x$ at which the tangent to the curve is parallel to the tangent at $\mathrm{P}\left(1, \frac{7}{6}\right)$ . x =      

  A curve is defined by bni7l h*u/18t4p ky6 terrep/$x^{2}+10 x y+y^{2}=48$ .
1. Show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{2 x+10 y}{10 x+2 y} $.  (代数式) 
2. Find the equation of the normal to the curve at the point $ \mathrm{P}(-2,22)$ .y =  (代数式) 
3 . Find the distance between the two points on the curve where each tangent is parallel to the line y=-x .   

  $\text { The double folium is a curve defined by the equation }\left(x^{2}+y^{2}\right)^{2}=8 x y^{2} \text {, shown in the diagram below. }$


\text { Determine the exact coordinates of the point } \mathrm{P} \text { on the curve where the tangent line is parallel to the } x \text {-axis. } (a,b) a =    b =   

  An oval-shaped superspeedway pfj;bbfvm 7s 3215 mxacan be described by the curve $x^{2}+12 y^{2}=1+10 x^{2} y^{2}$ , shown in the diagram below. A F1 racing car is moving along the track with $\frac{\mathrm{d} x}{\mathrm{~d} t}=240 \mathrm{kmh}^{-1}$ when x=0.67 $\mathrm{~km}$ .


$\text { Find the value of } \frac{\mathrm{d} y}{\mathrm{~d} t} \text {, giving your answer to the nearest } \mathrm{km}^{-1} \text {. }$≈   

The function f is d*u7bg*5 hol)zxs(r ;ha4 ue/c;tpk yk efined by $f(x)=e^{x} \cos x, x \in \mathbb{R}$ .
1. By finding a suitable number of derivatives of f , determine the Maclaurin series for f(x) as far as the term $x^{4}$ .
2. Hence, or otherwise, determine the exact value of $\lim _{x \rightarrow 0} \frac{e^{x} \cos x-x-1}{x^{3}}$ .

  Consider

$\lim _{x \rightarrow 0} \frac{2 \arctan \left(e^{x}\right)-c}{3 x}$

where $ c \in \mathbb{R}$ .
1. Show that a finite limit only exists for $c=\frac{\pi}{2}$ .   
2. Using l'Hôpital's rule, show algebraically that the value of the limit is $\frac{1}{3}$ .   

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

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 .

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

A particle moves such that aw.;4 jn- nly 5jchek 69d*exvits velocity v$ \mathrm{~m} \mathrm{~s}^{-1} $ is related to its displacement s $\mathrm{~m}$ by the equation v(s)=2$ \arctan (\cos s), 0 \leq s \leq \pi$ .
1. Find the particle's acceleration a $\mathrm{~m} \mathrm{~s}^{-2}$ in terms of s .
2. Using an appropriate graph, find the particle's displacement when its acceleration is 0.5$ \mathrm{~m} \mathrm{~s}^{-2}$ .

  1. Use L'Hôpital's rule to determine txtf ac0zr0 94h6e ng/ihe value of $\lim _{x \rightarrow 0} \frac{e^{-5 x^{2}}-2 \cos (5 x)+1}{4 x^{2}} $.   
2. Hence find $\lim _{x \rightarrow 0} \frac{\int_{0}^{x}\left(e^{-5 t^{2}}-2 \cos (5 t)+1\right) \mathrm{d} t}{\int_{0}^{x} 4 t^{2} \mathrm{~d} t}$ .   

1. By successive differentiation find the first five non-zero ter(krp7v 7r o5fdms in the Maclaurin series 5 or7krpf7dv(for $f(x)=(2-2 x) \ln (1-x)+2 x $.
2. Deduce that, for $n \geq 2$ , the coefficient of $x^{n}$ in this series is $\frac{2}{n(n-1)}$ .

Consider the functio6h)x lxop-lu 466smwkn $f(x)=\sqrt{\frac{10}{x^{2}}-1}$ , 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 a0gbx8zfw8lk. ivx,iy 2f) rp,nd g defined on the domain $0\lt x \lt 2$ $\pi$ by

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


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 .   

A water truck tank which is 3 metres long has a uniform cross-sectrotb:jsoni;-qd , )4cion in the shape of a major segment. The tank is divided into two equal parts and is partially filled with water as shown in the following diagram of the cross-sect qoord4nic)s j;tb-: ,ion. The centre of the circle is O , the angle AOB is $\alpha$ radians, and the angle AOF is $\beta$ radians.

1. Given that $\alpha=\frac{\pi}{4}$ , calculate the amount of water, in litres, in the right part of the water tank. Give your answer correct to the nearest integer.
2. Find an expression for the volume of water V , in $ \mathrm{m}^{3}$ , in the left part of the water tank in terms of $\beta$ .
he left part of the tank is now being filled with water at a constant rate of 0.001 $\mathrm{~m}^{3}$ per second.
3. Calculate $\frac{\mathrm{d} \beta}{\mathrm{d} t}$ when $\beta=\frac{3 \pi}{5} $. Round your answer to 3 significant figures.
4. Calculate the amount of time it will take for the left part of the tank to be fully filled with water. Give your answer in minutes and correct to the nearest integer.

Consider the function)k )9b)pmndmms $f(x)=\sin x,-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$ 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 $\frac{1}{2} \ln 2$ .

A curve is defined by4-lysb f i ggawy(v5wdb. t/89 $x^{2}+y^{2}-3 x y+45=0 $.
1. Show that $ \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{3 y-2 x}{2 y-3 x}$ .
2. Find the equation of the normal to the curve at the point (21,9) .
3. Find the distance between the two points on the curve where each tangent is perpendicular to the y -axis.

The curve C is defined by the equat 4n )tm3nrjeuwf*1rmk:6hcw *ion $y^{2}+4 x y+e^{x}=10$ . The point $ \mathrm{P}(0, b)$ lies on C where b$\lt$0 .
1. Find the value of b .
2. Show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{4 y+e^{x}}{2 y+4 x}$ .
3. Find the equation of the normal to C at the point P .
4. Find the coordinates of the second point at which the normal found in part (c) intersects C .
5. Given that $ u=y^{4}$, y$\lt$0 , find $\frac{\mathrm{d} u}{\mathrm{~d} x} $ at x=0 .

Let $ f(x)=e^{x} \cos x$ .
1. Show that $f^{\prime \prime}(x)=2\left(f^{\prime}(x)-f(x)\right)$ .
2. By further differentiation of the result in part (a), find the Maclaurin expansion of f(x) , as far as the term in $x^{5}$ .

Consider the function dd b .py2 7p uxn;1/h* k11cbazom)ztp$ g(x)=a x^{3}+b x^{2}+c x+d$ , where x$ \in \mathbb{R} $ and a, b, c, d $\in \mathbb{R}$ .
1. 1. Write down an expression for $ g^{\prime}(x)$ .
2. Hence, given that $g^{-1}$ does not exist, show that $ b^{2}-3$ a c$\lt $0 .

Consider the function $f(x)=\frac{x^{3}}{2}+3 x^{2}+6 x+\frac{9}{2}$
2. 1. Show that $f^{-1}$ exists.
2. f(x) can be written in the form p(x+2)^{3}+q , where p, q$ \in \mathbb{R}$ . Find the value of p and the value of q .
3. Hence, find $ f^{-1}(x)$ .

The graph of f(x) may be obtained by transforming the graph of $y=x^{3}$ using a sequence of three transformations.
3. State each of the transformations in the order in which they are applied.
4. Sketch the graphs of y=f(x) and $y=f^{-1}(x)$ on the same set of axes, indicating the points where each graph crosses the coordinate axes.

Let $f(x)=(x-1) e^{\frac{t}{3}}$, for $x \in \mathbb{R}$
1. Find $f^{\prime}(x) $.
2. Prove by induction that $ \frac{\mathrm{d}^{n} f}{\mathrm{~d} x^{n}}=\left(\frac{3 n+x-1}{3^{n}}\right) e^{\frac{x}{3}} $ for all $n \in \mathbb{Z}^{+}$ .
3. Find the coordinates of any local maximum and minimum points on the graph of y=f(x) . Justify whether such point is a maximum or a minimum.
4. Find the coordinates of any points of inflexion on the graph of y=f(x) . Justify whether such point is a point of inflexion.
5. Hence sketch the graph of y=f(x) , indicating clearly the points found in parts (c) and (d) and any intercepts with the axes.

Let $f(x)=x^{4}-0.4 x^{3}-2.85 x^{2}+0.9 x+1.35$ , for $x \in \mathbb{R}$ .
1. Find the solutions for $f(x)\lt 0$ .
2. For the graph of y=f(x) ,
1. find the coordinates of local minimum and maximum points.
2. find the x -coordinates of the points of inflexion.

The domain of f is now restricted to [a, b] where a, b $\in \mathbb{R}^{+}$.
3. 1. Write down the smallest value of $a\lt 0$ and the largest value of b>0 for which f has an inverse. Give your answers correct to three significant figures.
2. For these values of a and b , sketch the graphs of y=f(x) and $y=f^{-1}(x)$ on the same set of axes, showing clearly the coordinates of the end points of each curve.
3. Solve $f^{-1}(x)=0.5$ .

Let $g(x)=\frac{2}{3} \sin (2 x-1)+\frac{1}{2}, \frac{1}{2}-\frac{\pi}{4} \leq x \leq \frac{1}{2}+\frac{\pi}{4}$ .
4. 1. Find an expression for $g^{-1}$ and state its domain.
2. Solve $ \left(f^{-1} \circ g\right)(x) \lt 0.5$ .

Consider the functiond9 8-kj8c cw h(gm(spv $f(x)=\frac{\sqrt{x}}{2 \cos x}, \frac{\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 functiox.g*ku eipok5;gst*3 n 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 $f^{\prime}(x)$ .
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 $t \mathrm{sec}^{2}$ , 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. 1. 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 $\frac{\mathrm{d} h}{\mathrm{~d} t}$.
2. Find the value of $\frac{\mathrm{d} h}{\mathrm{~d} t} $ when $h=45 \mathrm{~cm}$ .
3. 1. Find $\frac{\mathrm{d}^{2} h}{\mathrm{~d} t^{2}}$
2. Find the values of h for which $\frac{\mathrm{d}^{2} h}{\mathrm{~d} t^{2}}=0 $.
3. 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 functio;x9qio 3twrj 6n $f(x)=\frac{a e^{-x}}{b-a e^{-x}}$ where $\lt 0, b\lt0$ .
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:
1. the vertical asymptote of f ;
2. the horizontal asymptote of f .
4. Draw a sign diagram for $f^{\prime}(x)$ .
5. If a=3 and b=1 ,
1. sketch the graph of f labelling all asymptotes;
2. find the area of the region enclosed by f , the x and y axes and the line $x=\ln 2 $.