Note: In this question, displ u-i3s2ri3(nhi q2v(pb d* ptance is in metres and time 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 $t^2+20t+4$,$\text{ for }0\le t\le 4$.

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

  Let f(x)=$e^{3x}$ . 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) 

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

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

  Let $f(x)=e^{-3x}$ .
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 $x^3-3.2x$. 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 $f^{\prime }(x)=x^3-9$ $x^2+24x+3$.
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 $f(x)=x^3-\frac{p}{x}$, $x\ne 0$ , 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 straight line with velocity +f,+ lu cck1mpqx-/ wf$v(t)=2t-0.3t^3+2$ , for $t\ge 0$ , 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 are at poifr-k j;t. :z5jx iwlj6nts jrl5- j 6fzk jt.i:wx;A and C. There is a minimum at point B and a maximum 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 sho-0e4hsm/a5a c1ehb gqw part of 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: $f(-2),f^{\prime }(8)$, $f^{\prime \prime }(-2)$.

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

  The following diagram shows the grap vxe779brx.i lh 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 $\mathrm{P}(2,8k)$ 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 $f(x)=4x\sin (2x)$ .
1. Find $f^{\prime }(x)$ .  (代数式) 
2. Find the gradient of the curve when $x=\frac{\pi }{2}$ .   

  Let $f(x)=6x-\ln x$, 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 $y=\frac{2x}{1+x}$,$x\in \mathbb{R}$,$x\ne -1$.
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) hatert dx ta0uuza-5o r35*:o/ls derivative $f^{\prime }(x)=6x^2-24x$ . 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 $g(x)=4x-x^2$ .
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 rectangul ki9orkw x,j (nl2qw),*(ff k*uwdq:lar enclosure for his chickens. The area of the enclosure must be 350ilrx*:*, l2nd((jqqkfwk ,w9fkw ) o u ${\mathrm{m}}^2$ . 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 defined for vt6h / h cf8hvdn30/gqall $x\in \mathbb{R}$ . 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 $g(x)=11-2x^2$ 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) , where g(3)=6, h(3)91dnf(uyv. /d 6w y ni5mwty5 lp13uur=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 -lrj j +tg 49ybra3xy 4ua2(bq$y=\frac{9}{5-x}+\frac{1}{x-1}$ .
Find the x -coordinates of the points on the curve where the gradient is zero.      

  Consider the function f(x)=ln (2 x-1) . Let poi1c 1eugot8i 5ynt A be the point on the curve whe8e1ocut1 iyg 5re 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 $f(x)=\frac{5x}{e^{3x}}$ , for $0\le x\le 10$.
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 nq;,l6/ s wogjrda+oaxm :w0/defined by $f(x)=\ln (4x^2-9)$ 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 derivativ*8 f29qait d5sem h*fae 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 line tr1 ;x. 9zuoxa7a vd/gand its velocity, v ${\mathrm{ms}}^{-1}$, at time t seconds, is given by $v(t)={(t^2-2)}^2$ , for $0\le t\le 2$.
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 $t^3-8t$ .
5. Find the values of t for which the velocity is positive and the acceleration is negative.

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



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 anzdg r38-5 +yd bp,rxikd g and their derivatives for x=3 and x=7 are shown in the r yzd+b8-g5,x3idkpr 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 .

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



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 thez5 s/03;nvvhvljk 6ca shape of a cuboid with square base, as shown in the foal ;56nj3v/ v0szvh kcllowing diagram.


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 $f(x)=24-2x^2$ , 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 function .kp27z*,ez4 ( npnvrar9r sns$f(x)=\frac{4x^2+px}{x+1}$ , where $x\ne -1$ 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 $f(x)={(x^2+a)}^5$.
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)=$\sin (3x)$ and g(x)=$\ln (2x+1)$ .
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+(dr*sn r/flmul f)i8 rfzo5z)go uw 3y/d/d9 and x=6 are shown in t9 )yuw3d g z8sm frr/dru(n5)do+*offl /l/zihe 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) 

  The displacement, s , in metres, of a particle t seconds afteb xe8b)8 paf-c7*3ih 7dnolpqr it passes through the origin is given by the expressionpx*q e7hdci8of )7p3n 8b-abl $s=\ln (3+t-2e^{-t})$,$t\ge 0$ .
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 sf3o q:i c/z44rt ) *qxm1nithytraight line so that its velocity, $v{\mathrm{ms}}^{-1}$ , after t seconds, is given by $v=\sin 3t-2\cos t-2$ , for $0\le t\le 6$ . 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 $y=(kx-1)\ln (2x)$ 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 $(\genfrac{}{}{0ex}{}{d}{0})$ .
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 particle t secondjjb6hv. y yiah7r1)p 6,ae 6pts after it passes through the origin is gia67)y,jhj6 1apit6v hbr p.y even by the expression $s=\ln (1+t{e}^{-t})$,$t\ge 0$ .
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 .   

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

  Use L'Hôpital's rule t *0 rd6w 41pdq5e. wsclo1wkylo find

$\underset{x\to 0}{lim}\frac{\arctan (3x)}{\tan (4x)}$

  

  Note: In this question, viic/v- ls 8*s+df:zx 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 $a(t)=-2t^2+13t-15$, for $0\le t\le 6$ .
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.$a<t<b$ 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.   

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



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 $f(x)=\sqrt{2x+1}$ for $x\ge -0.5$ .
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 $f(x)=\sin x-\sqrt{3}\cos x,0\le x\le 2\pi$ .
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 $f(x)=\frac{2x+1}{x-3}$,$x\in \mathbb{R}$,$x\ne 3$
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 $\mathrm{A}\mathrm{\&}\mathrm{B}$ on the x -axis and $\mathrm{C}\mathrm{\&}\mathrm{D}$ on the graph of f . Let $\mathrm{OA}=a$ .
2. Show that the area of ABCD is $32a-2$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 defined by the equation k1cf zqx 2x,r+( r)jvm+;f rtc b11tyz $x^{2}y+\ln (xy)=1$, x $<$0, y $<$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 def1-db.fn ybh*q m1zt 6j4ury6zined by




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 the vb skz-(aif5gs4,0m/65 iz feldma7 nralue of$\underset{x\to 0}{lim}\frac{2{\sin }^2(x)}{\ln (1+x^2)}$   

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



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

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

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]

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



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 centim/95,py s,s;bkqx9m v)foyf t zt.7 dcxeters, of a particle from an origin, O , at time t seconds, is given f;ftmxq ,.oybd y9tc5pvs97/, )z skxby $s(t)=t\sin 2t-7\sin t\cos t,0\le t\le 3$ .
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 $f(x)=\frac{1}{3}{x}^3+2x^2-5x+10$ .
1. Find $f^{\prime }(x)$ .

The graph of f has horizontal tangents at the points where x=a and x=b, a$<$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 $f(x)=(x+1)e^{-2x},x\in \mathbb{R}$
1. Find $f^{\prime }(x)$ .
2. Prove by induction that $\frac{{\mathrm{d}}^{n}f}{\mathrm{d}{x}^n}=[n(-2{)}^{n-1}+(-2{)}^n(x+1)]e^{-2x}$ for all $n\in {\mathbb{Z}}^{+}$ .

  Consider the ellipse9owx16j * fwtc defined by the equation $x^2+3y^2=12$ .
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 $x\ge 0$ and the y -axis for $y\ge 0$ is rotated through $2\pi$ about the y -axis.   

Air is being pumped into a spherical balloon so that its volu*xc8of) y4-r /zpext ume is increasing at u) x z-/xo*y tc4f8pera constant rate of $15{\mathrm{cm}}^3{\min }^{-1}$ . 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 . Ty x4go ;t) se/tz :fqg5dip..phe 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 $g(x)=f(x^2-1)$ 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 $y={(x^2+x)}^{\frac{s}{2}}$ , for $x\ge 0$ .
1. Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ .
2. Hence find $\int (2x+1)\sqrt{x^2+x}\mathrm{d}x$.

Consider the functions $f(x)=\frac{1}{4}\sqrt{x^2+x}$ and $g(x)=5-\frac{x}{2}\sqrt{x^2+x}$ , for $x\ge 0$ . 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 .