1.	Let g(x) = , for x > 0.
(a)	Use the quotient rule to show that .
(4)
 
(b)	The graph of g has a maximum point at A. Find the x-coordinate of A.
(3)
(Total 7 marks)

2.	The velocity v m s–1 of a particle at time t seconds, is given by v = 2t + cos2t, for 0 ≤ t ≤ 2.
(a)	Write down the velocity of the particle when t = 0.
(1)
 
	When t = k, the acceleration is zero.
(b)	(i)	Show that k = .
(ii)	Find the exact velocity when t = .
(8)
 
(c)	When t < ,  > 0 and when t > ,  > 0.
	Sketch a graph of v against t.
(4)
 
(d)	Let d be the distance travelled by the particle for 0 ≤ t ≤ 1.
(i)	Write down an expression for d.
(ii)	Represent d on your sketch.
(3)
(Total 16 marks)

3.	Let h(x) = . Find h′(0).
(Total 6 marks)

4.	The following diagram shows part of the graph of the function f(x) = 2x2.
		
diagram not to scale
	The line T is the tangent to the graph of f at x = 1.
 
(a)	Show that the equation of T is y = 4x – 2.
(5)
 
(b)	Find the x-intercept of T.
(2)
 
(c)	The shaded region R is enclosed by the graph of f, the line T, and the x-axis.
(i)	Write down an expression for the area of R.
(ii)	Find the area of R.
(9)
(Total 16 marks)

5.	The following diagram shows part of the graph of a quadratic function f.
		
	The x-intercepts are at (–4, 0) and (6, 0) and the y-intercept is at (0, 240).
 
(a)	Write down f(x) in the form f(x) = –10(x – p)(x – q).
(2)
 
(b)	Find another expression for f(x) in the form f(x) = –10(x – h)2 + k.
(4)
 
(c)	Show that f(x) can also be written in the form f(x) = 240 + 20x – 10x2.
(2)
 
	A particle moves along a straight line so that its velocity, v m s–1, at time t seconds is given by
v = 240 + 20t – 10t2, for 0 ≤ t ≤ 6.
(d)	(i)	Find the value of t when the speed of the particle is greatest.
(ii)	Find the acceleration of the particle when its speed is zero.
(7)
(Total 15 marks)

6.	The graph of f(x) = , for –2 ≤ x ≤ 2, is shown below.
		
	The region enclosed by the curve of f and the x-axis is rotated 360° about the x-axis.
Find the volume of the solid formed.
(Total 6 marks)

7.	Let f(x) = . Part of the graph of f is shown below.
		
	There is a maximum point at A and a minimum point at B(3, –9).
 
(a)	Find the coordinates of A.
(8)
 
(b)	Write down the coordinates of
(i)	the image of B after reflection in the y-axis;
(ii)	the image of B after translation by the vector ;
(iii)	the image of B after reflection in the x-axis followed by a horizontal stretch with scale factor .
(6)
(Total 14 marks)

8.	Let f(x) = , for sin x ≠ 0.
(a)	Use the quotient rule to show that f′(x) = .
(5)
 
(b)	Find f′′(x).
(3)
 
	In the following table, f′ = p and f′′ = q. The table also gives approximate values of f′(x) and f′′(x) near x = .

x				
f′(x)	–1.01	p	–1.01	
f″(x)	0.203	q	–0.203	
 
(c)	Find the value of p and of q.
(3)
 
(d)	Use information from the table to explain why there is a point of inflexion on the graph of f where x = .
(2)
(Total 13 marks)

9.	Let f(x) = kx4. The point P(1, k) lies on the curve of f. At P, the normal to the curve is parallel to y = . Find the value of k.
(Total 6 marks)

10.	A function f is defined for –4 ≤ x ≤ 3. The graph of f is given below.
		
	The graph has a local maximum when x = 0, and local minima when x = –3, x = 2.
 
(a)	Write down the x-intercepts of the graph of the derivative function, f′.
(2)
 
(b)	Write down all values of x for which f′(x) is positive.
(2)
 
(c)	At point D on the graph of f, the x-coordinate is –0.5. Explain why f′′(x) < 0 at D.
(2)
(Total 6 marks)

11.	Consider the function f with second derivative f′′(x) = 3x – 1. The graph of f has a minimum point at A(2, 4) and a maximum point at B.
(a)	Use the second derivative to justify that B is a maximum.
(3)
 
(b)	Given that f′ =  – x + p, show that p = –4.
(4)
 
(c)	Find f(x).
(7)
(Total 14 marks)

12.	Let f(x) = 6 + 6sinx. Part of the graph of f is shown below.
		
	The shaded region is enclosed by the curve of f, the x-axis, and the y-axis.
 
(a)	Solve for 0 ≤ x < 2π.
(i)	6 + 6sin x = 6;
(ii)	6 + 6 sin x = 0.
(5)
 
(b)	Write down the exact value of the x-intercept of f, for 0 ≤ x < 2.
(1)
 
(c)	The area of the shaded region is k. Find the value of k, giving your answer in terms of π.
(6)
 
	Let g(x) = 6 + 6sin. The graph of f is transformed to the graph of g.
(d)	Give a full geometric description of this transformation.
(2)
 
(e)	Given that  = k and 0 ≤ p < 2π, write down the two values of p.
(3)
(Total 17 marks)

13.	Let g(x) = 2x sin x.
(a)	Find g′(x).
(4)
 
(b)	Find the gradient of the graph of g at x = π.
(3)
(Total 7 marks)

14.	The graph of the function y = f(x) passes through the point . The gradient function of f is given as f′(x) = sin (2x – 3). Find f(x).
(Total 6 marks)

15.	Let f(x) = x3. The following diagram shows part of the graph of f.
		
diagram not to scale
	The point P (a, f(a)), where a > 0, lies on the graph of f. The tangent at P crosses the x-axis at the point Q. This tangent intersects the graph of f at the point R(–2, –8).
 
(a)	(i)	Show that the gradient of [PQ] is .
 
(ii)	Find f′(a).
 
(iii)	Hence show that a = 1.
(7)

	The equation of the tangent at P is y = 3x – 2. Let T be the region enclosed by the graph of f, the tangent [PR] and the line x = k, between x = –2 and x = k where –2 < k < 1. This is shown in the diagram below.
		
diagram not to scale
 
(b)	Given that the area of T is 2k + 4, show that k satisfies the equation k4 – 6k2 + 8 = 0.
(9)
(Total 16 marks)

16.	Let f(x) = ex cos x. Find the gradient of the normal to the curve of f at x = π.
(Total 6 marks)

17.	The following diagram shows the graphs of the displacement, velocity and acceleration of a moving object as functions of time, t.

(a)	Complete the following table by noting which graph A, B or C corresponds to each function.

Function	Graph	
displacement		
acceleration		
(4)
 
(b)	Write down the value of t when the velocity is greatest.
(2)
(Total 6 marks)

18.	The graph of y =  between x = 0 and x = a is rotated 360° about the x-axis.
The volume of the solid formed is 32π. Find the value of a.
(Total 7 marks)

19.	A rectangle is inscribed in a circle of radius 3 cm and centre O, as shown below.
		
	The point P(x, y) is a vertex of the rectangle and also lies on the circle. The angle between (OP) and the x-axis is θ radians, where 0 ≤ θ ≤ .
 
(a)	Write down an expression in terms of θ for
(i)	x;
(ii)	y.
(2)
 
	Let the area of the rectangle be A.
(b)	Show that A = 18 sin 2θ.
(3)

(c)	(i)	Find .
(ii)	Hence, find the exact value of θ which maximizes the area of the rectangle.
(iii)	Use the second derivative to justify that this value of θ does give a maximum.
(8)
(Total 13 marks)

20.	Let f(x) = , –8 ≤ x ≤ 8, a  . The graph of f is shown below.
		
	The region between x = 3 and x = 7 is shaded.
 
(a)	Show that f(–x) = –f(x).
(2)

(b)	Given that f′′(x) = , find the coordinates of all points of inflexion.
(7)

(c)	It is given that .
(i)	Find the area of the shaded region, giving your answer in the form p ln q.
(ii)	Find the value of .
(7)
(Total 16 marks)

21.	A function f has its first derivative given by f′(x) = (x – 3)3.
(a)	Find the second derivative.
(2)
 
(b)	Find f′(3) and f′′(3).
(1)
 
(c)	The point P on the graph of f has x-coordinate 3. Explain why P is not a point of inflexion.
(2)
(Total 5 marks)

22.	Let f(x) = e–3x and g(x) = .
(a)	Write down
(i)	f′(x);
(ii)	g′(x).
(2)
 
(b)	Let h(x) = e–3x sin. Find the exact value of h′.
(4)
(Total 6 marks)

23.	Consider f(x) = x2 + , x ≠ 0, where p is a constant.
(a)	Find f′(x).
(2)
 
(b)	There is a minimum value of f(x) when x = –2. Find the value of p.
(4)
(Total 6 marks)

24.	Let f(x) = 3 + , for x ≠ ±2. The graph of f is given below.
		
diagram not to scale
	The y-intercept is at the point A.
 
(a)	(i)	Find the coordinates of A.
(ii)	Show that f′(x) = 0 at A.
(7)
 
(b)	The second derivative f′′(x) = . Use this to
(i)	justify that the graph of f has a local maximum at A;
(ii)	explain why the graph of f does not have a point of inflexion.
(6)
 
(c)	Describe the behaviour of the graph of f for large │x│.
(1)
 
(d)	Write down the range of f.
(2)
(Total 16 marks)

25.	Let f(x) = . Line L is the normal to the graph of f at the point (4, 2).
(a)	Show that the equation of L is y = –4x + 18.
(4)
 
(b)	Point A is the x-intercept of L. Find the x-coordinate of A.
(2)
 
	In the diagram below, the shaded region R is bounded by the x-axis, the graph of f and the line L.

(c)	Find an expression for the area of R.
(3)
 
(d)	The region R is rotated 360° about the x-axis. Find the volume of the solid formed, giving your answer in terms of π.
(8)
(Total 17 marks)

26.	Let f(x) = cos 2x and g(x) = ln(3x – 5).
(a)	Find f′(x).
(2)
 
(b)	Find g′(x).
(2)
 
(c)	Let h(x) = f(x) × g(x). Find h′(x).
(2)
(Total 6 marks)

27.	(a)	Find 
(2)
(b)	Given that  = ln , find the value of P.
(4)
(Total 6 marks)

28.	A particle moves along a straight line so that its velocity, v m s−1 at time t seconds is given by
v = 6e3t + 4. When t = 0, the displacement, s, of the particle is 7 metres. Find an expression for s in terms of t.
(Total 7 marks)

29.	Consider f (x) = x3 + 2x2 – 5x. Part of the graph of f is shown below. There is a maximum point at M, and a point of inflexion at N.

(a)	Find f ′(x).
(3)
(b)	Find the x-coordinate of M.
(4)
(c)	Find the x-coordinate of N.
(3)
(d)	The line L is the tangent to the curve of f at (3, 12). Find the equation of L in the form y = ax + b.
(4)
(Total 14 marks)

30.	Let 
(a)	Show that 
(2)
(b)	Find the value of 
(5)
(Total 7 marks)

31.	Let f : x  sin3 x.
(a)	(i)	Write down the range of the function f.
(ii)	Consider f (x) =1, 0 £ x £ 2p. Write down the number of solutions to this equation. Justify your answer.
(5)
(b)	Find f ′ (x), giving your answer in the form a sinp x cosq x where a, p, q Î .
(2)
(c)	Let g (x) =  for 0 £ x £ . Find the volume generated when the curve of g is revolved through 2p about the x-axis.
(7)
(Total 14 marks)

32.	The diagram below shows part of the graph of the gradient function, y = f ′ (x).

(a)	On the grid below, sketch a graph of y = f ² (x), clearly indicating the x-intercept.

(2)
 
(b)	Complete the table, for the graph of y = f (x).
	x-coordinate	
(i)	Maximum point on f		
(ii)	Inflexion point on f		
(2)
 
(c)	Justify your answer to part (b) (ii).
(2)
(Total 6 marks)

33.	The diagram shows part of the graph of y = f′(x). The x-intercepts are at points A and C.
There is a minimum at B, and a maximum at D.

(a)	(i)	Write down the value of f′(x) at C.
 
(ii)	Hence, show that C corresponds to a minimum on the graph of f, i.e. it has the same x-coordinate.
(3)
 
(b)	Which of the points A, B, D corresponds to a maximum on the graph of f?
(1)
 
(c)	Show that B corresponds to a point of inflexion on the graph of f.
(3)
(Total 7 marks)

34.	The acceleration, a m s–2, of a particle at time t seconds is given by a = 2t + cost.
(a)	Find the acceleration of the particle at t = 0.
(2)

(b)	Find the velocity, v, at time t, given that the initial velocity of the particle is 2 m s–1.
(5)
 
(c)	Find , giving your answer in the form p – q cos 3.
(7)
 
(d)	What information does the answer to part (c) give about the motion of the particle?
(2)
(Total 16 marks)

35.	Find the equation of the tangent to the curve y = e2x at the point where x = 1.
Give your answer in terms of e2.
(Total 6 marks)

36.	(a)	Find dx.
(4)
 
(b)	Find .
(3)
(Total 7 marks)

37.	The velocity v m s–1 of a moving body at time t seconds is given by v = 50 – 10t.
(a)	Find its acceleration in m s–2.
(2)

(b)	The initial displacement s is 40 metres. Find an expression for s in terms of t.
(4)
(Total 6 marks)

38.	Let g(x) = x3 – 3x2 – 9x + 5.
(a)	Find the two values of x at which the tangent to the graph of g is horizontal.
(8)
 
(b)	For each of these values, determine whether it is a maximum or a minimum.
(6)
(Total 14 marks)

39.	The diagram below shows part of the graph of y = sin 2x. The shaded region is between x = 0 and x = m.

(a)	Write down the period of this function.
(2)
 
(b)	Hence or otherwise write down the value of m.
(2)

(c)	Find the area of the shaded region.
(6)
(Total 10 marks)

40.	The function f is given by f(x) = 2sin(5x – 3).
(a)	Find f′′(x).
(4)
 
(b)	Write down .
(2)
(Total 6 marks)

41.	The velocity v of a particle at time t is given by v = e−2t + 12t. The displacement of the particle at time t is s. Given that s = 2 when t = 0, express s in terms of t.
(Total 6 marks)

42.	The graph of the function y = f (x), 0 £ x £ 4, is shown below.

(a)	Write down the value of
(i)	f ′ (1);
(ii)	f ′ (3).

(b)	On the diagram below, draw the graph of y = 3 f (−x).

(c)	On the diagram below, draw the graph of y = f (2x).

(Total 6 marks)

43.	It is given that  f (x)dx = 5.
(a)	Write down  2 f (x)dx.
(b)	Find the value of  (3x2 + f (x))dx.
(Total 6 marks)

44.	Let f ′ (x) = 12x2 − 2.
Given that f (−1) =1, find f (x).
(Total 6 marks)

45.	Let f (x) = 3 cos 2x + sin2 x.
(a)	Show that f ′ (x) = −5 sin 2x.
(b)	In the interval  £ x £ , one normal to the graph of f has equation x = k.
Find the value of k.
(Total 6 marks)

46.	Differentiate each of the following with respect to x.
(a)	y = sin 3x
(1)
(b)	y = x tan x
(2)
(c)	y = 
(3)
(Total 6 marks)

47.	On the axes below, sketch a curve y = f (x) which satisfies the following conditions.
x	f (x)	f ′ (x)	f ′′ (x)	
−2 £ x < 0		negative	positive	
0	–1	0	positive	
0 < x <1		positive	positive	
1	2	positive	0	
1 < x £ 2		positive	negative	

(Total 6 marks)

48.	(a)	Let f (x) = e5x. Write down f ′ (x).
(b)	Let g (x) = sin 2x. Write down g′ (x).
(c)	Let h (x) = e5x sin 2x. Find h′ (x).
(Total 6 marks)

49.	The following diagram shows part of the curve of a function ƒ. The points A, B, C, D and E lie on the curve, where B is a minimum point and D is a maximum point.

(a)	Complete the following table, noting whether ƒ′(x) is positive, negative or zero at the given points.
A	B	E	
f ′ (x)				
 
(b)	Complete the following table, noting whether ƒ′′(x) is positive, negative or zero at the given points.
A	C	E	
ƒ′′ (x)				
(Total 6 marks)

50.	The velocity, v m s−1, of a moving object at time t seconds is given by v = 4t3 − 2t.
When t = 2, the displacement, s, of the object is 8 metres.
Find an expression for s in terms of t.
(Total 6 marks)

51.	The graph of a function g is given in the diagram below.

	The gradient of the curve has its maximum value at point B and its minimum value at point D. The tangent is horizontal at points C and E.
(a)	Complete the table below, by stating whether the first derivative g′ is positive or negative, and whether the second derivative g′′ is positive or negative.
Interval	g′	g′′	
a < x < b			
e < x < ƒ			
 
(b)	Complete the table below by noting the points on the graph described by the following conditions.
Conditions	Point	
g′ (x) = 0, g′′ (x) < 0		
g′ (x) < 0, g′′ (x) = 0		
(Total 6 marks)

52.	Consider the function ƒ : x  3x2 – 5x + k.
(a)	Write down ƒ′ (x).
The equation of the tangent to the graph of ƒ at x = p is y = 7x – 9. Find the value of
(b)	p;
(c)	k.
(Total 6 marks)

53.	The following diagram shows the graph of a function f.

Consider the following diagrams.

Complete the table below, noting which one of the diagrams above represents the graph of
(a)	f ′(x);
(b)	f ′′(x).
	Graph	Diagram	
(a)	f ′ (x)		
(b)	f " (x)		
(Total 6 marks)

54.	The velocity v in m s−1 of a moving body at time t seconds is given by v = e2t−1. When t = 0 5. the displacement of the body is 10 m. Find the displacement when t =1.
(Total 6 marks)

55.	The function f is given by f (x) = 2sin (5x – 3).
(a)	Find f " (x).
(b)	Write down .
(Total 6 marks)

56.	The velocity v m s–1 of a moving body at time t seconds is given by v = 50 – 10t.
(a)	Find its acceleration in m s–2.
(b)	The initial displacement s is 40 metres. Find an expression for s in terms of t.
(Total 6 marks)

57.	Let f (x) = (3x + 4)5. Find
(a)	f ¢(x);
(b)	òf (x)dx.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 6 marks)

58.	The curve y = f (x) passes through the point (2, 6).
	Given that  = 3x2 – 5, find y in terms of x.
Working:		
	Answer:
....……………………………………..........	
(Total 6 marks)

59.	The table below shows some values of two functions, f and g, and of their derivatives f ¢ and g ¢.
x	1	2	3	4	
f (x)	5	4	–1	3	
g (x)	1	–2	2	–5	
f ¢(x)	5	6	0	7	
g¢ (x)	–6	–4	–3	4	

	Calculate the following.
(a)	(f (x) + g (x)), when x = 4;
(b)	.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 6 marks)

60.	Let f (x) = .
(a)	Write down the equation of the vertical asymptote of y = f (x).
(1)
(b)	Find f ′ (x). Give your answer in the form  where a and b Î .
(4)
(Total 5 marks)

61.	Given dx = ln 7, find the value of k.

(Total 6 marks)

62.	Let f (x) = (2x + 7)3 and g (x) cos2 (4x). Find
(a)	f ′ (x);
(b)	g′ (x).

(Total 6 marks)

63.	The following diagram shows a rectangular area ABCD enclosed on three sides by 60 m of fencing, and on the fourth by a wall AB.

Find the width of the rectangle that gives its maximum area.

(Total 6 marks)

64.	Let f (x) = 6. Find f ¢ (x).
Working:		
	Answer:
…………………………………………........	
(Total 6 marks)

65.	The derivative of the function f is given by f ¢ (x) = e–2x + , x < 1.
	The graph of y = f (x) passes through the point (0, 4). Find an expression for f (x).
Working:		
	Answer:
…………………………………………........	
(Total 6 marks)

66.	Let f  be a function such that.
(a)	Deduce the value of
(i)	
(ii)	
(b)	If , write down the value of c and of d.
Working:		
	Answers:
(a)	(i) ........................................................
	(ii) .......................................................
(b)	c = ......................., d = .......................	
(Total 6 marks)

67.	The function f (x) is defined as f (x) = –(x – h)2 + k. The diagram below shows part of the graph of f (x). The maximum point on the curve is P (3, 2).

(a)	Write down the value of
(i)	h;
(ii)	k.
(2)
 
(b)	Show that f (x) can be written as f (x) = –x2 + 6x – 7.
(1)
 
(c)	Find f ¢ (x).
(2)
 
	The point Q lies on the curve and has coordinates (4, 1). A straight line L, through Q, is perpendicular to the tangent at Q.
(d)	(i)	Calculate the gradient of L.
(ii)	Find the equation of L.
(iii)	The line L intersects the curve again at R. Find the x-coordinate of R.
(8)
(Total 13 marks)

68.	Let f (x) = .
(a)	Write down the equation of the horizontal asymptote of the graph of f.
(1)
 
(b)	Find f ¢ (x).
(3)
 
(c)	The second derivative is given by f ²(x) = .
	Let A be the point on the curve of f where the gradient of the tangent is a maximum. Find the x-coordinate of A.
(4)
 
(d)	Let R be the region under the graph of f, between x = – and x = ,
	as shaded in the diagram below

	Write down the definite integral which represents the area of R.
(2)
(Total 10 marks)

69.	Let y = g (x) be a function of x for 1 £ x £ 7. The graph of g has an inflexion point at P, and a minimum point at M.
	Partial sketches of the curves of g¢ and g² are shown below.

	Use the above information to answer the following.
(a)	Write down the x-coordinate of P, and justify your answer.
(2)
 
(b)	Write down the x-coordinate of M, and justify your answer.
(2)
 
(c)	Given that g (4) = 0, sketch the graph of g. On the sketch, mark the points P and M.
(4)
(Total 8 marks)

70.	Let f (x) =  + 5 cos 2x. Find f¢ (x).
Working:		
	Answer:
..................................................................	
(Total 6 marks)

71.	Given that  = 10, deduce the value of
(a)	
(b)	
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 6 marks)

72.	It is given that = x3+2x – 1 and that y = 13 when x = 2.
	Find y in terms of x.
Working:		
	Answer:
..................................................................	
(Total 6 marks)

73.	The diagram shows the graph of y = f (x).

	On the grid below sketch the graph of y = f¢ (x).

(Total 6 marks)

74.	The derivative of the function f is given by f ¢(x) =  – 0.5 sin x, for x ¹ –1.
	The graph of f passes through the point (0, 2). Find an expression for f (x).
Working:		
	Answer:
......................................................................	
(Total 6 marks)

75.	Figure 1 shows the graphs of the functions f1,  f2,  f3,  f4.
	Figure 2 includes the graphs of the derivatives of the functions shown in Figure 1, eg the derivative of f1 is shown in diagram (d).
	Figure 1	Figure 2

	Complete the table below by matching each function with its derivative.
Function	Derivative diagram	
f 1	(d)	
f 2		
f 3		
f 4		

Working:	
(Total 6 marks)

76.	Consider functions of the form y = e–kx
(a)	Show that  =  (1 – e–k).
(3)
 
(b)	Let k = 0.5
(i)	Sketch the graph of y = e–0.5x, for –1 £ x £ 3, indicating the coordinates of the
y-intercept.
(ii)	Shade the region enclosed by this graph, the x-axis, y-axis and the line x = 1.
(iii)	Find the area of this region.
(5)
 
(c)	(i)	Find  in terms of k, where y = e–kx.
	The point P(1, 0.8) lies on the graph of the function y = e–kx.
(ii)	Find the value of k in this case.
(iii)	Find the gradient of the tangent to the curve at P.
(5)
(Total 13 marks)

77.	Let  f (x) =  Find
(a)	f¢ (x);
(b)	
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 6 marks)

78.	The diagram shows part of the curve y = sin x. The shaded region is bounded by the curve and the lines y = 0 and x = 

	Given that sin  =  and cos  = – , calculate the exact area of the shaded region.
Working:		
	Answer:
......................................................................	
(Total 6 marks)

79.	In this question, s represents displacement in metres, and t represents time in seconds.
(a)	The velocity v m s–1 of a moving body may be written as v =  = 30 – at, where a is a constant. Given that s = 0 when t = 0, find an expression for s in terms of a and t.
(5)

	Trains approaching a station start to slow down when they pass a signal which is 200 m from the station.
(b)	The velocity of Train 1 t seconds after passing the signal is given by v = 30 – 5t.
(i)	Write down its velocity as it passes the signal.
(ii)	Show that it will stop before reaching the station.
(5)
 
(c)	Train 2 slows down so that it stops at the station. Its velocity is given by
v =  = 30 – at, where a is a constant.
(i)	Find, in terms of a, the time taken to stop.
(ii)	Use your solutions to parts (a) and (c)(i) to find the value of a.
(5)
(Total 15 marks)

80.	The diagram below shows the shaded region R enclosed by the graph of y = 2x, the
x-axis, and the vertical line x = k.

(a)	Find .
(3)
(b)	Using the substitution u = 1 + x2 or otherwise, show that
	dx = (1 + x2) + c.
(3)
(c)	Given that the area of R equals 1, find the value of k.
(3)
(Total 9 marks)

81.	Consider the function h (x) = x.
(i)	Find the equation of the tangent to the graph of h at the point where x = a, (a ¹ 0). Write the equation in the form y = mx + c.
(ii)	Show that this tangent intersects the x-axis at the point (–4a, 0).
(Total 5 marks)

82.	Find
(a)	
(b)	.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

83.	Consider the function f (x) = k sin x + 3x, where k is a constant.
(a)	Find f ¢(x).
(b)	When x = , the gradient of the curve of f (x) is 8. Find the value of k.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

84.	Let f¢ (x) = 1 – x2. Given that f (3) = 0, find f (x).
Working:		
	Answer:
....................................................................	
(Total 4 marks)

85.	Given the function f (x) = x2 – 3bx + (c + 2), determine the values of b and c such that f (1) = 0 and f¢ (3) = 0.
Working:		
	Answer:
....................................................................	
(Total 4 marks)

86.	If f ¢(x) = cos x, and f  = – 2, find f (x).
Working:		
	Answer:
......................................................................	
(Total 4 marks)

87.	Let f (x) = x3.
(a)	Evaluate  for h = 0.1.
(b)	What number does  approach as h approaches zero?
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

88.	The diagram shows part of the graph of the curve with equation
y = e2x cos x.

(a)	Show that  = e2x (2 cos x – sin x).
(2)
 
(b)	Find .
(4)
 
	There is an inflexion point at P (a, b).
(c)	Use the results from parts (a) and (b) to prove that:
(i)	tan a = 
(3)
(ii)	the gradient of the curve at P is e2a.
(5)
(Total 14 marks)

89.	A curve with equation y =f (x) passes through the point (1, 1). Its gradient function is 
f¢ (x) = –2x + 3.
	Find the equation of the curve.
Working:		
	Answer:
......................................................................	
(Total 4 marks)

90.	Given that f (x) = (2x + 5)3 find 
(a)	f¢ (x);
(b)	
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

91.	Find the equation of the normal to the curve with equation
y = x3 + 1
	at the point (1, 2).
Working:		
	Answer:
.........................................................................	
(Total 4 marks)

92.	The graph represents the function
f : x  p cos x, p Î .

	Find
(a)	the value of p;
(b)	the area of the shaded region.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

93.	Differentiate with respect to x
(a)	
(b)	esin x
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

94.	The function f is given by
f (x) = , x Î , x ¹ 3.
(a)	(i)	Show that y = 2 is an asymptote of the graph of y = f (x).
(2)
(ii)	Find the vertical asymptote of the graph.
(1)
(iii)	Write down the coordinates of the point P at which the asymptotes intersect.
(1)
 
(b)	Find the points of intersection of the graph and the axes.
(4)
(c)	Hence sketch the graph of y = f (x), showing the asymptotes by dotted lines.
(4)
 
(d)	Show that f¢ (x) =  and hence find the equation of the tangent at 
the point S where x = 4.
(6)
 
(e)	The tangent at the point T on the graph is parallel to the tangent at S.
	Find the coordinates of T.
(5)
(f)	Show that P is the midpoint of [ST].
(l)
(Total 24 marks)

95.	The function f  is such that f ² (x) = 2x – 2.
	When the graph of f  is drawn, it has a minimum point at (3, –7).
(a)	Show that f¢ (x) = x2 – 2x – 3 and hence find f (x).
(6)
(b)	Find f (0), f (–1) and f¢ (–1).
(3)
(c)	Hence sketch the graph of f, labelling it with the information obtained in part (b).
(4)
	(Note: It is not necessary to find the coordinates of the points where the graph 
cuts the x-axis.)
(Total 13 marks)

96.	The diagram shows part of the graph of y = 

(a)	Find the coordinates of the point P, where the graph meets the y-axis. 
(2)
	The shaded region between the graph and the x-axis, bounded by x = 0 
and x = ln 2, is rotated through 360° about the x-axis.
(b)	Write down an integral which represents the volume of the solid obtained.
(4)
(c)	Show that this volume is p.
(5)
(Total 11 marks)

97.	The parabola shown has equation y2 = 9x.

(a)	Verify that the point P (4, 6) is on the parabola.
(2)
	The line (PQ) is the normal to the parabola at the point P, and cuts the x-axis at Q.
(b)	(i)	Find the equation of (PQ) in the form ax + by + c = 0.
(5)
(ii)	Find the coordinates of Q.
(2)

	S is the point 
(c)	Verify that SP = SQ.
(4)
(d)	The line (PM) is parallel to the x-axis. From part (c), explain why
(QP) bisects the angle 
(3)
(Total 16 marks)

98.	The diagram shows part of the graph of y = 12x2(1 – x).

(a)	Write down an integral which represents the area of the shaded region.

(b)	Find the area of the shaded region.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

99.	Differentiate with respect to x: 
(a)	(x2 + l)2.
(b)	1n(3x – 1).
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

100.	The diagram shows part of the graph of y = . The area of the shaded region is 2 units.

	Find the exact value of a.
Working:		
	Answer:
......................................................................	
(Total 4 marks)