1.	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)

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.	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)

4.	The following diagram shows the graph of f(x) = .
		
	The points A, B, C, D and E lie on the graph of f. Two of these are points of inflexion.
 
(a)	Identify the two points of inflexion.
(2)
 
(b)	(i)	Find f′(x).
(ii)	Show that f″(x) = (4x2 – 2).
(5)
 
(c)	Find the x-coordinate of each point of inflexion.
(4)
 
(d)	Use the second derivative to show that one of these points is a point of inflexion.
(4)
(Total 15 marks)

5.	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)

6.	Let f(x) = x(x – 5)2, for 0 ≤ x ≤ 6. The following diagram shows the graph of f.
		
	Let R be the region enclosed by the x-axis and the curve of f.
 
(a)	Find the area of R.
(3)
 
(b)	Find the volume of the solid formed when R is rotated through 360° about the x-axis.
(4)

(c)	The diagram below shows a part of the graph of a quadratic function g(x) = x(a – x). The graph of g crosses the x-axis when x = a.
		
	The area of the shaded region is equal to the area of R. Find the value of a.
(7)
(Total 14 marks)

7.	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)

8.	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)

9.	Let f(x) = ex sin 2x + 10, for 0 ≤ x ≤ 4. Part of the graph of f is given below.
		
	There is an x-intercept at the point A, a local maximum point at M, where x = p and a local minimum point at N, where x = q.
 
(a)	Write down the x-coordinate of A.
(1)
 
(b)	Find the value of
(i)	p;
(ii)	q.
(2)
 
(c)	Find . Explain why this is not the area of the shaded region.
(3)
(Total 6 marks)

10.	Consider f(x) = x ln(4 – x2), for –2 < x < 2. The graph of f is given below.

(a)	Let P and Q be points on the curve of f where the tangent to the graph of f is parallel to the x-axis.
(i)	Find the x-coordinate of P and of Q.
(ii)	Consider f(x) = k. Write down all values of k for which there are exactly two solutions.
(5)
 
	Let g(x) = x3 ln(4 – x2), for –2 < x < 2.
(b)	Show that g′(x) = .
(4)
 
(c)	Sketch the graph of g′.
(2)
 
(d)	Consider g′(x) = w. Write down all values of w for which there are exactly two solutions.
(3)
(Total 14 marks)

11.	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)