1.	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)

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

3.	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)

4.	Consider the function f (x) = 4x3 + 2x. Find the equation of the normal to the curve of f at the point where x =1.
(Total 6 marks)

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

6.	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)

7.	The following diagram shows part of the graph of a quadratic function, with equation in the form y = (x − p)(x − q), where p, q Î .

(a)	Write down
(i)	the value of p and of q;
(ii)	the equation of the axis of symmetry of the curve.
(3)
 (b)	Find the equation of the function in the form y = (x − h)2 + k, where h, k Î .
(3)
(c)	Find .
(2)
(d)	Let T be the tangent to the curve at the point (0, 5). Find the equation of T.
(2)
(Total 10 marks)
8.	The function f is defined by f : x  –0.5x2 + 2x + 2.5.
	Let N be the normal to the curve at the point where the graph intercepts the y-axis.
(a)	Show that the equation of N may be written as y = –0.5x + 2.5.
(4)
 
(b)	Find the coordinates of the other point of intersection of the normal and the curve.
(5)
 
(c)	Let R be the region enclosed between the curve and N. Find the area of R.
(4)
(Total 13 marks)

9.	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)

10.	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)

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.	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)

13.	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)

14.	The function f is defined by 
(a)	Write down
(i)	f ¢(x);
(ii)	f ¢(0).
(2)
(b)	Let N be the normal to the curve at the point where the graph intercepts the y-axis. Show that the equation of N may be written as y = –0.5x + 2.5.
(3)
	Let 
(c)	(i)	Find the solutions of f (x) = g (x).
(ii)	Hence find the coordinates of the other point of intersection of the normal and the curve.
(6)
(d)	Let R be the region enclosed between the curve and N.
(i)	Write down an expression for the area of R.
(ii)	Hence write down the area of R.
(5)
(Total 16 marks)

15.	Let f(x) = x3. The following diagram shows part of the graph of f.		
	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)