DIFFERENTIATION – PAST YEAR QUESTIONS PROGRAM BIJAK BELAJAR MGC KULAIJAYA KULAIJAYA (9 APRIL 2011) PAPER 1 x), calculate 1 Given that y = 14 x (5 – x x when y is a maximum, 2003 a) the value of x y. [3 marks] b) the maximum value of y
2 2003 [3 marks]
2
Given that y = x + 5 x, use differentiation to find the small change in y when x increases from 3 to 3.01
1
3 2004 [3 marks]
4 2004 [3 marks]
2 4 Differentiate 3 x (2 x – 5) with respect to x.
Two variables, x and y, are related by the equation y
3 x
2
x
. Given that y
increases at a constant constant rate of 4 units per second, second, find the rate of change change of x when x = 2.
2
5 2005 [4 marks]
6 2005 [3 marks]
x) = Given that h( x
1 (3 x 5)
2
, evaluate
(1). h’’ (1).
3 The volume of water, V cm , in a container is given by V =
1 3
h
3
8h, where h cm
is the height of the water in the container. Water is poured into the container at the 3 – 1 – 1 rate of 10 cm s . Find the rate rate of change of the height of water, in cm s , at the instant when when its height height is 2 cm.
3
7 2006 [3 marks]
8 2006 [4 marks]
2 x – 5) . It is given that the gradient of the normal The point P lies on the curve y = ( x
at P is
1
4
. Find the coordinates of P.
It is given that y = Find
dy dx
2 3
u 7 , where u = 3 x – 5.
in terms of x.
4
9 2006 [4 marks]
2 Given that y = 3 x + x – 4,
a) find the value of
dy dx
when x = 1,
b) express the approximate change in y, in terms of p, when x changes from 1 to 1 + p, where p is a small value.
10 2007 [2 marks]
x) is such that The curve y = f ( x
dy dx
= 3 kx + 5, where k is a constant.
The gradient of the curve at x = 2 is 9. Find the value of k .
5
11 2007 [3 marks]
12 2008 [3 marks]
2
The curve y = x – 32 x + 64 has a minimum point at x = p, where p is a constant. Find the value of p.
Two variables, x and y , are related by the equation y =
16 x
2
.
Express, in terms oh h, the approximate change in y, when x changes from 4 to 4 + h, where h is a small value.
6
2 13 The normal to the curve y = x 5 x at point P is parallel to the straight line 2008 y = x + 12. Find the equation of the normal to the curve at point P. [4 marks]
14 2009 [3 marks]
3
A block of ice in the form of a cube with sides x cm, melts at a rate of 9.72cm per minute. x at the instant when x = 12 cm. Find the rate of change of x
7
15 2010 [3 marks]
x – 6), find Given y = 2 x ( x dy (a) dx x when y is minimum, (b) the value of x y (c) the minimum value of y
16 2010 [3 marks]
The volume of a sphere is increasing at constant rate of 12.8 cm s . Find the -1 radius of the sphere at the instant when the radius is increasing at a rate of 0.2 cm s . 4 3 [Volume of sphere, V r ] 3
3
-1
8
1 2010 [10marks]
PAPER 2 The curve y x 3 6 x 2 9 x 1 passes through the point A(2, 3) and has two turning points, P(3, 1) and Q. Find (a) the gradient of the curve at A [3 marks] (b) the equation of the normal to the curve at A [3 marks] (c) the coordinates of Q and determine whether Q is the maximum or the minimum point. [4 marks]
9
Answer Year 2003
Paper 1 5
1 a) x
b)
2
175 2
2) y = 0.11 2004
3) 6 x(6 x 5)(2 x 5)3 4)
2005
2006
2007
5)
8
unit per second
5 27
8
6) 0.8333 cm s 7) P (7, 4) 2 8) 14 ( 3 x – 5) 9 a) 7 10) k
-1
b) 7 p
2
3
11) p = 16 2008
2009 2010 Year 2010
12) y
1 2
h
13) y = - x - 3 14) 0.0225 cm per minute 15 a) 4 x – 12 16) 4
b) 3
c) – 18
Paper 2 1a)
10
