1. MODUL BIMBINGAN EMaS 2007 ADDITIONAL MATHEMATICS FORM 4
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ADDITIONAL MATHEMATICS
FORM 4
MODULE 5
DIFFERENTIATIONS

2. ADDITIONAL MATHEMATICS FORM 4
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9 DIFFERENTIATIONS
 PAPER 1
1 Given y = 4(1 – 2x)3
, find
dy
dx
.
Answer : …………………………………
2 Differentiate 3x2
(2x – 5)4
with respect to x.
Answer : …………………………………
3 Given that 2
1
(3 5)
( )
x
h x

 , evaluate h’’(1).
Answer : …………………………………

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4 Differentiate the following expressions with respect to x.
(a) (1 + 5x2
)3
(b)
2
43
4


x
x
Answer : (a) …………………………………
(b) …………………………………
5 Given a curve with an equation y = (2x + 1)5
, find the gradient of the curve at the point x = 1.
Answer : …………………………………
6 Given y = (3x – 1)5
, solve the equation
2
2
12 0
d y dy
dx dx

Answer : …………………………………

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7 Find the equation of the normal to the curve 53 2
 xy at the point (1, 2).
Answer : …………………………………
8 Given that the curve qxpxy  2
has the gradient of 5 at the point (1, 2), find the values of
p and q.
Answer : p = ………………………………
q = ………………………………
9 Given (2, t) is the turning point of the curve 142
 xkxy . Find the values of k and t.
Answer : k = ………………………………
t = ………………………………
10 Given 22
yxz  and xy 21 , find the minimum value of z.
Answer : …………………………………

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11 Given 12
tx and 54  ty . Find
(a)
dx
dy
in terms of t , where t is a variable,
(b)
dx
dy
in terms of y.
Answer : (a) ……………………………
(b) ……………………………
12 Given that y = 14x(5 – x), calculate
(a) the value of x when y is a maximum,
(b) the maximum value of y.
Answer : (a) …………………………………
(b) …………………………………
13 Given that y = x2
+ 5x, use differentiation to find the small change in y when x increases from
3 to 301.
Answer : …………………………………

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14 Two variables, x and y, are related by the equation y = 3x +
2
x
. Given that y increases at a constant
rate of 4 units per second, find the rate of change of x when x = 2.
Answer : …………………………………
15 The volume of water, V cm3
, in a container is given by 31
8
3
V h h  , where h cm is the height of
the water in the container. Water is poured into the container at the rate of 10 cm3
s1
.
Find the rate of change of the height of water, in cm s1
, at the instant when its height is 2 cm.
Answer : ……………………………

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 PAPER 2
16 (a) Given that graph of function 2
3
)(
x
q
pxxf  , has gradient function 2
3
192
( ) 6f x x
x
  
where p and q are constants, find
(i) the values of p and q ,
(ii) x-coordinate of the turning point of the graph of the function.
(b) Given 3 29
( 1)
2
p t t   .
Find
dt
dp
, and hence find the values of t where 9.
dp
dt

17 The gradient of the curve 4
k
y x
x
  at the point (2, 7) is
1
2
4 , find
(a) value of k,
(b) the equation of the normal at the point (2, 7),
(c) small change in y when x decreases from 2 to 197.
18 The diagram above shows a piece of square zinc with 8 m sides. Four squares with 2x m sides are
cut out from its four vertices.The zinc sheet is then folded to form an open square box.
(a) Show that the volume, V m3
, is V = 128x – 128x2
+ 32x3
.
(b) Calculate the value of x when V is maximum.
(c) Hence, find the maximum value of V.
8 m
8 m
2x m
2x m2x m
2x m
2x m
2x m
2x m
2x m

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19 (a) Given that 12p q  , where 0p and 0.q  Find the maximum value of .2
qp
(b) The above diagram shows a conical container of diameter 8 cm and height 6 cm. Water
is poured into the container at a constant rate of 3 cm3
s1
. Calculate the rate of change of the
height of the water level at the instant when the height of the water level is 2 cm.
[Use = 3142 ; Volume of a cone = hr2
3
1
 ]
20 (a) The above diagram shows a closed rectangular box of width x cm and height h cm. The length
is two times its width and the volume of the box is 72 cm3
.
(i) Show that the total surface area of the box, A cm2
is
x
xA
216
4 2
 ,
(ii) Hence, find the minimum value of A.
(b) The straight line 4y + x = k is the normal to the curve y = (2x – 3)2
– 5 at point E. Find
(i) the coordinates of point E and the value of k,
(ii) the equation of tangent at point E.
6 cm
8 cm
h cm
x cm
2x cm