This document provides 10 problems related to differentiating functions. The problems cover differentiating various functions with respect to variables like x and y, finding maximum and minimum values, rates of change, and determining equations of tangents and normals. Solutions are not provided. The document is from an Additional Mathematics module for Form 4 students and focuses on differentiation techniques.

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