MODUL BIMBINGAN BIMBINGAN EMa EMaS S 2007
ADDITIONAL ADDITIONAL MATHEMATI MATHEMATICS CS FORM 4
ADDITIO ADDITIONAL NAL MATHEM MATHEMATI ATICS CS FORM 4
MODULE 5 DIFFERENTIATIONS
1
ADDITIONAL MATHEMATICS FORM 4
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DIFFERENTIATIONS
PAPER 1
1
Given y = 4(1 – 2 x) 3 , find
y dx
.
Answer : …………………………………
2
2
4
Differentiate 3 x (2 x – 5) with respect to x.
Answer : ………………………………… 3
Given that h( x)
1 3 x 5)
, evaluate h’’(1).
Answer : …………………………………
2
ADDITIONAL MATHEMATICS FORM 4
4
Differentiate the following expressions with respect to x. (a) (1 + 5 x2 )3 (b)
3 x 4 4
2
Answer : (a) …………………………………
(b) …………………………………
5
Given a curve with an equation y = (2 x + 1)5 , find the gradient of the curve at the point x = 1.
Answer : …………………………………
2
6
5
Given y = (3 x – 1) , solve the equation
d y dx
2
12
dy dx
0
Answer : …………………………………
3
ADDITIONAL MATHEMATICS FORM 4
7
Find the equation of the normal to the curve y
3x
2
5
at the point (1, 2).
Answer : ………………………………… 8
Given that the curve y p and q.
px 2
qx has the gradient of 5 at the point (1, 2), find the values of
Answer : p = ……………………………… q = ………………………………
9
Given ( 2, t ) is the turning point of the curve y
kx 2
4 x 1 . Find the values of k and t.
Answer : k = ……………………………… t = ………………………………
10
Given z x 2
y
2
and
1
2 , find the minimum value of z.
Answer : ………………………………… 4
ADDITIONAL MATHEMATICS FORM 4
11
Given x (a)
(b)
2
t 1
dy dx dy dx
and
t . Find
in terms of t , where t is a variable,
in terms of y.
Answer : (a) ……………………………
(b) …………………………… 12
Given that y = 14 x(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 = x 2 + 5x, use differentiation to find the small change in y when x increases from 3 to 301.
Answer : …………………………………
5
ADDITIONAL MATHEMATICS FORM 4
14
Two variables, x and y, are related by the equation y = 3 x +
2
. 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 V
1
3
8h , where h cm is the height of 3 the water in the container. Water is poured into the container at the rate of 10 cm 3 s1. Find the rate of change of the height of water, in cm s 1 , at the instant when its height is 2 cm.
h
Answer : ……………………………
6
ADDITIONAL MATHEMATICS FORM 4
PAPER 2 16
(a) Given that graph of function f ( x )
px3
q 2
, has gradient function f ( x) 6 x2
192 x
3
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 p (t 1) 3 Find
17
dp dt
9 2
t 2 .
, and hence find the values of t where
The gradient of the curve y
4x
x
dp dt
at the point (2, 7) is
.
1 2
, 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.
2 x m 2 x m 2 x m
2 x m 8m
2 x m
2 x m 2 x m
2 x m
8m 18
The diagram above shows a piece of square zinc with 8 m sides. Four squares with 2 x 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 = 128 x – 128 x2 + 32 x3. (b) Calculate the value of x when V is maximum. (c) Hence, find the maximum value of V.
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ADDITIONAL MATHEMATICS FORM 4
19
(a) Given that p
q 12 , where p
0 and q 0. Find the maximum value of p 2 q.
8 cm
6 cm
(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 cm 3 s1. 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. 1 [Use = 3 142 ; Volume of a cone = r 2 h ] 3
h cm
x cm 2 x cm 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 cm 3 . 216 (i) Show that the total surface area of the box, A cm2 is A 4 x 2 , x (ii) Hence, find the minimum value of A. (b) The straight line 4 y + x = k is the normal to the curve y = (2 x – 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 .
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