MODUL BIMBINGAN EMaS 2007 ADDITIONAL MATHEMATICS FORM 4

2007mozac 1

ADDITIONAL MATHEMATICSFORM 4

MODULE 5DIFFERENTIATIONS

ADDITIONAL MATHEMATICS FORM 4

2007mozac 2

9 DIFFERENTIATIONS

PAPER 1

1 Given y = 4(1 – 2x)3, finddy

dx.

Answer : …………………………………

2 Differentiate 3x2(2x – 5)4 with respect to x.

Answer : …………………………………

3 Given that 2

1

(3 5)( )

xh x

, evaluate h’’(1).

Answer : …………………………………

ADDITIONAL MATHEMATICS FORM 4

2007mozac 3

4 Differentiate the following expressions with respect to x.

(a) (1 + 5x2)3

(b)243

4

xx

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 equation2

2 12 0d y dy

dx dx

Answer : …………………………………

ADDITIONAL MATHEMATICS FORM 4

2007mozac 4

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 ofp 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 : …………………………………

ADDITIONAL MATHEMATICS FORM 4

2007mozac 5

11 Given 12 tx and 54 ty . Find

(a)dxdy

in terms of t , where t is a variable,

(b)dxdy

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 301.

Answer : …………………………………

ADDITIONAL MATHEMATICS FORM 4

2007mozac 6

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 318

3V 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 cm3s1.Find the rate of change of the height of water, in cm s1, at the instant when its height is 2 cm.

Answer : ……………………………

ADDITIONAL MATHEMATICS FORM 4

2007mozac 7

PAPER 2

16 (a) Given that graph of function2

3)(xq

pxxf , has gradient function 23

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)

2p t t .

Finddtdp

, and hence find the values of t where 9.dpdt

17 The gradient of the curve 4k

y xx

at the point (2, 7) is1

24 , 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 197.

18 The diagram above shows a piece of square zinc with 8 m sides. Four squares with 2x m sides arecut 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

ADDITIONAL MATHEMATICS FORM 4

2007mozac 8

19 (a) Given that 12p q , where 0p and 0.q Find the maximum value of .2qp

(b) The above diagram shows a conical container of diameter 8 cm and height 6 cm. Wateris poured into the container at a constant rate of 3 cm3 s1. Calculate the rate of change of theheight of the water level at the instant when the height of the water level is 2 cm.

[Use= 3142 ; Volume of a cone = hr 2

31 ]

20 (a) The above diagram shows a closed rectangular box of width x cm and height h cm. The lengthis 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 isx

xA216

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 cm2x cm