2. DIFFERENTIATION text

10. The diagram on the right shows graphs dy and d 2y for d–d–yx / d–dx–2y2
dx dx 2 6
function y = f (x). It is given that the function y = f (x)
–1 0 1
passes through (–1, 6) and (1, 2). Without finding the –3
equation of the function y = f (x), PL 4 –6

(a) determine the coordinates of the maximum and x
minimum points of the graph function y = f (x),
(b) sketch the graph for the function y = f (x).

11. The diagram on the right shows a part of the curveKEMENTERIAN PENDIDIKAN MALAYSIA y y = 3x 3 – 4x + 2
y = 3x 3 – 4x + 2. Find PL 3
(a) the equation of the tangent at point A(2, 1), 2
(b) the coordinates of another point on the curve such
that the tangent at that point is parallel to the A(2, 1)
tangent at A.
x
0
A

12. In the diagram on the right, ∆ ADB is a right-angled
triangle with a hypotenuse of 6! 3 cm. The triangle is
rotated about AD to form a cone ABC. Find PL 4 6�3 cm

(a) the height, (b) the volume of the cone, B DC

such that the volume generated is maximum.

13. In the diagram on the right, Mukhriz rows his canoe from A
point A to C where A is 30 m from the nearest point B,

which is on the straight shore BD, and C is x m from B.
He then cycles from C to D where BD is 400 m. Find 30 m

the distance from B to C if he rows with a velocity of C D
40 mmin–1 and cycles at 50 mmin–1. PL 5 B xm

400 m

14. The sides of a cuboid expand at a rate of 2 cms–1. Find the rate of change of the total surface
area when its volume is 8 cm3. PL 3

15. The diagram on the right shows a part of the curve y
y = 6x – x 2 which passes through the origin and P(x, y)

point P(x, y). PL 3

(a) If Q is point (x, 0), show that the area, A of triangle
1
POQ is given by A = 2  (6x 2 – x 3). y = 6x – x2

(b) Given that x increases at a rate of x

2 units per second, find 0 Q(x, 0) 6

(i) the rate of increase for A when x = 2,

(ii) the rate of decrease for A when x = 5.

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16. The diagram on the right shows an inverted cone with a Differentiation
12 cm
base radius of 12 cm and a height of 20 cm. PL 6

(a) If the height of water in the cone is h cm, show that the
3
volume of water, V cm3, in the cone is V = 25  π h3. r cm 20 cm
h cm
(b) Water leaks out through a small hole at the tip of PTER
KEMENTERIAN PENDIDIKAN MALAYSIA
CHAthe cone;2

(i) find the small change in the volume of water when

the height, h decreases from 5 cm to 4.99 cm,

(ii) show that a decrease of p% in the height of the water

will cause a decrease of 3p% in its volume.

MATHEMATICAL EXPLORATION

A multinational beverage company holds a competition to design a suitable container
for its new product, a coconut-flavoured drink.

DESIGNING A DRINK CONTAINER
COMPETITION

Criteria for the design of the drink container are Great prize
as follows: awaits you!
• The capacity of the container is 550 cm3.
• The shapes of the containers to be considered

are cylinders, cone, pyramid, prism, cuboid or
cubes. Spherical shape is not allowed.
• Material required to make the tin must
be minimum.
• The container must be unique and attractive.

Join this competition with your classmates. Follow the criteria given and follow the
steps given below:
1. Suggest three possible shapes of the containers.
2. For each container with a capacity of 550 cm3, show the dimensions of the

containers with their minimum surface areas. State each minimum surface area.
3. Choose the best design from the three designs to be submitted for the competition

by listing down the advantages of the winning design.

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