Page 53 - Focus SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 2 Differentiation

14. In an agricutural project, Ali is given the task of PAPER 2
fencing a piece of rectangular land with the size of 2
8x m × (6 – x) m. Determine the length, in m, the 1. (a) Given that y = 4x – 3x + 5, determine the first
fence that Ali needs to make, if the area of the land derivative by using the first principle.
is maximum. (b) Find
(i) d 1 1 2
15. The curve y = hx + 3x has a turning point at (2, k). dx 5x – 2
4
SPM Find (ii) d 1 x 2
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2019 (a) the value of h and k. dx 5x – 2
dy
(b) the value of when x = –1. (c) By using differentiation method, estimate the
dx 8
value of 3 .
16. The diagram below shows a piece of cardboard (2.01)
which is used to form a cylindrical-shape funnel 2. The gradient function of a curve is px – qx, such
2
with opening at both ends. that p and q are constants. Given the gradient of
the normal to the curve at the point x = 1 is –5 and
the point (5, –3) is the turning point of the curve.
(a) Find the value of p and q.
y cm
(b) Determine the nature of the turning point (5, –3)
by using the second derivative.
x cm
3. Give the equation of a curve is
3
2
Given that the perimeter of the cardboard is 50 cm. y = 4 – x(2 – x ).
Find the length x cm and width y cm such that the (a) Find the gradient function of the curve.
volume of the cylinder formed is maximum. (b) Find the coordinates of the turning points.
(c) Hence, determine whether each of the turning
17. The diagram below shows a closed right cylinder points is maximum or minimum.
with a fixed height of 8 cm and a changing radius
of base. 4. Given the equation of a curve is y = 3x(2x – 1) .
4
Find
(a) the gradient of the curve at the point where the
x-coordinate = 1.
8 cm (b) the equations of the tangent and normal to the
curve at the point where the x-coordinate = 1.

5. Given that the equation of a curve is y = – 4 2 .
Given that the radius of the base changes with a SPM dy x
rate of 0.3 cm s when the radius is 5 cm. 2017 (a) Find the value of dx when x = 5.
–1
Estimate (b) Hence, estimate the value of – 4 .
(a) the rate of change of the total surface area of (5.02) 2
the cylinder,
(b) the rate of change of the volume of the cylinder. 6. The diagram below shows a filter funnel in a cone-
shape with height 20 cm and a top circular surface
18. Two variables x and y are related by the equation radius of 8 cm. Ezza pours oil into the filter funnel
3
–1
yx = 25. Express in terms of p, the small change with a constant rate of 8 cm s .
2
in y when x changes from 5 to 5 – p, such that p is
Form 5
a small value.
x
19. Given that the equation of a curve is y = √x .
SPM dy Oil h
2017 (a) Find the value of dx when x = 16.
(b) Hence, estimate the value of √15.96 . Given that the surface radius of the oil in the filter
funnel is x cm and the height of the oil is h cm.
20. Given that A = 3q – 2q and x = 4q – 3. (a) Express the volume of the oil, V cm , in terms of
2
3
SPM dA h.
2018 (a) Express in terms of x.
dx (b) Hence, calculate the rate of change of the
(b) Find the small change in x when A changes height of the oil in the cone when the surface
from 4 to 3.9 when q = –1. radius is 2 cm.

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