PANITIA MATEMATIK TAMBAHAN

TAJUK:DIFFERENTIATION -

OLEH:CG. SHAIFUR AZURA BIN SUHAIMI

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RELATED RATES OF CHANGESBy: CG. Shaifur Azura Suhaimi

SMK Sultan Ibrahim (1), Pasir Mas, Kelantan

dd

THE RATES BOX

columnfor time

column for he ight,radius & x

column for Area,Volume & y

t

h, r, x

h, r, x

A, v, y

Briefing about Rates Box:

symbol X = times

respect to tdivide or by

differentiate

dd

THE RATES BOX

t

h, r, x

h, r, x

A, v, y

d dA

r

r

t

column for Area

column for radius columnfor time

THE RATES BOX

2

Refer to general

1)

dd

THE RATES BOX

columnfor time

column for radius

column forvolume

t

r

r

v

2)

d d hv

h t

column forvolume

column for he ight columnfor time

THE RATES BOX

3)

d dcolumnfor time

THE RATES BOX

column for he ight

column forvolume

tx

v x

4)

3

Refer to sphere

Refer to cylinder

Refer to cube

ddh,r,x

h,r,x t

S,A, V, Y

2: Find the value

ddh,r,x

h,r,x t

S,A, V, Y

3: Find the value

Example: SPM 2005, Q20

= h2 + 8

= (2)2 + 8

, when h = 2 = 12

4

12

10dd

h

h t

V

3: Find the value

= 0.833cms-1

Example 2: Sasbadi Nexus, pg 152

, spherical balloon

r = 7 cm. Find

Solution:

V sphere =

= = 4π (7)2

= 196π

2196dd

r

r t

V

= 392π

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KOLEKSI SOALAN MATEMATIK TAMBAHAN SPM 2004 – 2009

TAJUK : DIFFERENTIATIONArranged By: Cg. Shaifur Azura Suhaimi

2004: PAPER 1

Q20: Differentiate with respect to x. [3marks]

Q21: Two variables, m and n, are related by the equation . Given that m

increases at a constant rate of 5 units per second, find the rate of change of n when n = 2. [3marks]

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2005: PAPER 1

Q19: Given that , evaluate

[4marks]

Q20: The volume of water, V cm3, in a container is given by , where h cm

is the height of the water in the container. Water is poured into the container at the

rate of 10 cm3s-1. Find the rate of change of the height of water, in cms-1, at the

instant when its height is 2 cm.[3marks]

2006: PAPER 1

Q17: The point P lies on the curve y = (2x – 1)2. It is given that the gradient of normal is

. Find the coordinates of P.

[3marks]

Q18: Given y = , where u = 5x – 2. Find in terms of x.

[3marks]2006: PAPER 1

Q19: Given y = 4x2 + x – 3

a) Find the value of when x = 2

b) Express the approximate change in y, in terms of p, when x changes from 2 to 2 + p, where p is a small value.

[4marks]

2007: PAPER 1

Q19: The curve is such that , where k is a constant. If the

gradient of the curve at x = 3 is 2, find the value of k.[2marks]

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Q20: The curve has a maximum point at x = p, where p is aconstant. Find the value of p.

[3marks]

2008: PAPER 1

Q19: Two variables, x and y, are related by the equation , express, in terms of

h , the approximate change in y when x changes from 5 to 5 + h. [3marks]

Q20: The normal to the curve at point P is parallel to the straight line

. Find the equation of normal to the curve at point P.

[4marks]

2009: PAPER 1

Q19: The gradient function of a curve is , where k is a constant. It is given

that the curve has a turning point at (2, 1). Finda) the value of kb) the equation of the curve

[4marks]

Q20: A block of ice in the form of a cube with sides x cm, melts at rate of 9.72 cm3 per minute. Find the rate of change of x at the instant when x = 12 cm.

[3marks]

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2004: PAPER 2

Q5: The gradient function of a curve which passes through A(1, -12) is . Finda) the equation of the curve, [3marks]b) the coordinates of the turning points of the curve and determine whether

each of the turning points is a maximum or a minimum. [5marks]

2005: PAPER 2

Q2: A curve has a gradient function , where p is a constant. The tangent to a curve at the point (1, 3) is parallel to the straight line y + x – 5 = 0. Finda) the value of p, [3marks]b) the equation of the curve [3marks]

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2007: PAPER 2

Q4: A curve with gradient function has a turning point at (k, 8)

a) Find the value of k, [3marks]b) Determine whether the turning point is a maximum or a minimum point. [2marks]c) Find the equation of the curve. [3marks]

2009: PAPER 2

Q3: The gradient function of a curve is hx2 – kx, where h and k are constants. Thecurve has a turning point at (3, -4). The gradient of tangent to the curve at thepoint x = -1 is 8. Finda) the value of h and k, [5marks]b) the equation of the curve. [3marks]

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