Koleksi Soalan Matematik Tambahan Spm 2004-2009-Differentiation

PANITIA MATEMATIK TAMBAHAN TAJUK: DIFFERENTIATION -

 dy    dx  

OLEH: CG. SHAIFUR AZURA BIN SUHAIMI

1

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RELATED RATES OF CHANGES

By: CG. Shaifur Azura Suhaimi SMK Sultan Ibrahim (1), Pasir Mas, Kelantan THE RATES BOX

column for Area, Volume & y

A, v, y

h, r, x d

d column for he ight, radiu s & x

t

h, r, x

column for time time

Briefing about Rates Box:

THE RATES B

d iv i v id id e o r

r e s p e c t to to

A , v , y h , r, x d

d h , r, x

differentia

sym bol X = ti

t

THE RATES BOX

column for Are Are a

d

A

Refer to general

r d





dA

1)

dt

dA

=

dr

dr

X

dt

THE RATES BOX

column for volume

r

v

d

2)

dv dt

=

dv dr

X

d t

r

column for radius

r

Ref Refer to s he e

column for t ime ime

dr dt

THE RATES BOX

Refer to cylinder column for volume

v

h d

d h

column for height

dv

3)

dt

=

dv dh

t X

column for time

dh dt

THE RATES BOX

column for volume

v

Refer to cube

x d

d x

column column for he ight

d

d

d

t

column for time





2: Find Find the value

S,A, V, Y

h,r,x

d

d h,r,x

t

d ( s , A, v , y ) dt

= [ s , A, v, y ] X [ h, r , x ]

3: Find the value

S,A, V, Y

h,r,x

d

d t

h,r,x

d ( h, r , x ) dt

=

[ s , A, v, y ] [ h, r , x]

Example: SPM 2005, Q20 v

=

dv dt

1 3

h3

+

8h

dv dh

=

= 10

1 3

(3)h + 8 2

= h2 + 8 = (2)2 + 8

dh dt

= ? , when h = 2

dv dh

3: Find Find the v alue

V 10

h

= 12





dh dt

∴

=

10 12

= 0.833cms-1 Example 2: Sasbadi Nexus, pg 152 dr = 2cms dt

−1

, spherical balloon

r = 7 cm. Find

dv dt

=?

Solution: V sphere =

4

r 3

π

3

dv 4 = (3)π r 2 dr 3 = 4π r 2

= 4π (7)2 = 196π

V d

dv dt

196

r 2

r

t

= 196 π X 2 = 392π

d





KOLEKSI SOALAN MATEMATIK TAMBAHAN SPM 2004 – 2009 TAJUK : DIFFERENTIATION Arranged By: Cg. Shaifur Azura Suhaimi

2004: PAPER 1

Q20: Differentiate 2 x 3 ( 3 x − 7) with respect to x to x.. 5

[3marks] Q21: Two variables, m and n, are related by the equation m

=

4n +

3 n

. Given that m

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

2005: PAPER 1







Q20: The volume of water, V cm3, in a container is given by V =

1 3

h3

+

8h , 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

− 1 . Find the coordinates of P. 2

[3marks] Q18: Given y =

3 5

u 8 , where u = 5 x – 2. Find

dy dx

in terms of x. x. [3marks]

2006: PAPER 1

Q19:

Given y Given y = 4 x2 + x – 3 dy

a)

Find the value of

b)

Express the ap approxi oximate ch chang ange in y, y, in terms of p, p, when x when x changes from 2 to 2 + p + p,, where p is a small value. [4marks]

2007: PAPER 1

dx

when x when x = 2





h , the approximate change in y in y when x when x changes from 5 to 5 + h. [3marks] Q20: Q20:

2 The The nor normal mal to the the cur curve ve y = x − 4 x + 7 at point P is parallel to the straight line

y = −

1 2

x + 9 . Find the equation of normal n ormal to the curve at point P.

[4marks]

2009: PAPER 1

Q19: Q19:

The The gradi gradien entt func functi tion on of of a curv curvee is

dy dx

=

kx − 6 , where k is a constant. co nstant. It is given

that the curve has a turning point at (2, 1). Find a) the va value of k f k b) b) the the equat equatio ion n of the the curv curvee [4marks] Q20:

A block block of ice ice in in the the form form of of a cube cube with with side sidess x cm, melts at rate of 9.72 cm3 per minute. Find the rate of change of x x at the instant when x = 12 cm. [3marks]





2004: PAPER 2

Q5: Q5: Find

The The grad gradie ient nt func functi tion on of a curv curvee whic which h pass passes es thro throug ugh h A(1, A(1, -12) -12) is 3 x 2 a) b) b)

−

6 x .

the equation of the curve, [3marks] the the coord coordin inat ates es of the the turn turnin ing g point pointss of the the curve curve and and deter determi mine ne whet whethe her r each each of of the the tur turni ning ng poi point ntss is a maxi maximu mum m or a mini minimu mum. m. [5ma [5mark rks] s]

2005: PAPER 2

Q2: Q2:

px 2 − 4 x , where p is a constant. The tangent to a A cur curve ve has has a gra gradi dien entt fun funct ctiion px curve at the point (1, 3) is parallel to the straight line y + x – 5 = 0. Find





2009: PAPER 2

Q3: Q3:

The The gra gradi dien entt fun funct ctio ion n of of a curv curvee is is hx2 – kx, kx, where h and k are constants. The curve has a turning point at (3, -4). The gradient of tangent to the curve at the point x point x = -1 is 8. Find a) the va value of h f h and k , [5marks] b) the equation of the curve. [3marks]

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Koleksi Soalan Matematik Tambahan Spm 2004-2009-Differentiation

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