Integration and Differentiation

(1)

PAPER 1(SET 1) PAPER 1(SET 1)

JOHOR 2009 JOHOR 2009

1. Given

1. Given y y 2424₄₄ x x



, find the approximate value of , find the approximate value of y y , when, when x x changes from 2 to 2.02.changes from 2 to 2.02. [4 marks] [4 marks]

2.

2. Find Find the the coordinates coordinates of of the the turning turning points points on on the the curvecurve y y 44x x 11 x x





and determineand determine the

the maximum maximum point. point. [4 [4 marks]marks]

3. Given

3. Given y y 2 2 x x ₂₂ 33 x x





andand dy dy 33g g x

_{  }

x dx

dx



wherewhere g g x

  

x is a function of is a function of x x . Find the value of . Find the value of

  

1 1 1 1 g x d x g x d x  



. . [4 [4 marks]marks] KEDAH 2009 KEDAH 2009 4.

4. Find Find the the gradient gradient of of the the curvecurve 44 22 3 3 55 2 2 22 y y x x x x







at the pointat the point

  

1,31,3 [3 marks][3 marks]

5. Differentiate 5. Differentiate 2 2 4 4 11 2 2 11 x x x x





with respect towith respect to x x . . [2 [2 marks]marks]

6.

6. Given Given that that the the gradient gradient function function of of a a curve curve passing passing through through the the pointpoint

  

1,21,2 isis

 



22 3 3 2 2 2 2 11 x x x x





, determine , determine the the equation equation of of the the curve. curve. [4 [4 marks]marks]

7.

7. Given Given thatthat

 



5 5 2 2 33 10 10 x x y

y





andand x x is increasing at the rate of is increasing at the rate of 2 units per second, find the2 units per second, find the rate of change of

rate of change of y y whenwhen 11 2 2 x x  . . [3 [3 marks]marks] KELANTAN 2009 KELANTAN 2009 8.

  

 



33 4 4 2 2 11 g g x x x x





, find, find

  

"" 1 1 g g . . [4 [4 marks]marks] 9.

9. Two Two variablesvariables r r andand ss are related by the equationare related by the equation r r ss 33₂₂ ss



 



.Given that r .Given that r increase atincrease at a constant rate of 5 units per second, find the rate of change of

a constant rate of 5 units per second, find the rate of change of ss whenwhen ss



22.. [3 marks] [3 marks]

(2)

10.

  

3 3 1 1 10 10 g x d x g x d x





, find the value of m, find the value of m if if

  

3 3 1 1 2 2 g g x x m



mx x ddx x



66mm











. . [3 [3 marks]marks] MELAKA 2009 MELAKA 2009 11. Differentiate 11. Differentiate 22

_{ }

_

66 1 1 22 x

x



x x with respect towith respect to x x . . [3 [3 marks]marks] 12.

12. The The radius radius of of a a sphere sphere increases increases at at the the rate rate of of 0.10.1 cmscms-1-1. When its radius is 5 cm, find. When its radius is 5 cm, find the rate of increase of its surface area. [Given

the rate of increase of its surface area. [Given the surface area of a sphere isthe surface area of a sphere is

2 2

4 4 A

A



  r r ]. ]. [3 [3 marks]marks]

13.

  

3 3 1 1 5 5 f f x dx dx x





. Find the value of the constant. Find the value of the constant mm if if



 





 

2 2 33 1 1 22 15 15 f f x x ddx x



_



f f x x



mmx x d



_

dx x



 



[3 marks][3 marks] MRSM 2009 MRSM 2009 14.

14. Given Given thatthat y y t



 

t



22t t 22 andand x x



 

4 4 t t



11, find, find dy dy dx

dx in terms of x in terms of x . . [3 [3 marks]marks]

15.

15. It It is is given given thatthat y y 2 2 x x ₂₂ 11 x x





andand dy dy 22g g x

  

x dx

dx



, where, where g x g

  

x is a function in terms of is a function in terms of x x .. Find the value of

Find the value of

  

2 2 1 1 g x d x g x d x



. . [3 [3 marks]marks] 16. Given

16. Given dy dy 1414 dx dx



andand 3 3 0 0 9 9 ydx ydx





, find, find y y in terms of x in terms of x . . [3 [3 marks]marks] 17.

17. Diagram Diagram 1 1 shows shows part part of of the the graph graph of of a a curve.curve.

Diagram 1 Diagram 1

(3)

(a)

(a) State State the the value value of of

6 6 88 0 0 33 y yddx x



xxddy y

 



(b)

(b) Given Given thatthat

8 8 3 3 9 9 xdy xdy





, , find find the the area area of of the the shaded shaded region. region. [3 [3 marks]marks] PAHANG 2009

PAHANG 2009

18.

18. Find Find the the value value of of

2 2 2 2 2 2 3 3 lim lim 4 4 33 x x x x x x x x x x  





[2 marks][2 marks] 19.

19. Volume, Volume, V V cmcm33, of a solid is given by, of a solid is given by 88 2 2 22 33 3 3 V

V



  r r



  r r ,, r r is the radius. Find theis the radius. Find the approximate change in V if

approximate change in V if r r increases from 3 cm to increases from 3 cm to 3.005 cm.(Give your answers in3.005 cm.(Give your answers in terms of

terms of   ). ). [4 marks][4 marks]

20.

20. Find Find the the value value of of



 

 

1 1 4 4 1 1 6 6 x x 66 x x dx dx x x  







. . [3 [3 marks]marks] 21.

21. The The gradient gradient function function of of a a curve curve passing passing throughthrough

  

1,21,2 is given byis given by

 



22 1 1 3 3 x x



44 . Find the . Find the equation

equation of of the the curve. curve. [3 marks][3 marks]

PENANG 2009 PENANG 2009

22.

22. The curveThe curve y y



f x f

  

x is such thatis such that dy dy 2 2 kx kx 88 dx

dx





wherewhere k k is a constant. The gradient atis a constant. The gradient at 3

3 x

x



is 4. Find the value of is 4. Find the value of k k . . [2 [2 marks]marks] 23.

23. Given thatGiven that y y



4 4 x x 3 3



7 7 x x 22



11, find the value of , find the value of dy dy dx

dx at pointat point

  

2,52,5 . Hence, find the. Hence, find the small change in

small change in x x whenwhen y y increases increases from from 2 2 to to 2.1. 2.1. [3 [3 marks]marks]

24.

24. It It is is given given thatthat

  

4 4 3 3 f f x x ddx x m



m



, find the value of m, find the value of mif if

  

3 3 4 4 2 2 f f x x



7 7 ddx x



1177











. . [4 [4 marks]marks] PERAK 2009 PERAK 2009 25.



f x f

  

x is such thatis such that dy dy 2 2 px px 33 dx

dx





, where, where p p is a constant. The gradient of is a constant. The gradient of curve at

curve at x x



44 isis p



p. Find the value of . Find the value of p p. . [2 [2 marks]marks] 26.

  



 

2 2 x x 22 2424x x r



r has a maximum point athas a maximum point at x x r



r , where, where r r is a constant.is a constant. Find the value of

(4)

27.

  

3 3 1 1 5 5 g x d x g x d x  





, find, find (a) (a)

  

1 1 3 3 ,, g x d x g x d x  



(b)(b)

  

3 3 1 1 2 2 g g x x 33x x ddx x  













[4 marks][4 marks] PERLIS 2009 PERLIS 2009 28.

28. It It is is given given thatthat y y



2 2 3 x x 22

 

3 4 x x



4 ..



22 FindFind dy dy dx

dx whenwhen x x



11. . [3 [3 marks]marks]

29.

29. The The surface surface area, area, A A cmcm22, of a solid is given by, of a solid is given by

2 2 3 3 66 2 2 r r A A r r    





. It is given that the rate. It is given that the rate of change of the surface area is 5 cm

of change of the surface area is 5 cm22ss-1-1. Find. Find

(a)

(a) dAdA,, dr dr

(b)

(b) the the rate rate of of change change of of the the radius, radius, in in cm cm ss-1-1, when the , when the radius is 3 radius is 3 cm. cm. [4 marks][4 marks]

30.

30. Given Given that that the the gradient gradient function function of of the the curve curve at at point point (1,10) (1,10) isis 3 3 x x



5.5.Find theFind the equation

equation of of the the curve. curve. [3 marks][3 marks]

SABAH 2009 SABAH 2009

31. Differentiate

31. Differentiate x x 2 2 x x



11 with respect towith respect to x x . . [3 [3 marks]marks]

32.

32. A point A point P P lies lies on on the the curvecurve 11

 

2 2 55



22 2

2 y

y



x x



. Given that the tangent to the curve at P is. Given that the tangent to the curve at P is parallel to the straight line

parallel to the straight line 2 2 x x y



 

y

 

1 1 0



0. . Find Find the the coordinates coordinates of of P. P. [3 [3 marks]marks]

33.

  

4 4 1 1 5, 5, g x d x g x d x





findfind (a) (a)

  

1 1 4 4 ,, g x d x g x d x



(b)(b)

  

4 4 1 1 2 2 g g x x



3 3 x x ddx x ..



_









[4 marks][4 marks] SARAWAK ZON A 2009 SARAWAK ZON A 2009 34.

34. Find Find the the coordinates coordinates of of the the minimum minimum point point of of the the curvecurve 22 33 10.10. 2

2 y

y x



 

x



x x



[3 marks][3 marks]

35.

35. Two variables,Two variables, x x andand y y , are related by the equation, are related by the equation y y 3 3 x x 1616.. x x





Given thatGiven that y y increases at a constant rate of 10

increases at a constant rate of 10 unit per second, find the rate unit per second, find the rate of change of of change of x x whenwhen



(5)

36.

  

6 6 3 3 2 2 g x d x g x d x





, find, find (a) (a)

  

3 3 6 6 3 3 g x d x g x d x ,,



(b) (b) the the value value of of k k if if

  

6 6 3 3 10. 10. g g x x



kkx x ddx x













[3 marks][3 marks] SARAWAK ZON C 2009 SARAWAK ZON C 2009 37.

37. Given Given thatthat g x g



 

x

 

 





5 5 3



3x x



44, find, find gg" 2"

  

2 . . [3 [3 marks]marks] 38.

38. A A circle circle has has a a radius radius of of 5 c5 cm. m. The The radius radius of of the cthe circle ircle is is decreasing decreasing at at the the rate rate of of 0.10.1 cm

cm per per second. second. Find Find the the rate rate of of change change of of the the area area of of the the circle. circle. [4 [4 marks]marks]

39.

  

3 3 1 1 5 5 f f x dx dx x





, find the value of , find the value of k k if if

  

3 3 1 1 10, 10, f x f x



kkx x ddx x













wherewhere k k is ais a constant.

constant. [3 [3 marks]marks]

SBP 2009 SBP 2009

40.

40. Given Given thatthat f x x f



 

x x

 



33



5 5 3



3 x x



22,, findfind f f ' ' 2

  

2 .. [3 marks][3 marks]

41.

41. Two variables,Two variables, PP andand x x are related by the equationare related by the equation P P 3 3 x x 22.. x x





GivenGiven x x increases atincreases at a constant rate of 4

a constant rate of 4 units per second whenunits per second when x x



22, find the rate of change of , find the rate of change of PP.. [3 marks] [3 marks] 42. Given 42. Given

 

2 2 55



33 h h y y x x





andand

  

,, dy dy g g x x dx

dx



find the value of find the value of hh if if

  

3 3 2 2 1 1 7.7. g g x x



ddx x













[3 marks] [3 marks] SELANGOR 2009 SELANGOR 2009 43.

43. Two variables,Two variables, x x andand y y , are related by the equation, are related by the equation y y x



x

 

3 3



x x



22..Given thatGiven that y y increases at a constant rate of 4

increases at a constant rate of 4 units per second, find the rate of units per second, find the rate of change of change of x x whenwhen 2

2 x

x



. . [3 [3 marks]marks]

44.

44. The gradient The gradient function function of of a a curve curve isis px px



33, where, where p p is a constant. The straight lineis a constant. The straight line 5

5 22 y

y



 



x x is a tangent to the curve at the point (-2,1). Findis a tangent to the curve at the point (-2,1). Find (a)

(6)

45.

 



22 1 1 1 1 44 ,, 5 5 4 4 k k dx dx x x  







find the value of find the value of k k .. [3 marks][3 marks] KLANG 2009 KLANG 2009 46. Differentiate 46. Differentiate

 



22 3 3 4 4 2



2 x x with respect to

with respect to x x . . [3 [3 marks]marks]

47.

47. Given that Given that the the gradient gradient of of a a normal normal to to the the curvecurve y y k



kx x x

 

x



22



22 at the point (1,-6) isat the point (1,-6) is 1 1 6 6   . Find. Find (a)

(a) the the value value of of k k

(b)

(b) the the equation equation of of the the tangent tangent to to the the curve curve at at the the point point wherewhere x x



11. [4 marks]. [4 marks]

48.

  

3 3 0 0 ,, h x d x h x d x



evaluateevaluate (a) (a)

  

3 3 0 0 1 1 ,, 5 5h x d x h x d x



(b)(b)

  

3 3 0 0 3 3



h h x x ddx x ..



_









[3 marks][3 marks] TERENGGANU 2009 TERENGGANU 2009 49.

49. Given Given that that the the equation equation of of a a curve curve which which passes passes through through pointpoint PPisis y y



 

2 2 x x



1 1 ..



22 TheThe normal gradient to the curve at

normal gradient to the curve at pointpoint PPisis 11 2 2



 . Find the coordinates of . Find the coordinates of PP [3 marks].. [3 marks]

50.

  

4 4 1 1 7 7 g x d x g x d x





, find the value of , find the value of (a) (a)

  

1 1 4 4 ,, g x d x g x d x



(b)(b) p p if if

  

4 4 1 1 25. 25. g g x x



p dp dx x













[4 marks][4 marks] 51.

51. A curve A curve which which has has gradient gradient functionfunction kx kx



2,2, wherewhere k k is a constant, passes throughis a constant, passes through points (0,10) and (2,0). Find

points (0,10) and (2,0). Find

(a)

(a) the the value value of of k k ,, (b) (b) the the equation equation of of the the curve. curve. [4 [4 marks]marks]

WP 2009 WP 2009

52.

52. Given Given an an equation equation of of a a curvecurve y y



2 2 x x 3 3



6 6 x x 22



11. Find the value of . Find the value of x x whenwhen y y isis maximum.

(7)

53.

53. The The area area of of a a circle circle increases increases at at the the rate rate of of 1616  cmcm22ss-1-1. Find the rate of change of . Find the rate of change of

the

the radius radius when when the the radius radius is is 4 4 cm. cm. [3 [3 marks]marks]

54.

  

4 4 1 1 7 7 g x d x g x d x





andand



 





 

4 4 1 1 2 2 22 55 2299 k k k k g g x x ddx x





_



g x g x d



_

dx x



 



wherewhere 1 1



 

k k



44 andand

  

0.0. g

g x x



Find Find the the value value of of k. k. [3 [3 marks]marks] TIMES 2009

TIMES 2009

55.

 



22 3 3 ,, 2 2 33 y y x x





findfind (a)

(a) dy dy dx

dx in terms of in terms of x x ,, (b)

(b) the the approximate approximate change change inin y y given thatgiven that x x decreases from 2 to 1.98decreases from 2 to 1.98

[3 marks] [3 marks]

56.

56. The curveThe curve y x y x k

 



22

 

kx x



44 has a minimum point athas a minimum point at x x



 



33, where, where k k is a constant. Findis a constant. Find (a)

(a) the the value value of of k, k, (b) (b) the the equation equation of of the the tangent tangent to to the the curve curve atat x x



0.0. [4 marks] [4 marks]

57.

57. Diagram 2 Diagram 2 shows shows the the curvecurve y y



f f x

  

x . The curve intersects the x-axis at. The curve intersects the x-axis at x x



22..

Diagram 2 Diagram 2 Given that Given that

  

3 3 2 2 4 4 f f x dx dx x





units. units. Find Find the the area area of of the the shaded shaded region. region. [2 [2 marks]marks] 58.

 



2 2 1 1 2 2 5.5. k kx x ddx x  



