# Integration and Differentiation

July 24, 2018 | Author: Hayati Aini Ahmad | Category: Gradient, Derivative, Integral, Tangent, Equations

#### Description

DIFFERENTIATION & INTEGRATION

SMK BBSL_2010

PAPER 1(SET 1) JOHOR 2009 1.

Given y  

24 4

x

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

2.

Find the coordinates of the turning points on the curve y  4 x 

1

x

the maximum point. 3.

Given y  

2 x   3

[4 marks] dy

and

x 2

and determine

dx

 3g  x  where g  x  is a function of   x  . Find the value of

1

 g x d x  .

[4 marks]

1

KEDAH 2009 4.

Find the gradient of the curve y  4 x 2 

3 2 x

5 2

at the point 1,3 

1 with respect to  x  . 2 x   1

[3 marks]

2

4 x

5.

Differentiate

6.

Given that the gradient function of a curve passing through the point 1,2  is 3 2

 2 x   1 7.

[2 marks]

 2 x , determine the equation of the curve.

Given that y  

2 x  3

[4 marks]

5

10

and  x  is increasing at the rate of 2 units per second, find the

rate of change of  y  when  x

1 

2

.

[3 marks]

KELANTAN 2009 4

8.

Given that g  x  

9.

Two variables r  and s are related by the equation r  s 

, find g 1 . "

3

2 x  1

[4 marks]

3

s2

.Given that r  increase at

a constant rate of 5 units per second, find the rate of change of  s when s  2 . [3 marks]

1

DIFFERENTIATION & INTEGRATION

3

10.

SMK BBSL_2010

3

1

1

Given that g x d x   10 , find the value of  m if  2g  x   mx dx  6m . [3 marks]

MELAKA 2009 6

11.

Differentiate  x 2 1  2x  with respect to  x  .

12.

The radius of a sphere increases at the rate of  0.1 cms . When its radius is 5 cm, find

[3 marks] -1

the rate of increase of its surface area. [Given the surface area of a sphere is 2  A  4  r  ].

[3 marks] 3

13.

Given that

dx   5 . Find the value of the constant m if   f  x dx 1

2

3

 f  x dx    f  x  mxdx   15 1

[3 marks]

2

MRSM 2009 14.

Given that y  t  2t  and  x  4t   1 , find

15.

It is given that y  

2

2 x  1 2

x

and

dy  dx

dy  dx

in terms of  x  .

[3 marks]

 2g  x  , where g  x  is a function in terms of   x  .

2

Find the value of  g  x d x  .

[3 marks]

1

16.

17.

Given

dy  dx

3

 14 and  ydx   9 , find y  in terms of  x  . 0

Diagram 1 shows part of the graph of a curve.

Diagram 1

2

[3 marks]

DIFFERENTIATION & INTEGRATION

6

(a)

SMK BBSL_2010

8

0

3

State the value of  ydx  xdy

8

(b)

Given that  xdy   9 , find the area of the shaded region.

[3 marks]

3

PAHANG 2009  x  3x  2

18.

Find the value of  lim

19.

Volume, V cm , of a solid is given by V  8 r 2 

x 2

[2 marks]

x  4 x   3 2

3

2

3

r  , r  is the radius. Find the



3

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

[4 marks] 1

20.

Find the value of

 6  x   6  x 

1

21.

4

x

dx  .

[3 marks]

The gradient function of a curve passing through 1,2  is given by

1 2

 3 x   4 

equation of the curve.

. Find the

[3 marks]

PENANG 2009 22.

The curve y  f  x  is such that

dy  dx

 2kx  8 where k  is a constant. The gradient at

x   3 is 4. Find the value of  k  .

23.

[2 marks]

Given that y  4 x 3  7 x 2  1 , find the value of

dy  dx

at point  2,5  . Hence, find the

small change in  x  when y  increases from 2 to 2.1. 4

24.

It is given that

[3 marks] 3

 f  x dx  m, find the value of  m if   2 f x  7 dx  17 . 3

[4 marks]

4

PERAK 2009 25.

The curve y  f  x  is such that

dy  dx

 2 px  3 , where  p is a constant. The gradient of

curve at  x   4 is  p . Find the value of  p . 26.

[2 marks]

2 The curve y  2 x  24 x  r has   a maximum point at  x  r  , where r  is a constant.

Find the value of  r  .

[3 marks]

3

DIFFERENTIATION & INTEGRATION

SMK BBSL_2010

3

27.

Given that

 g x d x   5 , find

1 1

(a)

3

 g x d x ,

(b)

 2g  x   3x  dx

[4 marks]

1

3

PERLIS 2009 28.

2

2 It is given that y  2x  3x   4  . Find

dy  dx

when  x   1 .

[3 marks] 2

29.

2

The surface area, A cm , of a solid is given by  A  2

3 r  2

6

. It is given that the rate

-1

of change of the surface area is 5 cm s . Find (a) (b) 30.

dA dr

, -1

the rate of change of the radius, in cm s , when the radius is 3 cm.[4 marks]

Given that the gradient function of the curve at point (1,10) is 3 x  5. Find the equation of the curve.

[3 marks]

SABAH 2009 31.

Differentiate  x 2x  1 with respect to  x  .

32.

A point P lies on the curve y 

1 2

2x  5

2

[3 marks]

. Given that the tangent to the curve at P is

parallel to the straight line 2 x  y  1  0 . Find the coordinates of P.

[3 marks]

4

33.

Given that g x d x   5, find 1

1

(a)

 g x d x ,  4

4

(b)

 2g  x   3x  dx.

[4 marks]

1

SARAWAK ZON A 2009 3

34.

2 Find the coordinates of the minimum point of the curve y  x  x   10. [3 marks]

35.

Two variables,  x  and y  , are related by the equation y  3x 

2

16

x

. Given that y

increases at a constant rate of 10 unit per second, find the rate of change of   x  when  x   2.

[3 marks] 4

DIFFERENTIATION & INTEGRATION

SMK BBSL_2010

6

36.

Given that g  x d x   2 , find 3

3

(a)

6

 3g x d x ,

the value of  k  if  g  x   kx  dx   10.

(b)

6

[3 marks]

3

SARAWAK ZON C 2009 4

37.

Given that g  x    5  3x  , find g " 2 .

38.

A circle has a radius of 5 cm. cm. The radius of the circle circle is decreasing at the rate of 0.1

[3 marks]

cm per second. Find the rate of change of the area of the circle. 39.

Given that

3

3

1

1

[4 marks]

dx   5 , find the value of  k  if    f  x  kx dx    10, where k  is a  f  x dx

constant.

[3 marks]

SBP 2009 2

40.

Given that f x  x  5  3 x   , find  f '  2 .

41.

Two variables, P and  x  are related by the equation P  3x  . Given  x  increases at  x

3

[3 marks] 2

a constant rate of 4 units per second when  x   2 , find the rate of change of  P. [3 marks] 42.

Given y  

h

2 x  5

3

and

dy  dx

3

 g  x  , find the value of  h if   g  x   1 dx   7. 2

[3 marks] SELANGOR 2009 43.

2

Two variables,  x  and y  , are related by the equation y  x  3  x  . Given that y  increases at a constant rate of 4 units per second, find the rate of change of   x  when  x   2 .

44.

[3 marks]

The gradient function of a curve is  px  3 , where  p is a constant. The straight line y  5  2x  is a tangent to the curve at the point (-2,1). Find (a)

the value of  p,

(b)

the equation of the curve.

5

[3 marks]

45.

Given that

1

  4  x 

DIFFERENTIATION & INTEGRATION

dx   2

1

4 5

SMK BBSL_2010

, find the value of  k .

[3 marks]

KLANG 2009 3

46.

Differentiate

47.

Given that the gradient of a normal to the curve y  kx  x   2 at the point (1,-6) is

2

 4  2 x 

with respect to  x  .

[3 marks]

2

1 

6

. Find

(a)

the value of  k

(b)

the equation of the tangent to the curve at the point where  x   1 . [4 marks] 3

48.

Given that

 h x d x ,  evaluate 0

3

(a)

1

 5 h x d x ,

3

 3  h  x  dx.

(b)

0

[3 marks]

0

TERENGGANU 2009 49.

2

Given that the equation of a curve which passes through point P is y   2x   1 . The 1

normal gradient to the curve at point P is

2

. Find the coordinates of  P. [3 marks]

4

50.

Given that g  x d x   7 , find the value of  1

1

(a)

 g x d x ,

4

p if  g  x   pdx   25.

(b)

4

51.

[4 marks]

1

A curve which has gradient function kx  2, where k  is a constant, passes through points (0,10) and (2,0). Find (a)

the value of  k ,

(b)

the equation of the curve.

[4 marks]

WP 2009 52.

Given an equation of a curve y  2 x  6 x   1 . Find the value of   x  when y  is 3

2

maximum.

[3 marks]

6

53.

DIFFERENTIATION & INTEGRATION

2

-1

The area of a circle increases at the rate of 16    cm s . Find the rate of change of  the radius when the radius is 4 cm.

[3 marks]

4

54.

SMK BBSL_2010

4

1

1

29 where 1  k  4 and Given that g  x d x   7 and 2g  x  dx  2 5  g  x   dx  g  x   0. Find the value of k.

[3 marks]

TIMES 2009 55.

Given that y  

(a) (b)

dy  dx

3 2

 2 x  3

, find

in terms of  x ,

the approximate change in y  given that  x  decreases from 2 to 1.98 [3 marks]

56.

The curve y  x  kx   4 has a minimum point at  x   3 , where k  is a constant. Find 2

(a)

the value of k,

(b) the equation of the tangent to the curve at  x   0. [4 marks]

57.

Diagram 2 shows the curve y  f  x  . The curve intersects the x-axis at  x   2 .

Diagram 2 3

Given that

dx   4 units. Find the area of the shaded region.  f  x dx

[2 marks]

2

2

58.

Given that

  kx  2 dx   5. Find the value of k.

1

7

[3 marks]