Integration and Differentiation- Core 2 Revision
1.
Given that y = x
(a) express y as a single power of x x ,
(1) d y
(b) find , d x
(2)
(c) find
 y x
(2)
(d) evaluate
4

0 y d x .
(2)
(Total 7 marks)
2.
(a) Write x
2 x in the form x k
, where k is a fraction.
(1)
(b) The gradient of a curve at the point ( x , y ) is given by d y d x

7 x
2 x
Use integration to find the equation of the curve, given that the curve passes through the point (1,1).
(3)
(Total 4 marks)
3.
Given that y =
5 x 2 ,
(a) write down an expression for d y d x
,
(1) d y
(b) show that, when x = 2, the value of d x integer to be determined.
can be written in the form p

2, where p is an
(3)
(Total 4 marks)
South Wolds Comprehensive School 1

4.
A curve is defined for x > 0 by the equation y = x +
2
. x d y
(a) (i) Find d x
. (3)
1
(ii) Hence show that the gradient of the curve at the point P where x = 2 is
2
.
(b) Find an equation of the normal to the curve at this point P .
(1)
(4)
(Total 8 marks)
5.
Find the equation of the tangent to the curve with equation y
 x
2
x at the point P (1, –1). (3)
Determine the coordinates of the point where the tangent at P intersects the curve again. (5)
(Total 8 marks)
6.
A curve has equation y = 12 x 2
1
(a) Evaluate

0
2 determined. y d x , giving your answer in the form p 2 , where p is an integer to be
(3)
(b) Find the value of dy dx to be determined.
at x = 2, giving your answer in the form q 2 , where q is an integer
(3)
(Total 6 marks)
7.
A curve is defined for x > 0 by the equation y



  2 x



2
The point P lies on the curve where x = 2.
(a) Find the y -coordinate of P .
(b) Expand


  2 x 


2
.
(1)
(2)
(c) Find d y d x
(d) Hence show that the gradient of the curve at P is

2.
(3)
(2)
(e) Find the equation of the normal to the curve at P , giving your answer in the form x + by + c = 0, where b and c are integers. (4)
(Total 12 marks)
South Wolds Comprehensive School 2

8.
Given that y = x
2
– x
–2
, d y
(a) find the value of d x
at the point where x = 2,
(3)
(b) find
 y d x.
(2)
(Total 5 marks)
9.
Use the trapezium rule with four ordinates (three strips) to find an approximate value for

0
3
2 x d x giving your answer to three decimal places.
(Total 4 marks)
10.
At the point ( x , y), where x > 0, the gradient of a curve is given by d y d x

3 x
1
2
 16 x
2

7 d y
(a) (i) Verify that d x
= 0 when x = 4.
(1)
(ii) Write
16 x
2 in the form 16 x k
, where k is an integer.
(1)
(iii) Find d
2 y d x
2
.
(3)
(iv) Hence determine whether the point where x = 4 is a maximum or a minimum, giving a reason for your answer.
(2)
(b) The point P (1, 8) lies on the curve.
(i) Show that the gradient of the curve at the point P is 12.
(1)
(ii) Find an equation of the normal to the curve at P.
(3)
(c) (i) Find

( 3 x
1
2
 16 x
2

7 ) d x
(ii) Hence find the equation of the curve which passes through the point P (l, 8).
(3)
(Total 17 marks)
South Wolds Comprehensive School 3

11.
(a) Show that the equation
3
2 x 2 – 9 x + 6 = 0 has a root between 0 and 1.
(3)
(b) A curve has equation
3 y = 2 x 2 – 9 x . d y
(i) Find d x
and d
2 y d x
2
.
(5)
(ii) Calculate the coordinates of the stationary point on the curve.
(3) d
2 y
(iii) Find the value of at the stationary point and hence determine whether this d x
2 point is a maximum or a minimum.
(2)
(Total 13 marks)
12.
Use the trapezium rule with four ordinates (three strips) to find an approximation to

1
2 .
5
( 2 x 
1 ) d x giving your answer to 3 significant figures.
(Total 4 marks)
13.
(a) Find
 1 x 2 d x
(2)
(b) Hence find the value of

0
2
1 x 2 d x , giving your answer in the form p 2 , where p is a rational number.
(3)
(Total 5 marks)
South Wolds Comprehensive School 4

14.
The diagram shows the graph of
3 y = x 2 , 0
 x

4, and a straight line joining the origin to the point P which has coordinates (4, 8). y
8 P
O 4 x
(a) (i) Find

2 x 3 d x .
(2)
(ii) Hence find the value of
 4
0 x
3
2 d x.
(2)
(b) Calculate the area of the shaded region.
(2)
(Total 6 marks)
15.
The graph of y = x + 4 x
–2 has one stationary point. d y
(a) Find d x
.
(2)
(b) Find the coordinates of the stationary point.
(3)
(c) Find the value of d
2 y d x
2
at the stationary point, and hence determine whether the stationary point is a maximum or a minimum.
(4)
(Total 9 marks)
South Wolds Comprehensive School 5

1
16. It is given that y = x 3 . d y
(a) Find d x
(b) (i) Find
 y d x .
(ii) Hence evaluate
 8
0 y d x .
17.
(a) Express x
2 x in the form x p
.
(b) Given that y
 x
2 x , d y find the value of d x
at the point where x = 9.
18.
Calculate the gradient of the curve y
 x
4
3 at the point where x = 8. d y find the value of d x
19.
(a) Expand


1 x 

 2
(b) Hence find



1 x 


2 d x
South Wolds Comprehensive School
(2)
(Total 6 marks)
(1)
(2)
(2)
(3)
(Total 4 marks)
(Total 3 marks)
(1)
(3)
(Total 4 marks)
6

y
20.
0 x
1
The diagram shows a part of the curve y = x – x
3/2
(a) Show by differentiation that the curve is steeper at the point where x = 0 than it is at the point where x = 1.
(4)
(b) (i) Find
 y x
(2)
(ii) Hence find the area of the shaded region.
(2)
(Total 8 marks)
21.
A wire of length 10 cm is cut into two pieces. One of these pieces is bent to form an equilateral triangle of side x cm and the other piece is bent to form a sector of a circle of angle

radians and radius x cm as shown below. x cm
(a) Show that 5 x + x

= 10.
 x cm
(2)
(b) The sum of the areas of the triangle and sector is denoted by A cm
2
.
(i) Show that A

4
3 x
2 –
5
2 x
2 
5 x . (5)
(ii) Find d A d x
and hence find the value of x for which A has a stationary value. (3)
(iii) Find d
2
A
and hence determine whether this stationary value is a maximum or a d x
2 minimum. (2)
(Total 12 marks)
South Wolds Comprehensive School 7

y
5
22.
O
The diagram shows a sketch of the curve. y
 x ²

4 x ²
and the line y = 5.
(a) Find the coordinates of the two stationary points on the curve x
(6)
(b) (i) Show that the curve intersects the line when x
4
– 5 x
2
+ 4 = 0
(2)
(ii) By writing u = x ² in the equation x
4
– 5 x
2
+ 4 = 0 form an equation for u . Solve this equation for u and hence find the corresponding values for x .
(3)
(iii) Show that the shaded region has area
2
3
.
(6)
(Total 17 marks)
South Wolds Comprehensive School 8