# Integration and Differentiation- Core 2 Revision

**Integration and Differentiation- Core 2 Revision**

1.Given that

y = x,

(a)express y as a single power of x

(1)

(b)find

(2)

(c)find,

(2)

(d)evaluate.

(2)

(Total 7 marks)

2.(a)Write in the form xk, where k is a fraction.

(1)

(b)The gradient of a curve at the point (x, y) is given by

Use integration to find the equation of the curve, given that the curve passes through thepoint (1,1).

(3)

(Total 4 marks)

3.Given that

y =,

(a)write down an expression for ,

(1)

(b)show that, when x = 2, the value of can be written in the form p2, where p is an integer to be determined.

(3)

(Total 4 marks)

4.A curve is defined for x > 0 by the equation y = x + .

(a)(i)Find .(3)

(ii)Hence show that the gradient of the curve at the pointP where x = 2 is .(1)

(b)Find an equation of the normal to the curve at this point P.(4)

(Total 8 marks)

5.Find the equation of the tangent to the curve with equation

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 = 12x

(a)Evaluate y dx, giving your answer in the form , where p is an integer to be determined. (3)

(b)Find the value of at x = 2, giving your answer in the form , where q is an integer to be determined. (3)

(Total 6 marks)

7.A curve is defined for x > 0 by the equation

The point P lies on the curve where x = 2.

(a)Find the y-coordinate of P.(1)

(b)Expand .(2)

(c)Find (3)

(d)Hence show that the gradient of the curve at P is 2.(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)**

8.Given that

y = x2 – x–2,

(a)find the value of at the point where x = 2,

(3)

(b)find y dx.

(2)

(Total 5 marks)

9.Use the trapezium rule with four ordinates (three strips) to find an approximate value for

giving your answer to three decimal places.

(Total 4 marks)

10.At the point (x, y), where x0, the gradient of a curve is given by

(a)(i)Verify that = 0 when x = 4.

(1)

(ii)Write in the form 16xk, where k is an integer.

(1)

(iii)Find .

(3)

(iv)Hence determine whether the point where x = 4 is a maximum or a minimum, givinga 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

(ii)Hence find the equation of the curve which passes through the point P(l, 8).

(3)

**(Total 17 marks)**

11.(a)Show that the equation

2 – 9 x + 6 = 0

has a root between 0 and 1.

(3)

(b)A curve has equation

y = 2 – 9x.

(i)Find and .

(5)

(ii)Calculate the coordinates of the stationary point on the curve.

(3)

(iii)Find the value of at the stationary point and hence determine whether this 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

giving your answer to 3 significant figures.

(Total 4 marks)

13.(a)Find

(2)

(b)Hence find the value of ,

giving your answer in the form p, where p is a rational number.

(3)

(Total 5 marks)

14.The diagram shows the graph of

y =, 0 x 4,

and a straight line joining the origin to the point P which has coordinates (4, 8).

(a)(i)Find .

(2)

(ii)Hence find the value of dx.

(2)

(b)Calculate the area of the shaded region.

(2)

(Total 6 marks)

15.The graph of

y = x + 4x–2

has one stationary point.

(a)Find .

(2)

(b)Find the coordinates of the stationary point.

(3)

(c)Find the value of at the stationary point, and hence determine whether the stationary point is a maximum or a minimum.

(4)

(Total 9 marks)

16.It is given that y = x.

(a)Find

(2)

(b)(i)Find dx.

(2)

(ii)Hence evaluate dx.

(2)

(Total 6 marks)

17.(a)Express x2 in the form xp.

(1)

(b)Given that

,

find the value of at the point where x = 9.

(3)

(Total 4 marks)

18.Calculate the gradient of the curve

at the point where x = 8.

(Total 3 marks)

find the value of 19.(a)Expand

(1)

(b)Hence find

(3)

(Total 4 marks)

20.

The diagram shows a part of the curve

y = x – x3/2

(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 .

(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.

(a)Show that 5x + x = 10.

(2)

(b)The sum of the areas of the triangle and sector is denoted by A cm2.

(i)Show that .(5)

(ii)Find and hence find the value of x for which A has a stationary value.(3)

(iii)Find and hence determine whether this stationary value is a maximum or a minimum. (2)

**(Total 12 marks)**

22.

The diagram shows a sketch of the curve.

and the line y = 5.

(a)Find the coordinates of the two stationary points on the curve

(6)

(b)(i)Show that the curve intersects the line when

x4 – 5x2 + 4 = 0

(2)

(ii)By writing u = x² in the equation

x4 – 5x2 + 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 .

(6)

**(Total 17 marks)**

South Wolds Comprehensive School1