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