Pure Mathematics Year 1 (AS) Unit Test 6: Differentiation
1 Prove, from first principles, that the derivative of is (4 marks)
2
Find the set of values of x for which f(x) is increasing. (5 marks)
3 A curve C has equation
The point P has x-coordinate 2
a Find in terms of x. (2 marks)
b Find the equation of the tangent to the curve C at the point P. (4 marks)
The normal to C at P intersects the x-axis at A.
c Find the coordinates of A. (4 marks)
4
Sketch the graph of the gradient function, y = f ʹ(x). Use algebraic methods to determine any points
where the graph cuts the coordinate axes and mark these on the graph. Using calculus, find the
coordinates of any turning points on the graph. (9 marks)
5 A fish tank in the shape of a cuboid is to be made from 1600 cm2 of glass.
The fish tank will have a square base of side length x cm, and no lid. No glass is wasted.
The glass can be assumed to be very thin.
a Show that the volume, V cm3, of the fish tank is given by (5 marks)
b Given that x can vary, use differentiation to find the maximum or minimum value of V.
(4 marks)
c Justify that the value of V you found in part b is a maximum. (2 marks)
© Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free. 1
1 Prove, from first principles, that the derivative of is (4 marks)
2
Find the set of values of x for which f(x) is increasing. (5 marks)
3 A curve C has equation
The point P has x-coordinate 2
a Find in terms of x. (2 marks)
b Find the equation of the tangent to the curve C at the point P. (4 marks)
The normal to C at P intersects the x-axis at A.
c Find the coordinates of A. (4 marks)
4
Sketch the graph of the gradient function, y = f ʹ(x). Use algebraic methods to determine any points
where the graph cuts the coordinate axes and mark these on the graph. Using calculus, find the
coordinates of any turning points on the graph. (9 marks)
5 A fish tank in the shape of a cuboid is to be made from 1600 cm2 of glass.
The fish tank will have a square base of side length x cm, and no lid. No glass is wasted.
The glass can be assumed to be very thin.
a Show that the volume, V cm3, of the fish tank is given by (5 marks)
b Given that x can vary, use differentiation to find the maximum or minimum value of V.
(4 marks)
c Justify that the value of V you found in part b is a maximum. (2 marks)
© Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free. 1
