Differentiation (Cambridge O Level Additional Maths)

A curve has equation  where . The normal to the curve at the point where , cuts the -axis, at the point .
Find the equation of the normal and the -coordinate of .

Variables  and  are such that , where  is in radians. Use differentiation to find the approximate change in  as  increases from 1 to , where  is small.

The tangent to the curve , at the point where , meets the -axis at the point . Find the exact coordinates of .

A container is a circular cylinder, open at one end, with a base radius of  cm and a height of  cm. The volume of the container is 1000 cm³. Given that  and  can vary and that the total outer surface area of the container has a minimum value, find this value.

Find the -coordinates of the stationary points of the curve .

A curve has equation  and has exactly two stationary points. Given that , , use the second derivative test to determine the nature of each of the stationary points of this curve.

In this question all lengths are in centimetres.

The diagram shows a solid cuboid with height  and a rectangular base measuring  by . The volume of the cuboid is . Given that  and  can vary and that the surface area of the cuboid has a minimum value, find this value.

Find the equation of the tangent to the curve at the point where .
Give your answer in the form , where  and  are integers.

This tangent intersects the -axis at  and the -axis at . Find the length of .

Given that , show that , where  and  are constants.

Find the exact coordinates of the stationary point of the curve .

Determine the nature of this stationary point.

Differentiate  with respect to .

Variables  and  are such that . Use differentiation to find the approximate change in  as  increases from , where  is small.

It is given that .

Find the exact value of  when .

Hence find the approximate change in  as  increases from where  is small.

Given that  is increasing at the rate of 3 units per second, find the corresponding rate of change in when  , giving your answer in its simplest surd form.

It is given that .

Find .

Find the value of  for which .

Given that , find .

Hence, given that  is increasing at the rate of 2 units per second, find the exact rate of change of  when .

In this question, all lengths are in centimetres.

The diagram shows a cone of base radius , height  and sloping edge . The volume of the cone is .

Show that the curved surface area, , of the cone is given by .

Given that  can vary and that  has a minimum value, find the value of  for which  is a minimum.

Variables and are such that  .Use differentiation to find the approximate change in as  increases from , where is small.

In this question all lengths are in centimetres.
The volume, , of a cone of height  and base radius  is given by 

The diagram shows a large hollow cone from which a smaller cone of height 180 and base radius 90 has been removed. The remainder has been fitted with a circular base of radius 90 to form a container for water. The depth of water in the container is w and the surface of the water is a circle of radius .

Find an expression for  in terms of  and show that the volume of the water in the container is given by .

Water is poured into the container at a rate of . Find the rate at which the depth of the water is increasing when . 

A curve has equation 

Find the equation of the normal to the curve at the point where .

Find the approximate change in  as  increases from , where  is small.

The rectangle  represents a ploughed field where  and . Joseph needs to walk from  to  in the least possible time. He can walk at  on the ploughed field and at on any part of the path  along the edge of the field. He walks from  to  and then from  to . The distance .

Find, in terms of , the total time, , Joseph takes for the journey.

Given that  can vary, find the value of  for which  is a minimum and hence find the minimum value of .