Differentiation (CIE IGCSE Additional Maths)

Find the -coordinate of the stationary point on the curve , giving your answer in the form ,where  and  are rational numbers.

The radius, , of a circle is increasing at the rate of . Find, in terms of , the rate at which the area of the circle is increasing when .

The volume, , of a sphere of radius is given by .

The radius, , of a sphere is increasing at the rate of . Find, in terms of , the rate of change of the volume of the sphere when .

Given that , show that , where  and  are integers.

Find the coordinates of the stationary point of the curve . Give each coordinate correct to 2 significant figures.

Determine the nature of this stationary point.

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

The equation of a curve is .

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

Find and hence evaluate the area enclosed by the curve  and the lines .

A curve has equation .

Show that , where  and  are constants.

Hence write down the -coordinate of the stationary point of the curve.

Determine the nature of this stationary point.

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

Find the coordinates of the point where this tangent meets the curve again.

It is given that  for .

Find 

Find the value of  when , giving your answer in the form , where  is an integer.

Find the values of  for which .

A curve has equation .

Find .

Find the equation of the normal to the curve at the point where , giving your answer in the form .

Find the equation of the tangent to the curve  at the point where . Give your answer in the form, where  and  are constants correct to 3 decimal places.

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 .