Applications of Differentiation (Edexcel IGCSE Further Pure Maths): Exam Questions

Figure 1 shows the curve with equation

The point , with coordinate −2, lies on and line is the tangent to at the point .

Find an equation for

The point lies on and the line is the tangent to at the point .

Given that and are parallel,

find an equation for

The normal to at meets at the point .

Find the coordinates of .

Figure 2 shows a right circular cone with a base radius of cm. The slant height of the cone is cm and the height of the cone is cm. The vertex of the cone is and the points and , on the base of the cone, are such that is a diameter of the base.

The cone is increasing in size in such a way that the size of the angle is constant at and the total surface area of the cone is increasing at a constant rate of 10 cm² /s.

Find the exact rate of increase of the volume of the cone when

Given that can be written in the form , where and are integers,

(i) Hence, or otherwise, find the value of for which is a maximum

(ii) find the maximum value of

The surface area of a sphere with radius cm is increasing at a constant rate of cm² /s

Find, in cm³ , the exact volume of the sphere at the instant when the rate of increase of is cm/s

Figure 1 shows a sketch of part of the curve with equation

The point lies on and has coordinates

The line is the normal to at the point

Show that an equation of is

The finite region is bounded by the curve , the line , the -axis and the line with equation

Use calculus to find the exact area of

Figure 4 shows a solid right triangular prism

The cross section of the prism is an isosceles triangle.

• 

• cm

• cm

• 

The triangular faces of the prism are vertical and the edges and are horizontal.
The volume of the prism is cm³
The total external surface area of the prism is cm²

Show that satisfies the equation

Given that can vary, use calculus, to find to 3 significant figures, the value of for which is a minimum.

Justify that this value of gives a minimum value of

Hence find, to 2 significant figures, the minimum value of

Figure 2 shows a shape

is a quarter circle with radius cm
and are congruent rectangles of length cm and width cm

The total area of the shape is 100cm²
The perimeter of the shape is cm

Show that

Use calculus to find the value of for which is a minimum, justifying that this value of gives a minimum value of

Find the minimum value of

Given that can be expressed in the form where , and are positive constants

Hence write down the maximum value of

The height of liquid in a vessel is
The volume, , of the liquid in is given by
Liquid is leaking from at a constant rate of

Find the exact rate of change, in cm/s, of when

A curve has equation

Point lies on such that the gradient of at is parallel to the line with equation

The normal to at intersects the -axis at point and the -axis at point
Given that the coordinate of is positive,

find, in its simplified form, the exact length of line

A solid cuboid has width cm, length 4 cm and height cm.
The volume of the cuboid is 75cm³ and the surface area of the cuboid is

Show that

Given that can vary, using calculus,

(i) find to 3 significant figures, the value of for which is a minimum,

(ii) justify that this value of gives a minimum value of

Find, to 3 significant figures, the minimum value of

The equation of a curve is

When is increased to , increases to where and are small.

Show that

The equation of a curve is

When is increased to , increases to where and are small.

Given that = 2.5

find an estimate, to 2 significant figures, of the value of when the value of increases by 0.2%

Figure 3 shows a right triangular prism . A cross section of the prism is a triangle in which and radians.

The volume of the prism is increasing in such a way that the size of and the size of remain constant and the length of , the length of and the length of remain constant.
The lengths of and are each increasing at a constant rate of 0.2cm / s

Find the exact rate of increase, in cm³ / s, of the volume of the prism when the area of the rectangular face is 60 cm²

Find

(i) the minimum value of

(ii) the value of at which this minimum occurs.

Hence, or otherwise, write down

(i) the minimum value of

(ii) the value of at which this minimum occurs.

Figure 1 shows part of the curve with equation
where , and are constants.

The points , and with coordinates (– 2, 0), (6, 0) and (4, – 6) respectively lie on

The line is the normal to at the point

Show that an equation of is

Figure 1 shows part of the curve with equation
where , and are constants.

The finite region shown shaded in Figure 1 is bounded by and

Use algebraic integration to find the exact area of the shaded region.

The volume of oil in a container is cm³ when the height of the oil is cm.
Oil is pouring into the container at a constant rate of 12 cm³ /s.
Given that

find the exact rate, in cm/s, at which the height of the oil is increasing when

Two numbers and are such that

Given that can vary, use calculus to find the value of for which is a minimum, justifying that this value of gives a minimum value of

Find the minimum value of

Figure 1 shows a badge, shown shaded, made from two identical rectangles, and , and a sector of a circle with centre .

Each rectangle measures cm by cm.
The radius of the sector is cm and the angle is 0.5 radians.

The area of the badge is 50 cm²
The perimeter of the badge is cm.

Show that

Given that can vary,

use calculus, to find the exact value of for which is a minimum.
Justify that this value of gives a minimum value for

Find the minimum value of
Give your answer in the form , where is an integer to be found.

The equation of a curve is

Find an equation of the tangent to the curve at the point on the curve where

Give your answer in the form

A cube has edges of length cm.

The total surface area, , of the cube is increasing at a constant rate of 0.45cm² /s

Find the rate of increase, in cm³ /s, of the volume of the cube at the instant when the total surface area of the cube is 384cm²

Figure 3 shows part of the curve with equation
The curve crosses the -axis at the point and the -axis at the point .

The line is the normal to at the point .

Find an equation for , giving your answer in the form